Reasoning about Action: An Argumentation - Theoretic Approach
JJournal of Artificial Intelligence Research 24 (2005) 465-518 Submitted 09/04; published 10/05
Reasoning about Action:An Argumentation-Theoretic Approach
Quoc Bao Vo [email protected]
School of Computer Science and Information TechnologyRMIT UniversityGPO Box 2476V, Melbourne, VIC 3001, Australia
Norman Y. Foo [email protected]
Knowledge Systems GroupArtificial Intelligence LaboratorySchool of Computer Science and EngineeringUniversity of New South Wales, Sydney, NSW 2052, Australia
Abstract
We present a uniform non-monotonic solution to the problems of reasoning about actionon the basis of an argumentation-theoretic approach. Our theory is provably correct relativeto a sensible minimisation policy introduced on top of a temporal propositional logic.Sophisticated problem domains can be formalised in our framework. As much attention ofresearchers in the field has been paid to the traditional and basic problems in reasoningabout actions such as the frame, the qualification and the ramification problems, approachesto these problems within our formalisation lie at heart of the expositions presented in thispaper.
1. Motivation and Introduction
The need for a good reasoning about action formalism is apparent for research in artificialintelligence (AI). Alongside the logicist point of view to artificial intelligence, more recently,there emerges the cognitivist and situated action-based approaches(see Kushmerick, 1996and the references therein). The latter approaches provide some immediate and practicalanswers to certain issues of AI. The current problem domains for (Soccer) Robot Cupseem to be an area where these approaches promise to gain fruitful results. On the otherhand, the logicist approach aims at long term solutions for the general problems of AI.From a logicist approach, formalising dynamic domains for reasoning about action canbe realised within a logical knowledge representation. The general idea is that intelligentagents should be able to represent all kinds of knowledge in a uniform way such that somegeneral problem solver can fully employ and find a solution based on their knowledge. Asit turns out, there are difficulties with such a general approach to AI. Consider the task offormalising dynamic domains in some logical language. To formalise the dynamics of anaction (or event) in a language with n fluents , one will need to axiomatise not only aboutthe fluents that are effected by the action but also about those that are not. Essentially, itrequires that n axioms be asserted. Such a formalisation can hardly be considered a good fluent is a technical term referring to functions or predicates whose values can be varied relative to time. c (cid:13) o & Foo representation. Hence, there is the need to solve this problem in logic-based reasoning aboutaction formalisms. This is the well known frame problem as introduced by McCarthy andHayes (1969). Moreover, there is still a problem in axiomatising the effects of an action,called the effect axioms . A logical axiomatisation requires that the conditions under whichthe effects will take place after executing the action be precisely specified. However, thereare potentially infinitely many such conditions, some of which the reasoner may never havethought about. No realistic formalisation would ever be able to exhaustively enumerate allof those conditions. Nonetheless, to start a car, most people only worry about whether theyhave the key to that car. They never bother checking whether there is something blockingthe tailpipe or checking all electric circuits to make sure that they are all well connected.Such a story has long been well-known within the community of commonsense reasoning,in particular reasoning about action. This is known as the qualification problem and wasintroduced by McCarthy (1977).While there have been a number of solutions to the frame problem (e.g., Shanahan, 1997;Reiter, 1991; Castilho, Gasquet, & Herzig, 1999), the qualification problem has largely beenignored with the notable exception of Thielscher’s (2001) solution within the Fluent Cal-culus and Doherty and Kvarnstr¨om’s (1998) circumscription-based solution using fluentdependency constraints. Some people argue that the frame problem is already very chal-lenging and it would be a good approach to thoroughly solve the frame problem beforecomplicating a formalism with the qualification problem. We argue that there is a dangerof approaching these problems from that point of view for (at least) two reasons:1. It may be very hard to come up with a uniform solution for all problems: while manyexisting solutions for the frame problem are monotonic (e.g., Reiter, 1991; Castilhoet al., 1999), the qualification problem inherently requires a non-monotonic solution.This is the case with the original qualification problem as stated by McCarthy (1977)for the dynamics of actions/events need to be finitely axiomatisable and when anunexpected qualification for an action arises, the agent must necessarily retract hisinitial expectation that the effects caused by the action would take place, making theunderlying reasoning machinery non-monotonic (see section 1.2 for a discussion onthe qualification problem).2.
Many solutions to the frame problem can only succeed under some precise assumptions.
For instance, • Actions always succeed. This is the action omniscience assumption. More pre-cisely, this assumption dictates that the qualification problem is skipped. Thisis the case with all monotonic solutions to the frame problem. • Fluents change if and only if the reasoner knows that there exists an actionthat possibly changes its value. This can be termed as domain omniscience
2. The argument that any solution to the frame problem which works with nondeterministic action is notsubject to this assumption does not stand. The quick fix of allowing such an approach to a fortiori expressactions that may fail by representing failure as a possible effect is an invalid one. It is because we canno longer infer that, in the absence of evidences that suggest otherwise, actions would normally succeed.It’s also worth noting that Lin’s (1996) extension of Reiter’s (1991) solution to the frame problem in theSituation Calculus to deal with nondeterministic action is based on circumscription. easoning about Action: An Argumentation-Theoretic Approach assumption. It assumes that the reasoner has complete (ontological) knowledgeof the domain about which he is reasoning.The above two reasons are of course closely related as the former arises due to theunderlying assumptions in the latter which no longer holds once the qualification problemis taken into consideration.In the remainder of this section, we review several works on this topic before introducingthe reader to our approach.
In the late 1960s, the frame problem had been recognised as a major obstacle to formalisingdynamic domains (see the discussions and exposition by McCarthy & Hayes, 1969; Green,1969). Several alternative responses to the frame problem have been proposed along way.To respond to the explosive number of axioms required for theorem proving-based plannersas proposed by Green (1969), Fikes and Nilsson (1971) introduce procedures that operateon special data structures used to represent dynamic domains. However, for complex andsophisticated problem domains, e.g. those with domain constraints, concurrent actions,observations at different time points, etc. STRIPS quite often fails to express the domainknowledge. In fact, the expressivity of STRIPS is quite limited as has been pointed outby Lifschitz (1987). Another response attributing the frame problem as an artefact ofthe situation calculus has proved to be ungrounded. There are attempts to distinguisha logical or epistemological aspect of the frame problem from the computational aspect(e.g., McDermott, 1987; Kowalski, 1992). While the computational inefficiency associatedwith a representation of dynamic domains in the situation calculus can be attributed tothe explosive number of global situations required by the situation calculus as argued byKowalski and Sergot (1986), the logical aspect of the frame problem is inherent to anylogic-based representation of dynamic domains. It is thus essential that a logical approachto AI and knowledge representation have a decent solution to the frame problem.Later, with the introduction of the qualification problem by McCarthy (1977), it isreckoned that formalising dynamic domains not only is about solving the frame problembut would require systematic studies to fundamental issues of knowledge representation. Inthe early 1980s, the frame and the qualification problem were considered to be instancesof commonsense reasoning problems. In particular, many believed that a non-monotonicreasoning framework would solve the frame problem. It was argued that the ‘principle ofinertia’ which is considered to be the key to the frame problem can be formalised in termsof default rules or default axioms in default reasoning. Moreover, it was also argued that tosolve the qualification problem, the following common sense law should be rendered: “anaction, by default, would qualify to succeed and bring about the intended effects unless thereis known reason for it not to,” in formalisations of dynamic domains.With the introduction of several non-monotonic reasoning formalisms, e.g. truth main-tenance systems (TMSs) by Doyle (1979), default logics by Reiter (1980), circumscriptiveapproaches by McCarthy (1980), modal non-monotonic logic by McDermott and Doyle
3. The principle of inertia or the common sense law of inertia basically states that “By default a fluentis assumed to persist over time unless there is evidence to believe otherwise.”
The reader is referred toShanahan’s (1997) book for more details on this principle and the issues around it. o & Foo (1980), autoepistemic logic by Moore (1985), etc., it was believed that these problems weresolved. The solutions to these problems were illustrated as examples for the proposed non-monotonic reasoning frameworks. For instance, McCarthy (1986) showed how circumscrip-tion is used to solve the frame problem relative to the blocks world domain. Unfortunately,Hanks and McDermott (1987) show that these formalisations do not work correctly in asimple dynamic domain known as the Yale Shooting Problem (YSP). We will now reviewsuccessful attempts to solve the frame problem:1. Baker (1989) successfully modifies the original (and incorrect) circumscription policyproposed by McCarthy (1986) to deal with the Yale Shooting Problem. In the tradi-tional circumscriptive policy, the predicate
Abnormal is minimised with the predicate
Holds allowed to vary. Baker suggests that, instead of allowing
Holds to vary, thefunction
Result should be the one to be varied. While this does not solve the frameproblem in its full generality, this initiates the line of research which brings manyfruitful results to reasoning about action community. For a more detailed discussionabout these solutions and the follow up works, the reader is referred to Shanahan’s(1997) book. Furthermore, Foo, Zhang, Vo, and Peppas (2001) present an exposi-tion on the issue from an automata and system theory point of view. In order forBaker’s (1989) solution to work correctly, additional axioms need to be introduced,e.g. domain closure axioms, axioms about the existence of situations, etc. This em-phasises that: (i) the circumscriptive approach to reasoning about action only worksunder careful designation and considerations of the domain; and more importantly,(ii) circumscription is domain dependent. That is, a domain dependent circumscrip-tive policy is required to correctly render the common sense of a particular problemdomain.2. A number of researchers argue that in many cases, a monotonic solution to the frameproblem will be sufficient. Pednault (1989) assumes that the effect of actions on fluentsare specified by effect axioms of the following forms: ε + F ( ~x, ~y, s ) ⊃ F ( ~x, do ( A ( ~y ) , s )) , (1) ε − F ( ~x, ~y, s ) ⊃ ¬ F ( ~x, do ( A ( ~y ) , s )) , (2)Here, A ( ~y ) and F ( ~x, s ) are the parameterised action and fluent, respectively; ε + F ( ~x, ~y, s )and ε − F ( ~x, ~y, s ) are first order formulas whose free variables are among ~x, ~y, s . Pednault(1989) makes the following Causal Completeness Assumption : The axioms (1) and (2) specify all the causal laws relating the action A and the fluent F . Note that the Causal Completeness Assumption is a stronger form of the domainomniscience assumption presented above. Under this assumption, the following frameaxioms can be introduced: F ( ~x, s ) ∧ ¬ ε − F ( ~x, ~y, s ) ⊃ F ( ~x, do ( A ( ~y ) , s )) . easoning about Action: An Argumentation-Theoretic Approach and ¬ F ( ~x, s ) ∧ ¬ ε + F ( ~x, ~y, s ) ⊃ ¬ F ( ~x, do ( A ( ~y ) , s )) . Schubert (1990), elaborating on a proposal of Haas (1987), employs the so-called
Explanation Closure Axioms of the following forms: F ( ~x, s ) ∧ ¬ F ( ~x, do ( A, s )) ⊃ α F ( ~x, A, s ) , (3) ¬ F ( ~x, s ) ∧ F ( ~x, do ( A, s )) ⊃ β F ( ~x, A, s ) , (4)Or, equivalently, we can rewrite the above two axioms as follows: F ( ~x, s ) ∧ ¬ α F ( ~x, A, s ) ⊃ F ( ~x, do ( A, s )) . and ¬ F ( ~x, s ) ∧ ¬ β F ( ~x, A, s ) ⊃ ¬ F ( ~x, do ( A, s )) . Schubert’s proposal is correct under the following assumption, called the
Explana-tion Closure Assumption : α F completely characterises all those actions A that can cause the fluent F ’s truth value to change from true to false ; similarly for β F . Reiter (1991) then combines the merits of the above two proposals by systematicallygenerating the frame axioms as proposed by Pednault (1989) with the quantifiers overthe set of actions as proposed by Haas (1987) and Schubert (1990).Other researchers who also propose monotonic solution to the frame problem includeCastilho et al. (1999), and Zhang and Foo (2002).3. Attempts to solve the frame problem using default logic (Reiter, 1980) also encountersome problematic issues. Hanks and McDermott (1987) show that a natural formu-lation of the Yale Shooting Problem in default logic suffers the same problem as thatwith circumscriptive approaches, viz. the existence of anomalous extensions. Morris(1988) proposes a slight modification on Hanks and McDermott’s original formulationto the Yale Shooting Problem in an attempt to avoid the anomalous extensions. Aspointed out by Turner (1997), Morris’ formulation is complete, and thus eliminates theanomalous extensions in the Yale Shooting Problem, but unsound. More importantly,from Morris’ formulation it is not clear how dynamic domains should be formulatedin general. Turner (1997) himself then proposes a way to formalise dynamic domainsusing default logic (and also logic programming). His solution is based on the fol-lowing observation: In the Yale Shooting domain and similar dynamic domains, theanomalous extensions arise because undesired effects of an action can be derived byreasoning “backward in time.” For instance, in the Yale Shooting domain, by making o & Foo the counterintuitive supposition that the victim of the shooting is somehow still aliveafter the shooting, an anomalous extension come up in the following way. First, itallows the default saying that the victim persists to be alive regarding the shootingaction to be applicable. As a consequence, the gun must not be loaded before theshooting action. Therefore, it blocks the application of the default saying that thegun persists to be loaded regarding the waiting action. In other words, the loadedgun would get unloaded (magically) during the waiting action which is an undesir-able conclusion. To block these lines of “backward” reasoning, Turner appeals to thenon-contrapositivity of inference rules and replaces the implications by inference rules.To guarantee that this formulation work correctly, additional techniques are requiredsuch as a fact ϕ will be formulated as an inference rule: ¬ ϕ false and enforcing the completeness of the initial situation by adding the following rules:: Holds ( f, S ) Holds ( f, S ) : ¬ Holds ( f, S ) ¬ Holds ( f, S )for every fluent f .However, Turner’s (1997) formulation is still fairly ad hoc as different techniques areadded to fix the known issues. For example, the inference rules are used in place ofimplications to block the application of the undesirable “backward” reasoning, therules completing the initial situations are added to overcome the unsoundness issue inMorris’ (1988) formulation, etc. This also shows the problematic side of default logic asa uniform formalisation to various problems of common sense reasoning. This becomesa serious issue if one proceeds with the question of how the qualification problem, atypical problem of default reasoning, is solved in Turner’s (1997) formalisation ofdynamic domains. This is the case, for example, when instead of asserting that thevictim dies whenever it is shot by a loaded gun the reasoner can only maintain that asa default proposition as there may be many hidden possible conditions under whichthe victim may not die. Thus the reasoner is able to deal with ‘surprising’ situationsin which the victim is observed to be still alive after the shooting action (of a loadedgun). Another, symmetric, case is with ‘surprises’ regarding the persistence of fluents.For instance, after a waiting action, which is not supposed to unload a gun, the gun,which was loaded before the wait action, is observed to be unloaded. Such a scenariowas first introduced by Kautz (1986) in a scenario called the Stolen Car Problem.In both cases, reasoning “backward in time” is necessary. It is not clear how thesewill be rendered in Turner’s formulation which explicitly intends to block “backward”reasoning. These scenarios will be analysed in the solution we present later in thispaper. While there have been several solutions to the qualification problem, none of these addressedthe original qualification problem introduced by McCarthy (1977) and later formalised byGinsberg and Smith (1988). easoning about Action: An Argumentation-Theoretic Approach
1. Lin and Reiter (1994) propose a formalisation for action theories in the situation cal-culus (SC). Their formalism is an extension of Reiter’s (1991) solution to the frameproblem (sometimes) by incorporating state constraints. They discover that there areat least two different kinds of state constraints which they call ramification and quali-fication constraints. They then go further to claim a solution to the qualification (andthe ramification) problem. The basic idea behind their solution to the qualificationproblem is that certain state constraints imply implicit preconditions of some actions.Thus an action may not be qualified even though it appears (from the explicit actiondescription) to be. While this is of course a special case of the qualification prob-lem, the classical qualification problem as introduced by McCarthy (1977) has a muchbroader extent. In this setting, the qualification problem is more a pragmatic issuethan a technical issue. Similar to the frame problem, it is impractical, and sometimesimpossible, to axiomatise all the possible preconditions of an action. For example, inaddition to the requirement that the gun be loaded, to guarantee that performing theaction shoot would kill the victim, many preconditions must also be included such as:the gun is not malfunctioning, the shooter does not miss the victim, the victim doesnot wear a bullet-proof jacket, etc. among which some may be very improbable suchas: “no alien interferes with the bullet.” The reasoner simply does not want to con-sider these conditions by assuming that they are not the case unless there are explicitevidences stating otherwise. In other words, the qualification problem in its originalform requires that the reasoner be able to tolerate the mistaken conclusions possiblyjumped to by previous inferences and to correct them appropriately. Henceforth, wewill always refer to the qualification problem in this original form. It is this similarityto the frame problem that led John McCarthy to conjecture that:The frame problem may be a sub-case of what we call the qualificationproblem , and a good solution of the qualification problem may solve theframe problem also. (McCarthy, 1977, p. 1040, italics are original .)Roughly 10 years after his introduction of the qualification problem, McCarthy (1986)presented his solution to the problem using his non-monotonic formalism of circum-scription. However, this solution suffers an almost identical flaw as its counterpartregarding the frame problem: simple minimisation of abnormalities sanctions anoma-lous models (e.g., Thielscher, 2001).2. McCain and Turner (1995) propose a solution to the problem described by Lin andReiter (1994), viz. the problem of deriving the implicit preconditions from stateconstraints. McCain and Turner’s solution is posed in a model-based representationof action theories.3. Similar to McCain and Turner’s (1995) result, Baral (1995) offers a solution to theproblem defined by Lin and Reiter using a state-based representation. Baral extendsthe language of disjunctive logic programs for state specification and as an actiondescription language. o & Foo
4. Doherty and Kvarnstr¨om (1998) make a careful investigation to the qualification prob-lem. They are aware of the shortcomings present in the definition of the “qualificationproblem” introduced by Lin and Reiter (1994). They proceed one step further to dis-tinguish between the weak and strong forms of the qualification problem. To dealcomprehensively with the qualification problem in its full extent, Doherty and Kvarn-str¨om apply circumscription on a predicate which plays a similar role to the predicate
P oss used by Lin and Reiter (1994).Even though Doherty and Kvarnstr¨om’s (1998) solution is closest in spirit to theoriginal form of the qualification problem, there is still a serious problem with theirapproach. The intended designation on predicate
P oss and its variants is action-oriented. That is, it would qualify on the executability condition for the action underconsideration, not towards the effects that action is supposed to cause. In otherwords, only circumscribing
P oss does not guarantee to capture the full extent ofthe qualification problem. For example, a single action of shooting a gun may causeseveral effects: killing the victim, making a loud noise, emptying the cartridge, etc.The conditions for such an action to be executable is: having the gun, the gun is notbroken, the gun is loadable, etc. Once the action is executable, it is not necessarythat all effects will take place. It may be the case that there is a loud noise and thecartridge is emptied but the victim is still alive since the victim was wearing a bullet-proof jacket. Assuming that the reasoner is somehow aware about this possibility,should he include the requirement that the victim not wear a bullet-proof jacket as aqualification for the action shoot? Perhaps he should not for he would still expect tohear a loud noise and the cartridge to be emptied after the shoot action.These are not the end of all the troubles though. The presence of both qualificationand ramification constraints causes several complications. Firstly, they are not syntac-tically distinguishable. Secondly, as has been mentioned by Doherty and Kvarnstr¨om(1998), qualification constraints may cause indirect effects to arise and vice versa, i.e.ramification constraints may reveal implicit action preconditions.
Remark:
The terms ramification constraints and qualification constraints were firstintroduced by Lin and Reiter (1994) when a careful examination to state constraintswas taken. As discussed by Doherty and Kvarnstr¨om (1998), these two kinds of con-straints might interact in several ways. Consider the example introduced by Dohertyand Kvarnstr¨om (1998): the only preconditions of the action board a plane are having-ticket and at-gate . However, to a passenger who places a gun into his pocket at homebefore travelling to the airport and proceeding to the gate, a new qualification forthe action board materialises. Because one ramification of putting an object in yourpocket is that it will stay with you as you travel from location to location (i.e. result ofa ramification constraint ), a reasoner could easily conjecture that our passenger failsto board the plane. On the other hand, the fact that a passenger who possesses a gunwhen trying to board the plane must fail to board the plane is a result of a qualificationconstraint . Now, not only the fact that a passenger possesses a gun disqualifies theaction of boarding a plane but it also brings about an indirect effect when the actionof boarding a plane is executed: the passenger being put under arrest. Note that forthis indirect effect to take place, both requirements must be present: the action of easoning about Action: An Argumentation-Theoretic Approach boarding a plane being executed, and the above qualification constraint is present. Inother words, a qualification constraint might also bring about indirect effects.We believe that all these problems are too sophisticated for any circumscription pol-icy to successfully address in most situations. Furthermore, such a policy would beextremely hard to understand and error-prone. Recall the failure of non-monotonicreasoning formalisms regarding the frame problem (in its simplest form viz. withoutthe ramification and the qualification problems). Researchers had failed to point outthis bug for several years before Hanks and McDermott discovered it in their awardwinning paper (1987).5. More recently, Thielscher (2001) gives an exposition on the qualification problem.Thielscher discusses the problem sustained by McCarthy’s (1986) simplistic circum-scription policy, viz. the anomalous models. He then introduces a default logic basedformalisation for the qualification problem in the Fluent Calculus and shows thathis formalisation does not suffer the problem of anomalous models. Note that inThielscher’s formalisation, circumscription is still required to generate the initial the-ories of the default theories (in addition to the set of default rules). Nevertheless,Thielscher’s solution still suffers the following drawbacks: (i) Thielscher’s use of thepredicate
P oss is in the same way as has been formulated by Doherty and Kvarnstr¨om(1998). Thus, qualifications are taken over the executability conditions for the actionsrather than over different effects of the actions; and (ii) while Thielscher shows thatthe problem of anomalous models sustained by McCarthy’s (1986) circumscriptionpolicy is overcome in his formalisation, it’s not entirely clear whether Thielscher’sformalisation which is based on both circumscription and default logic will not sufferfrom other anomalies.
In the context of reasoning about action, the ramification problem is mainly related toindirect effects. Finding a solution to this problem may not be easy as indirect effectsindicate exceptions to frame assumptions and require special treatment. While there havebeen several formalisms dealing with the ramification problem, e.g., see (Lin, 1995; McCain& Turner, 1995; Thielscher, 1997), there are still several issues that need a more carefulconsideration. We consider three examples to motivate our discussion.
Example 1
Consider Thielscher’s (1997) circuit:This example is interesting because it gives a counterexample for the minimalistic ap-proaches e.g. in the work of McCain and Turner (1995). In this domain, the intendedrelationship between relay and sw is that when relay is on, it would make sw jump off.Thus, when sw and sw are both closed, sw can not be also closed as that is preventedby relay . However, there is certainly a duration (no matter how short it is) before sw is forced to jump off by relay . In the state given in Figure 1, after performing the actionof closing switch sw , two next states are equally possible: one in which detect is on, inanother it is off. Only the latter is sanctioned in a minimalistic account. Through this
4. The reason for nondeterminism in this case is due to insufficiency of domain information: depending onthe sensitivity of relay , light and detect , when light could get lit quickly and detect is very sensitive to o & Foo -sw1 sw2sw3-relay -light -detect Figure 1: Thielscher’s circuitexample, Thielscher pointed out the need for keeping track of the chains of applications ofindirect effects.Thielscher (1997) proposes a way to remedy this problem by keeping track of the appli-cations of the domain constraints which are re-expressed in terms of causal relationships.Thus, given the above example, his formalism is able to arrive at a next state in which the detect is on. Following such chains of causal relationships, the dynamic system undergoesseveral intermediate states before arriving at the next state. ✷ In this paper, we proceed one step further from Thielscher’s (1997) position by formallyrepresenting the intermediate states as possible states of the world. We believe that anintelligent agent should be able to reason about these intermediate states even thoughthey may not satisfy all domain constraints. This capability is especially important if thereasoner needs to explain certain observations about the world in a systematic way. Wenote that given an observation, there may be several chains of causal relationships thatbring about that observation. Unless intermediate states are explicitly represented andreasoned about, there is no way for an agent to have a full insight to the system in handand certain information would be missing.
Example 2
Consider Lin’s (1995) spring-loaded suitcase with two latches. Let’s assumethat the latches can be toggled if the suitcase is closed. The following state constraint issupposed to apply in this domain: up ( Latch ) ∧ up ( Latch ) ⊃ open ( Suitcase ).The question is: how does a robot close the above suitcase back after opening it?
McCainand Turner (1997) also consider this problem and their answer is: detect any glimpse of light and relay is not sensitive enough to make the switch sw jump off quicklyenough then detect will be on; otherwise it will stay off.5. Note that this point of view also corresponds to the traditional definition of states as snapshots of theworld.6. For example, given detect is not on in the next state, it can be that either the light has never been brightor the light may have been bright but the detect is not sufficiently sensitive to detect its momentarybrightness. easoning about Action: An Argumentation-Theoretic Approach In general, when both latches are up, it is impossible to perform only the actionof closing the suitcase; one must also concurrently toggle at least one of thelatches. (McCain & Turner, 1997, p. 464, italic is original .)The problem now is how to represent the action of holding the suitcase closed suchthat it would overcome the above indirect effect caused by the loaded spring. This alsosuggests another kind of actions whose direct effects are to keep the world unchanged.These actions have usually been formalised by other researchers as fluents, e.g. holding .Our main objection to this approach is that agents also need to reason about these actionssince they may also require certain preconditions such as the agent is strong enough tohold the object. Moreover, under certain (abnormal) circumstances, agents may also fail toperform such actions. That is, these actions are also subject to the qualification problemdiscussed in the previous subsection. ✷ Example 3
Consider the circuit in Figure 2: sw2-sw1 -relay1-relay2 Figure 2: A dynamic domain with a (potentially) infinite sequence of indirect effectsIt is quite obvious that after performing the action f lip whose direct effect is having sw closed, the following circular sequence of indirect effects will take place: { relay , ¬ relay } →¬ sw → {¬ relay , relay } → sw → { relay , ¬ relay } . This sequence of course wouldpotentially carry on the above sequence of indirect effects indefinitely unless sw is flippedopen or some device stopped functioning correctly, e.g. when the battery is out of charge.In other words, this action domain requires some action to be inserted in between a seriesof on going indirect effects which can not be captured by the above representation. Notealso that none of the causation-based representations proposed by Lin (1995), McCain andTurner (1995) or Thielscher (1997) is able to deal with the above action domain. ✷ To address the problems discussed in the previous sections, we argue that in order to find auniform solution to these problems one should avoid cryptic formalisms whose consequences
7. This example is an instance of the so-called “vicious cycles” scenarios, e.g., see (Shanahan, 1999). o & Foo can not be seen clearly from the formalisation of the problem domains. As a consequence,we propose a uniform non-monotonic solution to the main problems of reasoning about ac-tion. Essentially, when performing commonsense reasoning, the reasoner relies on a numberof plausible assumptions, e.g., assuming that an instance of birds flies, or assuming thatshooting a turkey with a loaded gun causes it to die, etc. In traditional default reasoningformalisms such as circumscriptive approaches or default logic, these assumptions are madeimplicit. For example, these are the instances of predicates which are minimised away bycircumscription or the implicitly asserted justifications in default rules when they are stillconsistent with the extension under consideration in default logic. The proposed represen-tation formalism aims at making these assumptions explicit so that an automated reasoneris conscious (at least) about what assumptions it relies on when performing reasoning.Then the reasoner can always manipulate these assumptions independently of each other.It is also the basic idea of assumption-based frameworks which are at heart of Bondarenko,Dung, Kowalski, and Toni’s (1997) argumentation-theoretic approach.We then proceed to consider the ramification problems and domain theories with concur-rent and non-deterministic events. Among the major results, we show that our frameworkcaptures the essence of the causation-based approaches regarding the ramification problem.Moreover, we also show the expressiveness of our formalism through two examples in whichindirect effects also need qualifications and infinite sequence of indirect effects. To the bestof our knowledge, none of the existing formalisms are able to cope with these scenarios.Based on the basic idea of assumption-based frameworks, our approach comprises thefollowing major aspects of representation:1. We introduce different types of assumptions to render various laws of common sensein dynamic domains. For instance, frame assumptions are introduced to capturethe common sense law of inertia whilst (two types of) qualification assumptions areintroduced to overcome the qualification problem.2. We introduce a special class of (system-generated) dummy actions to allow the ex-planation problem, i.e. when some actions or events occur outside of the reasoner’sknowledge, to be dealt with in a uniform manner.3. Being based on Bondarenko et al.’s (1997) argumentation-theoretic framework, ourapproach makes use of the inference rules to represent domain knowledge.4. Lying at heart of our approach is an argumentation-theoretic semantics, called plausi-bility semantics , which is argued to best render common sense knowledge in dynamicdomains. This semantics consists in a particular policy of resolving conflicting as-sumptions when computing the argumentation to be accepted.To summarise, in this paper we formalise an expressive representation scheme in orderto cope with sophisticated action domains. We believe that such a formalisation sometimesrequires certain advanced knowledge to be encoded in a precise and well-engineered way.The representation of action theories proposed in this paper can be considered as the in-termediate level between commonsense and scientific knowledge. The expressiveness of theformalism is improved through several independent steps by adding further assumptions easoning about Action: An Argumentation-Theoretic Approach into the domain descriptions. This also shows one advantage of our solution: a simple rep-resentation can be achieved by simply removing the involved assumptions. This is arguablya desirable feature as the reasoner has the option of either increasing the expressibility of therepresentation formalism or improving the simplicity and, as a consequence, the efficiencyof the reasoning system.The rest of the paper is organised as follows: Section 2 summarises relevant features ofthe abstract argumentation framework proposed by Bondarenko et al. (1997), its seman-tics and concrete instances. In Section 3 we present the syntax and semantics of the basictemporal logic and the extension for reasoning about action. In Section 4 we present ourformalisation for reasoning about action based on the argumentation-theoretic approachintroduced by Bondarenko et al. (1997). Our approach to reasoning about action, in par-ticular a uniform solution to the frame and the qualification problems, as well as the mainresults of the paper are presented in Section 5. In Section 6, we show how the proposedformalism is extended to deal with more complex dynamic domains, including those withconcurrent and non-deterministic events, and indirect effects. Related work and futureresearch directions are discussed in Section 7. We will defer most proofs of the resultspresented in the paper to the Appendix.This paper is an extended version of two earlier conference papers (Vo & Foo, 2001,2002). The main differences are that in this version all proofs are included, lemmas that areused in the proofs of the theorems are introduced to help the reader more easily comprehendthese results, and the presentation has been improved and extended with more examples ofthe various constructions.
2. Defeasible Reasoning by Argumentation
Let a deductive system hL , Ri be given, where L is some formal language with countablymany sentences and R is a set of inference rules. Given a theory Γ ⊆ L and a sentence α ∈ L , we write Γ ⊢ hL , Ri α if there is a deduction from Γ whose last element is α . T h hL , Ri (Γ)denotes the set { α ∈ L | Γ ⊢ hL , Ri α } . Since the language L is generally kept fixed whereasthe set of inference rules R is likely to vary depending on the description of the domain,when there is no possible confusion we will abbreviate ⊢ hL , Ri and T h hL , Ri as ⊢ R and T h R ,respectively. Thus the classical inference relation ⊢ can also be written as ⊢ R C where R C isthe set of inference rules of classical propositional logic. Note also that every set of inferencerules considered in this paper will be a super set of R C .Given a deductive system hL , Ri , an assumption-based framework with respect to hL , Ri consists of a theory Γ representing the current knowledge of the reasoner about the domain,an assumption base AB and a contrariness operator − , i.e. given an assumption δ ∈ AB , δ denotes the contrary of δ . Remark:
The notion of the contrary of an assumption is intended to generalise the classicalnegation ¬ δ . Note that in general assumptions may be constructed by special operators(e.g. negation-as-failure in the case of logic programming, or the modal operator L in thecase of autoepistemic logic, Moore, 1985), thus the contrariness operator must also be suf-ficiently general. o & Foo The hardest part in reasoning with assumption-based frameworks is computing the setof assumptions to augment the given theory Γ. In an argumentation-theoretic approach,this is realised by the attack relation. To determine which assumptions to be accepted,assumptions are put together to form arguments. The assumptions behind the best argu-ments are considered to be acceptable. Several semantics for best arguments are presentedby Bondarenko et al. (1997) based on the notions of attack: Given an assumption-basedframework h Γ , AB , − i and an assumption set ∆ ⊆ AB : • ∆ attacks an assumption δ ∈ AB iff δ ∈ T h (Γ ∪ ∆). • ∆ attacks an assumption set ∆ ′ ∈ AB iff ∆ attacks some assumption δ ∈ ∆ ′ . • ∆ is closed iff ∆ = AB ∩
T h (Γ ∪ ∆). • ∆ is conflict-free iff there does not exist any δ ∈ AB such that Γ ∪ ∆ ⊢ R δ, δ .Assumption-based frameworks in which assumption sets are always closed are referredto as flat . In a flat assumption-based framework, the conflict-free property of a set ofassumptions ∆ is equivalent to the property that ∆ does not attack itself. The majorargumentation-theoretic semantics defined by Bondarenko et al. (1997) for assumption-based frameworks include: • Stability semantics : an assumption set ∆ ⊆ AB is stable iff1. ∆ is closed,2. ∆ does not attack itself, and3. ∆ attacks each assumption δ / ∈ ∆.Bondarenko et al. (1997) show that the above stability semantics corresponds to thestandard semantics of extensions of Theorist (Poole, 1988), minimal models of (manycases of) circumscription (McCarthy, 1980, 1986), extensions of Default Logic (Reiter,1980), stable expansions of Autoepistemic Logic (Moore, 1985), and stable models oflogic programming. In other words, from a complexity-theoretic perspective, anyapproach based on the existing formalisms to default reasoning can be rendered in acorresponding assumption-based argumentation framework with no loss in terms ofcomputational complexity. • Admissibility and Preferability semantics : Bondarenko et al. (1997) go further toextend these existing formalisms by generalising the semantics of admissible and pre-ferred arguments which were originally proposed for logic programming only. Thenew semantics are defined in terms of “admissible” and “preferred” sets of assump-tions/extensions. An assumption set ∆ ⊆ AB is admissible iff1. ∆ is closed,2. ∆ does not attack itself, and3. for all closed sets of assumptions ∆ ′ ⊆ AB if ∆ ′ attacks ∆ then ∆ attacks ∆ ′ .Maximal (with respect to set inclusion) admissible assumption sets are called preferred . easoning about Action: An Argumentation-Theoretic Approach Throughout this paper assumptions are expressed in terms of usual propositions. Thus,we will replace the notion of contrariness − in Bondarenko et al.’s (1997) system with theclassical negation ¬ and omit it from the specification of assumption-based frameworks.That is, an assumption-based framework h Γ , ABi consists of a theory Γ ⊆ L , and theassumption base AB which contains the assumptions to be used in the reasoning.
3. Domain Descriptions
We introduce a propositional action description language based on a more comprehensiverepresentation formalism proposed by Sandewall (1994). In particular, we extend Draken-gren and Bj¨a reland’s (1999) language so that it is possible to describe narratives in ourframework.
Following Sandewall (1994), the underlying representation of time is a ( discrete ) time struc-ture T = h T , <, + , −i consisting of • a time domain T whose members are called time points which are integers in thispaper (except in a later part of the paper where the distinction will be made explicit); • <, + , − are as usual for integers.Given a time structure T = h T , <, + , −i , a signature σ with respect to T is a tuple= hT , F , Ai , where T is a set of countably infinitely many time-point variables , F is a setof propositional fluent names , and A is a set of action names . Since the time structure T isfixed in the rest of this paper, T will be taken implicitly whenever a signature is introduced.We assume that all sets in σ are countable. We denote F ∗ = { [ ¬ ] f | f ∈ F } . A memberof F ∗ is a fluent literal . Moreover, A = A ∪ DA , where A is the set of domain dependentaction names, called basic actions , e.g. load , shoot , etc. and DA = { da l | l ∈ F ∗ } is the setof dummy actions . As will be explained later in this paper, our solution to the problemsof reasoning about action is based on the basic guideline of attributing changes to events.Given the reasoner’s ignorance about certain events that bring about changes in the world,the dummy actions are to be used to make up for these gaps in the reasoner’s belief state.That is why we need to associate the dummy actions with the fluent literals from F ∗ .For each fluent literal l ∈ F ∗ , we introduce the following two symbols: AQ l , and F A l : • AQ l is associated with the assumed qualifications upon the preconditions of an actionregarding the fluent literal l . Essentially, AQ l when used in the description of thedynamics of an action α allows the reasoner to describe only the main preconditionsof α (with regards to the fluent literal l ) while leaving other possible (but less probable)qualifications to be rendered by a single assumption AQ l . • F A l is associated with the frame assumptions regarding l . F A l , when coupled witha particular frame inference rule , allows the reasoner to infer that the fluent literal l continues to hold in future time points unless there is a reason that defeats F A l .
8. The notation [ ¬ ] means that the formula following it may, or may not, be negated. o & Foo Given a set of fluent literals Γ ⊆ F ∗ , we denote F A Γ def = { F A l | l ∈ Γ } and AQ Γ def = { AQ l | l ∈ Γ } .A time-point expression is one of the following: • a member of T , • a time-point variable in T , • an expression formed from time-point expressions using + and − . For convenience,we will also write τ + and τ − instead of τ + 1 and τ −
1, respectively.We denote the set of time-point expressions by
T E . Definition 3.1
Let a signature σ = hT , F , Ai be given and τ, υ ∈ T E , f ∈ F , α ∈ A , R ∈ { = , < } , ⊗ ∈ {∧ , ∨ , → , ↔} . Define the basic (domain description) language Λ over σ by: Λ ::= true | false | f | τ R υ | ¬ Λ | Λ ⊗ Λ | [ τ ] Λ ,Λ ::= Λ | [ τ, υ ] α | ¬ Λ | Λ ⊗ Λand the assumption base AB by: AB = AB AQ ∪ AB F A , where AB AQ = { [ τ, υ ] AQ l | τ, υ ∈ T E and l ∈ F ∗ } , and AB F A = { [ τ ] F A l | τ ∈ T E and l ∈ F ∗ } .The domain description language L D (over σ ) is defined: L D = Λ ∪ AB . [ τ, υ ] α means the action α has a duration corresponding to the interval [ τ, υ ]. [ τ, υ ] AQ l means the fluent literal l is assumed to be qualified to hold by the end of the interval [ τ, υ ].[ τ ] F A l means the fluent literal l is assumed by default to persist from the time point τ to the next, i.e. the principle of inertia . Notice the difference between [ τ ] F A l and [ τ ] l for some fluent literals l , l ∈ F ∗ . [ τ ] l indicates that the fluent literal l holds at τ while[ τ ] F A l indicates that the fact that the fluent literal l persists during the interval [ τ, τ + ]is true.For example, in the blocks world domain, to say that block A is on block B at the timepoint 2, we write: [2] on ( A, B ); or, to say that an action pickup the block A occurs betweentime points t + 3 and t − < holds between t + 5 and t , we write[ t + 3 , t − pickup ( A ) ∧ ( t + 5 < t ).A formula that does not contain any connectives (i.e. ∧ , ∨ , → , ↔ , ¬ , and [ . ]) is atomic .If γ is atomic and τ ∈ T E , then the formulas γ , [ τ ] γ , ¬ γ , ¬ [ τ ] γ , and [ τ ] ¬ γ are literals .Let γ be a formula. A fluent f ∈ F occurs free in γ iff it does not occur within thescope of a [ τ ] expression in γ . τ ∈ T E binds f in γ if a formula [ τ ] ψ occurs as a subformulaof γ , and f is free in ψ . If no fluent occurs free in γ , γ is closed . If γ does not contain anyoccurrence of [ τ ] for any τ ∈ T E , then γ is propositional .
9. It would be more precise to denote the domain description language over σ by L σ . However, as thesignature is usually clear from the context and in order to avoid the mention of σ every time we have toformalise something with the domain description language, we choose to denote it by L D . easoning about Action: An Argumentation-Theoretic Approach Let σ = hT , F , Ai be a signature. A state over σ is a function from F tothe set { true , false } of truth values. A history over σ is a function h from T to the set ofstates. A valuation is a function φ from T E to T . A narrative assignment is a function η from T × A × T to the set { true , false } . In addition, we define ε q : T × AQ F ∗ × T → { true , false } and ε f : T × FA F ∗ → { true , false } . An interpretation over σ is a tuple h h, φ, η, ε q , ε f i where h is a history, φ is a valuation, η is a narrative assignment and ε q , ε f are defined as above. Example 4
Consider Hanks and McDermott’s (1987) Yale Shooting Problem (YSP) :There are three possible actions: load (the gun), wait , and shoot (the victim with thegun). Normally, waiting does not cause any change in the world, but shooting leads to thevictim’s death, provided that, of course, the gun is loaded. Assume that all three actionsare performed, in the given order.We define the signature σ ysp to be a tuple h{ t, t , t , . . . , u, u , u , . . . } , { loaded, alive } , { load,wait, shoot }i . Then the Yale Shooting problem can be formulated in the domain descriptionlanguage L ysp as the following theory: Γ ysp , = { [0] alive, [0 , load, [1 , wait, [2 , shoot } .The following two histories h and h are corresponding to the well-known models inthe literature of reasoning about action: h to the intended model and h to the anomalousmodel most frameworks would produce. h AL AL AL AL h AL AL AL AL
Figure 3: The two histories for the YSP action description.Each oval in Figure 3 represents a state over σ ysp . A narrative assignment complyingwith the above action description would map the three tuples (0 , load, , (1 , wait, , shoot,
3) to true and other tuples to false (relative to the assumption that ‘normally,given any action and any time point, there is no instance of that action at that time pointunless specified otherwise’ ). Definition 3.3
Let γ, δ ∈ Λ and I = h h, φ, η, ε q , ε f i an interpretation. Assume τ, υ ∈ T E , f ∈ F , A ∈ A , R ∈ { = , < } , l ∈ F ∗ , ⊗ ∈ {∧ , ∨ , → , ↔} , and χ ∈ { true , false } . Define thetruth value of γ in I for a time point t ∈ T , denoted I ( γ, t ) as follows: o & Foo I ( χ, t ) = χ I ( f, t ) = h ( t )( f ) I ([ τ, υ ] A, t ) = η ( φ ( τ ) , A, φ ( υ )) I ([ τ, υ ] AQ l , t ) = ε q ( φ ( τ ) , AQ l , φ ( υ )) I ([ τ ] F A l , t ) = ε f ( φ ( τ ) , F A l ) I ( τ Rυ, t ) = φ ( τ ) Rφ ( υ ) I ( ¬ γ, t ) = ¬ I ( γ, t ) I ( γ ⊗ δ, t ) = I ( γ, t ) ⊗ I ( δ, t ) I ([ τ ] γ, t ) = I ( γ, φ ( τ ))Two formulas γ and δ are equivalent iff I ( γ, t ) = I ( δ, t ) for all I and t . An interpretation I is a model of a set Γ ⊆ Λ of formulas, denoted I | = Γ, iff I ( γ, t ) = true for every t ∈ T and γ ∈ Γ. A formula γ ∈ Λ is entailed by a set Γ ⊆ Λ of formulas, denoted Γ | = γ , iff γ istrue in all models of Γ. Definition 3.4
Let I = h h, φ, η, ε q , ε f i be an interpretation. The set
Occ I = { ( t, A, u ) ∈ T × A × T | η ( t, A, u ) = true } is called action occurrencedenotation of I .2. The set
F A I = { ( t, F A l ) ∈ T × FA F ∗ | ε f ( t, F A l ) = true } is called F A - denotation of I .3. The set AQ I = { ( t, AQ l , u ) ∈ T × AQ F ∗ × T | ε q ( t, AQ l , u ) = true } is called AQ - denotation of I .
4. Representing Dynamic Domains in the Argumentation-TheoreticApproach
We now proceed to showing an assumption-based framework for representing dynamic do-mains. We subsequently introduce a uniform framework for solving the frame and thequalification problems based on the argumentation-theoretic approach. General solutionsfor the frame and the qualification problems can be obtained by computing plausible setsof assumptions which guarantee that extensions computed from these sets of assumptionswill be consistent when the given theory is consistent. We now introduce some additionalnotations: Given an inference rule r ∈ R , we denote by prem ( r ) and cons ( r ) the premiseand the consequence of rule r , respectively. Definition 4.1
A deductive system hL D , Ri is well-defined iff for each subset S ⊆ R , ifthe set S r ∈ S prem ( r ) is consistent then the set CON S ( S ) = { cons ( r ) | r ∈ S } is alsoconsistent.Henceforth, we will assume that deductive systems are well-defined. Being formalisedin terms of the argumentation-theoretic approach, the representation requires an extendednotion of consistency. Definition 4.2
Let hL D , Ri be a deductive system,(i) a set of sentences Γ ⊆ L D is R - consistent iff Γ R false ; (ii) an assumption-based framework h Γ , ABi with respect to hL D , Ri is consistent iff Γ is R -consistent. easoning about Action: An Argumentation-Theoretic Approach Remark:
Observe that even when hL D , Ri is a well-defined deductive system, consistencyis not equivalent to R -consistency. For instance, let R = { b ¬ a } , the (logically) consistenttheory Γ = { a, b } is not R -consistent. Example 4 ( continued ) Returning to the Yale Shooting problem, the following inferencerules describe the actions of this domain:[ τ, υ ] load [ υ ] loaded ∧ ¬ [ τ ] F A ¬ loaded (5)[ τ, υ ] shoot, [ τ ] loaded ¬ [ υ ] alive ∧ ¬ [ τ ] F A alive (6)[ τ ] loaded, [ τ ] F A loaded [ τ + ] loaded (7)[ τ ] alive, [ τ ] F A alive [ τ + ] alive (8) ¬ [ τ ] loaded, [ τ ] F A ¬ loaded ¬ [ τ + ] loaded (9) ¬ [ τ ] alive, [ τ ] F A ¬ alive ¬ [ τ + ] alive (10)Rules (5) and (6) represent the descriptions of the actions load and shoot , respectively.Action wait does not cause any effect to the world, so there is no need to describe it. Otherrules render the common sense law of inertia: “at any time point, a fluent literal presumablypersists to the next time point.” Most argumentation-theoretic semantics, e.g. stability, admissibility, preferability, com-plete, well-founded semantics, etc. (Bondarenko et al., 1997) are based on the notion ofattack. However, to reason about problem domains with incomplete information, especiallyaction domains, this notion alone may not be sufficient as we may not always be able toconstruct explicit arguments to defeat unsound assumptions. For example, consider theYale Shooting Problem: By observing that a turkey is shot with a loaded gun at time point1, the reasoner infers plausibly that the turkey is dead at time point 2 using the assumptionthat the action shoot is qualified to bring about the effect of killing the victim. However,at time point 2, the reasoner could observe that the turkey is still alive. Existing solutionsto the frame problem, e.g. Reiter’s (1991), Thielscher’s (1997), Castilho et al.’s (1999), etc.fail to deal with such a surprise since they allow a contradiction to be derived. Observe thatthe reasoner does not have any explicit reason to defeat the above qualification assumption,i.e. she is not aware of any cause that prevents the application of this qualification assump-tion. She only knows that it is not acceptable in this case by common sense. To formalisesuch phenomena, we introduce the notion of rejected assumptions. o & Foo
Definition 4.3
Given an assumption-based framework h Γ , ABi , a set of assumptions ∆ ⊆AB rejects an assumption δ ∈ AB iff(a) ∆ is conflict-free, and(b) ∆ ∪ { δ } attacks itself.For instance, in example 4, the set of assumptions ∆ = { [0] F A alive , [1] F A alive , [1] F A loaded } attacks the assumption [2] F A alive . Moreover, relative to the given action description, anyset of assumptions attacks the assumption [0]
F A ¬ loaded . On the other hand, the set of as-sumptions ∆ = { [0] F A alive , [1] F A alive , [2] F A alive } rejects the assumption [1] F A loaded but∆ does not attack it. Observation 1
Given an assumption-based framework h Γ , ABi and a conflict-free set ofassumptions ∆ ⊆ AB , if ∆ attacks an assumption δ / ∈ ∆ then ∆ rejects δ . Then, why do we not generalise the contrariness notion of an assumption so that itwould be general enough to account for all rejected assumptions? The reason is because wewant to isolate the set of assumptions that are rejected but are not attacked as part of oursolution to the frame problem.
Definition 4.4
Given an assumption-based framework h Γ , ABi , a set of assumptions ∆ ⊆AB leniently rejects an assumption δ ∈ AB iff(a) ∆ rejects δ , and(b) ∆ does not attack δ .We denote Lr (∆) def = { δ ∈ AB | δ is leniently rejected by ∆ } .To show that our solution provides an intuitive account for problems of reasoning aboutaction, several scenarios should be considered. These include the projection problem, themost basic form of the frame problem, whose typical example is the infamous YSP. Anotherscenario concerns with the explanation problem which are usually discussed with the StolenCar Problem (Kautz, 1986) and the Stanford Murder Mystery (Baker, 1989). We will firstprovide an informal discussion of our approach through these examples.In the present solution, the frame assumptions are the essence of the principle of iner-tia,and their role in the argumentation approach is illustrated below by the Yale ShootingProblem. In this formulation we intentionally ignore the qualification problem (it is ad-dressed in the next section) to highlight how the frame problem is solved. We now recon-sider the well-worn example YSP to motivate our approach to the frame problem. Example 4 ( continued ) Given the theory Γ ysp , the argumentation-theoretic approach willyield the following preferred set of assumptions (Bondarenko et al., 1997): { [ t ] F A l | t ∈ T and l ∈ { loaded, alive, ¬ loaded, ¬ alive }} \ { [0] F A ¬ loaded , [2] F A alive } ,which corresponds to the intended model of this scenario in which the gun remains loadedat time point 2 and the victim is not alive at time point 3.This extension is also the stable extension and well-founded semantics (Bondarenkoet al., 1997) of the given theory under the argumentation-theoretic approach. Note that in
10. In fact, any set of assumptions containing the assumption [1]
F A loaded would attack [2]
F A alive . easoning about Action: An Argumentation-Theoretic Approach case one would like to be uncertain about whether the gun is still loaded after the shootingaction, one just simply needs to add an axiom: [ τ, υ ] shoot → ¬ [ τ ] F A loaded to dictate thatthe persistence of the fluent loaded after the action shooting is not guaranteed. In thatcase, we can still derive that [ τ ] loaded for τ = 1 ,
2, but we can no longer give a definiteassertion about [ τ ] loaded for τ ≥ ¬ [2] loaded can be (magically) derived, it cannot lead to ¬ [1] F A loaded . Therefore, theset of assumptions corresponding to this case does not satisfy the conditions of preferredset of assumptions, thus ruling out this unintended model. This shows one of the importantfeatures of assumption-based frameworks on its capability of making explicit the assump-tions used by the reasoner during the course of inference. Recall that defaults’ justificationsare accepted as long as they are consistent with some extension for credulous semanticsor all extensions for skeptical semantics (thus the name consistency-based approach.) Inlight of the inertia principle, it’s considered to be abnormal if a fluent does not persistfrom a state to the next state. To minimise the abnormality, (normal) default rules areintroduced to express the fact that if it’s consistent to believe that there is no abnormalitywith respect to a fluent f and an action a in a situation s then assert that. But then wewould fail to distinguish between the abnormalities brought about by reasonable causes andthose unintuitively generated to make them consistent with some possible extension. Thelatter is of course corresponding to the anomalous models. By using explicit assumptions,not only consistency is maintained (by preventing the accepted assumptions from attackingthemselves) but each rejection of assumptions must also be justified by the known factsfrom the given theory. Discussion:
1. Turner (1997) showed that an alternative representation of the YSP in default logiccan help solve the issue of anomalous models introduced by Hanks and McDermott’s(1987) representation. Turner formulates the Yale Shooting scenario as follows: ¬ Holds ( Alive, S ) False (11)
True
Holds ( Loaded, Result ( Load, s )) (12)
Holds ( Loaded, s ) ¬ Holds ( Alive, Result ( Shoot, s )) (13):
Holds ( f, S ) Holds ( f, S ) (14) o & Foo : ¬ Holds ( f, S ) ¬ Holds ( f, S ) (15) Holds ( f, s ) : Holds ( f, Result ( a, s )) Holds ( f, Result ( a, s )) (16) ¬ Holds ( f, s ) : ¬ Holds ( f, Result ( a, s )) ¬ Holds ( f, Result ( a, s )) (17)Notice that Turner also uses the inference rules to block the “backward” reasoningthat generates the anomalous models of the Yale Shooting scenario. However, this alsomeans that all kinds of useful backward reasoning will also be blocked. In other words,Turner’s formulation fails to deal with “surprising” observations about states at latertime points. As a consequence, Turner’s formulation only works when the domain isrestricted to qualification-free. As soon as the action descriptions, e.g. the one forthe shoot action in YSP, need to rely on some default justifications, e.g. qualificationassumptions, Turner’s formulation would also encounter the problem of undesirableextensions. Our approach offers solutions to both of the above issues.2. After Hanks and McDermott’s (1987) seminal paper in which early approaches tothe frame problems were exposed, besides new attempts to solve the frame problem,Sandewall (1994) should be accredited as the first who tried to approach the prob-lems of reasoning about action in a systematic way. As part of this effort, he alsoexamines the reason behind the failure of early approaches to the frame problem. Asdiscussed by Sandewall, early approaches to reasoning about action while attemptingto formulate the inertia principle have made the common mistake of making changesthe abnormality regarding this principle but failing to distinguish between normalchanges triggered by actions and anomalous changes. This important insight turnsout to be a consequence of a much more general law for reasoning about dynamicdomains discovered by researchers in the community in pursuit of solutions to variousproblems of reasoning about actions: “action dynamics are causality-based.” It is thisprinciple that underpins most solutions to the problems of reasoning about action.The anomalous models that arise in early approaches to the frame problem discov-ered by Hanks and McDermott (1987) or to the qualification problem as discussedby Thielscher (2001) are those in which the causes of abnormalities are not present.On the other hand, regarding the ramification problem, given a domain constraintinvolving a number of fluents, it’s important to know which of these fluents are thecauses influencing the other fluents, i.e. the causality direction between the involvedfluents.In light of the above analysis, Turner’s (1997) approach appears to be rather ad hoc.Note that Turner’s solution to the problem of anomalous models is to block “back-ward reasoning” by the use of inference rules without any motivation on why backwardreasoning is a bad thing. While his approach appears to share with solutions basedon chronological ignorance (which will be discussed in more details in Section 7) the easoning about Action: An Argumentation-Theoretic Approach notion of directedness: By minimizing chronologically or blocking backward reason-ing, one tends to minimize causes rather than effects. However, a more systematicapproach to various problems of reasoning about action is still very much desired.Nevertheless, while the preferability semantics copes successfully with the YSP, it cannot properly account for the explanation problem, e.g. the Stanford Murder Mystery (Baker,1989), the Stolen Car Problem (Kautz, 1986). The subtlety lies in the derivation of thecontrary of the frame assumptions. The contrary of a frame assumption is derived onlywhen both the occurrence of the event that brings about the change (absent in the StolenCar Problem) and the preconditions required to be satisfied for the change to actually takeplace (absent in the Stanford Murder Mystery) are explicitly derivable. This is where thenotion of (leniently) rejected assumptions is called into service.
Definition 4.5
Given an assumption-based framework
F A = h Γ , ABi , a set of assumptions∆ ⊆ AB is presumable wrt F A iff(a) ∆ = { δ ∈ AB | Γ ∪ ∆ ⊢ R δ } (in the terms given in Bondarenko et al., 1997, ∆ isclosed),(b) ∆ does not attack itself, and(c) for each assumption δ ∆, δ is rejected by ∆. Definition 4.6
Given an assumption-based framework
F A = h Γ , ABi , a set of assumptions∆ ⊆ AB is plausible wrt F A iff(a) ∆ is presumable, and(b) there exists no ∆ ′ ⊆ AB such that ∆ ′ is presumable and Lr (∆ ′ ) ⊂ Lr (∆).We now proceed to formalising action theories in our framework. Definition 4.7
Let σ = hT , F , Ai be a signature. Assume τ, υ ∈ T E , α ∈ A , Φ ⊆ Λ, and l ∈ F ∗ . A domain description D (over σ ) is a tuple hL D , R , AB , Γ i , where:1. L D is the domain description language and AB is an assumption base over σ ;2. R = R C ∪ R F ∪ R A ∪ R Q , where(a) R C is the set of inference rules of (classical) propositional logic;(b) R F is the set of frame-based inference rules of the form: [ τ ] l, [ τ ] F A l [ τ + ] l , i.e. thosethat represent the frame axioms in terms of inference rules;(c) R A is the set of action descriptions which are inference rules of the form: Φ , [ τ, υ ] α, [ τ, υ ] AQ l [ υ ] l ∧ ¬ [ τ ] F A ¬ l , i.e. those that represent the conditions for the action α to bring about l ; and(d) R Q is the set of qualification-based inference rules of the form: Φ ¬ [ τ, υ ] AQ l , i.e.those that represent the qualifications regarding the fluent literal l .3. The theory Γ ⊆ Λ.Given a set of assumptions ∆, we denote ∆
F A = ∆ ∩ AB
F A and ∆ AQ = ∆ ∩ AB AQ . Observation 2
Let D = hL D , R , AB , Γ i be a domain description, for each set of assump-tions ∆ ⊆ AB , either ∆ is closed or ∆ attacks itself. o & Foo
5. Reasoning about Action: The Frame and the Qualification Problems
In general, we adopt the following guidelines in seeking a uniform solution to the problemsof reasoning about action: • The derived pieces of information do not conflict with the given facts; • Occurrences of events are minimised; and • The inertia of fluents is maximised though the minimality of the event occurrenceswill be of higher priority.Aside from the trivial case of occurrences of actions causing the frame assumptions tobe rejected, two aspects of actions can be distinguished:1. An action happens but the change it is supposed to cause does not take place. Wecall this expectation failure and this is more or less the qualification problem; and2. No actions that are known to have happened and caused a change but the changedid take place. We call this surprise and this is usually known as the explanationproblem.The following assumption represents our underlying intuition behind reasoning aboutaction formalisms.
Assumption 1
Intuitive models contain minimal (with respect to set inclusion) sets ofsurprises.
Now we introduce some model-theoretic counterpart notions of the assumption-basednotions presented above.
Definition 5.1
Let σ = hT , F , Ai be a signature and D = hL D , R , AB , Γ i a domain de-scription over σ . An interpretation I = h h, φ, η, ε q , ε f i is a model of D iff1. I is a model of Γ;2. for each r ∈ R , if I | = prem ( r ) then I | = cons ( r ).The following definition captures one of several aspects of the (model-theoretic) solutionof the frame problem. This aspect is known as the action-oriented frame problem in Linand Shoham’s (1995) terms. The proposed minimisation policy formalises the intuition thatchange does not happen by itself but is caused by some kind of event. Thus, for each fluent,if its value is changed between two timepoints τ and υ , (at least) an occurrence of someevent must end at υ that brings about that change. Definition 5.2
Let D = hL D , R , AB , Γ i be a domain description and I a model of D . I isa coherent model of D iff1. for each basic action α ∈ A and τ, υ ∈ T E , if I | = [ τ, υ ] α then Γ | = [ τ, υ ] α ; and easoning about Action: An Argumentation-Theoretic Approach
2. for each l ∈ F ∗ and t ∈ T , if I | = [ t ] l ∧ ¬ [ t + ] l then either(a) there are α ∈ A and s ∈ T such that r = Φ , [ τ , τ ] α, [ τ , τ ] AQ ¬ l ¬ [ τ ] l ∧ ¬ [ τ ] F A l ∈ R and I | = prem ( r )[ τ /s, τ /t + ], or(b) I | = [ t, t + ] da ¬ l Thus, in a coherent model: (i) all satisfiable basic actions must follow from the giventheory, and (ii) all changes are attributable to events of one kind or another.Given an interpretation I , we want to extract the sets of assumptions satisfiable in I . Definition 5.3
Let σ = hT , F , Ai be a signature and I an interpretation over σ . The setof frame assumptions satisfiable in I , denoted ∆ IF A , is defined as follows:∆
IF A = { [ t ] F A l | ( t, F A l ) ∈ F A I } and the set of qualification assumptions satisfiable in I , denoted ∆ IAQ , is :∆
IAQ = { [ t , t ] AQ l | ( t , AQ l , t ) ∈ AQ I } We also write ∆
IQF = ∆
IAQ ∪ ∆ IF A .Conversely, given a theory Γ and a set of assumptions ∆, a reasoner can also constructhis models about the domain of interest.
Definition 5.4
Let D = hL D , R , AB , Γ i be a domain description and ∆ ⊆ AB . A model I = h h, φ, η, ε q , ε f i of D is ∆- relativised iff1. for each δ ∈ AB , I | = δ iff δ ∈ ∆; and2. Occ I = OA D ∪ DAS (∆), where:(a) OA D = { ( φ ( τ ) , α, φ ( τ )) ∈ T × A × T | (cid:0) | = [ τ , τ ] α } , and(b) DAS (∆) = { ( t, da ¬ l , t + ) ∈ T × DA × T | [ ≈ ] FA ⋖ / ∈ (cid:1) and there do not exist anyaction α ∈ A and s ∈ T such that r = Φ , [ τ , τ ] α, [ τ , τ ] AQ ¬ l ¬ [ τ ] l ∧ ¬ [ τ ] F A l ∈ R and I | = prem ( r )[ τ /s, τ /t + ] } .∆-relativised models are one of the central notions of our framework. Essentially, as-sumptions underpin our machinery to conjecture information based on common sense knowl-edge. As such, we will try to accept as many assumption as possible unless there is a goodreason not to. Therefore, given a set of assumptions, we will attribute every missing frameassumption to a possible change in the domain the agent is reasoning about which is causedby either a known action/event or some unknown action, called dummy actions in thispaper.The following observation is immediate from condition (1.) in the above definition. Observation 3
Let a domain description D = hL D , R , AB , Γ i and a set of assumptions ∆ ⊆ AB be given. If the model I of D is ∆ -relativised then ∆ IQF = I (∆ , t ) for every t ∈ T .
11. The notation ϕ [ v /t , . . . , v n /t n ] is standard in logic and meant to be the instantiation of the formula ϕ with the variables v , . . . , v n being replaced by the terms t , . . . , t n , respectively. o & Foo First we will address the frame problem in a simple setting viz. without qualificationassumptions, but we will lift the restrictions later.
Definition 5.5
Let D = hL D , R , AB , Γ i be a domain description. D is a simple domaindescription , or S-domain , iff R Q = ∅ and AQ does not occur anywhere in R or Γ. Definition 5.6
Let D = hL D , R , AB , Γ i a domain description. An interpretation I = h h, φ, η, ε q , ε f i is a simple model , or S-model , of D iff1. I is a model of D ; and2. ε q ( t, AQ l , u ) = true for every ( t, AQ l , u ) ∈ T × AQ F ∗ × T .This effectively isolates the frame problem from the qualification problem. Note alsothat if I is an S-model then ∆ IAQ = AB AQ . A coherent S-model is an S-model which iscoherent. Example 4 ( continued .) The following is part of one of the coherent models of D ysp : { [0 , load, ¬ [0] loaded, [1] loaded, [0] alive, [1] alive, [1 , wait, [1 , da ¬ loaded , ¬ [2] loaded, [2] alive, [2 , shoot, ¬ [3] loaded, [3] alive } ,which corresponds to one of the anomalous models of this scenario (the one pointed out byHanks and McDermott).But it is not desirable to admit the occurrence of an event when there is no evidencefor it. Thus we need to minimise the set of action occurrences in a given action theory. Definition 5.7
Let D be an S-domain. A coherent S-model I of D is a prioritised minimalmodel (or simply PMM ) of D iff there does not exist any coherent S-model I ′ of D suchthat Occ I ′ ⊂ Occ I .Note that the above model-theoretic minimisation policy is not based on the frameassumptions. This solution to the frame problem is thus amenable to well-known techniquessuch as circumscription , but we believe an argumentation-theoretic approach is not onlymore direct but also has wider applicability. In order to provide the connection betweenthe above (model-theoretic) minimisation policy and the (argumentation-theoretic) notionof plausible sets of assumptions we need to maximise the set of assumptions satisfiable in aPMM. Definition 5.8
Let D be an S-domain. A PMM I of D is a canonical prioritised minimalmodel (or simply CPMM ) of D iff there does not exist any PMM I ′ of D such that F A I ⊂ F A I ′ .We now want to see how the account of plausible sets of assumptions connects to thisaccount of minimality.
12. in combination with the introduction of occurrences of dummy actions. easoning about Action: An Argumentation-Theoretic Approach
Lemma 1
Let D = hL D , R , AB , Γ i be an S-domain. If I is a CPMM of D then for eachassumption δ ∈ AB : δ / ∈ ∆ IQF iff δ is rejected by ∆ IQF .Proof . ( ⇒ ) Suppose by way of contradiction that δ is not rejected by ∆ IQF , then ∆
IQF ∪{ δ } is R -consistent, i.e. Γ ∪ ∆ IQF ∪ { δ } 6⊢ R false . Since D is an S-domain, AB AQ ⊆ ∆ IQF . Assumethat δ = [ τ ] F A l for some τ ∈ T E and l ∈ F ∗ , we can construct an interpretation I ′ in such away that I ′ interprets everything except F A the same as I and F A I ′ = F A I ∪{ ( φ ( τ ) , F A l ) } .Since I is a PMM of D , from the above construction, I ′ is also a PMM of D . But F A I ⊂ F A I ′ . Contradiction.( ⇐ ) Obvious. ✷ Lemma 2
Let D = hL D , R , AB , Γ i be an S-domain and I a CPMM of D . If there are t ∈ T and l ∈ F ∗ such that [ t ] F A l ∈ Lr (∆ IQF ) then I | = [ t, t + ] da ¬ l .Proof . Let δ denote the assumption [ t ] F A l . First we observe that δ ∈ Lr (∆ IQF ) implies δ ∆ IQF since δ is rejected by ∆ IQF and ∆
IQF does not attack itself. This in turn impliesthat I | = { [ t ] l, ¬ [ t + ] l } as the denotation of F A in I is required to be maximal since I is aCPMM of D . From the condition that I is coherent, either (i) there are α ∈ A and s ∈ T such that r = Φ , [ τ , τ ] α, [ τ , τ ] AQ ¬ l ¬ [ τ ] l ∧ ¬ [ τ ] F A l ∈ R and I | = prem ( r )[ τ /s, τ /t + ], or (ii) I | = [ t, t + ] da ¬ l . Condition (i) guarantees that δ isattacked by ∆ IQF and thus δ can not be a member of Lr (∆ IQF ). Therefore, (ii) must be thecase. ✷ The converse of Lemma 2 does not hold. There are cases in which I | = [ t, t + ] da ¬ l andthe assumption [ t ] F A l is attacked by ∆ IQF as a basic action occurs that changes the fluentliteral l . Theorem 1
Let D = hL D , R , AB , Γ i be an S-domain. If I is a CPMM of D then ∆ IQF isplausible.
We now prove that not only can we derive a plausible set of assumptions from a givenCPMM but we can also construct CPMMs from a plausible set of assumptions of a givenS-domain.The set of ∆-relativised models of an S-domain D is denoted as M od S ∆ ( D ). Observation 4
Let D be an S-domain and ∆ a set of assumptions of D . For each I ∈ M od S ∆ ( D ) , ∆ = ∆ IQF .Proof . From the construction of ∆-relativised models:For each I ∈ M od S ∆ ( D ), δ = [ τ ] F A l ∈ ∆ iff I | = [ τ ] F A l iff δ ∈ ∆ IF A . (More precisely,we have the assumption [ φ ( τ )] F A l is in ∆ IF A , where φ is the valuation defined in I . Butrelative to I , it is identical to δ .)Therefore, ∆ = ∆ IQF . ✷ o & Foo Theorem 2
Let D = hL D , R , AB , Γ i be an S-domain and ∆ ⊆ AB . ∆ is plausible wrt D iff M od S ∆ ( D ) = ∅ and for each I ∈ M od S ∆ ( D ) , I is a CPMM of D . Theorem 3
Let D = hL D , R , AB , Γ i be an S-domain. Furthermore, suppose that CP M M ( D ) is the set of CPMMs of D and P laus ( D ) is the set of plausible sets of assumptions of D ,then CP M M ( D ) = S ∆ ∈ P laus ( D ) M od S ∆ ( D ) . So, how does our account of the frame problem relate to the existing approaches to the frameproblem? While there has been a long line of development behind monotonic approachesto the frame problem starting with Haas’s (1987) and Schubert’s (1990) early attempts andresulting in Reiter’s (1991) monotonic solution to the frame problem together with othersolutions proposed by others such as Castilho et al. (1999) and Zhang and Foo (2002) orThielscher’s (1999) Fluent Calculus-based monotonic solution to the frame problem, withthe notable exception of Thielscher’s (2001) attempt to address the qualification problem,few have tried to tackle the qualification within the framework they use to address the frameproblem.On the other hand, in the action languages (Gelfond & Lifschitz, 1998) and relatedapproaches such as those proposed by McCain and Turner (1995, 1997), Giunchiglia, Kartha,and Lifschitz (1997), and Giunchiglia and Lifschitz (1998), state transition systems areemployed as the underlying computation machinery which essentially provides the reasonerwith all possible complete states of the world. Furthermore, as domain decsriptions canbe uniquely translated to state transition systems, the reasoner could safely derive thesuccessor state(s) based on the current together with the transition function.
The results reported in the previous section are established in a simple setting. If we addthe following observation to the theory in example 4: [3] alive , i.e. after the shoot action,the victim is still alive, then like most existing formalisms, the above account of plausibilitywould come up with a contradiction. In fact, it would be more reasonable that such afailure is explained as an occurrence of some qualification. In this section, we remove certainrestrictions on the qualifications of actions in order to achieve a more general framework.There are some subtleties in the way action theories are represented in our proposedassumption-based framework. Note first that there is a potential difficulty if frame assump-tions and qualification assumptions are treated equally, which can be illustrated by a versionof the YSP. Consider the following action description: { [ τ ] alive, [ τ ] F A alive [ τ + ] alive , [ τ ] loaded, [ τ, υ ] shoot, [ τ, υ ] AQ ¬ alive ¬ [ υ ] alive ∧ ¬ [ τ ] F A alive } ⊆ R , { [0] loaded, [0] alive, [0 , shoot } ⊆ Γ.From this, we have (at least) two stable sets of assumptions: one contains the frameassumption [0]
F A alive which rejects the qualification assumption [0 , AQ ¬ alive and anothercontains [0 , AQ ¬ alive which attacks [0] F A alive . Only the latter is intuitive in this case butwe do not have any explicit criterion to prefer one over another. The following assumption easoning about Action: An Argumentation-Theoretic Approach asserts that in our solution to the frame problem in the presence of the qualification problem,an action is presumed to bring about its effects unless there is an explicit justification forits disqualification.
Assumption 2
When there is a direct conflict between a frame assumption and a qualifi-cation assumption (over a fluent literal), the qualification assumption takes precedence.
Given the presence of several kinds of assumptions, i.e. frame and qualification, wewill adopt the following convention: we will write Lr P (∆) instead of ( Lr (∆)) P for P ∈{ F A, AQ } . Since we no longer exclude qualification assumptions from our assumption-based domain descriptions, we will simply refer to assumption-based domain descriptionsas Q-domains . Definition 5.9
Let D = hL D , R , AB , Γ i be a Q-domain. A presumable set of assumptions∆ ⊆ AB is semi-Q-plausible wrt D iff Lr F A (∆) is minimal (with respect to set inclusion).
Definition 5.10
Let D = hL D , R , AB , Γ i be a Q-domain. A set of assumptions ∆ ⊆ AB is Q-plausible wrt D iff1. ∆ is semi-Q-plausible wrt D ,2. ∆ AQ is maximal, i.e. there does not exist any ∆ ′ ⊆ AB such that ∆ ′ is semi-Q-plausible (wrt D ) and ∆ AQ ⊂ ∆ ′ AQ , and3. ∆ F A is maximal relative to the above two conditions, i.e. there does not exist any∆ ′ ⊆ AB such that ∆ ′ satisfies the above two conditions and ∆ F A ⊂ ∆ ′ F A .We will now refer to models of a Q-domain as
Q-models . A coherent Q-model is a Q-model which is coherent. We minimise the set of action occurrences in coherent Q-modelsof a given action theory.
Definition 5.11
Let D be a Q-domain. A coherent Q-model I of D is a prioritised minimalQ-model (or simply PMQM ) of D iff there does not exist any coherent Q-model I ′ of D such that Occ I ′ ⊂ Occ I . Definition 5.12
Let D be an S-domain. A PMQM I of D is a canonical prioritised minimalQ-model (or simply CPMQM ) of D iff1. there does not exist any PMQM I ′ of D such that AQ I ⊂ AQ I ′ , and2. there does not exist any PMM I ′ of D such that F A I ⊂ F A I ′ .Now we can proceed to obtaining the main results for CPMQMs regarding Q-plausiblesets of assumptions which are similar to those for CPMMs regarding plausible sets of as-sumptions. The following lemma, which is a straightforward extension of Lemma 1 andLemma 2 proved in the previous section, is introduced to assist in the proof of Theorem 4 . Lemma 3
Let D = hL D , R , AB , Γ i be an Q-domain and I a CPMM of D , o & Foo
1. For each assumption δ ∈ AB : δ / ∈ ∆ IQF iff δ is rejected by ∆ IQF .2. If δ = [ τ ] F A l ∈ Lr (∆ IQF ) then I | = [ τ, τ + ] da ¬ l . Theorem 4
Let D be a Q-domain. If I is a CPMQM of D then ∆ IQF is Q-plausible wrt D . Similar to the previous section, we now prove that not only can we derive a plausible setof assumptions from a given CPMQM but we can also construct CPMQMs from a plausibleset of assumptions of a given domain description. The set of ∆-relativised models of aQ-domain D is denoted as M od Q ∆ ( D ).The following observation is obvious: Observation 5
Let D be a Q-domain and ∆ a set of assumptions of D . For each I ∈ M od Q ∆ ( D ) , ∆ = ∆ IQF . Theorem 5
Let D = hL D , R , AB , Γ i be a Q-domain and ∆ ⊆ AB . ∆ is Q-plausible wrt D iff M od Q ∆ ( D ) = ∅ and for each I ∈ M od Q ∆ ( D ) , I is a CPMQM of D . Theorem 6
Let D be a Q-domain. Furthermore, suppose that CP M QM ( D ) is the set ofCPMQMs of D and P laus Q ( D ) is the set of Q-plausible sets of assumptions of D , then CP M QM ( D ) = S ∆ ∈ P laus Q ( D ) M od Q ∆ ( D ) . Q-plausible sets of assumptions allow one to overcome scenarios in which expectationfailures (or, qualification surprises) arise, e.g. shooting a turkey with a loaded gun andobserving that the turkey is still alive. When such surprises arise, the reasoner knowswho’s to blame: qualification assumptions. She can then accordingly remove the “guilty”assumptions. Just as the anomalous models forming obstacle to early approaches to theframe problem, similar anomalous models can also arise in solutions to the qualificationproblem. This important issue related to the qualification problem has been thoroughlydiscussed by Thielscher (2001) and a solution was presented within the framework of theFluent Calculus. To give the reader a flavour of this problem within our framework, weinvite the reader to consider the following classical example:
Example 5
Consider the problem of starting a car whose tail pipe could possibly beblocked by a potato, formalised in our formalism as follows.1. The set of inference rules R is: [ τ ] BlockedT P ¬ [ τ, υ ] AQ GetStarted , [ τ ] HasP otato, [ τ, υ ] InsertP otato, [ τ, υ ] AQ BlockedT P [ υ ] BlockedT P ∧ ¬ [ τ ] F A ¬ BlockedT P , [ τ ] HasKey, [ τ, υ ] T urnOnIgnition, [ τ, υ ] AQ GetStarted [ υ ] GetStarted ∧ ¬ [ τ ] F A ¬ GetStarted .
13. Of course we also have the frame-based inference rules for
HasKey , BlockedT P , etc. but we omit themfrom this representation for sake of readability. easoning about Action: An Argumentation-Theoretic Approach
2. The theory Γ is: { [0] HasP otato, [0]
HasKey, ¬ [0] BlockedT P, ¬ [0] GetStarted, [0 , InsertP otato, [1 , T urnOnIgnition } .Of course, we have designed this example to try to avoid any possible troubles withthe frame problem. We have to consider two conflicting qualification assumptions in thiscase which are [0 , AQ BlockedT P and [1 , AQ GetStarted . Given the above action theory, anyQ-plausible set of assumptions would contain exactly one of them. Thus, there will be atleast two extensions, one in which the car can not get started since the tailpipe is blockedand it’s no longer consistent to assume [1 , AQ GetStarted . The other extension disqualifiesthe action of inserting the potato into the tailpipe and thus the action of starting the carbecomes successful. Only the former is intuitive in this case and the account of Q-plausiblesets of assumptions fails to deliver this desired solution.However, in exactly the same way the problem of anomalous models arising in the solu-tion to the frame problem is tackled, the above problem is easily addressed within our frame-work. The key insight is of course also underlined by the notion of causality: [1]
BlockT P was caused (by the action [0 , InsertP otato ) which in turn allows ¬ [1 , AQ GetStarted to bederived. On the other hand, there was no cause that allows ¬ [0 , AQ BlockT P to be derived.How is the above insight realised in our framework? The answer turns out to be rather sim-ple: Just as the distinction between leniently rejected frame assumptions and non-lenientlyrejected frame assumptions allows us to distinguish between normal changes (i.e. action-triggered) and anomalous changes, a distinction between leniently rejected qualification as-sumptions and non-leniently rejected qualification assumptions will allow us to distinguishbetween normal disqualifications (i.e. those underlined by a cause) and anomalous disqual-ifications. Thus, facing a collection of Q-plausible sets of assumptions, a reasoner simplyselects the set of assumptions that contains the smallest (with respect to set inclusion) setof leniently rejected qualification assumptions.The ability to introduce different argumentation-theoretic semantics for assumption-based frameworks is arguably the biggest advantage of our approach. The following exampleillustrates this critical point:
Example 6
We modify an example presented by Lin and Shoham (1995) which is in turna modification of Kautz’s (1986) Stolen Car Problem. A spy possessed a microfilm of a topsecret evidence which an organisation, A , tried to steal. For some reason, another organi-sation, B , wanted to murder this spy. The microfilm was in the safe at the spy’s home attime 0. The spy was not at home at time 0. A tried to steal the evidence between thetime points 0 and 1. The spy might return home at any time between 0 and 1. B tried to murder the spy at any time between 0 and 1. return cancels the effects of steal , murder cancels the effects of return and steal cancels the effects of murder . Any of the threeactions steal , return , and murder takes only one time step. The domain is formalised asfollows: Γ = {¬ [0] EvStolen, [0]
Alive, ¬ [0] AtHome } ; and R contains { [ τ, υ ] steal, [ τ, υ ] AQ EvStolen [ υ ] EvStolen ∧ ¬ [ τ ] F A ¬ EvStolen , [ τ, υ ] return, [ τ, υ ] AQ AtHome [ υ ] AtHome ∧ ¬ [ τ ] F A ¬ AtHome , [ τ, υ ] murder, [ τ, υ ] AQ ¬ Alive ¬ [ υ ] Alive ∧ ¬ [ τ ] F A
Alive , [ τ ] AtHome ∧ τ < υ ¬ [ τ, υ ] AQ EvStolen , ¬ [ τ ] Alive ∧ τ < υ ¬ [ τ, υ ] AQ AtHome , [ τ ] EvStolen ∧ τ < υ ¬ [ τ, υ ] AQ ¬ Alive } . o & Foo Given the above formalisation of the problem, traditional accounts of non-monotonicreasoning (i.e. Default Logic, circumscription, Autoepistemic Logic, etc.) can not provideone with a solution since these formalisms can not produce any extension under their stan-dard semantics. However, the argumentation-theoretic approach does gives several seman-tics for this problem including preferability semantics. Note, however, that all admissiblesets of assumptions and preferred sets of assumptions for this domain contain none of theassumptions: [0 , AQ EvStolen , [0 , AQ AtHome , and [0 , AQ ¬ Alive . This essentially meansthat, under the admissibility and the preferability semantics, the reasoner could only inferthat none of the above actions would succeed On the other hand, our proposed plausiblesemantics gives an alternative solution for this problem in which each consistent set of as-sumptions can contain at most one of the following three assumptions: [0 , AQ EvStolen ,[0 , AQ AtHome , and [0 , AQ ¬ Alive . Moreover, each plausible set of assumptions must con-tain exactly one of them. Among the other two assumptions which do not belong to aplausible set of assumptions, one is attacked and the other is leniently rejected.We believe that the above examples have underlined the major advantages of our ap-proach in which a reasoner is aware of the (defeasible) assumptions used in her reasoning aswell as being able to explicitly reason about these assumptions. The flexibility of allowinga reasoner to introduce different argumentation-theoretic semantics for assumption-basedframework by simply varying the notion of acceptability of sets of assumptions is certainlyanother advantage in favour of our approach.
6. More Complex Dynamic Domains and Indirect Effects
So far we haven’t taken into consideration the issues of concurrent actions and indirecteffects. To ensure that the formalisation introduced is expressive enough to deal withcomplex domains, we will show how these issues are coped with by our approach. Firstly,we motivate our formalisation with an informal discussion.
Given our temporal representation, formulating concurrent events is not a difficult issue inour framework. However, there are some subtleties which need to be carefully considered.Firstly, the use of assumptions. As presented earlier in this paper, qualification assumptionsare fluent oriented, i.e. we qualify over the effects of action rather than over the action itself.Whilst this manifests the capability of formulating actions with multiple effects, and thuseach effect should be qualified independently, it may fail to formalise concurrent eventswith the same effects. For example, both actions hit the vase with a hammer and shoot it with a loaded gun bring about the effect that the vase is broken . In other words, itis essential that qualification assumptions be dependent on the actions that bring aboutthe effect under consideration. Thus, given n actions and m fluents whose values canbe changed by those actions, we have to potentially introduce 2 × m × n qualificationsassumptions. Therefore, instead of subscripting the assumption symbols AQ with thefluent literals from F ∗ , we extend the set of subscripts of AQ , denoted as AF , to contain
14. In fact, as we will see later, there are potentially 2 × m × ( n + 1) qualification assumptions in this casesince there is one special event corresponding to all (natural) events that bring about the indirect effects. easoning about Action: An Argumentation-Theoretic Approach both the action and the corresponding fluent literals. For example, given the above twoactions hit and shoot and an additional action repair whose effect is change a broken vaseto being non-broken, viz. ¬ broken , we will need to introduce the following qualificationassumptions: AQ Hit - Broken , AQ
Shoot - Broken and AQ Repair - ¬ Broken . Therefore, syntacticallywe also extend the set AB AQ = { [ τ, υ ] AQ ϕ | τ, υ ∈ T E and ϕ ∈ AF } and semanticallythe function ε q : T × AQ AF × T → { true , false } . In the definition of an interpretation I = h h, φ, η, ε q , ε f i , we also have I ([ τ, υ ] AQ ϕ , t ) = ε q ( φ ( τ ) , AQ ϕ , φ ( υ )), where τ, υ ∈ T E and ϕ ∈ AF .Though this does increase the complexity of the framework, it is the price we have topay for the expressiveness of the resulting system. To the best of our knowledge, none ofthe existing formalisms possesses such an expressiveness.How about non-deterministic actions? The solution turns out to be quite simple: we cantreat action non-determinism as a special kind of action qualification. For example, to for-mulate the Russian shooting scenario (Sandewall, 1994) in which a gun non-deterministicallygets loaded or not after spinning its revolver, the following action descriptions can be as-serted: adr = [ τ, υ ] spin, [ τ, υ ] AQ Spin - Loaded [ υ ] loaded ∧ ¬ [ τ ] F A ¬ loaded and adr = [ τ, υ ] spin, [ τ, υ ] AQ Spin - ¬ Loaded ¬ [ υ ] loaded ∧ ¬ [ τ ] F A loaded together with the following two qualification rules: qr = [ υ ] loaded ¬ [ τ, υ ] AQ Spin - ¬ Loaded and qr = ¬ [ υ ] loaded ¬ [ τ, υ ] AQ Spin - Loaded .The above guarantee that either ¬ [ υ ] loaded or [ υ ] loaded will follow from [ τ, υ ] spin (re-mark that the set of qualification assumptions ∆ AQ is maximised), but not both. As aconsequence, two possible extensions will arise given the above non-deterministic action.Dealing with non-determinism could be more complicated when the information used toencode the action description is disjunctive. As we only restrict the conclusion of an actiondescription rule to contain only fluent literals, such disjunctive effect is not straightforwardlydealt with in our framework. However, note that the restriction is similar to that imposedon action language A (Gelfond & Lifschitz, 1998). An extension to action descriptionswhich is similar to the way the action language A is extended into the action language C can be done to allow a complex expression in the conclusion of action description rule.However, for the sake of presentation, we choose to use a simpler language to introduce ourframework to the reader. From the analysis in the introductory section regarding indirect effects, we observe thatthe ordering in which domain constraints are applied plays an essential role in a technicallysound framework. Moreover, it’s no longer guaranteed that domain constraints would bestrictly satisfied at every time point. To help the reader better understand this technicalsubtlety, it’s useful to think that changes are attributable to events. However, events
15. The reader is referred to section 3 for the general formalisation.16. Of course, it’s not necessary that the state of the world at every time point is observable to a reasoner.However, it is important that she be aware of such states and able to reason about them. o & Foo are further divided into two categories: external and internal. Basically, external events correspond to the direct effects of actions performed by some agents (including the reasoner.)On the other hand, internal events correspond to the indirect effects when certain conditionsabout the world are met and can be attributed to Nature. Like external events, internal events also happen in a certain order. Although it is notstraightforward to observe this order , it is important that a reasoner be able to reasonabout them. Hence we propose to add one more dimension into the set of assumptionsof a given domain description. Since these assumptions play essentially the same role asthat of qualification assumptions, we can subsume them under the set of qualificationassumptions AQ AF . As they are not associated with any specific action, we can ignore theinitial of the action name from the subscripts of AQ . For example we will write AQ broken in case we want to qualify over the fluent broken as an indirect effect, i.e. independently ofany action. Moreover, to avoid the axiomatiser from the confusion of determining the timespan taken by indirect effects, we will assume that it is atomic. It means all indirect effectsalways take place between one time point and the next one. This does not mean that allindirect effects have the same real-time duration, it simply means that relative to the giventime structure T they take place between two consecutive time points. This allows us toavoid the granularity problem as it is irrelevant from the viewpoint of the problems we aretrying to solve. One important remark is in place here. Until now, we have used integers and the corre-sponding expressions to denote time points. There are two reasons for this pratice: (i) Itsignificantly simplifies the presentation, and (ii) Without the complication of the ramifica-tion problem, all time points which are reasoned about are also observable to the reasoner.However, as will be discussed thoroughly in the next section, in the presence of ramificationsa time point at which some change takes place maight not be observable to the reasoner assuch a change could be one of the indirect effects after the execution of some action. Suchrepresentation issues make the integer-based time structure employed so far in this paperinappropriate. We do need a richer representation of time structure. Following Sandewall’s(1994) basic formulation, the only basic constant to be included in the representation ofthe time structure T is denoted by the symbol Θ and is referred to as the origin of T .The standard algebraic operations + and − are still employed to obtain time expressionsand bear their usual meanings. The relation symbol < will also be used to compare timeexpressions. However, we can no longer allow expressions such as Θ + 1 for integers are nolonger elements of the time structure.Given a time expression τ :
17. We avoid the use of the terms exogenous/endogenous events to label these two categories becausethroughout the literature of reasoning about action, exogenous events are used to refer to actions thatcarried out by agents that are different from the reasoner or outside events whose occurrences are beyondthe reasoner’s control.18. Unless indirect effects are somehow delayed and become observable to the reasoner, they usually takeplace immediately after direct effects.19. The only difference is that these assumptions are qualified over the indirect effects which can be consideredto be the effects of the actions of Nature. easoning about Action: An Argumentation-Theoretic Approach • We will continue to denote the next time point of τ by τ + . • We define τ to be τ + and, let n > τ n + to be ( τ ( n − ) + .Furthermore, as we will assume throughout that an indirect effect takes place betweentwo consecutive time points, we also simplify the presentation by using the following syn-tactical sugar: instead of writing [ τ, τ + ] AQ l (for some τ ∈ T E and l ∈ F ∗ ), we will write[ τ ] AQ l . This also provides a simple way to distinguish between qualification assumptionsfor ramificational effects and those caused directly by an action or event.How are domain descriptions affected from this augmentation? The only change is forthe set R A of action descriptions which consists of rules either of the formΦ , [ τ, υ ] α, [ τ, υ ] AQ α - l [ υ ] l ∧ ¬ [ τ ] F A ¬ l or of the form Φ , [ τ ] AQ l [ τ + ] l ∧ ¬ [ τ ] F A ¬ l . For convenience, we will refer to the set of inference rules having the latter form as R I ,i.e. R I = { r ∈ R A | r = Φ , [ τ ] AQ l [ τ + ] l ∧ ¬ [ τ ] F A ¬ l for some fluent literal l ∈ F ∗ } . Now, regardingthe ramification problem, the basic idea is that the state obtained by updating the previousstate may not necessarily be stable due to the presence of indirect effects. By representingindirect effects as causal rules (using inference rules in our framework) we can reason aboutwhich causal rules have fired and (relatively) when. Moreover, since these causal rulesmay not take the same amount of time to fire, we should be able to reason about differentpossible orders in which they fire. This will be achieved by a distinction between stable andunstable states. Definition 6.1
A time-point expression θ ∈ T E is stable wrt a given domain description D = hL D , R , AB , Γ i and ∆ ⊆ AB iff there does not exist any fluent literal l ∈ F ∗ such that1. Φ , [ τ ] AQ l [ τ + ] l ∧ ¬ [ τ ] F A ¬ l ∈ R I and Γ ∪ ∆ ⊢ R Φ , [ θ ] AQ l , and2. Γ ∪ ∆ R [ θ ] l . Definition 6.2
Let D = hL D , R , AB , Γ i be a domain description. A set of assumptions ∆is said to be ramification-compliant iff1. there does not exist any δ ∈ AB AQ such that T h R (∆ ∪ { δ } ) = T h R (∆) ∪ { δ } .2. there does not exist any unstable time-point expression τ ∈ T E such that [ τ ] F A l ∈ ∆for every l ∈ F ∗ .We note that by using unstable time points, we have conceptually isolated the ramifi-cation problem from the task of reasoning about the (explicit) actions. o & Foo Definition 6.3
Let D = hL D , R , AB , Γ i be a domain description. A set of assumptions ∆is generally plausible for action domains , or simply AD-plausible , iff • ∆ is ramification-compliant, and • ∆ is Q-plausible relative to the above condition. Remarks:
1. From the above definition, it can be seen that a solution for the ramification is (ina sense) independent from the frame and the qualification problems regarding actionoccurrences.2. Given the above temporal ontology, it is worth noting that the traditional notionof state constraints may no longer hold in this representation regarding time points.More precisely, only states associated with stable time points are subject to theseconstraints.
The question is, of course, whether the above computational mechanism gives satisfactoryconclusions for problems in reasoning about action. While many formalisms have beenproposed, a general criterion for reasoning about action formalisms still seems to be missing.The major stream of research towards a solution to the ramification problem is based on thenotion of causality (e.g., Lin, 1995; McCain & Turner, 1995; Thielscher, 1997). As has beendiscussed above, none of these approaches can deal with domains in which (potentially)infinite sequences of indirect effects are present. Most of these approaches (including theabove three references), however, produce a successor state after the execution of an actionto capture changes that have taken place either as direct or as indirect effects of that action.In this sub-section, we show how our formalism captures the notion of successor states inthe absence of the infinite sequences of indirect effects.
Definition 6.4
A domain description D = hL D , R , AB , Γ i is non-stratified iff there existtwo sets Φ ⊆ Λ and { l , . . . , l n } ⊆ F ∗ such that1. for each 1 ≤ i ≤ n , Φ ∪ { [ τ ] l i } is R -consistent, for every τ ∈ T E ; and2. for each 1 ≤ i ≤ n , if k = ( i mod n ) + 1, then there exists a set Ψ k ⊆ Λ such thatΦ ∪ { [ τ ] l i } ⊢ R Ψ k , and Ψ k , [ τ ] AQ l k [ τ + ] l k ∧ ¬ [ τ ] F A ¬ l k ∈ R I . Of course, a domain description is stratified if and only if it is not non-stratified. Given adomain description D = hL D , R , AB , Γ i , we’ll denote E D (∆) def = T h R (Γ ∪ ∆), the extensionof an action theory D according to ∆. We will also write E D instead of E D ( ∅ ) for brevity. Definition 6.5
Let σ = hT , F , Ai be a signature. easoning about Action: An Argumentation-Theoretic Approach A set S ⊆ F ∗ is an instantwise state iff for every l ∈ F ∗ , either l ∈ S or ¬ l ∈ S . IS will be used to denote the set of instantwise states of Λ, Let Γ ⊆ F ∗ and τ ∈ T E , we denote Γ def = {¬ l | l ∈ Γ } and [ τ ]Γ def = { [ τ ] l | l ∈ Γ } , Let D = hL D , R , AB , Γ i be a domain description. D is simplistic iff Γ = ∅ .The motivation behind the introduction of instantwise states is to allow the reasonerto reason about the intermediate states which may not be practically observable to her.That is, when indirect effects (following an occurrence of an action or event) take place,the world might transit through a number of intermediate states before becoming sta-bilised in a final state in which all domain constraints necessarily hold. The ability toexplicitly reason about such (unstable) intermediate states is one of the advantages of-fered by our approach. For instance, let’s consider the scenario in Example 1 whichwas originally introduced by Thielscher (1997). Thielscher’s (1997) approach allows twopossible successor states as the results of closing the switch sw in the initial state S = {¬ sw , sw , sw , ¬ relay, ¬ light, ¬ detect } . These are T = { sw , ¬ sw , sw , relay, ¬ light, ¬ detect } and T = { sw , ¬ sw , sw , relay, ¬ light, detect } . However, to an outside observer who hasno idea about the tricky internal mechanism of this circuit, it could be quite difficult toexplain why T should be one of the possible outcomes of the action of closing the switch sw : starting from an initial state in which detect and light are both off, after performingan action toggle ( sw ), light remains off but somehow detect becomes on. In other words,Thielscher’s (1997) framework fails to render the intermediate (and unstable) state in which light was on, albeit for only a short instant, before it was turned off again. On the otherhand, such states are explicitly represented and reasoned about as instantwise states in ourframework. As a consequence, domain constraints do not necessarily hold in an instantwisestate.In the following, we abbreviate simplistic and stratified domain descriptions as SSD s.Let Ω ⊆ R I , we denote: CON S R (Ω) def = { l ∈ F ∗ | [ τ ]Ψ , [ τ ] AQ l [ τ + ] l ∧ ¬ [ τ ] F A ¬ l ∈ Ω } . Similarly, let Ω ⊆ R A , we denote CON S A (Ω) def = { l ∈ F ∗ | [ τ ]Ψ , [ τ , τ ] α, [ τ , τ ] AQ α - l [ τ ] l ∧ ¬ [ τ ] F A ¬ l ∈ Ω } . The action description [ τ ]Ψ , [ τ , τ ] α, [ τ , τ ] AQ α - l [ τ ] l ∧ ¬ [ τ ] F A ¬ l ∈ R A is applicable in S iff Ψ ⊆ S .We use Υ Sα to denote the set of action descriptions that is applicable (in S ) regarding theaction α . Ω ⊆ Υ Sα is a possible application of α in S iff CON S A (Ω) is consistent and theredoes not exist any Ω ′ ⊆ Υ Sα such that Ω ⊂ Ω ′ and CON S A (Ω ′ ) is consistent. Definition 6.6
Let σ = hT , F , Ai be a signature and D = hL D , R , AB , ∅i a SSD. Suppose S ∈ IS and α ∈ A . We formalise the direct effects of an action α using a relation Res D :for all S, S ′ ∈ IS , ( S, α, S ′ ) ∈ Res D iff there is a possible application Ω of α in S such that: o & Foo (i) CON S A (Ω) ⊆ S ′ , and(ii) there does not exist any instantwise state S ′′ such that S ′′ satisfies (i) and S ′′ \ S ⊂ S ′ \ S . Definition 6.7
Let σ = hT , F , Ai be a signature and D = hL D , R , AB , ∅i a SSD. The causation relation according to D , denoted as Causes D , is defined as follows: for all S, S ′ ∈ IS , Causes D ( S, S ′ ) iff there exists a non-empty set Ω ⊆ R I of ramification inference rulessuch that(i) for each [ τ ]Ψ , [ τ ] AQ l [ τ + ] l ∧ ¬ [ τ ] F A ¬ l ∈ Ω, Ψ ⊆ S and(ii) CON S R (Ω) ⊆ S ′ , and(iii) there does not exist any instantwise state S ′′ such that S ′′ satisfies (i) and (ii) and S ′′ \ S ⊂ S ′ \ S .Given a domain description D , a state S ∈ IS is stable regarding D iff there does notexist any S ′ ∈ IS such that Causes D ( S, S ′ ). Definition 6.8
Let σ = hT , F , Ai be a signature and D a SSD. Suppose w ∈ IS and α ∈ A .The state transition from w in D according to α , denoted T rans αD ( w ), is the transitive clo-sure of Causes D regarding the state ω ∈ IS satisfying ( w, α, ω ) ∈ Res D . Formally, T rans αD ( w ) = { ̟ ∈ IS | there exists a sequence ω , . . . , ω n ∈ IS such that ( w, α, ω ) ∈ Res D and ̟ = ω n and ( ω i , ω i +1 ) ∈ Causes D for each 1 ≤ i < n and ̟ is stable regarding D } .Let I be an interpretation for L D and t ∈ T , • we use [ I ] t to denote the instantwise state specified by I at time point t : [ I ] t def = { l ∈F ∗ | I ( l, t ) = true } . If [ I ] t is stable then t is said to be a stable time point in I . • we use N I to denote a function that maps a time point t ∈ T to the next stable timepoint in I : N I ( t ) ∈ T such that [ I ] N I ( t ) is stable and for every u ∈ T , if t < u < N I ( t )then [ I ] u is not stable. Theorem 7
Let σ = hT , F , Ai be a signature and D = hL D , R , AB , ∅i a SSD and α ∈ A ′ .Suppose that w ∈ IS . Define a domain description D = hL D , R , AB , Γ i , where Γ = { [Θ] ϕ | ϕ ∈ w }∪{ [Θ , Θ + ] α } . A set ∆ ⊆ AB of assumptions is AD-plausible wrt D iff for each model M of E D (∆) , [ M ] N M (Θ) ∈ T rans αD ([ M ] Θ ) , i.e. [ M ] N M (Θ) belongs to the state transitionfrom [ M ] Θ in D according to α . We now re-consider the motivating examples introduced in Section 1.3.
Example 1 ( continued .) We re-formulate the action theory for this example in terms ofour formalism: easoning about Action: An Argumentation-Theoretic Approach [ τ ] sw , [ τ ] sw , [ τ ] AQ light [ τ + ] light ∧ ¬ [ τ ] F A ¬ light , [ τ ] sw , [ τ ] sw , [ τ ] AQ relay [ τ + ] relay ∧ ¬ [ τ ] F A ¬ relay , ¬ [ τ ] sw i , [ τ ] AQ ¬ light ¬ [ τ + ] light ∧ ¬ [ τ ] F A light ( i = 1 , , ¬ [ τ ] sw i , [ τ ] AQ ¬ relay ¬ [ τ + ] relay ∧ ¬ [ τ ] F A relay ( i = 1 , , [ τ ] relay, [ τ ] AQ ¬ sw ¬ [ τ + ] sw ∧ ¬ [ τ ] F A sw , [ τ ] light, [ τ ] AQ detect [ τ + ] detect ∧ ¬ [ τ ] F A ¬ detect , ¬ [ τ ] sw i , [ τ, υ ] toggle i [ υ ] sw i ∧ ¬ [ τ ] F A ¬ sw i ( i = 1 , , , [ τ ] sw i , [ τ, υ ] toggle i ¬ [ υ ] sw i ∧ ¬ [ τ ] F A sw i ( i = 1 , , , [ τ ] ζ, [ τ ] F A ζ [ τ + ] ζ , where ζ ∈ F ∗ . The theory is described as follows:Γ = {¬ [Θ] sw , [Θ] sw , [Θ] sw , ¬ [Θ] relay, ¬ [Θ] light, ¬ [Θ] detect } ∪ { [Θ , N (Θ)] toggle } . Consider ∆ such that ∆ AQ = AB AQ \{ [Θ ] AQ detect , [Θ ] AQ detect } and ∆ F A = AB F A \{ [Θ + ] F A ¬ light , [Θ + ] F A ¬ relay , [Θ ] F A sw , [Θ ] F A light } . ∆ is AD-plausible resulting inthe next stable state being ω = { sw , ¬ sw , sw , relay, ¬ light, ¬ detect } at [Θ ].In addition, the set ∆ ′ ⊆ AB , where ∆ ′ AQ = AB AQ \ { [Θ ] AQ detect } and ∆ ′ F A = AB F A \ { [Θ + ] F A ¬ light , [Θ + ] F A ¬ relay , [Θ ] F A sw , [Θ ] F A light , [Θ ] F A ¬ detect } , is alsoAD-plausible which results to the next stable state ω ′ = { sw , ¬ sw , sw , relay, ¬ light, detect } at [Θ ].Moreover, the set ∆ ′′ ⊆ AB , where ∆ ′′ AQ = AB AQ and ∆ ′′ F A = AB F A \{ [ N (Θ)] F A ¬ light , [Θ + ] F A ¬ relay , [Θ ] F A sw , [Θ ] F A ¬ detect , [Θ ] F A light } , is also AD-plausible which alsoresults to the next stable state ω ′ at [Θ ]. In other words, the model implied by ∆ ′ reflectsa domain in which it takes the same amount of time for the detect to be on and the switch sw to jump off. On the other hand, the model implied by ∆ ′′ reflects a domain in whichthe amount of time for the detect to be on is (approximately) equal to the amount of timefor switch sw to jump off and cause the light to be off as well. The detect implied by ∆ isso insensitive that even though the light and the relay are on at the same time (Θ ), the relay causes switch sw to jump off and then leads to the light to be off as well but the detect is yet on. Example 2 ( continued .) We re-formulate the action theory for this example in terms ofour formalism: o & Foo [ τ ] upL , [ τ ] upL , [ τ ] AQ open [ τ + ] open ∧ ¬ [ τ ] F A ¬ open , [ τ, υ ] f lip i , [ τ ] upL i ¬ [ υ ] upL i ∧ ¬ [ τ ] F A upL i ( i = 1 , , [ τ, υ ] f lip i , ¬ [ τ ] upL i [ υ ] upL i ∧ ¬ [ τ ] F A ¬ upL i ( i = 1 , , [ τ, υ ] close, [ τ ] upL , [ τ ] upL ¬ [ υ ] open ∧ ¬ [ τ ] F A open , [ τ , τ ] hold closed, ¬ [ τ ] open, τ ≤ τ ≤ τ [ τ ] held closed ∧ ¬ [ τ ] F A ¬ held closed , [ τ ] held closed ¬ [ τ ] AQ open , [ τ ] ζ, [ τ ] F A ζ [ τ + ] ζ , where ζ ∈ F ∗ . The theory is described as follows:Γ = { [Θ] upL , [Θ] upL , [Θ] open, [Θ] ¬ held closed }∪{ [ c , c ] close, [ c , c ] hold closed, [ c , c ] f lip }∪{ Θ ≤ c < c ≤ c < c ≤ c } .Consider ∆ ⊆ AB such that ∆ AQ = AB AQ \ { [ c ] AQ open | c ≤ c ≤ c } and ∆ F A = AB F A \ ( { [ c ] F A open } ∪ { [ c ] F A ¬ held closed | c ≤ c ≤ c } ∪ { [ c ] F A upL } ). ∆ is AD-plausibleand resulting in the following:[ c ] { upL , upL , open, ¬ held closed } , [ c ] { upL , upL , ¬ open, held closed } , [ c ] { upL , upL , ¬ open, held closed } , [ c ] {¬ upL , upL , ¬ open, held closed } , [ c ] {¬ upL , upL , ¬ open, held closed } . Example 3 ( continued .) The domain description:[ τ ] sw , [ τ ] sw , [ τ ] AQ relay [ τ + ] relay ∧ ¬ [ τ ] F A ¬ relay , [ τ ] sw , ¬ [ τ ] sw , [ τ ] AQ relay [ τ + ] relay ∧ ¬ [ τ ] F A ¬ relay , [ τ ] relay , [ τ ] AQ ¬ sw ¬ [ τ + ] sw ∧ ¬ [ τ ] F A sw , [ τ ] relay , [ τ ] AQ sw [ τ + ] sw ∧ ¬ [ τ ] F A ¬ sw , ¬ [ τ ] sw i , [ τ, υ ] toggle i [ υ ] sw i ∧ ¬ [ τ ] F A ¬ sw i ( i = 1 , , [ τ ] sw i , [ τ, υ ] toggle i ¬ [ υ ] sw i ∧ ¬ [ τ ] F A sw i ( i = 1 , , [ τ ] ζ, [ τ ] F A ζ [ τ + ] ζ , where ζ ∈ F ∗ . The state of the circuit in Figure 2 is captured by the following action theory:Γ = {¬ [Θ] sw , [Θ] sw , ¬ [Θ] relay , ¬ [Θ] relay }∪{ [ c , c ] toggle , [ c , c ] toggle }∪ { Θ ≤ c 7. Discussion and Future Work We developed a uniform framework for reasoning about action using an argumentation-theoretic approach (more precisely, assumption-based approach). We have also presentedhow our framework copes with the frame, the qualification and the ramification problems inseveral sophisticated settings. We have shown how our framework can be naturally extendedto become more expressive.We also explored a new abstraction level which we believe to be an intermediate layerbetween the common sense knowledge and the scientific knowledge. Sophisticated domainknowledge as well as representation are, we argue, required to achieve an adequate underly-ing representation and reasoning process. Among the merits of this approach, we emphasisethe following: • Non-monotonicity is handled by assumptions and argumentation-theoretic approach. 20. The authors would like to acknowledge an anonymous referee for pointing out this very subtle issue. o & Foo • The flexibility of working with different kinds of information representation since thereis no restriction on the syntax of the system. • Expressivity : temporal information is explicitly represented. Thus the system is ca-pable of capturing many important features of temporal reasoning.As reviewed in the introductory section of this paper, numerous approaches to reasoningabout action have been proposed. As such, much research related to these approaches andframeworks has evolved around the central problems addressed in this paper, namely theframe, qualification and the ramification problems. However, as non-monotonic reasoning-based formalisms for reasoning about actions were shown to be flawed by Hanks and McDer-mott (1987), many existing solutions of the frame problem are based on other approacheswhich are usually monotonic. Since one of the inherent properties of the argumentation-theoretic approach is monotonicity, few have attempted to solve the problems of reasoningabout actions using this approach.A framework for reasoning about actions under the argumentation-theoretic approach isindependently proposed by Kakas, Miller, and Toni (1999, 2000, 2001). In their approach,the admissibility semantics is also employed to resolve conflicts between adversary argu-ments. To represent common sense knowledge, e.g. the common sense law of inertia, anorder is imposed on the set of inference rules of the assumption-based framework. The com-putation of the arguments and their competing is performed on top of the so-called prooftrees . Nodes on the proof trees are arguments which are sets of propositions. Construc-tion of proof trees is on the same level of hardness as known frameworks for argumentation-theoretic computation. While their framework allows the persistence of (inertial) fluents tobe captured and dealt with, it is not very clear whether their formalisation can be extendedto deal with other problems of reasoning about actions such as the qualification and theramification problems.More recently, Dimopoulos, Kakas, and Michael (2004), and Bracciali and Kakas (2004)extended Kakas et al.’s framework to deal with the ramification and the qualification prob-lems. Essentially, in their solution to the ramification problem, the so-called r-propositions of the following form: L whenever C are added to the domain description. A resulting model for a given domain description willthen be computed by first computing all possible indirect effects as the fixed point of re-peated application of r-propositions, and then completing the model by allowing unaffectedfluent literals to persist over time. It should be clear that the above representation of indi-rect effects is quite restrictive as it doesn’t allow more complex expressions for the conditionsof an indirect effects and the indirect effects themselves. These restriction is essentially dueto their use of Answer Set Programming (ASP) as the underlying computation mechanismof their framework. On the other hand, their solution to the qualification problem lies in theuse of default rules for representing the effects of actions. This approach is fairly similar to 21. In Kakas et al.’s (1999, 2000) terminologies, each node is a set of arguments which is equivalent to thenotion of arguments in our exposition.22. L is a fluent literal and C is a set of fluent literals which is essentially equivalent to a conjunction offluent literals. easoning about Action: An Argumentation-Theoretic Approach Thielscher’s (2001) solution to the qualification problem discussed above. As their approachis essentially based on stability semantics, it’s not clear how the argumentation-theoreticapproach will bring about any benefit to their framework.As discussed earlier in the paper, the key insight behind the solutions implemented inour framework is the exploitation of causality to drive the inference. It is causality that hashelped throughout in our solutions to the frame and the qualification problems to providethe mechanisms to eliminate the anomalous models while retaining the intuitive modelsduring the process of reasoning. This insight is certainly a more general version of the con-clusions derived by Sandewall (1994) when investigating the reason behind the productionof anomalous models in early approaches to the frame problem, namely the failure to dis-tinguish between normal changes which are triggered by actions and abnomalous changes.Sandewall’s insight has certainly originated the occlusion-based solution to the frame prob-lem (Sandewall, 1994; Doherty, 1994; Gustafsson & Doherty, 1996). Roughly speaking, inthis approach each action type is associated with a subset of fluents that are influencedby the action. Also, a predicate Occlude is introduced to allow this subset of fluents tobe specified. When reasoning about dynamic domains, changes are minimised with theexception of the fluents specified by Occlude . In other words, Occlude distinguishes thenormal changes (associated with the action types) from the anomalous changes and thusavoids unintended models from arising.Regarding the qualification problem, Thielscher’s (2001) solution to the problem ofanomalous models in relation to the qualification problem shares the key insight of causality with our approach. However, Thielscher’s framework is based on the standard semanticsof Default Logic (with the initial theories are generated using circumscription). Thus, hisapproach does not deal with problem domains where standard semantics of Default Logicdoes not produce an extension. Furthermore, as discussed in the introductory section of ourpaper, in Thielscher’s framework qualifications are taken over the executability conditionsfor the actions instead of over different effects of the actions.In parallel to the causality-based insight to dynamic domains, other mechanisms forsolving various problems of reasoning about action exist. For instance, a number of ap-proaches employ the concept of chronological ignorance (Shoham, 1987, 1988) to tacklethe problem of anomalous models. In general, causality-based frameworks and approachesbased on chronological ignorance share the notion of directedness: when changes are min-imised chronologically, causes are minimised instead of effects as causes are likely to precedeeffects. However, as a consequence, backward reasoning is blocked which prevents chrono-logical ignorance-based approaches from dealing with surprises and expectation failures.Furthermore, non-deterministic actions or incomplete state knowledge are known to causedifficulty to chronological minimization. For a more detailed comparison between causality-based approaches and their counterparts that are based on chronological ignorance, thereader is referred to an article by Thielscher (2001). On the other hand, approaches thatare based on Motivated Action Theory (MAT) (Amsterdam, 1991; Stein & Morgenstern,1994) can be considered as a special case of the causality-based paradigm. MAT frame-works also advocate for the insight that an appropriate notion of causality is necessary whenassuming away abnormalities. MAT frameworks, however, don’t cope very well with theexplanation problem and the ramification problem as pointed out by Jr. (1999) who alsointroduce an approach to improve MAT and overcome these problems. o & Foo Nonetheless, several issues still remain within our framework and will need further treat-ment to achieve an optimal solution. Firstly, this formalism may not be very suitable forthe large-scale problems as too many assumptions will need to be taken into account. Toaddress this problem, a localisation procedure is invented to guarantee that only an ade-quate sub-language will be used to capture the circumstance the agent is reasoning about.As a consequence, the set of assumptions will be restricted to those which are necessary toinfer the conclusions the agent is interested. This idea is under development. Furthermore,while a uniform solution to all major problems of reasoning about action may be quiteattractive especially regarding the toy scenarios such as those considered in this paper aswell as in the literature, such a solution may not be pragmatic. By considering the localityaccount of reasoning, in particular of the assumptions used in default reasoning, promisingsolution can be achieved from both computational and representational points of view. Weare undertaking further investigation towards this research direction.A major limitation of our framework is the ability to deal with disjunctive axiomati-sation of action occurrences. For instance, when the domain axiomatisation constains adisjunction about the action occurrences such as [ t , t ] α ∨ [ t , t ] α , then no plausible setsof assumptions, and accordingly, no CPMMs (resp. CPMQMs) will be produced to accountfor the effects of these actions. While an initial formalisation of our framework allowedcomplex expressions for action occurrences in the premises of inference rules which over-came this problem, the formalisation appeared to be too complex and some of the technicalresults could not be established. Further investigation, therefore, is needed to overcomethis problem while avoiding to produce an overly complex formalisation.Provided that our formalism has been designed to provide a general framework for rea-soning about action, the following comes as a natural question: To what extent the proposedapproach can be used to formalise dynamic domains? Sandewall (1994) suggests that oneshould systematise a framework for reasoning about action against some standard criteriato provide the formal indications about the expressiveness and capacity of a formalism.This is in contrast to the more traditional approach that had prevailed for many years inreasoning about action in which one tries to come up with some examples to show that noexisting approaches are able to deal with such a scenario and claims that one’s approach isbetter than others in which it can solve the proposed scenario.Relative to Sandewall’s standard criteria, our approach enjoys the following properties:1. Our approach is non-inertia as it allows observations about the later state to cor-rect the system’s predictions about those states using some explanation mechanismthrough the use of assumptions.2. Our formalism is able to deal with non-deterministic and concurrent events.Another issue is the abduction problem which is also known in the literature as theexplanation problem. For example, in the Stolen car scenario, by observing that the cardisappeared from the parking lot where the reasoner had left it, an expectation failure arises.Most formalisms would try to accommodate this problem by introducing a stealing actionas part of the vocabulary and try to bind the disappearance of the car to this action. Thisarguably is not very intuitive as there is no good reason why we should include this actionin the vocabulary in the first place but not others such as the car being towed away by the easoning about Action: An Argumentation-Theoretic Approach police or a fairy turning the car into a pumpkin, etc. From a pragmatic point of view, somereasoners may just simply acquire more information (perhaps from the police) instead ofconfusing themselves with all kinds of explanations towards possible but uncertain causes.In other words, we effectively isolate the issues of deducing the new conclusions from theexisting knowledge base from abducing the possible causes of some observations. This isof course closely related to Shanahan’s approach in his IJCAI’95 paper (Shanahan, 1989)whose title clearly indicated that “ Prediction is Deduction but Explanation is Abduction ”.We are also working on an assumption-based framework to solve the abduction problem. Acknowledgements This work was performed when the first author was at the School of Computer Science andEngineering, University of New South Wales. The authors wish to thank other membersof the Knowledge Systems Group, in paricular Dongmo Zhang and Maurice Pagnucco, andthe anonymous reviewers of an earlier version of this paper for very helpful comments andsuggestions that have significantly improve the quality as well as the readability of thispaper. The first author was partially supported by an International Postgraduate ResearchScholarship (IPRS) sponsored by the Australian government. The first author is presentlysupported by a by a DEST IAP grant (2004-2006, grant CG040014). Appendix A Theorem 1 Let D = hL D , R , AB , Γ i be an S-domain. If I is a CPMM of D then ∆ IQF isplausible.Proof: Suppose that I is a CPMM of D ,(i) we prove that ∆ IQF is presumable: ∆ IQF is R -consistent since I is a model of D . From Observation 2, we have ∆ IQF isclosed and does not attack itself. From Lemma 1, for each assumption δ ∈ AB , if δ / ∈ ∆ IQF then δ is rejected by ∆ IQF .(ii) we prove that Lr (∆ IQF ) is minimal: Suppose by way of contradiction that there exists a presumable set of assumptions ∆(wrt D ) such that Lr (∆) ⊂ Lr (∆ IQF ). Let I ∆ be the ∆-relativised model of D . It is obviousthat I ∆ is coherent. We will derive a contradiction by proving that Occ I ∆ ⊂ Occ I :(ii.1) OA D ⊆ Occ I : obvious as OA D = { ( φ ( τ ) , α, φ ( τ )) ∈ T × A × T | Γ | = [ τ , τ ] α } ,and I is a model of D .(ii.2) DAS (∆) ⊆ Occ I : Let ( t, da ¬ l , t + ) ∈ DAS (∆), then there do not exist any action α ∈ A and s ∈ T such that r = Φ , [ τ , τ ] α, [ τ , τ ] AQ ¬ l ¬ [ τ ] l ∧ ¬ [ τ ] F A l ∈ R and I | = prem ( r )[ τ /s, τ /t + ] } . Thus, the assumption δ = [ t ] F A l ∈ Lr (∆). Thus, δ ∈ Lr (∆ IQF ) (from the hypothesis.) From Lemma 2, I | = [ t, t + ] da ¬ l . Thus ( t, da ¬ l , t + ) ∈ Occ I .(ii.3) Occ I Occ I ∆ : Let δ = [ t ] F A l ∈ Lr (∆ IQF ) \ Lr (∆) for some t ∈ T and l ∈ F ∗ .Since I is a CPMM of D , from Lemma 2, I | = [ t, t + ] da ¬ l . Thus ( t, da ¬ l , t + ) ∈ Occ I . o & Foo Moreover, δ / ∈ Lr (∆) iff either (a) δ ∈ ∆, or (b) there is α ∈ A such that r = Φ , [ τ , τ ] α, [ τ , τ ] AQ ¬ l ¬ [ τ ] l ∧ ¬ [ τ ] F A l ∈ R and Γ ∪ ∆ ⊢ R Φ[ τ /t, τ /t + ] , [ t, t + ] α, [ t, t + ] AQ ¬ l iff , following the construction of ∆-relativised models, I ∆ = [ t, t + ] da ¬ l . Thus ( t, da ¬ l , t + ) / ∈ Occ I ∆ . Therefore, Occ I ∆ ⊂ Occ I .Hence, Lr (∆ IQF ) is minimal and ∆ IQF is plausible. ✷ Theorem 2 Let D = hL D , R , AB , Γ i be an S-domain and ∆ ⊆ AB . ∆ is plausible wrt D iff M od S ∆ ( D ) = ∅ and for each I ∈ M od S ∆ ( D ) , I is a CPMM of D .Proof: ( ⇒ ) Suppose that ∆ ⊆ AB is plausible wrt D . Then Γ ∪ ∆ is R -consistent, i.e. Γ ∪ ∆ R false . From the construction of ∆-relativised models, M od S ∆ ( D ) = ∅ .For each I = h h, φ, η, ε q , ε f i ∈ M od S ∆ ( D ), we prove that I is a CPMM of D .(i) I is a coherent S-model of D : obvious from the definition of ∆-relativised models.(ii) Occ I is minimal: Assume the contrary, i.e. there is a non-empty set M I = { J | J is a coherent S-modelof D and Occ J ⊂ Occ I } . Let H ∈ M I such that there does not exist any model J ∈ M I and F A H ⊂ F A J .Consider the set of assumptions ∆ HQF :(ii.a) ∆ HQF is presumable wrt D : • ∆ HQF is closed and does not attack itself (since H is a model of D ); • Let δ ∈ AB , if δ = [ τ ] F A l ∆ HQF (for some τ ∈ T E and l ∈ F ∗ ) and δ isn’t rejected by∆ HQF then we can easily construct a model J that interprets everything except F A thesame as H and F A J = F A H ∪ { ( φ ( τ ) , F A l ) } . Obviously, J ∈ M I and F A J ⊂ F A H which is a contradiction. Thus δ is rejected by ∆ HQF .Therefore, ∆ HQF is presumable.(ii.b) Lr (∆ HQF ) ⊂ Lr (∆): Occ H ⊂ Occ I since H ∈ M I .Let δ ∈ Lr (∆ HQF ). Since AB AQ ⊆ ∆ HQF , δ = [ t ] F A l ∈ AB F A for some t ∈ T and l ∈ F ∗ . From the definition of leniently rejected assumptions, Γ ∪ ∆ HQF R false butΓ ∪ ∆ HQF ∪ { δ } ⊢ R false . But then, from the definition of coherent models, H | = [ t, t + ] da ¬ l (since H is a coherent model of D ). Thus, I | = [ t, t + ] da ¬ l .From the construction of ∆-relativised models, [ t ] F A l ∆. Thus, δ is rejected by ∆(since ∆ is plausible wrt D ). Also from the construction of ∆-relativised models, Γ ∪ ∆ R ¬ δ . Therefore, δ ∈ Lr (∆), or Lr (∆ HQF ) ⊆ Lr (∆). Now, Occ I = OA D ∪ DAS (∆). OA D ⊆ Occ H since H is a model of D . Thus, Occ I \ Occ H = DAS (∆) \ Occ H . Let ( t, da ¬ l , t + ) ∈ DAS (∆) \ Occ H . From the construction of ∆-relativised models, δ = [ t ] F A l ∈ Lr (∆).Suppose further that δ ∈ Lr (∆ HQF ). Then, δ / ∈ ∆ HQF (as ∆ HQF is presumable). Weconstruct a model J in such a way that J interprets everything except F A the same as H . easoning about Action: An Argumentation-Theoretic Approach From the definition of coherent models, ( t, da ¬ l , t + ) / ∈ Occ H iff either (1) H = [ t ] l ∧ ¬ [ t + ] l ;or (2) there is α ∈ A , and r = Φ , [ τ , τ ] α, [ τ , τ ] AQ ¬ l ¬ [ τ ] l ∧ ¬ [ τ ] F A l ∈ R and Γ ∪ ∆ | = prem ( r )[ τ /t, τ /t + ], i.e., Γ ∪ ∆ | = Φ[ τ /t, τ /t + ] , [ t, t + ] α, [ t, t + ] AQ ¬ l . Since δ = [ t ] F A l ∈ Lr (∆), condition (2) can not be satisfied.Thus, ( t, da ¬ l , t + ) / ∈ Occ H iff H = [ t ] l ∧ ¬ [ t + ] l . In other words, it is consistent to addthe assumption [ t ] F A l to the set of assumptions ∆ HQF . Or, from a model-theoretic point ofview, to augment the denotation of F A in H with ( t, F A l ) and we can still obtain a coherentS-model J such that F A H ⊂ F A J . But this is a contradiction. Hence, δ / ∈ Lr (∆ HQF ).We have shown that Lr (∆ HQF ) ⊂ Lr (∆).From (ii.a) and (ii.b) we are led to the conclusion that Occ I is minimal. As a conse-quence, I is a CPMM of D (from (i) and (ii)).( ⇐ ) Suppose that M od S ∆ ( D ) = ∅ and for each I ∈ M od S ∆ ( D ), I is a CPMM of D .We prove that ∆ is plausible wrt D . Take an arbitrary model I ∈ M od S ∆ ( D ). FollowingObservation 4, ∆ = ∆ IQF . From Theorem 1 and the hypothesis that I is a CPMM of D ,we conclude that ∆ is plausible wrt D . ✷ Theorem 3 Let D = hL D , R , AB , Γ i be an S-domain. Furthermore, suppose that CP M M ( D ) is the set of CPMMs of D and P laus ( D ) is the set of plausible sets of assumptions of D ,then CP M M ( D ) = S ∆ ∈ P laus ( D ) M od S ∆ ( D ) .Proof: ( ⊇ ) Let ∆ ∈ P laus ( D ), then M od S ∆ ( D ) ⊆ CP M M ( D ) (Following Theorem 2). There-fore, S ∆ ∈ P laus ( D ) M od S ∆ ( D ) ⊆ CP M M ( D ).( ⊆ ) Let I ∈ CP M M ( D ), then ∆ IQF ∈ P laus ( D ) (Following Theorem 1).We prove that I ∈ M od ∆ IQF ( D ) based on Definition 5.4:1) I is an S-model of D ,2) for each δ = [ τ ] F A l ∈ AB F A (for some τ ∈ T E and l ∈ F ∗ ), δ ∈ ∆ IQF iff ( φ ( τ ) , F A l ) ∈ F A I (Following the definition of ∆ IQF - Definition 5.3)3) We prove that Occ I = OA D ∪ DAS (∆ IQF ):(3. ⊇ ) • OA D = { ( φ ( τ ) , α, φ ( τ )) ∈ T × A × T | Γ | = [ τ , τ ] α } ⊆ Occ I (as I is a model of D ), • ( t, da ¬ l , t + ) ∈ DAS (∆ IQF ).From Definition 5.4,(i) δ = [ t ] F A l / ∈ ∆ IQF , and(ii) there exists no action α ∈ A such that: r = Φ , [ τ , τ ] α, [ τ , τ ] AQ ¬ l ¬ [ τ ] l ∧ ¬ [ τ ] F A l ∈ R o & Foo and Γ ∪ ∆ | = Φ[ τ /t, τ /t + ] , [ t, t + ] α, [ t, t + ] AQ ¬ l .From Lemma 1, δ / ∈ ∆ IQF iff δ is rejected by ∆ IQF iff — as I is a coherent model —either (a) there are α ∈ A and s ∈ T such that r = Φ , [ τ , τ ] α, [ τ , τ ] AQ ¬ l ¬ [ τ ] l ∧ ¬ [ τ ] F A l ∈ R and I | = prem ( r )[ τ /s, τ /t + ],or (b) I | = [ t, t + ] da ¬ l .Since (a) violates condition (ii) above, (b) must be the case. Thus ( t, da ¬ l , t + ) ∈ Occ I . ⇒ DAS (∆ IQF ) ⊆ Occ I . ⇒ Occ I ⊇ OA D ∪ DAS (∆ IQF ).(3. ⊆ ) Suppose that Occ I OA D ∪ DAS (∆ IQF ). From (3. ⊇ ), we have Occ I ⊃ OA D ∪ DAS (∆ IQF ). Since ∆ IQF is plausible, there exists a model J ∈ M od S ∆ IQF ( D ) such that J is a CPMM of D . Following Definition 5.4, Occ J = OA D ∪ DAS (∆ IQF ) which is a propersubset of Occ I . Contradiction! Therefore, Occ I ⊆ OA D ∪ DAS (∆ IQF ).Thus we have shown that I ∈ M od S ∆ IQF ( D ), ⇒ M ∈ S ∆ ∈ P laus ( D ) M od S ∆ ( D ) (Since ∆ IQF ∈ P laus ( D )) ⇒ CP M M ( D ) ⊆ S ∆ ∈ P laus ( D ) M od S ∆ ( D ).Therefore, CP M M ( D ) = S ∆ ∈ P laus ( D ) M od S ∆ ( D ). ✷ Theorem 4 Let D be a Q-domain. If I is a CPMQM of D then ∆ IQF is Q-plausible wrt D .Proof: Suppose that I is a CPMQM of D ,(i) it’s easy to verify that ∆ IQF is semi–Q-plausible wrt D : similar to the proof of theorem1, but using Lemma 3 instead of Lemma 1 and Lemma 2.(ii) ∆ IAQ is maximal:Assume the contrary, i.e. there exists a set of assumptions ∆ such that ∆ is semi–Q-plausible wrt D and ∆ IAQ ⊂ ∆ AQ . Since ∆ is presumable, Γ ∪ ∆ is R -consistent. Let I ∆ bethe ∆-relativised model of D . Obviously, I ∆ is a coherent Q-model of D . It’s easy to verifythat Occ I ∆ is minimal because otherwise a coherent Q-model J (of D ) can be constructedsuch that Occ J ⊂ Occ I ∆ . We have Occ I ∆ = OA D ∪ DAS (∆) and OA D ⊆ Occ J since J is amodel of D . Thus there exists ( t, da ¬ ϕ , t + ) ∈ DAS (∆) \ Occ J . But then ∆ JQF is presumablewrt D and Lr (∆ JQF ) ⊂ Lr (∆) which is a contradiction. Thus, Occ I ∆ is minimal. But thenthe model I ∆ is a PMQM of D and AQ I ∆ ⊂ AQ I which contradicts with the fact that I isa CPMQM of D . Therefore, ∆ IAQ is maximal.(iii) ∆ IF A is maximal (relative to (i) and (ii)):Assume the contrary, i.e. there exists a set of assumptions ∆ such that ∆ is semi–Q-plausible wrt D and ∆ AQ is maximal and ∆ IF A ⊂ ∆ F A . Let I ∆ be the ∆-relativised model easoning about Action: An Argumentation-Theoretic Approach of D . Similar to the proof in part (ii), we can easily verify that I ∆ is a PMQM of D . As thefact that ∆ AQ is maximal and ∆ IF A ⊂ ∆ F A contradicts with the given hypothesis that I isa CPMQM of D , we conclude that ∆ IF A is maximal. Therefore, ∆ IQF is Q-plausible wrt D . ✷ Theorem 5 Let D = hL D , R , AB , Γ i be a Q-domain and ∆ ⊆ AB . ∆ is Q-plausible wrt D iff M od Q ∆ ( D ) = ∅ and for each I ∈ M od Q ∆ ( D ) , I is a CPMQM of D .Proof: ( ⇒ ) Suppose that ∆ is Q-plausible wrt D .As ∆ is presumable, Γ ∪ ∆ is R -consistent. Following the construction of ∆-relativisedmodels, M od Q ∆ ( D ) = ∅ .Let I ∈ M od Q ∆ ( D ):(i) it’s easy to verify that I is coherent from the construction of ∆-relativised models.(ii) Occ I is minimal:Assume the contrary, i.e. the set Σ I = { σ | σ is a coherent Q-model of D and Occ σ ⊂ Occ I } is non-empty.Let J ∈ Σ I such that the following conditions are satisfied:1. there does not exist any model σ ∈ Σ I such that Occ σ ⊂ Occ J .2. AQ J is maximal relative to 1.3. F A J is maximal relative to 1. and 2.Consider the set of assumptions ∆ JQF : Obviously, ∆ JQF is closed and does not attachitself. For each δ ∈ AB , if δ / ∈ ∆ JQF then δ is rejected by ∆ JQF , otherwise it would violatethe maximality of ∆ JAQ and ∆ JF A . Thus, ∆ JQF is presumable.Remark that Occ J ⊂ Occ I . Besides, Occ I = OA D ∪ DAS (∆). But OA D ⊆ Occ J as J is a model of D . Thus there exists ( t, da ¬ ϕ , t + ) ∈ DAS (∆) \ Occ J . But then ∆ JQF ispresumable wrt D and Lr (∆ JQF ) ⊂ Lr (∆) which is a contradiction. Thus, Occ I is minimal.As a consequence, I is a PMQM of D .(iii) AQ I is maximal (relative to (i) and (ii)):Assume the contrary, i.e. the set Σ I = { σ | σ is a PMQM of D and AQ I ⊂ AQ σ } isnon-empty.Let J ∈ Σ I such that the following conditions are satisfied:1. F A J is maximal; and2. AQ J is maximal relative to 1.We can prove that that ∆ JQF is semi-Q-plausible (i.e. presumable wrt D and Lr F A (∆ JQF )is minimal):That ∆ JQF is presumable wrt D is easy to verify.It’s also easy to verify that Lr F A (∆ JQF ) is minimal as J is coherent and Occ J is minimal.This would guarantee that the set of occurrences of dummy actions in J is minimised and asa consequence the set of leniently rejected frame assumptions is also minimised. Formally,if a presumable set of assumptions Π is such that Lr F A (Π) ⊂ Lr F A (∆ JQF ) then the ∆-relativised model I Π is a coherent Q-model and Occ I Π ⊂ Occ J , which is a contradictionwith the fact that J ∈ Σ I and thus J is a PMQM of D .Thus, ∆ JQF is semi-Q-plausible wrt D . But, ∆ AQ ⊂ ∆ JAQ which is a contradiction.Therefore, we have shown that AQ I is maximal (relative to (i) and (ii)). o & Foo (iv) Now, we can prove that F A I is maximal (relative to (i), (ii) and (iii)):Assume the contrary, i.e. there exists a PMQM J of D such that F A I ⊂ F A J .Among those models satisfying the above condition, we choose a model K such thatthere does not exist any PMQM K ′ of D such that F A K ⊂ F A K ′ . Thus, K is a CPMQMof D .Following Theorem 4, we have the set of assumptions ∆ KQF is Q-plausible and ∆ F A ⊂ ∆ KF A since F A I ⊂ F A K . Contradiction with the hypothesis that ∆ is Q-plausible.From (i), (ii), (iii) and (iv) we are led to the conclusion that I is a canonical prioritisedminimal Q-model of D .( ⇐ ) Suppose that M od Q ∆ ( D ) = ∅ and I is a canonical prioritised minimal model of D for each I ∈ M od Q ∆ ( D ), we prove that ∆ is plausible.Take an arbitrary model I ∈ M od Q ∆ ( D ). Following Observation 5, ∆ = ∆ IQF . FromTheorem 4 and the hypothesis that I is a canonical prioritised minimal Q-model of D , weconclude that ∆ is Q-plausible. ✷ Theorem 6 Let D be a Q-domain. Furthermore, suppose that CP M QM ( D ) is the set ofCPMQMs of D and P laus Q ( D ) is the set of Q-plausible sets of assumptions of D , then CP M QM ( D ) = S ∆ ∈ P laus Q ( D ) M od Q ∆ ( D ) .Proof: Similar to the proof of Theorem 3. ✷ Theorem 7 Let σ = hT , F , Ai be a signature and D = hL D , R , AB , ∅i a SSD and α ∈ A .Suppose that w ∈ IS . Define a domain description D = hL D , R , AB , Γ i , where Γ = { [Θ] ϕ | ϕ ∈ w }∪{ [Θ , Θ + ] α } . A set ∆ ⊆ AB of assumptions is AD-plausible wrt D iff for each model M of E D (∆) , [ M ] N M (Θ) ∈ T rans αD ([ M ] Θ ) , i.e. [ M ] N M (Θ) belongs to the state transitionfrom [ M ] Θ in D according to α .Proof: ( ⇒ ) Suppose that ∆ ⊆ AB is AD-plausible wrt D , we prove that for each model M of E D (∆), [ M ] N M (Θ) ∈ T rans αD ([ M ] Θ ).Let M ∈ M od ( E D (∆)), as M | = Γ, we have M | = [Θ] ϕ iff ϕ ∈ w . Thus w = [ M ] Θ .We prove that [ M ] N M (Θ) ∈ T rans αD ( w ), i.e. there exists a sequence ω , . . . , ω n ∈ IS suchthat ( w, α, ω ) ∈ Res D and [ M ] N M (Θ) = ω n and ( ω i , ω i +1 ) ∈ Causes D for each 1 ≤ i < n .If Υ wα = ∅ : to the reasoner’s knowledge, α is not applicable in the instantwise state w due to either non-executability of α in w , or α does not bring about any effects concernedto the reasoner. Thus, [ M ] Θ + = w . Of course, ( w, α, [ M ] Θ + ) ∈ Res D .If Υ wα = ∅ : let Ω = { r ∈ R A | M | = prem ( r ) and M | = cons ( r ) } , we prove that Ω is apossible application of α in w .Apparently, Ω ⊆ Υ wα . Ω is maximal since otherwise we can construct a model M ′ suchthat M ′ satisfies the qualification assumption of the additional action description rule. Butit means M ′ is a PMQM model of D such that AQ M ⊂ AQ M ′ . Thus M is not a CPMQMof D . Following Theorem 5, ∆ is not Q-plausible wrt D . Contradiction.We now prove that there does not exists any instantwise state S such that CON S A (Ω) ⊆ S and [ M ] Θ + \ w ⊂ S \ w . easoning about Action: An Argumentation-Theoretic Approach Assume the contrary, i.e. there exists a fluent literal ϕ ∈ F ∗ such that ϕ ∈ ( S \ w ) \ ([ M ] Θ + \ w ).Then M = [Θ] F A ϕ since M is a model of D .Construct a model M ′ in such a way that M ′ interprets everything the same as M except F A , and F A M ′ = F A M ∪ { (Θ , F A ϕ ) } . 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