A Qualitative Theory of Cognitive Attitudes and their Change
AA Qualitative Theory of Cognitive Attitudesand their Change
EMILIANO LORINIIRIT-CNRS, Toulouse University, France
Abstract
We present a general logical framework for reasoning about agents’ cognitiveattitudes of both epistemic type and motivational type. We show that it allows us toexpress a variety of relevant concepts for qualitative decision theory including theconcepts of knowledge, belief, strong belief, conditional belief, desire, conditionaldesire, strong desire and preference. We also present two extensions of the logic,one by the notion of choice and the other by dynamic operators for belief changeand desire change, and we apply the former to the analysis of single-stage gamesunder incomplete information. We provide sound and complete axiomatizationsfor the basic logic and for its two extensions.The paper is “under consideration in Theory and Practice of Logic Program-ming (TPLP)”.
Since the seminal work of Hintikka on epistemic logic [28], of Von Wright on the logicof preference [55, 56] and of Cohen & Levesque on the logic of intention [19], manyformal logics for reasoning about cognitive attitudes of agents such as knowledge andbelief [24], preference [32, 48], desire [23], intention [44, 30] and their combination[38, 54] have been proposed. Generally speaking, these logics are nothing but formalmodels of rational agency relying on the idea that an agent endowed with cognitiveattitudes makes decisions on the basis of what she believes and of what she desires orprefers.The idea of describing rational agents in terms of their epistemic and motivationalattitudes is something that these logics share with classical decision theory and gametheory. Classical decision theory and game theory provide a quantitative account ofindividual and strategic decision-making by assuming that agents’ beliefs and desirescan be respectively modeled by subjective probabilities and utilities. Qualitative ap-proaches to individual and strategic decision-making have been proposed in AI [16, 22]to characterize criteria that a rational agent should adopt for making decisions when shecannot build a probability distribution over the set of possible events and her preferenceover the set of possible outcomes cannot be expressed by a utility function but only bya qualitative ordering over the outcomes. For example, going beyond expected utilitymaximization, qualitative criteria such as the maxmin principle (choose the action that1 a r X i v : . [ c s . A I] F e b ill minimize potential loss) and the maxmax principle (choose the action that willmaximize potential gain) have been studied and axiomatically characterized [18, 17].The aim of this paper is to present an expressive logical framework for representingboth the static and the dynamic aspects of a rich variety of agents’ cognitive attitudesin a multi-agent setting. In agreement with philosophical theories [41, 43, 29, 34], ourlogic allows us to distinguish two general categories of cognitive attitudes: epistemic attitudes, including belief and knowledge, and motivational ones, including desire andpreference. Moreover, in agreement with rational choice theory, it allows us to capturea notion of choice which depends on what an agent believes and prefers. The example depicted in Figure 1 brings to the fore the epistemic and motivationalattitudes that are involved in everyday situations whereby artificial agents are supposedto interact. There are two autonomous agents meeting at a crossroad: agent 1 and agent2. The two agents could be either two mobile robots or two autonomous vehicles. Eachagent can decide either to stop or to continue. If an agent stops, then it will lose time. Ifboth agents decide to continue, they will collide and, consequently, each of them willlose time. Therefore, for an agent not to lose time, it has to continue, while the otheragent decides to stop.In this situation, each agent is identified with the set of cognitive attitudes it en-dorses. For instance, it is reasonable to suppose that the two agents know that in thesituation they face necessarily some of them will lose time and that if one of them losestime by letting the other pass, there will be no collision. On the motivational side, it isreasonable to suppose that each agent is strongly motivated by two desires, namely, thedesire not to lose time and the desire to avoid a collision.
1 2
Figure 1: Crossroad gameOn the dynamic side, we consider two basic forms of cognitive attitude change,namely, belief change and desire change. While belief change has been extensivelystudied in the area of belief revision [1, 15, 20, 46, 42, 13] and dynamic epistemic logic(DEL) [47, 10, 51, 4], desire change is far less studied and understood. We will studytwo basic forms of cognitive attitude revision, namely, radical attitude revision and conservative attitude revision. While the distinction between radical and conservativebelief revision has been drawn before (see, e.g., [47]), the distinction between radical Rational choice theory (RCT) is a umbrella term for a family of theories prescribing that an agent shouldchoose the course of action that, according to her beliefs, leads to the most desirable (or most preferred)consequences. In other words, RCT relies on the general assumption that agents make optimal choices in thelight of her beliefs, desires and preferences. See [40] for more details on RCT. ϕ makesall states at which ϕ is true more plausible than all states at which ϕ is false, whereasconservative belief revision by ϕ simply promotes the most plausible states in which ϕ is true to the highest plausibility rank, but apart from that, it keeps the old plausibilityordering. For example, suppose in the crossroad game of Figure 1, agent 1 and agent 2can communicate. Agent 1 informs agent 2 that “if they both lose time, then there willno collision” and agent 2 trusts what agent 1 says. Then, by performing a conservativebelief revision, agent 2 will promote the most plausible situations in which the formulaannounced by 1 is true to the highest plausibility rank. As a consequence, agent 2 willstart to believe what 1 has just said.Symmetrically, radical desire revision by ϕ makes all states at which ϕ is true moredesirable than all states at which ϕ is false, whereas conservative desire revision by ϕ simply demotes the least desirable states in which ϕ is false to the lowest desirabilityrank, but apart from that, it keeps the old desirability ordering. For example, supposein the crossroad game agent 1 has just learnt that agent 2 is an ambulance which hasto transport a patient to the hospital as quickly as possible. Consequently, 1 starts tobe altruistically motivated by the fact that 2 does not lose time. Thus, by performinga radical desire revision, agent 1 will start to consider all situations in which 2 doesnot lose time more desirable than the situations in which it does. This radical desirerevision operation leads agent 1 to strongly desire that agent 2 does not lose time.The paper is organized as follows. In Section 2, we present the semantics andsyntax of our logic, called Dynamic Logic of Cognitive Attitudes (DLCA). At the se-mantic level, it exploits two orderings that capture, respectively, an agent’s comparativeplausibility and comparative desirability over states. At the syntactic level, it uses pro-gram constructs of dynamic logic (sequential composition, non-deterministic choice,intersection, complement, converse and test) to build complex cognitive attitudes fromsimple ones. Following [39, 25], it also exploits nominals in order to axiomatize inter-section and complement of programs. In Section 3, we illustrate the expressive powerof our logic by using it to formalize a variety of cognitive attitudes of agents includingknowledge, belief, strong belief, conditional belief, desire, strong desire, conditionaldesire and preference. We instantiate some of these concepts in the crossroad gamedepicted in Figure 1. In Section 4, we present a sound and complete axiomatizationfor our logic. In Section 5 we present the first extension of our logic by the notion ofchoice and apply it to the analysis of single-stage games under incomplete information.Section 6 presents the second extension of our logic by dynamic operators for beliefand desire change. In Section 7 we conclude. Formal proofs are given in a technicalannex at the end of the paper. This paper is an extended and improved version of [35]. The JELIA’19 paper did not include the twoextensions of Section 5 and Section 6, or the detailed proof of the completeness theorem for the logic DLCA.Also, the logical analysis of the cognitive attitudes in Section 3 has been extended: (i) we included the notionof conditional desire which was not considered in the JELIA’19 paper, and (ii) we added new logical validitieswhich describe interesting properties of cognitive attitudes. Dynamic Logic of Cognitive Attitudes
Let
Atm be a countable infinite set of atomic propositions, let
Nom be a countableinfinite set of nominals disjoint from
Atm and let
Agt be a finite set of agents.
Definition 1 (Multi-agent cognitive model)
A multi-agent cognitive model (MCM) isa tuple M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) where: • W is a set of worlds or states; • for every i ∈ Agt, (cid:22) i , P and (cid:22) i , D are preorders on W and ≡ i is an equivalencerelation on W such that for all τ ∈ { P , D } and for all w , v ∈ W : (C1) (cid:22) i , τ ⊆≡ i , (C2) if w ≡ i v then w (cid:22) i , τ v or v (cid:22) i , τ w; • V : W −→ Atm ∪ Nom is a valuation function such that for all w , v ∈ W : (C3) V Nom ( w ) (cid:54) = /0 , (C4) if V Nom ( w ) ∩ V Nom ( v ) (cid:54) = /0 then w = v;where V Nom ( w ) = Nom ∩ V ( w ) .w (cid:22) i , P v means that, according to agent i , v is at least as plausible as w , whereas w (cid:22) i , D v means that, according to agent i , v is at least as desirable as w . Finally, w ≡ i v means that w and v are indistinguishable for agent i . For every w ∈ W , ≡ i ( w ) is alsocalled agent i ’s information set at state w . According to Constraint C1, an agent canonly compare the plausibility (resp. desirability) of two states in her information set.According to Constraint C2, the plausibility (resp. desirability) of two states in anagent’s information set are always comparable. Constraints C3 and C4 capture the twobasic properties of nominals: every state is associated with at least one nominal andthere are no different states associated with the same nominal.Note that there is no connection between binary relations (cid:22) i , P and (cid:22) i , D . In ac-cord with classical decision and game theory in which an agent’s subjective probabilityand utility function do not interact, we adopt a normative view of epistemic and moti-vational attitudes according to which an agent’s epistemic plausibility and desirabilityare assumed to be independent. Therefore, we do not consider cognitive biases typicalof human reasoning such as wishful thinking. as the tendency to form beliefs accordingto what is desired in the absence of a clear evidence against it [37]. Nonetheless, aswe will show in Section 3.3, the primitive relations (cid:22) i , P and (cid:22) i , D can be combined toobtain a notion of realistic preference which is essential for elucidating the connectionbetween an agent’s beliefs and desires and her choices.We introduce the following modal language L DLCA ( Atm , Nom , Agt ) , or simply L DLCA , for the Dynamic Logic of Cognitive Attitudes DLCA: The normative view is usually opposed to the descriptive view. The normative view is aimed at de-scribing the reasoning and decision-making of ideal agents conforming to standards of rationality, whilethe descriptive view is concerned with psychologically realistic cognitive agents who systematically violatestandards of rationality and exhibit different types of cognitive bias. :: = ≡ i |(cid:22) i , P |(cid:22) i , D |(cid:22) ∼ i , P |(cid:22) ∼ i , D | π ; π (cid:48) | π ∪ π (cid:48) | π ∩ π (cid:48) | − π | ϕ ? ϕ :: = p | x | ¬ ϕ | ϕ ∧ ϕ (cid:48) | [ π ] ϕ where p ranges over Atm , x ranges over Nom and i ranges over Agt . The other Booleanconstructions (cid:62) , ⊥ , ∨ , → and ↔ are defined from p , ¬ and ∧ in the standard way.The propositional language built from the set of atomic propositions Atm is noted L PL ( Atm ) . Note that the sets Atm , Nom and
Agt define the signature of the language L DLCA . They are not part of the model since every atomic proposition p , nominal x and modal formula [ π ] ϕ should be interpretable relative to any MCM.Elements π are called cognitive programs or, more shortly, programs . The set ofall programs is noted P ( Atm , Nom , Agt ) , or simply, P .Cognitive programs correspond to the basic constructions of Propositional Dy-namic Logic (PDL) [26]: atomic programs of type ≡ i , (cid:22) i , P , (cid:22) i , D , (cid:22) ∼ i , P and (cid:22) ∼ i , D , se-quential composition (;), non-deterministic choice ( ∪ ), intersection ( ∩ ), converse ( − )and test (?). A given cognitive program π corresponds to a specific configuration of theagents’ cognitive states including their epistemic states and their motivational states.The formula [ π ] ϕ has to be read “ ϕ is true, according to the cognitive program π ”.As usual, we define (cid:104) π (cid:105) to be the dual operator of [ π ] , that is, (cid:104) π (cid:105) ϕ = def ¬ [ π ] ¬ ϕ .The atomic program ≡ i represents the standard S5, partition-based and fully intro-spective notion of knowledge [24, 5]. [ ≡ i ] ϕ has to be read “ ϕ is true according to whatagent i knows” or more simply “agent i knows that ϕ is true”, which just means that“ ϕ is true in all worlds that agent i envisages”.The atomic programs (cid:22) i , P and (cid:22) i , D capture, respectively, agent i ’s plausibility or-dering and agent i ’s desirability ordering over facts. In particular, [ (cid:22) i , P ] ϕ has to be read“ ϕ is true at all states that, according to agent i , are at least as plausible as the currentone”, while [ (cid:22) i , D ] ϕ has to be read “ ϕ is true at all states that, according to agent i , are atleast as desirable as the current one”. The atomic programs (cid:22) ∼ i , P and (cid:22) ∼ i , D are the com-plements of the atomic programs (cid:22) i , P and (cid:22) i , D , respectively. In particular, [ (cid:22) ∼ i , P ] ϕ hasto be read “ ϕ is true at all states that, according to agent i , are not at least as plausibleas the current one”, while [ (cid:22) ∼ i , D ] ϕ has to be read “ ϕ is true at all states that, accordingto agent i , are not at least as desirable as the current one”. The program constructs ;, ∪ , ∩ , − and ? are used to define complex cognitive programs from the atomic cognitiveprograms. For example, the formula [ (cid:22) i , P ∪ (cid:22) i , D ] ϕ has to be read “ ϕ is true at all statesthat, according to agent i , are either at least as plausible or at least as desirable as thecurrent one”, whereas the formula [ (cid:22) i , P ∩ (cid:22) i , D ] ϕ has to be read “ ϕ is true at all statesthat, according to agent i , are at least as plausible and at least as desirable as the currentone”.The following definition provides truth conditions for formulas in L DLCA : Definition 2 (Truth conditions)
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be MCM and let w ∈ W . Then:M , w | = p ⇐⇒ p ∈ V ( w ) , M , w | = x ⇐⇒ x ∈ V ( w ) , M , w | = ¬ ϕ ⇐⇒ M , w (cid:54)| = ϕ , M , w | = ϕ ∧ ψ ⇐⇒ M , w | = ϕ and M , w | = ψ , M , w | = [ π ] ϕ ⇐⇒ ∀ v ∈ W : if wR π v then M , v | = ϕ , where the binary relation R π on W is inductively defined as follows, with τ ∈ { P , D } :wR ≡ i v iff w ≡ i v , wR (cid:22) i , τ v iff w (cid:22) i , τ v , wR (cid:22) ∼ i , τ v iff w ≡ i v and w (cid:54)(cid:22) i , τ v , wR π ; π (cid:48) v iff ∃ u ∈ W : wR π u and uR π (cid:48) v , wR π ∪ π (cid:48) v iff wR π v or wR π (cid:48) v , wR π ∩ π (cid:48) v iff wR π v and wR π (cid:48) v , wR − π v iff vR π w , wR ϕ ? v iff w = v and M , w | = ϕ . For notational convenience, we use wR π v and ( w , v ) ∈ R π as interchangeable nota-tions.We can build a variety of cognitive programs capturing different types of plau-sibility and desirability relations between possible worlds. For instance, for every τ ∈ { P , D } , we can define: (cid:23) i , τ = def − (cid:22) i , τ , (cid:31) i , τ = def (cid:23) i , τ ∩ (cid:22) ∼ i , τ , (cid:23) ∼ i , τ = def − (cid:22) ∼ i , τ , ≺ i , τ = def (cid:22) i , τ ∩ (cid:23) ∼ i , τ , ≈ i , τ = def (cid:22) i , τ ∩ (cid:23) i , τ . The five definitions denote respectively “at most as plausible (resp. desirable) as”, “lessplausible (resp. desirable) than”, “not at most as plausible (resp. desirable) as”, “moreplausible (resp. desirable) than” and “equally plausible (resp. desirable) as”.For every formula ϕ in L DLCA we say that ϕ is valid, noted | = MCM ϕ , if and onlyif for every multi-agent cognitive model M and world w in M , we have M , w | = ϕ .Conversely, we say that ϕ is satisfiable if ¬ ϕ is not valid.For a given multi-agent cognitive model M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , N , V ) ,we define || ϕ || M = { v ∈ W : M , v | = ϕ } to be the truth set of ϕ in M . Moreover, for ev-ery w ∈ W and for every i ∈ Agt , we define || ϕ || i , w , M = { v ∈ W : M , v | = ϕ and w ≡ i v } to be the truth set of ϕ from i ’s point of view at state w in M .6 Formalization of Cognitive Attitudes
In this section, we show how the logic DLCA can be used to model the variety ofcognitive attitudes of agents that we have briefly discussed in the introduction.
We start with the family of epistemic attitudes by defining a standard notion of belief.We say that an agent believes that ϕ if and only if ϕ is true at all states that the agentconsiders maximally plausible. Definition 3 (Belief)
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM andlet w ∈ W . We say that agent i believes that ϕ at w, noted M , w | = B i ϕ , if andonly if Best i , P ( w ) ⊆ || ϕ || M where Best i , P ( w ) = { v ∈ W : w ≡ i v and ∀ u ∈ W , if w ≡ i u then u (cid:22) i , P v } . As the following proposition highlights, the previous notion of belief is expressible inthe logic DLCA by means of the cognitive program ≡ i ; [ ≺ i , P ] ⊥ ?. Proposition 1
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and letw ∈ W . Then, we haveM , w | = B i ϕ iff M , w | = (cid:2) ≡ i ; [ ≺ i , P ] ⊥ ? (cid:3) ϕ . It is worth noting that the set
Best i , P ( w ) in Definition 3 might be empty, sinceit is not necessarily the case that the relation (cid:22) i , P is conversely well-founded. Asa consequence, the belief operator B i does not necessarily satisfy Axiom D, i.e., theformula B i ϕ ∧ B i ¬ ϕ is satisfiable in the logic DLCA.In the literature on epistemic logic [11], mere belief of Definition 3 is usually dis-tinguished from strong belief. Specifically, we say that an agent strongly believes that ϕ if and only if, according to agent i , all ϕ -worlds are strictly more plausible than all ¬ ϕ -worlds. Definition 4 (Strong belief)
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCMand let w ∈ W . We say that agent i strongly believes that ϕ at w, noted M , w | = SB i ϕ ,if and only if ∀ v ∈ || ϕ || i , w , M and ∀ u ∈ ||¬ ϕ || i , w , M : u ≺ i , P v . As the following proposition highlights, the previous notion of strong belief is ex-pressible in the logic DLCA by means of the cognitive program ≡ i ; ϕ ?; (cid:22) i , P . Proposition 2
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and letw ∈ W . Then, we haveM , w | = SB i ϕ iff M , w | = (cid:2) ≡ i ; ϕ ?; (cid:22) i , P (cid:3) ϕ . This means that there could be a world v such that w ≡ i v and there is a (cid:22) i , P -infinite ascending chainfrom v . ϕ implies belief that ϕ , if the agent envisages at least one state inwhich ϕ is true. This property is expressed by the following validity: | = MCM (cid:0) SB i ϕ ∧ (cid:104)≡ i (cid:105) ϕ (cid:1) → B i ϕ (1)Conditional belief is another notion which has been studied by epistemic logiciansgiven its important role in belief dynamics [47]. We say that an agent believes that ϕ conditional on ψ , or she would believe that ϕ if she learnt that ψ , if and only if,according to the agent, all most plausible ψ -worlds are also ϕ -worlds. Definition 5 (Conditional belief)
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) bea MCM and let w ∈ W . We say that agent i would believe that ϕ if she learnt that ψ atw, noted M , w | = B i ( ψ , ϕ ) , if and only if Best i , P ( ψ , w ) ⊆ || ϕ || M , where Best i , P ( ψ , w ) = { v ∈ || ψ || i , w , M : ∀ u ∈ || ψ || i , w , M , u (cid:22) i , P v } . Note that
Best i , P ( (cid:62) , w ) = Best i , P ( w ) .As for belief and strong belief, we have a specific cognitive program ≡ i ; ( ψ ∧ [ ≺ i , P ] ¬ ψ ) ? corresponding to the belief that ϕ conditional on ψ , so that the latter can berepresented in in the language of the logic DLCA. Proposition 3
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and letw ∈ W . Then, we haveM , w | = B i ( ψ , ϕ ) iff M , w | = (cid:2) ≡ i ; ( ψ ∧ [ ≺ i , P ] ¬ ψ ) ? (cid:3) ϕ . The first kind of motivational attitude we consider is desire. Following [23], we saythat an agent desires that ϕ if and only if all states that the agent envisages at which ϕ is true are not minimally desirable for her. In other words, desiring that ϕ consists inhaving some degree of attraction for all situations in which ϕ is true, since minimallydesirable states are those to which the agent is not attracted at all. Definition 6 (Desire)
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM andlet w ∈ W . We say that agent i desires that ϕ at w, noted M , w | = D i ϕ , if and only ifWorst i , D ( w ) ∩ || ϕ || M = /0 , where Worst i , D ( w ) = { v ∈ W : w ≡ i v and ∀ u ∈ W , if w ≡ i u then v (cid:22) i , D u } . As the following proposition highlights, the previous notion of desire is characterizedby the cognitive program ≡ i ; [ (cid:31) i , D ] ⊥ ?. Proposition 4
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and letw ∈ W . Then, we haveM , w | = D i ϕ iff M , w | = (cid:2) ≡ i ; [ (cid:31) i , D ] ⊥ ? (cid:3) ¬ ϕ . Best i , P ( w ) in Definition 3, the set Worst i , D ( w ) in Definition6 might be empty, since it is not necessarily the case that the relation (cid:22) i , D is well-founded. As a consequence, desires are not necessarily consistent and an agent maydesire the tautology, i.e., the formulas D i ϕ ∧ D i ¬ ϕ and D i (cid:62) are satisfiable in the logicDLCA. As emphasized by [23], this notion of desire satisfies the following property: | = MCM D i ϕ → D i ( ϕ ∧ ψ ) (2)Indeed, if an agent has some degree of attraction for all situations in which ϕ is truethen, clearly, it should have some degree of attraction for all situations in which ϕ ∧ ψ is true, since all ϕ ∧ ψ -situations are also ϕ -situations.Note that there is no counterpart of this property for belief, as the formula B i ϕ → B i ( ϕ ∧ ψ ) is clearly not valid. It is a property that the notion of desire shares with the open reading of the conceptof permission studied in the area of deontic logic (see, e.g., [3, 31]). One way ofblocking this inference is by strengthening the notion of desire. We say that an agentstrongly desires that ϕ if and only if, according to agent i , all ϕ -worlds are strictly moredesirable than all ¬ ϕ -worlds. Definition 7 (Strong desire)
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCMand let w ∈ W . We say that agent i strongly desires that ϕ at w, noted M , w | = SD i ϕ , ifand only if ∀ v ∈ || ϕ || i , w , M and ∀ u ∈ ||¬ ϕ || i , w , M : u ≺ i , D v . As for desire, there exists a cognitive program which characterizes strong desire, namely,the program ≡ i ; ϕ ?; (cid:22) i , D . Proposition 5
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and letw ∈ W . Then, we haveM , w | = SD i ϕ iff M , w | = (cid:2) ≡ i ; ϕ ?; (cid:22) i , D (cid:3) ϕ . We have that strong desire implies desire, when the agent envisages at least one statein which ϕ is false: | = MCM ( SD i ϕ ∧ (cid:104)≡ i (cid:105)¬ ϕ ) → D i ϕ (3)Unlike desire, it is not necessarily the case that strongly desiring that ϕ implies stronglydesiring that ϕ ∧ ψ , i.e., SD i ϕ ∧ ¬ SD i ( ϕ ∧ ψ ) is satisfiable in the logic DLCA. Indeed,strongly desiring that ϕ is compatible with envisaging a situation in which ϕ ∧ ψ holdsand another situation in which ϕ ∧¬ ψ holds such that the first situation is less desirablethan the second. This means that there could be a world v such that w ≡ i v and there is a (cid:22) i , D -infinite descending chainfrom v . See [23] for more details about the differences between the notion of belief and the notion of desire. According to deontic logicians, there are at least two candidate readings of the statement “ ϕ is permit-ted”: (i) every instance of ϕ is OK according to the normative regulation, and (ii) at least one instance of ϕ (but possibly not all) is OK according to the normative regulation. The former is the so-called open reading of permission. ϕ con-ditional on ψ , or she would desire that ϕ if she started to desire that ψ , if and onlyif, according to agent i , there is no least desirable ¬ ψ -world which is also a ϕ -world.The idea behind this notion is the following. If the agent started to desire that ψ , all ψ -worlds would start to have some degree of attraction for her and the least desirable ¬ ψ -worlds would become the minimally desirable worlds. Therefore, the fact thatthere is no least desirable ¬ ψ -world which is also a ϕ -world guarantees that, if theagent started to desire that ψ , no ϕ -world would be included in the set of minimallydesirable worlds for the agent. The latter means that, if the agent started to desire that ψ , all ϕ -worlds would have some degree of attraction for her and she would desire that ϕ . Definition 8 (Conditional desire)
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) bea MCM and let w ∈ W . We say that agent i would desire that ϕ if she started to desirethat ψ at w, noted M , w | = D i ( ψ , ϕ ) , if and only if Worst i , D ( ¬ ψ , w ) ∩ || ϕ || M = /0 , withWorst i , D ( ¬ ψ , w ) = { v ∈ ||¬ ψ || i , w , M : ∀ u ∈ ||¬ ψ || i , w , M , v (cid:22) i , D u } . As for the other cognitive attitudes, there is a specific cognitive program whichcharacterizes conditional desire.
Proposition 6
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and letw ∈ W . Then, we haveM , w | = D i ( ψ , ϕ ) iff M , w | = (cid:2) ≡ i ; ( ¬ ψ ∧ [ (cid:31) i , D ] ψ ) ? (cid:3) ¬ ϕ . In Section 2, we emphasized that the relations (cid:22) i , P and (cid:22) i , D do not interact sinceour logic is aimed at modeling ideal rational agents with no wishful thinking and, moregenerally, with no cognitive biases. We conclude this section by showing how theassumption of independence between epistemic plausibility and desirability could berelaxed and, consequently, how wishful thinking could be modeled in our framework.A wishful thinker is nothing but an agent who systematically believes what shestrongly desires in the absence of a reason to believe the contrary. Such a connectionbetween the agent’s beliefs and desires is captured by the following “wishful thinking”(WT) constraint on MCMs: ∀ w ∈ W : Best i , P ( w ) ⊆ Best i , D ( w ) or Best i , P ( w ) ⊆ Worst i , D ( w ) , where Best i , P ( w ) and Worst i , D ( w ) are defined as in Definitions 3 and 6, and Best i , D ( w ) = { v ∈ W : w ≡ i v and ∀ u ∈ W , if w ≡ i u then u (cid:22) i , D v } . It is routine to verify that if theMCM M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) satisfies the previous constraintWT, then the following holds for every w ∈ W : M , w | = ( SD i ϕ ∧ ¬ B i ¬ ϕ ) → B i ϕ . We leave for future work an in-depth analysis of the variant of our logic in whichwishful thinking is enabled. 10 .3 Motivational Attitudes II: Preferences
We consider two views about comparative statements between formulas of the form“agent i prefers ϕ to ψ ” or “the state of affairs ϕ is for agent i at least as good as thestate of affairs ψ ”. According to the optimistic view, when assessing whether ϕ is atleast as good as ψ , an agent focuses on the best ϕ -situations in comparison with the best ψ -situations. Specifically, an “optimistic” agent i prefers ϕ to ψ if and only if, for every ψ -situation envisaged by i there exists a ϕ -situation envisaged by i such that the latteris at least as desirable as the former. According to the pessimistic view, she focuseson the worst ϕ -situations in comparison with the worst ψ -situations. Specifically, a“pessimistic” agent i prefers ϕ to ψ if and only if, for every ϕ -situation envisaged by i there exists a ψ -situation envisaged by i such that the former is at least as desirable asthe latter.Let us first define a dyadic operator for preference according to the optimistic view. Definition 9 (Preference: optimistic view)
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and let w ∈ W . We say that, according to agent i’s optimistic assess-ment, ϕ is at least as good as ψ at w, noted M , w | = P Opti ( ψ (cid:22) ϕ ) , if and only if ∀ u ∈ || ψ || i , w , M , ∃ v ∈ || ϕ || i , w , M : u (cid:22) i , D v . As the following proposition highlights, it is expressible in the language L DLCA . Proposition 7
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and letw ∈ W . Then, we haveM , w | = P Opti ( ψ (cid:22) ϕ ) iff M , w | = (cid:2) ≡ i ; ψ ? (cid:3) (cid:104)(cid:22) i , D (cid:105) ϕ . Let us now define preference according to the pessimistic view.
Definition 10 (Preference: pessimistic view)
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and let w ∈ W . We say that, according to agent i’s pessimistic as-sessment, ϕ is at least as good as ψ at w, noted M , w | = P Pessi ( ψ (cid:22) ϕ ) , if and only if ∀ v ∈ || ϕ || i , w , M , ∃ u ∈ || ψ || i , w , M : u (cid:22) i , D v . As for the optimistic view, the pessimistic view is also expressible in the language L DLCA . Proposition 8
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and letw ∈ W . Then, we haveM , w | = P Pessi ( ψ (cid:22) ϕ ) iff M , w | = (cid:2) ≡ i ; ϕ ? (cid:3) (cid:104)(cid:23) i , D (cid:105) ψ . Thanks to the totality of the relation (cid:22) i , D (Constraint C2 in Definition 1), dyadicpreference over formulas is total too. This fact is illustrated by the following validity.For every x ∈ { Opt , Pess } : | = MCM P xi ( ψ (cid:22) ϕ ) ∨ P xi ( ϕ (cid:22) ψ ) (4)To see this suppose M , w | = ¬ P Opti ( ψ (cid:22) ϕ ) for an arbitrary model M and world w in M .Because of Constraint C2 in Definition 1, the latter implies that ∃ u ∈ || ψ || i , w , M , ∀ v ∈ | ϕ || i , w , M : v ≺ i , D u . Therefore, ∀ v ∈ || ϕ || i , w , M , ∃ u ∈ || ψ || i , w , M : v (cid:22) i , D u which is equiv-alent to M , w | = P Opti ( ϕ (cid:22) ψ ) . The case x = Pess can be proved in an analogous way.The previous notion of (optimistic and pessimistic) preference does not depend onwhat the agent believes. This means that, in order to assess whether ϕ is at least as goodas ψ , an agent also takes into account worlds that are implausible (or, more generally,not maximally plausible). Realistic preference requires that an agent compares twoformulas ϕ and ψ only with respect to the set of most plausible states. This idea hasbeen discussed in the area of qualitative decision theory by different authors [16, 18,17].The following definition introduces realistic preference according to the optimisticview. Definition 11 (Realistic preference: optimistic view)
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and let w ∈ W . We say that, according to agent i’soptimistic assessment, ϕ is realistically at least as good as ψ at w, noted M , w | = RP Opti ( ψ (cid:22) ϕ ) , if and only if ∀ u ∈ Best i , P ( w ) ∩ || ψ || i , w , M , ∃ v ∈ Best i , P ( w ) ∩ || ϕ || i , w , M : u (cid:22) i , D v . The idea is that an “optimistic” agent i considers ϕ realistically at least as good as ψ if and only if, for every ψ -situation in agent i ’s belief set there exists a ϕ -situationin agent i ’s belief set such that the latter is at least as good as the former.The previous notion as well is expressible in the language L DLCA . Proposition 9
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and letw ∈ W . Then, we haveM , w | = RP Opti ( ψ (cid:22) ϕ ) iff M , w | = (cid:2) ≡ i ; [ ≺ i , P ] ⊥ ?; ψ ? (cid:3) (cid:104)(cid:22) i , D ∩ ( ≡ i ; [ ≺ i , P ] ⊥ ? ) (cid:105) ϕ . The following definition introduces realistic preference according to the pessimisticview.
Definition 12 (Realistic preference: pessimistic view)
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and let w ∈ W . We say that, according to agent i’spessimistic assessment, ϕ is realistically at least as good as ψ at w, noted M , w | = RP Pessi ( ψ (cid:22) ϕ ) , if and only if ∀ v ∈ Best i , P ( w ) ∩ || ϕ || i , w , M , ∃ u ∈ Best i , P ( w ) ∩ || ψ || i , w , M : u (cid:22) i , D v . The idea is that a “pessimistic” agent i considers ϕ realistically at least as good as ψ if and only if, for every ϕ -situation in agent i ’s belief set there exists a ψ -situation inagent i ’s belief set such that the former is at least as good as the latter.It is also expressible in the language L DLCA . Proposition 10
Let M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and letw ∈ W . Then, we haveM , w | = RP Pessi ( ψ (cid:22) ϕ ) iff M , w | = (cid:2) ≡ i ; [ ≺ i , P ] ⊥ ?; ϕ ? (cid:3) (cid:104)(cid:23) i , D ∩ ( ≡ i ; [ ≺ i , P ] ⊥ ? ) (cid:105) ψ . x ∈ { Opt , Pess } , we have: | = MCM RP xi ( ψ (cid:22) ϕ ) ∨ RP xi ( ϕ (cid:22) ψ ) (5)The following abbreviations define strict variants of dyadic preference operators: P Opti ( ψ ≺ ϕ ) = def ¬ P Opti ( ϕ (cid:22) ψ ) P Pessi ( ψ ≺ ϕ ) = def ¬ P Pessi ( ϕ (cid:22) ψ ) RP Opti ( ψ ≺ ϕ ) = def ¬ RP Opti ( ϕ (cid:22) ψ ) RP Pessi ( ψ ≺ ϕ ) = def ¬ RP Pessi ( ϕ (cid:22) ψ ) P Opti ( ψ ≺ ϕ ) (resp. P Pessi ( ψ ≺ ϕ ) ) has to be read “according to i ’s optimistic (resp.pessimistic) assessment, ϕ is better than ψ ”. RP Opti ( ψ ≺ ϕ ) (resp. RP Pessi ( ψ ≺ ϕ ) )has to be read “according to agent i ’s optimistic (resp. pessimistic) assessment, ϕ isrealistically better than ψ ”.We conclude this section by defining two notions of monadic preference and cor-responding two notions of realistic monadic preference, respectively noted P Opti ϕ , P Pessi ϕ , RP Opti ϕ and RP Pessi ϕ : P Opti ϕ = def P Opti ( ¬ ϕ ≺ ϕ ) P Pessi ϕ = def P Pessi ( ¬ ϕ ≺ ϕ ) RP Opti ϕ = def RP Opti ( ¬ ϕ ≺ ϕ ) RP Pessi ϕ = def RP Pessi ( ¬ ϕ ≺ ϕ ) An optimistic (resp. pessimistic) agent has a preference for ϕ , noted P Opti ϕ (resp. P Pessi ϕ ), if and only if, according to her optimistic (resp. pessimistic) assessment, ϕ is better than ¬ ϕ . An optimistic (resp. pessimistic) agent has a realistic preferencefor ϕ , noted RP Opti ϕ (resp. RP Pessi ϕ ), if and only if, according to her optimistic (resp.pessimistic) assessment, ϕ is realistically better than ¬ ϕ .The following validity illustrates the relationship between the notion of desire de-fined in Definition 6 and the previous notion of pessimistic monadic preference: | = MCM ¬ D i (cid:62) → ( D i ϕ ↔ P Pessi ϕ ) (6)This means that if there exists at least a minimally desirable state for agent i (condition ¬ D i (cid:62) ), then i desires that ϕ if and only if, according to her pessimistic assessment, ϕ is better than ¬ ϕ . In the previous sections, we have defined a variety of cognitive attitudes of epistemicand motivational type. Let us illustrate them with the help of the crossroad scenariosketched in the introduction. For simplicitly, we assume that
Agt = { , } and that the13et of atomic propositions Atm includes the following elements with their correspond-ing meaning: co (“agent 1 and agent 2 collide”), lo (“agent 1 loses time”) and lo (“agent 2 loses time”).We are going to make different hypotheses about the agents’ cognitive attitudes andpresent a number of conclusions that can be drawn from them. Our initial hypothesisconcerns the agents’ knowledge: ϕ = def (cid:94) i ∈{ , } [ ≡ i ] (cid:16)(cid:0) ( lo ∧ ¬ lo ) → ¬ co (cid:1) ∧ (cid:0) ( ¬ lo ∧ lo ) → ¬ co (cid:1) ∧ ¬ ( ¬ lo ∧ ¬ lo ) (cid:17) . According to hypothesis ϕ , agents 1 and 2 know (i) that there will be no collision ifone of them loses time by letting the other pass, and (ii) that necessarily one of themwill lose time (since if they both pass, there will be a collision so that they will bothlose time).Our second hypothesis concerns what the agents merely envisage: ϕ = def (cid:94) i ∈{ , } (cid:16) (cid:104)≡ i (cid:105) co ∧ (cid:104)≡ i (cid:105) ( lo ∧ ¬ lo ) ∧ (cid:104)≡ i (cid:105) ( ¬ lo ∧ lo ) ∧ (cid:104)≡ i (cid:105) ( lo ∧ lo ∧ ¬ co ) (cid:17) . According to hypothesis ϕ , agents 1 and 2 envisage four possible situations: (i) thesituations in which they collide, (ii) the two situations in which one of them losesits time while the other does not, and (iii) the situation in which they both lose timebecause of a collision.We conclude with the following hypothesis about the agents’ motivations, accord-ing to which each agent strongly desires not to collide and strongly desires not to losetime: ϕ = def (cid:94) i ∈{ , } (cid:0) SD i ¬ lo i ∧ SD i ¬ co (cid:1) . As the following validities highlight, the previous hypotheses lead to different con-clusions about the agents’ epistemic and motivational attitudes: | = MCM ϕ → (cid:94) i ∈{ , } (cid:16) [ ≡ i ] (cid:0) co → ( lo ∧ lo ) (cid:1) ∧ B i ( ¬ lo , lo ) ∧ B i ( ¬ lo , lo ) (cid:17) (7) | = MCM ( ϕ ∧ ϕ ) → (cid:94) i ∈{ , } (cid:0) SD i ( ¬ lo i ∧ ¬ co ) ∧ D i ¬ lo i ∧ D i ¬ co (cid:1) (8) | = MCM ( ϕ ∧ ϕ ∧ ϕ ) → (cid:94) i ∈{ , } (cid:0) D i ( lo ∧ ¬ lo ) ∧ D i ( ¬ lo ∧ lo ) (cid:1) (9) | = MCM ( ϕ ∧ ϕ ) → (cid:94) i ∈{ , } (cid:0) ¬ SD i ( lo ∧ ¬ lo ) ∧ ¬ SD i ( ¬ lo ∧ lo ) (cid:1) (10)The single hypothesis ϕ leads to the conclusion (i) that the agents know that a collisionimplies that they both lose time, and (ii) that they believe that an agent loses time14onditional on the fact that the other does not. Thanks to the set of hypotheses { ϕ , ϕ } ,we can conclude (i) that each agent strongly desires not to lose time and to avoid acollision, and (ii) each agent has both the desire not to lose time and the desire to avoida collision. Finally, thanks to the set of hypotheses { ϕ , ϕ , ϕ } , we can conclude thateach agent finds desirable the situations in which only one of them loses time by lettingthe other pass. As the last validity indicates, such situations are merely desirable forthe agent but not strongly desirable. In this section, we provide a sound and complete axiomatization for the Dynamic Logicof Cognitive Attitudes (DLCA). The first step consists in precisely defining this logicwhich includes several axioms and rule of necessitation for the modalities [ π ] as wellas one non-standard rule of inference for nominals. Definition 13 (Logic
DLCA ) We define
DLCA to be the extension of classical propo-sitional logic given by the following axioms and rules with τ ∈ { P , D } : ([ π ] ϕ ∧ [ π ]( ϕ → ψ )) → [ π ] ψ ( K π ) [ ≡ i ] ϕ → ϕ ( T ≡ i ) [ ≡ i ] ϕ → [ ≡ i ][ ≡ i ] ϕ ( ≡ i ) ¬ [ ≡ i ] ϕ → [ ≡ i ] ¬ [ ≡ i ] ϕ ( ≡ i ) [ (cid:22) i , τ ] ϕ → ϕ ( T (cid:22) i , τ ) [ (cid:22) i , τ ] ϕ → [ (cid:22) i , τ ][ (cid:22) i , τ ] ϕ ( (cid:22) i , τ ) [ ≡ i ] ϕ → [ (cid:22) i , τ ] ϕ ( Inc (cid:22) i , τ , ≡ i ) (cid:0) (cid:104)≡ i (cid:105) ϕ ∧ (cid:104)≡ i (cid:105) ψ (cid:1) → (cid:0) (cid:104)≡ i (cid:105) ( ϕ ∧ (cid:104)(cid:22) i , τ (cid:105) ψ ) ∨ (cid:104)≡ i (cid:105) ( ψ ∧ (cid:104)(cid:22) i , τ (cid:105) ϕ ) (cid:1) ( Conn (cid:22) i , τ , ≡ i ) [ π ; π (cid:48) ] ϕ ↔ [ π ][ π (cid:48) ] ϕ ( Red ; ) [ π ∪ π (cid:48) ] ϕ ↔ ([ π ] ϕ ∧ [ π (cid:48) ] ϕ ) ( Red ∪ ) ([ π ] ϕ ∧ [ π (cid:48) ] ψ ) → [ π ∩ π (cid:48) ]( ϕ ∧ ψ ) ( Add1 ∩ ) ( (cid:104) π (cid:105) x ∧ (cid:104) π (cid:48) (cid:105) x ) → (cid:104) π ∩ π (cid:48) (cid:105) x ( Add2 ∩ ) ϕ → [ π ] (cid:104)− π (cid:105) ϕ ( Conv1 − ) ϕ → [ − π ] (cid:104) π (cid:105) ϕ ( Conv2 − ) ([ (cid:22) i , τ ] ϕ ∧ [ (cid:22) ∼ i , τ ] ϕ ) ↔ [ ≡ i ] ϕ ( Comp1 ∼ ) (cid:104)(cid:22) i , τ (cid:105) x → [ (cid:22) ∼ i , τ ] ¬ x ( Comp2 ∼ ) [ ? ϕ ] ψ → ( ϕ → ψ ) ( Red ? ) (cid:104) π (cid:105) ( x ∧ ϕ ) → [ π (cid:48) ]( x → ϕ ) ( Most x ) ϕ [ π ] ϕ ( Nec π ) [ π ] ¬ x for all x ∈ Nom [ π ] ⊥ ( Cov )15ote that the primitive operators [ (cid:22) i , P ] and [ (cid:22) i , D ] are S4 (or KT4), while [ ≡ i ] isS5. The only interaction principles between these three operators are the “inclusion”Axiom Inc (cid:22) i , τ , ≡ i and the “connectedness” Axiom Conn (cid:22) i , τ , ≡ i . Operators [ (cid:22) i , P ] and [ (cid:22) i , D ] do not interact since, as we have emphasized in Section 2, epistemic plausibilityand desirability are assumed to be independent notions.For every ϕ ∈ L DLCA , we write (cid:96) ϕ to denote the fact that ϕ is a theorem of DLCA,i.e., there exists an at most countably infinite sequence ψ , ψ , . . . such that ψ = ϕ andfor all k ≥ ψ k is an instance of some axiom or ψ k can be obtained from some latermembers of the sequence by an application of some inference rule.The rest of this section is devoted to prove that the logic DLCA is sound and com-plete for the class of multi-agent cognitive models.Soundness, namely checking that the axioms are valid and the the rules of infer-ences preserve validity, is a routine exercise. Notice that the admissibility of the ruleof inference Cov is guaranteed by the fact that the set of nominals
Nom is infinite.As for completeness, the proof is organized in several steps. We use techniquesfrom dynamic logic and modal logic with names [39, 25].In the rest of this section, we denote sets of formulas from L DLCA by Σ , Σ (cid:48) , . . . . Let ϕ ∈ L DLCA and Σ ⊆ L DLCA , we define: Σ + ϕ = { ψ ∈ L DLCA : ϕ → ψ ∈ Σ } . Let us start by defining the concepts of theory and maximal consistent theory.
Definition 14 (Theory)
A set of formulas Σ is said to be a theory if it contains alltheorems of DLCA and is closed under modus ponens and rule
Cov . It is said to bea consistent theory if it is a theory and ⊥ (cid:54)∈ Σ . It is said to be a maximal consistenttheory (MCT) if it is a consistent theory and, for each consistent theory Σ (cid:48) , we havethat if Σ ⊆ Σ (cid:48) then Σ = Σ (cid:48) . We have the following property for theories.
Proposition 11
Let Σ be a theory and let ϕ ∈ L DLCA . Then, Σ + ϕ is a theory. More-over, if Σ is consistent then either Σ + ϕ is consistent or Σ + ¬ ϕ is consistent. P ROOF . Let us first prove that if Σ is a theory then Σ + ϕ is a theory as well. Suppose Σ is a theory. Then, Σ + ϕ clearly contains all theorems of DLCA. Moreover, suppose ψ → ψ (cid:48) , ψ ∈ Σ + ϕ . Thus, by definition of Σ + ϕ , we have ϕ → ψ , ϕ → ( ψ → ψ (cid:48) ) ∈ Σ .Since Σ is closed under modus ponens and contains all theorems of DLCA, the latterimplies ( ϕ → ψ ) ∧ (cid:0) ϕ → ( ψ → ψ (cid:48) ) (cid:1) ∈ Σ . Consequently, since Σ is closed under modusponens, ϕ → ψ (cid:48) ∈ Σ . Hence, ψ (cid:48) ∈ Σ + ϕ . This means that Σ + ϕ is closed under modusponens. Finally, let us show that Σ + ϕ is closed under Cov . Suppose [ π ] ¬ x ∈ Σ + ϕ for all x . Thus, by definition of Σ + ϕ , ϕ → [ π ] ¬ x ∈ Σ for all x . Since Σ is a theory, thelatter implies that [ ? ϕ ; π ] ¬ x ∈ Σ for all x . Thus, since Σ is a theory, [ ? ϕ ; π ] ⊥ ∈ Σ and,consequently, ϕ → [ π ] ⊥ ∈ Σ . It follows that [ π ] ⊥ ∈ Σ + ϕ .Let us show that if Σ is consistent then either Σ + ϕ is consistent or Σ + ¬ ϕ isconsistent. Suppose the antecedent is true while the consequent is false. Then, ϕ → ∈ Σ and ¬ ϕ → ⊥ ∈ Σ . Since Σ is a theory, we have ( ϕ → ⊥ ) ∧ ( ¬ ϕ → ⊥ ) ∈ Σ . Thus, ⊥ ∈ Σ which is in contradiction with the fact that Σ is consistent. (cid:4) The following proposition highlights some standard properties of MCTs.
Proposition 12
Let Σ be a MCT. Then, for all ϕ , ψ ∈ L DLCA : • ϕ ∈ Σ or ¬ ϕ ∈ Σ , • ϕ ∨ ψ ∈ Σ iff ϕ ∈ Σ or ψ ∈ Σ . P ROOF . We only prove the first item by reductio ad absurdum. Suppose Σ is a MCT, ϕ (cid:54)∈ Σ and ¬ ϕ (cid:54)∈ Σ . We clearly have Σ ⊆ Σ + ϕ and Σ ⊆ Σ + ¬ ϕ . Moreover, ϕ ∈ Σ + ϕ and ¬ ϕ ∈ Σ + ¬ ϕ . Thus, Σ ⊂ Σ + ϕ and Σ ⊂ Σ + ¬ ϕ . By Proposition 11, Σ + ϕ and Σ + ¬ ϕ are theories. Moreover, either Σ + ϕ is consistent or Σ + ¬ ϕ is consistent. Thiscontradicts the fact that Σ is a MCT. (cid:4) The following variant of the Lindenbaum’s lemma is proved in the same way as[39, Lemma 4.15].
Lemma 1
Let Σ be a consistent theory and let ϕ (cid:54)∈ Σ . Then, there exists a MCT Σ + such that Σ ⊆ Σ + and ϕ (cid:54)∈ Σ + . The following lemma highlights a fundamental property of MCTs.
Lemma 2
Let Σ be a MCT. Then, there exists x ∈ Nom such x ∈ Σ . P ROOF . We prove the lemma by reductio ad absurdum. Let Σ be a MCT. Moreover,suppose that, for all x ∈ Nom , x (cid:54)∈ Σ . By Proposition 12, it follows that, for all x ∈ Nom , ¬ x ∈ Σ .By Axiom Red ? , we have ¬ x ↔ [ ? (cid:62) ] ¬ x ∈ Σ for all x ∈ Nom . Thus, for all x ∈ Nom , [ ? (cid:62) ] ¬ x ∈ Σ . Hence, since Σ is closed under Cov , [ ? (cid:62) ] ⊥ ∈ Σ . By Axiom Red ? , thelatter is equivalent to ⊥ ∈ Σ . The latter is contradiction with the fact that Σ is a MCT. (cid:4) Let us now define the canonical model for our logic.
Definition 15 (Canonical model)
The canonical model is the tuple M c = ( W c , ( (cid:22) ci , P ) i ∈ Agt , ( (cid:22) ci , D ) i ∈ Agt , ( ≡ ci ) i ∈ Agt , V c ) such that: • W c is the set of all MCTs, • for all i ∈ Agt, for all τ ∈ { P , D } , for all w , v ∈ W c , w (cid:22) ci , τ v iff, for all ϕ ∈ L DLCA ,if [ (cid:22) i , τ ] ϕ ∈ w then ϕ ∈ v, • for all i ∈ Agt, for all w , v ∈ W c , w ≡ ci v iff, for all ϕ ∈ L DLCA , if [ ≡ i ] ϕ ∈ w then ϕ ∈ v, • for all w ∈ W c , V c ( w ) = ( Atm ∪ Nom ) ∩ w. Let us now define the canonical relations for the complex programs π .17 efinition 16 (Canonical relation) Let M c = ( W c , ( (cid:22) ci , P ) i ∈ Agt , ( (cid:22) ci , D ) i ∈ Agt , ( ≡ ci ) i ∈ Agt , V c ) be the canonical model. Then, for all π ∈ P and for all w , v ∈ W c :wR c π v iff, for all ϕ ∈ L DLCA , if [ π ] ϕ ∈ w then ϕ ∈ v . The following Lemma 3 highlights one fundamental property of the canonicalmodel.
Lemma 3
Let M c = ( W c , ( (cid:22) ci , P ) i ∈ Agt , ( (cid:22) ci , D ) i ∈ Agt , ( ≡ ci ) i ∈ Agt , V c ) be the canonical model.Then, for all Σ , Σ (cid:48) ∈ W c , for all π ∈ P and for all x ∈ Nom, if x ∈ Σ , x ∈ Σ (cid:48) and Σ R c π Σ (cid:48) then Σ = Σ (cid:48) . P ROOF . Let us first prove that (i) if x ∈ Σ and ϕ ∈ Σ then [ π ]( x → ϕ ) ∈ Σ . Suppose x , ϕ ∈ Σ . Thus, x ∧ ϕ ∈ Σ since Σ is a MCT. Moreover, ( x ∧ ϕ ) → [ π ]( x → ϕ ) ∈ Σ ,because of Axiom Most x . Hence, [ π ]( x → ϕ ) ∈ Σ .Now let us prove by absurdum that (ii) if x ∈ Σ , Σ (cid:48) and Σ R c π Σ (cid:48) then Σ = Σ (cid:48) . Suppose x ∈ Σ , Σ (cid:48) , Σ R c π Σ (cid:48) and Σ (cid:54) = Σ (cid:48) . The latter implies that there exists ϕ such that ϕ ∈ Σ and ϕ (cid:54)∈ Σ (cid:48) . By item (i) above, it follows that [ π ]( x → ϕ ) ∈ Σ . Since Σ R c π Σ (cid:48) , thelatter implies that x → ϕ ∈ Σ (cid:48) . Since x ∈ Σ (cid:48) , it follows that ϕ ∈ Σ (cid:48) which leads to acontradiction. (cid:4) The next step consists in proving the following existence lemma.
Lemma 4
Let M c = ( W c , ( (cid:22) ci , P ) i ∈ Agt , ( (cid:22) ci , D ) i ∈ Agt , ( ≡ ci ) i ∈ Agt , V c ) be the canonical model,let w ∈ W c and let (cid:104) π (cid:105) ϕ ∈ L DLCA . Then, if (cid:104) π (cid:105) ϕ ∈ w then there exists v ∈ W c suchthat wR c π v and ϕ ∈ v. P ROOF . Suppose w is a MCT and (cid:104) π (cid:105) ϕ ∈ w . It follows that [ π ] w = { ψ : [ π ] ψ ∈ w } is aconsistent theory. Indeed, it is easy to check that [ π ] w contains all theorems of DLCA,is closed under modus ponens and rule Cov . Let us prove that it is consistent byreductio ad absurdum. Suppose ⊥ ∈ [ π ] w . Thus, [ π ] ⊥ ∈ w . Hence, [ π ] ¬ ϕ ∈ w . Since (cid:104) π (cid:105) ϕ ∈ w , ⊥ ∈ w . The latter contradicts the fact that w is a MCT. Let us distinguishtwo cases.Case 1: ϕ ∈ [ π ] w . Thus, ¬ ϕ (cid:54)∈ [ π ] w since w is consistent. Thus, by Lemma 1, thereexists MCT v such that [ π ] w ⊆ v , ϕ ∈ v and ¬ ϕ (cid:54)∈ v . By definition of R c π , wR c π v .Case 2: ϕ (cid:54)∈ [ π ] w . By Proposition 11, [ π ] w + ϕ is a theory since [ π ] w is a theory. [ π ] w + ϕ is consistent. Suppose it is not. Thus, ϕ → ⊥ ∈ [ π ] w and, consequently, ¬ ϕ ∈ [ π ] w . Hence, [ π ] ¬ ϕ ∈ w . It follows that ⊥ ∈ w , since (cid:104) π (cid:105) ϕ ∈ w . But thiscontradicts the fact that w is a MCT. Thus, [ π ] w + ϕ is a consistent theory. Moreover, ϕ ∈ [ π ] w + ϕ , ¬ ϕ (cid:54)∈ [ π ] w + ϕ and [ π ] w ⊆ [ π ] w + ϕ . By Lemma 1, there exists MCT v such that [ π ] w ⊆ v , ϕ ∈ v and ¬ ϕ (cid:54)∈ v . By definition of R c π , wR c π v . (cid:4) The following truth lemma is proved in the usual way by induction on the structureof ϕ thanks to Lemma 4. Lemma 5
Let M c = ( W c , ( (cid:22) ci , P ) i ∈ Agt , ( (cid:22) ci , D ) i ∈ Agt , ( ≡ ci ) i ∈ Agt , V c ) be the canonical model,let w ∈ W c and let ϕ ∈ L DLCA . Then, M c , w | = ϕ iff ϕ ∈ w. ROOF . The proof is by induction on the structure of ϕ . We only prove the case inwhich ϕ is a modal formula [ π ] ψ . As for the right-to-left direction we have: [ π ] ψ ∈ w only if ∀ v ∈ R c π ( w ) : ψ ∈ v (by definition of R c π )iff ∀ v ∈ R c π ( w ) : M c , v | = ψ (by induction hypothesis)iff M c , w | = [ π ] ψ As for the left-to-right direction, we prove that if [ π ] ψ (cid:54)∈ w then M c , w (cid:54)| = [ π ] ψ that,given the property of MCSs, is equivalent to proving that if (cid:104) π (cid:105) ψ ∈ w then M c , w | = (cid:104) π (cid:105) ψ . Suppose (cid:104) π (cid:105) ψ ∈ w . Then, by Lemma 4, there exists v ∈ W c such that wR c π v and ψ ∈ v . Hence, by induction hypothesis, there exists v ∈ W c such that wR c π v and M c , v | = ψ . The latter is equivalent to M c , w | = (cid:104) π (cid:105) ψ . (cid:4) The pre-final stage of the proof consists in introducing an alternative semantics forthe language L DLCA which turns out to be equivalent to the original semantics basedon MCMs.
Definition 17 (Quasi multi-agent cognitive model)
A quasi multi-agent cognitive model(quasi-MCM) is a tuple M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) where W , (cid:22) i , P , (cid:22) i , D , ≡ i and V are as in Definition 1 except that Constraint C4 is replaced by thefollowing weaker constraint. For all w , v ∈ W : (C4 ∗ ) if V Nom ( w ) ∩ V Nom ( v ) (cid:54) = /0 and wR π v for some π ∈ P then w = v. By the generated submodel property, it is easy to show that the semantics in termsof MCMs and the semantics in terms of quasi-MCMs are equivalent with respect to thelanguage L DLCA . Proposition 13
Let ϕ ∈ L DLCA . Then, ϕ is valid relative to the class of MCMs if andonly if ϕ is valid relative to the class of quasi-MCMs. The following theorem highlights that the canonical model is indeed a structure ofthe right type.
Lemma 6
The canonical model M c is a quasi-MCM. P ROOF . The fact that M c satisfies Constraints C3 and C4 ∗ follows from Lemma 2 andLemma 3. To prove that ≡ i is an equivalence relation that (cid:22) ci , D and (cid:22) ci , D are preordersand that M c satisfies Constraints C1 and C2 is just a routine exercise. Indeed, Axioms T ≡ i , ≡ i , ≡ i , T (cid:22) i , τ , (cid:22) i , τ Inc (cid:22) i , τ , ≡ i and Conn (cid:22) i , τ , ≡ i are canonical for these semanticconditions.To conclude, we need to prove that the following six conditions hold, for i ∈ Agt τ ∈ { P , D } : ( w , v ) ∈ R c (cid:22) ∼ i , τ iff ( w , v ) ∈ R c ≡ i and ( w , v ) (cid:54)∈ R c (cid:22) i , τ ( w , v ) ∈ R c π ; π (cid:48) iff ∃ u ∈ W c : ( w , u ) ∈ R c π and ( u , v ) ∈ R c π (cid:48) ( w , v ) ∈ R c π ∪ π (cid:48) iff ( w , v ) ∈ R c π or ( w , v ) ∈ R c π (cid:48) ( w , v ) ∈ R c π ∩ π (cid:48) iff ( w , v ) ∈ R c π and ( w , v ) ∈ R c π (cid:48) ( w , v ) ∈ R c − π iff ( v , w ) ∈ R c π wR c ϕ ? v iff w = v and M c , w | = ϕ We only prove the second and fourth conditions which are the most difficult ones toprove.Let us start with the proof of the second condition. The right-to-left direction isstandard. We only prove the left-to-right direction. Suppose ( w , v ) ∈ R c π ; π (cid:48) . Let [ π ] w = { ψ : [ π ] ψ ∈ w } . Moreover, let (cid:104) π (cid:48) (cid:105) v = {(cid:104) π (cid:48) (cid:105) ψ : ψ ∈ v } . Finally, let (cid:104) π (cid:48) (cid:105) ψ , (cid:104) π (cid:48) (cid:105) ψ , . . . be an enumeration of the elements of (cid:104) π (cid:48) (cid:105) v . We define Σ = [ π ] w + (cid:104) π (cid:48) (cid:105) ψ and, forall k > Σ k = Σ k − + (cid:104) π (cid:48) (cid:105) ψ k . By Lemma 11 and the fact that [ π ] w is a theory, it canbe shown that every Σ k is a theory. Moreover, by induction on k , it can be shown thatevery Σ k is consistent. Since Σ k − ⊆ Σ k for all k >
1, it follows that Σ = (cid:83) k > Σ k − isa consistent theory. By Lemma 1 and the definition of Σ , there exists u ∈ W c such that Σ ⊆ u , ( w , u ) ∈ R c π and ( u , v ) ∈ R c π (cid:48) .Let us now prove the fourth condition. Suppose ( w , v ) ∈ R c π ∩ π (cid:48) . By Definition 16and Proposition 12, it follows that, for all ϕ , if ϕ ∈ v then (cid:104) π ∩ π (cid:48) (cid:105) ϕ ∈ w . The latterimplies that for all ϕ , if ϕ ∈ v then (cid:104) π ∩ π (cid:48) (cid:105) ( ϕ ∨ ⊥ ) ∈ w since (cid:96) (cid:104) π ∩ π (cid:48) (cid:105) ϕ → (cid:104) π ∩ π (cid:48) (cid:105) ( ϕ ∨ ⊥ ) . By Axiom K π , it follows that, for all ϕ , if ϕ ∈ v then (cid:104) π (cid:105) ϕ ∨ (cid:104) π (cid:48) (cid:105)⊥ ∈ w .Thus, for all ϕ , if ϕ ∈ v then (cid:104) π (cid:105) ϕ ∈ w , since (cid:96) ( (cid:104) π (cid:105) ϕ ∨ (cid:104) π (cid:48) (cid:105)⊥ ) → (cid:104) π (cid:105) ϕ . In a similarway, we can prove that, for all ϕ , if ϕ ∈ v then (cid:104) π (cid:48) (cid:105) ϕ ∈ w . By Definition 16 andProposition 12, it follows that ( w , v ) ∈ R c π and ( w , v ) ∈ R c π (cid:48) .Now suppose ( w , v ) ∈ R c π and ( w , v ) ∈ R c π (cid:48) . Thus, by Definition 16 and Proposition12, (i) for all ϕ , if ϕ ∈ v then (cid:104) π (cid:105) ϕ ∈ w and (cid:104) π (cid:48) (cid:105) ϕ ∈ w . By Proposition 12 and Lemma2, we have that (ii) there exists x ∈ Nom such that, for all ϕ , ϕ ∈ v iff x ∧ ϕ ∈ v . Item (i)and item (ii) together imply that (iii) there exists x ∈ Nom such that, for all ϕ , if ϕ ∈ v then (cid:104) π (cid:105) ( x ∧ ϕ ) ∈ w and (cid:104) π (cid:48) (cid:105) ( x ∧ ϕ ) ∈ w . We are going to prove the following theorem: (cid:96) ( (cid:104) π (cid:105) ( x ∧ ϕ ) ∧ (cid:104) π (cid:48) (cid:105) ( x ∧ ϕ )) → (cid:104) π ∩ π (cid:48) (cid:105) ( x ∧ ϕ ) By Axiom K π , (cid:104) π (cid:105) ( x ∧ ϕ ) ∧ (cid:104) π (cid:48) (cid:105) ( x ∧ ϕ ) implies (cid:104) π (cid:105) x ∧ (cid:104) π (cid:48) (cid:105) x . By Axiom Add2 ∩ , thelatter implies (cid:104) π ∩ π (cid:48) (cid:105) x . Moreover, by Axiom Inc (cid:22) i , τ , ≡ i and Axiom Most x , (cid:104) π (cid:105) ( x ∧ ϕ ) implies [ ≡ /0 ]( x → ϕ ) . By Axiom Inc (cid:22) i , τ , ≡ i , the latter implies [ π ∩ π (cid:48) ]( x → ϕ ) .By Axiom K π , [ π ∩ π (cid:48) ]( x → ϕ ) and (cid:104) π ∩ π (cid:48) (cid:105) x together imply (cid:104) π ∩ π (cid:48) (cid:105) ( x ∧ ϕ ) . Thus, (cid:104) π (cid:105) ( x ∧ ϕ ) ∧ (cid:104) π (cid:48) (cid:105) ( x ∧ ϕ ) implies (cid:104) π ∩ π (cid:48) (cid:105) ( x ∧ ϕ ) .From previous item (iii) and the previous theorem it follows that there exists x ∈ Nom such that, for all ϕ , if ϕ ∈ v then (cid:104) π ∩ π (cid:48) (cid:105) ( x ∧ ϕ ) . The latter implies that, for all ϕ , if ϕ ∈ v then (cid:104) π ∩ π (cid:48) (cid:105) ϕ . The latter implies that ( w , v ) ∈ R c π ∩ π (cid:48) . (cid:4) Let us conclude the proof by supposing (cid:54)(cid:96) ¬ ϕ . Therefore, by Lemma 1 and the factthat the set of DLCA-theorems is a consistent theory, there exists a MCT w such that20 ϕ (cid:54)∈ w . Thus, by Proposition 12, we can find a MCT w such that ϕ ∈ w . By Lemma5, the latter implies M c , w | = ϕ for some w ∈ W c . Since, by Lemma 6, M c is a quasi-MCM, it follows that ϕ is satisfiable relative to the class of quasi-MCMs. Therefore,by Proposition 13, ϕ is satisfiable relative to the class of MCMs.We can finally state the main result of this section. Theorem 1
The logic
DLCA is sound and complete for the class of multi-agent cog-nitive models.
In this section, we apply our logical framework to the analysis of single-stage gamesunder incomplete information in which agents only play once (i.e., interaction is non-repeated) and may not know some relevant characteristic of others including their pref-erences, choices and beliefs.Let
Act be a set of action names with elements noted a , b , . . . Let a joint action be afunction δ : Agt −→ Act and the set of joint actions be denoted by
JAct .For every coalition C ∈ Agt and for every δ ∈ JAct , let δ C be the C -restriction of δ ,that is, the function δ C : C −→ Act such that δ C ( i ) = δ ( i ) for all i ∈ C . For notationalconvenience, we write − i instead of Agt \ { i } , with i ∈ Agt .In order to model strategic interaction in our setting, we extend MCMs of Definition1 by agents’ choices. We call MCM with choices the resulting models.
Definition 18 (Multi-agent cognitive model with choices)
A multi-agent cognitive modelwith choices (MCMC) is a tuple M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , ( C i ) i ∈ Agt , V ) , where M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) is a MCMand every C i is a choice function C i : W −→ Act, which satisfies the following con-straint, for each i ∈ Agt and δ ∈ JAct: (C5) if ∀ j ∈ Agt, ∃ w j ∈ W such that w ≡ i w j and C j ( w j ) = δ ( j ) , then ∃ v ∈ W suchthat w ≡ i v and, ∀ j ∈ Agt, C j ( v ) = δ ( j ) . For every w ∈ W , C i ( w ) denotes agent i ’s actual choice at w . If w ≡ i v and C j ( v ) = a , then a is a potential choice of agent j from agent i ’s perspective.According to Constraint C5, agents’ choices are subjectively independent, in thesense that every agent i knows that an agent cannot be deprived of her choices due tothe choices made by the others. In other words, suppose that, from agent i ’s perspective, δ ( j ) is a potential choice of j for every agent j . Then, from agent i ’s perspective, thereshould be a state at which the agents choose the joint action δ . It is a subjective versionof the property of choice independence formulated in the “seeing to it that” (STIT)framework [12, 33, 7].At the syntactic level, we extend the language L DLCA by special constants forchoices of type play ( i , a ) , with i ∈ Agt and a ∈ Act , denoting the fact that “agent i plays (or chooses) action a ”. The resulting language is noted L DLCAG , where DLCAGstands for “Dynamic Logic of Cognitive Attitudes in Games” and a constant play ( i , a ) is interpreted relative to a MCMC M and a world w in M , as follows: M , w | = play ( i , a ) ⇐⇒ C i ( w ) = a . δ ∈ JAct and C ∈ Agt . We define: play ( δ C ) = def (cid:94) i ∈ C play (cid:0) i , δ C ( i ) (cid:1) . For every formula ϕ in L DLCAG we say that ϕ is valid, noted | = MCMC ϕ , if and onlyif for every multi-agent cognitive model with choices M and world w in M , we have M , w | = ϕ . Definition 19 (Logic
DLCAG ) We define
DLCAG to be the extension of logic
DLCA given by the following axioms: play ( i , a ) → ¬ play ( i , b ) if a (cid:54) = b ( MostAct ) (cid:95) a ∈ Act play ( i , a ) ( LeastAct ) (cid:16) (cid:94) j ∈ Agt (cid:104)≡ i (cid:105) play (cid:0) j , δ ( j ) (cid:1)(cid:17) → (cid:104)≡ i (cid:105) play (cid:0) δ Agt (cid:1) ( SIC )Axiom
MostAct means that an agent chooses at most one action from
Act while, ac-cording to Axiom
LeastAct , an agent chooses at least one action from
Act . Axiom
SIC is the syntactic counterpart of subjective choice independence expressed by ConstraintC5.We can adapt the techniques used for proving Theorem 1 in order to prove thefollowing Theorem 2.
Theorem 2
The logic
DLCAG is sound and complete for the class of multi-agent cog-nitive models with choices. P ROOF . Verifying that the logic DLCAG is sound for the class of MCMCs is a routineexercise. As for completeness, the proof is just a straightforward adaptation of theproof of completeness of the logic DLCA. First, we need to define correspondingnotions of theory and maximal consistent theory for the logic DLCAG which are akinto the ones for the logic DLCA and use them to define the canonical model and thecanonical relation for DLCAG. The canonical model for DLCAG is defined to be atuple M c = ( W c , ( (cid:22) ci , P ) i ∈ Agt , ( (cid:22) ci , D ) i ∈ Agt , ( ≡ ci ) i ∈ Agt , ( C ci ) i ∈ Agt , V c ) where W c is the setof all MCTs for DLCAG, (cid:22) ci , P , (cid:22) ci , D , ≡ ci and V c are defined as in the definition ofthe canonical model for DLCA (Definition 15), and C ci : W c −→ Act such that, for all a ∈ Act and w ∈ W c , a ∈ C ci ( w ) if and only if play ( i , a ) ∈ w . The canonical relationfor DLCAG is defined in the same way as the canonical relation for DLCA (Definition16).It is immediate to adapt the proof of the existence and truth lemma for DLCA(Lemma 4 and Lemma 5) to prove corresponding existence and truth lemma for DLCAG.Secondly, we need to define the notion of quasi multi-agent cognitive model withchoices (quasi-MCMC) which is analogous to the definition of quasi-MCM (Definition17). In particular, a quasi-MCMC is defined to be a tuple M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , ( C i ) i ∈ Agt , V ) where W , (cid:22) i , P , (cid:22) i , D , ≡ i , C i and V are as in Definition 18except that Constraint C4 is replaced by the weaker Constraint C4 ∗ of Definition 17. As22or MCMs, by the generated submodel property, it is easy to show that the semanticsin terms of MCMCs and the semantics in terms of quasi-MCMCs are equivalent withrespect to the language L DLCAG .The only property that has to be checked carefully is whether the canonical modelfor DLCAG is indeed a quasi-MCMC. To this aim, we need to extend the proof ofLemma 6 in order to verify that the canonical model for DLCAG satisfies ConstraintC5 of Definition 18 and that, for all i ∈ Act and for all w ∈ W c , C ci ( w ) is a singleton.Suppose a , b ∈ C ci ( w ) for a (cid:54) = b . The latter means that play ( i , a ) , play ( i , b ) ∈ w . Wehave play ( i , a ) → ¬ play ( i , b ) ∈ w , because of Axiom MostAct . Thus, ¬ play ( i , b ) ∈ w .Hence, ⊥ ∈ w which contradicts the fact that w is a consistent theory. Consequently,the set C ci ( w ) has at most one element. Now, let us prove that C ci ( w ) has at least oneelement. Because of Axiom LeastAct , we have (cid:87) a ∈ Act play ( i , a ) ∈ w . Thus, there exists a ∈ Act such that play ( i , a ) ∈ w . Hence, C ci ( w ) is non-empty. Now, let us prove that thecanonical model for DLCAG satisfies Constraint C5. Suppose ∀ j ∈ Agt , ∃ w j ∈ W c suchthat w ≡ ci w j and C cj ( w j ) = δ ( j ) . The latter means that ∀ j ∈ Agt , ∃ w j ∈ W c such that w ≡ ci w j and play (cid:0) j , δ ( j ) (cid:1) ∈ w j . Thus, we have that, ∀ j ∈ Agt , (cid:104)≡ i (cid:105) play (cid:0) j , δ ( j ) (cid:1) ∈ w .Hence, (cid:86) j ∈ Agt (cid:104)≡ i (cid:105) play (cid:0) j , δ ( j ) (cid:1) ∈ w . By Axiom SIC , (cid:16) (cid:86) j ∈ Agt (cid:104)≡ i (cid:105) play (cid:0) j , δ ( j ) (cid:1)(cid:17) →(cid:104)≡ i (cid:105) play (cid:0) δ Agt (cid:1) ∈ w . Consequently, (cid:104)≡ i (cid:105) play (cid:0) δ Agt (cid:1) ∈ w . By the existence lemma forDLCAG, the latter implies that ∃ v ∈ W c such that w ≡ ci v and play (cid:0) δ Agt (cid:1) ∈ v . Thus, ∃ v ∈ W c such that w ≡ ci v and C j ( v ) = δ ( j ) for every j ∈ Agt . (cid:4) With the support of the language L DLCAG , we can define a variety of notions fromthe theory of games under incomplete information. The first notion we consider is bestresponse, both from the perspective of an optimistic agent and from the perspective ofa pessimistic one: BR Opti ( a , δ − i ) = def (cid:94) b ∈ Act RP Opti (cid:16)(cid:0) play ( i , b ) ∧ play ( δ − i ) (cid:1) (cid:22) (cid:0) play ( i , a ) ∧ play ( δ − i ) (cid:1)(cid:17) , BR Pessi ( a , δ − i ) = def (cid:94) b ∈ Act RP Pessi (cid:16)(cid:0) play ( i , b ) ∧ play ( δ − i ) (cid:1) (cid:22) (cid:0) play ( i , a ) ∧ play ( δ − i ) (cid:1)(cid:17) . We say that playing action a is for agent i an optimistic (resp. pessimistic) best responseto the others’ joint action δ − i , noted BR Opti ( a , δ − i ) (resp. BR Pessi ( a , δ − i ) ) if and onlyif for every action b , according to agent i ’s optimistic (resp. pessimistic) assessment,playing a while the others play δ − i is realistically at least as good as playing b whilethe others play δ − i .As for best response, we can define two types of subjective Nash equilibrium, onefor optimistic agents and the other for pessimistic ones. Our notion of subjective Nashequilibrium corresponds to a qualitative variant of the notion of Bayesian Nash equi-librium (BNE): a similar qualitative variant of BNE is studied by [2] in the context ofpossibility theory. The joint action δ is said to be a subjective optimistic (resp. pes-simistic) Nash equilibrium, noted NE Opt ( δ ) (resp. NE Pess ( δ ) ), if no agent i wants tounilaterally deviate from the chosen strategy δ ( i ) , under that the assumption that i is23ptimistic (resp. pessimistic): NE Opt ( δ ) = def (cid:94) i ∈ Agt BR Opti (cid:0) δ ( i ) , δ − i (cid:1) , NE Pess ( δ ) = def (cid:94) i ∈ Agt BR Pessi (cid:0) δ ( i ) , δ − i (cid:1) . Note that assuming the finiteness of the set of agents
Agt is essential for definingNash equilibrium, since our language is finitary and does not allow universal quantifi-cation over infinite sets.Given the distinction between optimistic and pessimistic agent, two notions of ra-tionality can be defined. Agent i is said to be optimistic (resp. pessimistic) rational,noted Rat
Opti (resp.
Rat
Pessi ), if she cannot choose an action that, according to heroptimistic (resp. pessimistic) assessment, is better not to choose than to choose:
Rat
Opti = def (cid:94) a ∈ Act (cid:16) play ( i , a ) → RP Opti (cid:0) ¬ play ( i , a ) (cid:22) play ( i , a ) (cid:1)(cid:17) , Rat
Pessi = def (cid:94) a ∈ Act (cid:16) play ( i , a ) → RP Pessi (cid:0) ¬ play ( i , a ) (cid:22) play ( i , a ) (cid:1)(cid:17) . As the following proposition indicates, the action chosen by an optimistic (resp.pessimistic) rational agent is, according to the agent’s optimistic (resp. pessimistic)assessment, at least as good as the other actions she may choose.
Proposition 14
Let i ∈ Agt and x ∈ {
Opt , Pess } . Then, | = MCMC (cid:0)
Rat xi ∧ play ( i , a ) (cid:1) → (cid:94) b ∈ Act RP xi (cid:0) play ( i , b ) (cid:22) play ( i , a ) (cid:1) (11)P ROOF . Let us prove the case x = Opt . Let M be a MCMC and let w be a world in M . Suppose M , w | = Rat
Opti and M , w | = play ( i , a ) . Thus, M , w | = RP Opti (cid:0) ¬ play ( i , a ) (cid:22) play ( i , a ) (cid:1) . The latter means that ∀ u ∈ Best i , P ( w ) ∩||¬ play ( i , a ) || i , w , M , ∃ v ∈ Best i , P ( w ) ∩|| play ( i , a ) || i , w , M : u (cid:22) i , D v . Since | = MCMC play ( i , a ) → ¬ play ( i , b ) if a (cid:54) = b , the latter im-plies ∀ b ∈ Act , ∀ u ∈ Best i , P ( w ) ∩ || play ( i , b ) || i , w , M , ∃ v ∈ Best i , P ( w ) ∩|| play ( i , a ) || i , w , M : u (cid:22) i , D v . Thus, (cid:86) b ∈ Act RP Opti (cid:0) play ( i , b ) (cid:22) play ( i , a ) (cid:1) .The case x = Pess can be proved in an analogous way. (cid:4)
The following proposition elucidates the connection between the notions of belief,rationality and Nash equilibrium: if all agents are optimistic (resp. pessimistic) rationaland have a correct belief about the others’ actual choices, then the joint action theychoose is a subjective optimistic (resp. pessimistic) Nash equilibrium. Proposition 15
Let x ∈ { opt , pess } and δ ∈ JAct. Then: | = MCMC (cid:16) play ( δ ) ∧ (cid:94) i ∈ Agt (cid:0)
Rat xi ∧ B i play ( δ − i ) (cid:1)(cid:17) → NE xi ( δ ) (12) A similar epistemic characterization of Nash equilibrium is provided by Aumann & Brandenburger(A&B) [6] in the context of games with complete information. See also [45] for a similar result using aprobabilistic approach. ROOF . Let M be a MCMC and let w be a world in M . Suppose M , w | = play (cid:0) i , δ ( i ) (cid:1) , M , w | = Rat xi and M , w | = B i play ( δ − i ) , for all i ∈ Agt . By Proposition 14, it followsthat (cid:86) b ∈ Act RP xi (cid:16) play ( i , b ) (cid:22) play (cid:0) i , δ ( i ) (cid:1)(cid:17) and M , w | = B i play ( δ − i ) , for all i ∈ Agt .From the latter, we can conclude that M , w | = BR xi (cid:0) δ ( i ) , δ − i (cid:1) , for all i ∈ Agt . Thus, M , w | = NE xi ( δ ) . (cid:4) We conclude this section by illustrating the game-theoretic concepts involved in thecrossroad game described in Section 3.4. It is a game under incomplete informationsince an agent does not necessarily know the other agent’s beliefs and desires. It issingle-stage since that interaction is non-repeated and agents are supposed to choosesimultaneously.
Example (cont.)
Let us suppose that the set of actions that agents and can chooseis Act = { C , S } , where C is the action “to continue” and S is the action “to stop”.The following hypotheses capture the agents’ knowledge and beliefs about actions andtheir effects: ϕ = def (cid:94) i ∈{ , } [ ≡ i ] (cid:16)(cid:0) ( play ( , C ) ∧ play ( , C )) → co (cid:1) ∧ (cid:0) ( play ( , C ) ∧ play ( , S )) → ( ¬ lo ∧ lo ) (cid:1) ∧ (cid:0) ( play ( , S ) ∧ play ( , C )) → ( lo ∧ ¬ lo ) (cid:1) ∧ (cid:0) ( play ( , S ) ∧ play ( , S )) → ( lo ∧ lo ∧ ¬ co ) (cid:1)(cid:17) , ϕ = def (cid:16)(cid:0)(cid:98) B play ( (cid:55)→ C , (cid:55)→ C ) ↔ (cid:98) B play ( (cid:55)→ S , (cid:55)→ C ) (cid:1) ∧ (cid:0)(cid:98) B play ( (cid:55)→ C , (cid:55)→ S ) ↔ (cid:98) B play ( (cid:55)→ S , (cid:55)→ S ) (cid:1) ∧ (cid:0)(cid:98) B play ( (cid:55)→ C , (cid:55)→ C ) ↔ (cid:98) B play ( (cid:55)→ C , (cid:55)→ S ) (cid:1) ∧ (cid:0)(cid:98) B play ( (cid:55)→ S , (cid:55)→ C ) ↔ (cid:98) B play ( (cid:55)→ S , (cid:55)→ S ) (cid:1)(cid:17) , where (cid:98) B i ϕ = def ¬ B i ¬ ϕ . According to the hypothesis ϕ , the agents know that (i) ifthey both continue, they will collide, (ii) if one of them continues while the other stops,then the first will lose its time while the second will not, and (iii) if they both stop, eachof them will lose its time but there will be no collision. According to the hypothesis ϕ , the fact that an agent considers possible that the other will decide to continue(resp. to stop) does not depend on the agent’s choice. This hypothesis is justified bythe assumption that an agent’s beliefs are ex ante , i.e., relative to the instant before anagent makes its choice.As the following validity indicates, the previous hypotheses ϕ and ϕ together withthe hypotheses ϕ and ϕ stated in Section 3.4 lead to the conclusion that (i) an agent’saction of continuing is both an optimistic and a pessimistic best response to the otheragent’s action of stopping, and an agent’s action of stopping is both an optimisticand a pessimistic best response to the other agent’s action of continuing. For every ∈ { opt , pess } , we have: | = MCMC ( ϕ ∧ ϕ ∧ ϕ ∧ ϕ ) → (cid:0) BR x ( S , (cid:55)→ C ) ∧ BR x ( C , (cid:55)→ S ) ∧ BR x ( S , (cid:55)→ C ) ∧ BR x ( C , (cid:55)→ S ) (cid:1) (13) The logics we presented so far merely provide a static picture of the cognitive attitudesand choices of multiple agents in interactive situations. Following the tradition of dy-namic epistemic logic (DEL) [50], in this section we move from a static to a dynamicperspective and extend the language L DLCA by a variety of dynamic operators for cog-nitive attitude change. We consider two types of cognitive attitude change, namely,radical attitude and conservative attitude change. Radical attitude change, both in itsepistemic and in its motivational form, satisfies a strong form of success postulate.Particularly, if an agent forms the belief that ϕ , as a consequence of a radical beliefrevision by ϕ , then she should also form the strong belief that ϕ . Analogously, if anagent forms the desire that ϕ , as a consequence of a radical desire revision by ϕ , thenshe should also form the strong desire that ϕ . On the contrary, after a conservativebelief (resp. desire) revision by ϕ is performed, an agent may form the belief (resp.desire) that ϕ without forming the strong belief (resp. strong desire) that ϕ . While rad-ical and conservative belief revision have been studied before in the literature on DEL[47, 10], we are the first to apply DEL techniques to the analysis of desire revision andto oppose belief revision to desire revision in the DEL setting. In the rest of this section, we first define the semantics of radical belief revision anddesire revision operators (Section 6.1). Then, we turn to conservative attitude changeand define the semantics of conservative belief revision and desire revision operators(Section 6.2). Finally, we provide an axiomatics for the dynamic extension of our logicDLCA (Section 6.3).
Radical attitude revision operators are of the form [ ⇑ i , τ ϕ ] , with τ ∈ { P , D } . Theydescribe the consequences of a radical revision operation. In particular, the formula [ ⇑ i , P ϕ ] ψ is meant to stand for “ ψ holds, after agent i has radically revised her beliefswith ϕ ”, whereas [ ⇑ i , D ϕ ] ψ is meant to stand for “ ψ holds, after agent i has radicallyrevised her desires with ϕ ”. We assume that radical revision operations are public,i.e., if an agent radically revises her beliefs (resp. desires) with ϕ , then this is com-mon knowledge among all agents. This assumption could be easily relaxed by usingaction models as introduced in [8, 9], which would allow us to model private and semi-private attitude change operations. Radical revision operators are interpreted relative to Research in the DEL area has rather concentrated on preference change [49, 52], leaving desire changeunexplored.
26 MCM M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) and a world w in W , as follows: M , w | = [ ⇑ i , τ ϕ ] ψ ⇐⇒ M ⇑ i , τ ϕ , w | = ψ , where M ⇑ i , P ϕ = ( W , ( (cid:22) ⇑ i , P ϕ i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) , M ⇑ i , D ϕ = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) ⇑ i , D ϕ i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) , (cid:22) ⇑ i , τ ϕ i , τ = (cid:110) ( w , v ) ∈ W × W : (cid:0) ( M , w | = ϕ iff M , v | = ϕ ) and w (cid:22) i , τ v (cid:1) or ( M , w | = ¬ ϕ , M , v | = ϕ and w ≡ i v ) (cid:111) , and (cid:22) ⇑ i , τ ϕ j , τ = (cid:22) j , τ for all j ∈ Agt such that i (cid:54) = j .Radical belief and desire revision are completely symmetric from the point of viewof the plausibility and desirability ordering. Agent i ’s radical belief revision with ϕ transforms agent i ’s plausibility ordering (cid:22) i , P into the new plausibility ordering (cid:22) ⇑ i , P ϕ i , P .In particular, it makes all ϕ -worlds in i ’s information set more plausible than all ¬ ϕ -worlds and, within those two zones, it keeps the old plausibility ordering. Analogously,agent i ’s radical desire revision with ϕ transforms agent i ’s desirability ordering (cid:22) i , D into the new desirability ordering (cid:22) ⇑ i , D ϕ i , D . It makes all ϕ -worlds in i ’s informationset more desirable than all ¬ ϕ -worlds and, within those two zones, it keeps the olddesirability ordering.As emphasized above, radical revision satisfies a strong form of success principlewhich is formally expressed by the following two validities. Let ϕ ∈ L PL ( Atm ) . Then, | = MCM (cid:104)≡ i (cid:105) ϕ → [ ⇑ i , P ϕ ]( B i ϕ ∧ SB i ϕ ) (14) | = MCM (cid:104)≡ i (cid:105)¬ ϕ → [ ⇑ i , D ϕ ]( D i ϕ ∧ SD i ϕ ) (15)This means that (i) if ϕ is compatible with an agent’s knowledge then, after she hasradically revised her beliefs with ϕ , the agent will both believe that ϕ and stronglybelieve that ϕ , and (ii) if ¬ ϕ is compatible with an agent’s knowledge then, after shehas radically revised her desires with ϕ , the agent will both desire that ϕ and stronglydesire that ϕ .The two validities highlight that belief and desire behave in a slightly different wayunder radical revision, despite the fact that the plausibility and desirability ordering aremodified in the same way.We have the following additional validities, for ϕ ∈ L PL ( Atm ) : | = MCM [ ⇑ i , P ϕ ]( B i ϕ → SB i ϕ ) (16) | = MCM [ ⇑ i , D ϕ ]( D i ϕ → SD i ϕ ) (17)This means that the formation of a belief (resp. desire) through radical belief (resp.desire) revision necessarily entails the formation of a strong belief (resp. strong desire)with the same content. 27 xample (cont.) Let us go back to the crossroad game in order to illustrate the radicaldesire revision mechanism. Suppose agent performs a radical desire revision opera-tion with ¬ lo , since it learns that agent is an ambulance which has to lose no time atthe crossroad. By the previous validity (15), we can prove that, under the hypothesis ϕ stated in Section 3.4, will both desire and strongly desire that ¬ lo , after the radicaldesire revision operation with ¬ lo : | = MCM ϕ → [ ⇑ , D ¬ lo ]( D ¬ lo ∧ SD ¬ lo ) (18) Moreover, under the set of hypotheses { ϕ , ϕ , ϕ } , after the radical desire revisionoperation with ¬ lo , will not strongly desire anymore not to lose time, but it willmerely desire it: | = MCM ( ϕ ∧ ϕ ∧ ϕ ) → [ ⇑ , D ¬ lo ]( D ¬ lo ∧ ¬ SD ¬ lo ) (19)As the following proposition indicates, we have reduction axioms which allow usto eliminate radical attitude revision operators from a formula. Proposition 16
The following equivalences are valid: [ ⇑ i , τ ϕ ] p ↔ p [ ⇑ i , τ ϕ ] x ↔ x [ ⇑ i , τ ϕ ] ¬ ψ ↔ ¬ [ ⇑ i , τ ϕ ] ψ [ ⇑ i , τ ϕ ]( ψ ∧ ψ ) ↔ ([ ⇑ i , τ ϕ ] ψ ∧ [ ⇑ i , τ ϕ ] ψ )[ ⇑ i , τ ϕ ][ π ] ψ ↔ [ F ⇑ i , τ ϕ ( π )][ ⇑ i , τ ϕ ] ψ where for all j ∈ Agt and for all τ , τ (cid:48) ∈ { P , D } :F ⇑ i , τ ϕ ( ≡ j ) = ≡ j F ⇑ i , τ ϕ ( (cid:22) i , τ ) = ( ϕ ?; (cid:22) i , τ ; ϕ ? ) ∪ ( ¬ ϕ ?; (cid:22) i , τ ; ¬ ϕ ? ) ∪ ( ¬ ϕ ?; ≡ i ; ϕ ? ) F ⇑ i , τ ϕ ( (cid:22) j , τ (cid:48) ) = (cid:22) j , τ (cid:48) if i (cid:54) = j or τ (cid:54) = τ (cid:48) F ⇑ i , τ ϕ ( (cid:22) ∼ i , τ ) = ( ϕ ?; (cid:22) ∼ i , τ ; ϕ ? ) ∪ ( ¬ ϕ ?; (cid:22) ∼ i , τ ; ¬ ϕ ? ) ∪ ( ϕ ?; ≡ i ; ¬ ϕ ? ) F ⇑ i , τ ϕ ( (cid:22) ∼ j , τ (cid:48) ) = (cid:22) ∼ j , τ (cid:48) if i (cid:54) = j or τ (cid:54) = τ (cid:48) F ⇑ i , τ ϕ ( π ; π (cid:48) ) = F ⇑ i , τ ϕ ( π ) ; F ⇑ i , τ ϕ ( π (cid:48) ) F ⇑ i , τ ϕ ( π ∪ π (cid:48) ) = F ⇑ i , τ ϕ ( π ) ∪ F ⇑ i , τ ϕ ( π (cid:48) ) F ⇑ i , τ ϕ ( π ∩ π (cid:48) ) = F ⇑ i , τ ϕ ( π ) ∩ F ⇑ i , τ ϕ ( π (cid:48) ) F ⇑ i , τ ϕ ( − π ) = − F ⇑ i , τ ϕ ( π ) F ⇑ i , τ ϕ ( ψ ? ) = [ ⇑ i , τ ϕ ] ψ ? 28 .2 Conservative Attitude Revision Let us move from radical attitude change to conservative attitude change by introduc-ing radical revision operators of type [ ↑ i , τ ϕ ] , with τ ∈ { P , D } . The formula [ ↑ i , P ϕ ] ψ (resp. [ ↑ i , D ϕ ] ψ ) is meant to stand for “ ψ holds, after agent i has conservatively revisedher beliefs (resp. desires) with ϕ ”. As for radical revision, we assume that conservativerevision operations are public, i.e., if an agent conservatively revises her beliefs (resp.desires) with ϕ , then this is common knowledge among all agents. The semantic in-terpretation of such operators relative to a MCM M = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) be a MCM and a world w in W is as follows: M , w | = [ ↑ i , τ ϕ ] ψ ⇐⇒ M ↑ i , τ ϕ , w | = ψ , where: M ↑ i , P ϕ = ( W , ( (cid:22) ↑ i , P ϕ i , P ) i ∈ Agt , ( (cid:22) i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) , M ↑ i , D ϕ = ( W , ( (cid:22) i , P ) i ∈ Agt , ( (cid:22) ↑ i , D ϕ i , D ) i ∈ Agt , ( ≡ i ) i ∈ Agt , V ) , with: (cid:22) ↑ i , P ϕ i , P = (cid:110) ( w , v ) ∈ W × W : (cid:16)(cid:0) w ∈ Best i , P ( ϕ , w ) iff v ∈ Best i , P ( ϕ , w ) (cid:1) and w (cid:22) i , P v (cid:17) or (cid:0) w (cid:54)∈ Best i , P ( ϕ , w ) , v ∈ Best i , P ( ϕ , w ) and w ≡ i v (cid:1)(cid:111) , (cid:22) ↑ i , D ϕ i , D = (cid:110) ( w , v ) ∈ W × W : (cid:16)(cid:0) w ∈ Worst i , D ( ¬ ϕ , w ) iff v ∈ Worst i , D ( ¬ ϕ , w ) (cid:1) and w (cid:22) i , D v (cid:17) or (cid:0) w ∈ Worst i , D ( ¬ ϕ , w ) , v (cid:54)∈ Worst i , D ( ¬ ϕ , w ) and w ≡ i v (cid:1)(cid:111) , and (cid:22) ↑ i , τ ϕ j , τ = (cid:22) j , τ for all j ∈ Agt such that i (cid:54) = j .Unlike radical revision, plausibility update and desirability update are asymmetricunder conservative revision. Agent i ’s conservative belief revision with ϕ replaces thecurrent plausibility ordering (cid:22) i , P with the new plausibility ordering (cid:22) ↑ i , P ϕ i , P . It promotesthe most plausible ϕ -worlds to the highest plausibility rank, but apart from that, the oldplausibility ordering remains. Agent i ’s conservative desire revision with ϕ replacesthe current desirability ordering (cid:22) i , D with the new desirability ordering (cid:22) ↑ i , P ϕ i , P . Inparticular, it demotes the least desirable ¬ ϕ -worlds to the lowest desirability rank, butapart from that, the old desirability ordering remains.Conservative attitude revision satisfies a weak form of success principle whichguarantees the formation of a belief (resp. a desire), after a belief (resp. desire) re-vision is performed. Let ϕ L PL ( Atm ) . Then, | = MCM ¬ B i ( ϕ , ⊥ ) → [ ↑ i , P ϕ ] B i ϕ (20) | = MCM ¬ D i ( ϕ , (cid:62) ) → [ ↑ i , D ϕ ] D i ϕ (21)According to the previous validities, if an agent does not believe a contradiction con-ditional on ϕ then, after she has conservatively revised her beliefs with ϕ , she willbelieve that ϕ . If an agent does not desire a tautology conditional on ϕ then, after she29as conservatively revised her desires with ϕ , she will desire that ϕ . But, unlike rad-ical attitude revision, conservative attitude revision does not necessarily guarantee theformation of a strong belief (resp. a strong desire), after a belief (resp. desire) revisionis performed. Indeed, we have the following, for ϕ ∈ L PL ( Atm ) : (cid:54)| = MCM [ ↑ i , P ϕ ]( B i ϕ → SB i ϕ ) (22) (cid:54)| = MCM [ ↑ i , D ϕ ]( D i ϕ → SD i ϕ ) (23)This means that the formation of a belief (resp. desire) through conservative belief(resp. desire) revision does not necessarily entail the formation of a strong belief (resp.strong desire) with the same content. Example (cont.)
Let us illustrate the conservative belief revision mechanism with thehelp of the crossroad game. Suppose agent informs agent that “if they both losetime, then there will no collision” and, as a consequence, performs a conservativebelief revision operation with input ( lo ∧ lo ) → ¬ co. By the previous validity (20) wecan prove that, under the hypothesis ϕ stated in Section 3.4 and the assumption that does not believe a contradiction conditional on ’s assertion, believes that there willbe no collision, after its conservative belief operation: | = MCM (cid:16) ¬ B (cid:0) ( lo ∧ lo ) → ¬ co , ⊥ (cid:1) ∧ ϕ (cid:17) → [ ↑ , P ( lo ∧ lo ) → ¬ co ] B ¬ co (24)As for radical revision, we have reduction axioms which allow us to eliminateconservative attitude revision operators from a formula. Proposition 17
The following equivalences are valid: [ ↑ i , τ ϕ ] p ↔ p [ ↑ i , τ ϕ ] x ↔ x [ ↑ i , τ ϕ ] ¬ ψ ↔ ¬ [ ↑ i , τ ϕ ] ψ [ ↑ i , τ ϕ ]( ψ ∧ ψ ) ↔ ([ ↑ i , τ ϕ ] ψ ∧ [ ↑ i , τ ϕ ] ψ )[ ↑ i , τ ϕ ][ π ] ψ ↔ [ F ↑ i , τ ϕ ( π )][ ↑ i , τ ϕ ] ψ here for all j ∈ Agt and for all τ , τ (cid:48) ∈ { P , D } :F ↑ i , τ ϕ ( ≡ j ) = ≡ j F ↑ i , P ϕ ( (cid:22) i , P ) = (cid:0) ( ϕ ∧ [ ≺ i , P ] ¬ ϕ ) ?; (cid:22) i , τ ; ( ϕ ∧ [ ≺ i , P ] ¬ ϕ ) ? (cid:1) ∪ (cid:0) ¬ ( ϕ ∧ [ ≺ i , P ] ¬ ϕ ) ?; (cid:22) i , τ ; ¬ ( ϕ ∧ [ ≺ i , P ] ¬ ϕ ) ? (cid:1) ∪ (cid:0) ¬ ( ϕ ∧ [ ≺ i , P ] ¬ ϕ ) ?; ≡ i ; ( ϕ ∧ [ ≺ i , P ] ¬ ϕ ) ? (cid:1) F ↑ i , D ϕ ( (cid:22) i , D ) = (cid:0) ( ¬ ϕ ∧ [ (cid:31) i , D ] ϕ ) ?; (cid:22) i , τ ; ( ¬ ϕ ∧ [ (cid:31) i , D ] ϕ ) ? (cid:1) ∪ (cid:0) ¬ ( ¬ ϕ ∧ [ (cid:31) i , D ] ϕ ) ?; (cid:22) i , τ ; ¬ ( ¬ ϕ ∧ [ (cid:31) i , D ] ϕ ) ? (cid:1) ∪ (cid:0) ( ¬ ϕ ∧ [ (cid:31) i , D ] ϕ ) ?; ≡ i ; ¬ ( ¬ ϕ ∧ [ (cid:31) i , D ] ϕ ) ? (cid:1) F ↑ i , τ ϕ ( (cid:22) j , τ (cid:48) ) = (cid:22) j , τ (cid:48) if i (cid:54) = j or τ (cid:54) = τ (cid:48) F ↑ i , P ϕ ( (cid:22) ∼ i , P ) = (cid:0) ( ϕ ∧ [ ≺ i , P ] ¬ ϕ ) ?; (cid:22) i , τ ; ( ϕ ∧ [ ≺ i , P ] ¬ ϕ ) ? (cid:1) ∪ (cid:0) ¬ ( ϕ ∧ [ ≺ i , P ] ¬ ϕ ) ?; (cid:22) i , τ ; ¬ ( ϕ ∧ [ ≺ i , P ] ¬ ϕ ) ? (cid:1) ∪ (cid:0) ( ϕ ∧ [ ≺ i , P ] ¬ ϕ ) ?; ≡ i ; ¬ ( ϕ ∧ [ ≺ i , P ] ¬ ϕ ) ? (cid:1) F ↑ i , D ϕ ( (cid:22) ∼ i , D ) = (cid:0) ( ¬ ϕ ∧ [ (cid:31) i , D ] ϕ ) ?; (cid:22) i , τ ; ( ¬ ϕ ∧ [ (cid:31) i , D ] ϕ ) ? (cid:1) ∪ (cid:0) ¬ ( ¬ ϕ ∧ [ (cid:31) i , D ] ϕ ) ?; (cid:22) i , τ ; ¬ ( ¬ ϕ ∧ [ (cid:31) i , D ] ϕ ) ? (cid:1) ∪ (cid:0) ¬ ( ¬ ϕ ∧ [ (cid:31) i , D ] ϕ ) ?; ≡ i ; ( ¬ ϕ ∧ [ (cid:31) i , D ] ϕ ) ? (cid:1) F ↑ i , τ ϕ ( (cid:22) ∼ j , τ (cid:48) ) = (cid:22) ∼ j , τ (cid:48) if i (cid:54) = j or τ (cid:54) = τ (cid:48) F ↑ i , τ ϕ ( π ; π (cid:48) ) = F ↑ i , τ ϕ ( π ) ; F ↑ i , τ ϕ ( π (cid:48) ) F ↑ i , τ ϕ ( π ∪ π (cid:48) ) = F ↑ i , τ ϕ ( π ) ∪ F ↑ i , τ ϕ ( π (cid:48) ) F ↑ i , τ ϕ ( π ∩ π (cid:48) ) = F ↑ i , τ ϕ ( π ) ∩ F ↑ i , τ ϕ ( π (cid:48) ) F ↑ i , τ ϕ ( − π ) = − F ↑ i , τ ϕ ( π ) F ↑ i , τ ϕ ( ψ ? ) =[ ↑ i , τ ϕ ] ψ ? The modal language L DLCAC ( Atm , Nom , Agt ) , or simply L DLCAC , for the DynamicLogic of Cognitive Attitudes and their Change (DLCAC) extends the language L DLCA of the logic DLCA by dynamic operators of type [ ⇑ i , τ ϕ ] and [ ↑ i , τ ϕ ] . It is defined bythe following grammar: ϕ :: = p | x | ¬ ϕ | ϕ ∧ ϕ (cid:48) | [ π ] ϕ | [ ⇑ i , τ ϕ ] ψ | [ ↑ i , τ ϕ ] ψ where π ranges over the language of cognitive programs P , p ranges over Atm , x ranges over Nom , i ranges over Agt and τ ranges over { P , D } . Definition 20
We define
DLCAC to be the extension of
DLCA given by the reductionprinciples of Proposition 16 and Proposition 17 and the following rule of replacementof equivalents ψ ↔ ψ ϕ ↔ ϕ [ ψ / ψ ] ( REP )31 here ϕ [ ψ / ψ ] is the formula that results from ϕ by replacing zero or more occur-rences of ψ , in ϕ , by ψ . As the rule of replacement of equivalents preserves validity, the equivalences of Propo-sitions 16 and 17 together with this allow to reduce every formula of the language L DLCAC to an equivalent formula of the language L DLCA . Call red the mapping whichiteratively applies the above equivalences from the left to the right, starting from one ofthe innermost modal operators. red pushes the dynamic operators inside the formula,and finally eliminates them when facing an atomic formula.
Proposition 18
Let ϕ be a formula in the language of L DLCAC . Then • red ( ϕ ) has no dynamic operators [ ⇑ i , τ ϕ ] or [ ↑ i , τ ϕ ] , and • red ( ϕ ) ↔ ϕ is valid relative to the class of MCMs. The first item of Proposition 18 is clear. The second item is proved using the equiva-lences of Propositions 16 and 17 and the rule of replacement of equivalents.The following theorem is a direct consequence of Theorem 1 and Proposition 18.
Theorem 3
The logic
DLCAC is sound and complete for the class of multi-agent cog-nitive models.
We have presented a logical framework for modelling a rich variety of cognitive atti-tudes of both epistemic type and motivational type. We have presented two extensionsof the basic setting, one by the notion of choice and the other by dynamic operatorsfor belief change and desire change. We have applied the former to the analysis ofgames under incomplete information. We have provided sound and complete axioma-tizations for the basic setting and for its two extensions. Directions of future researchare manifold and are briefly discussed in the rest of this section.
Decidability and complexity
The present paper is devoted to study the proof-theoreticaspects of the proposed logics. In future work, we plan to investigate their computa-tional aspects including decidability of their satisfiability problems and, at a later stage,complexity. In order to prove decidability, we expect to be able to use existing filtrationtechniques from modal logic. Note that once we have proved decidability of the staticsetting DLCA, we can use the reduction axioms of Propositions 16 and 17 to provedecidability of the dynamic setting DLCAC.We plan to study complexity of the satisfiability problems for interesting fragmentsof the language L DLCA by reducing them to satisfiability problems of existing logics.For instance, consider the following single-agent ( sa ) fragment of the language L DLCA where only atomic programs ( ap ) are allowed, noted L sa , ap DLCA : ϕ :: = p | x | ¬ ϕ | ϕ ∧ ϕ (cid:48) | [ (cid:22) , P ] ϕ | [ (cid:22) , D ] ϕ | [ ≡ ] ϕ Agt . We can observe that the satisfiability problem forthis fragment is EXPTIME-hard. Indeed, because of Constraint C1 in Definition 1, themodality [ ≡ ] plays the role of the universal modality with respect to the modalities [ (cid:22) , P ] and [ (cid:22) , D ] . As shown in [27], adding the universal modality to a multimodallogic with independent modalities, such as the S4-modalities [ (cid:22) , P ] and [ (cid:22) , D ] , causesEXPTIME-hardness.Consider now the following intersection-free ( if ) and complement-free ( cf ) frag-ment of L DLCA , noted L if , cf DLCA : π :: = ≡ i |(cid:22) i , P |(cid:22) i , D | π ; π (cid:48) | π ∪ π (cid:48) | − π | ϕ ? ϕ :: = p | x | ¬ ϕ | ϕ ∧ ϕ (cid:48) | [ π ] ϕ Our first conjecture is that we can find a polysize reduction of the satisfiability prob-lem for L if , cf DLCA to the satisfiability problem of converse propositional dynamic logic(PDL) with nominals, also called converse combinatory propositional dynamic logic(CcPDL). The latter problem is known to be EXPTIME-complete [21]. Therefore,if our conjecture is true, we will be able to conclude that the satisfiability problems forthe fragments L sa , ap DLCA and L if , cf DLCA are both EXPTIME-complete.We also intend to study complexity of the nominal-free ( nf ) fragment of L DLCA ,noted L nf DLCA . Nominals play a technical role in the logic DLCA by making it easierthe task of axiomatizing intersection and complement of programs (Axioms
Add2 ∩ and Comp1 ∼ in Definition 13). Our second conjecture is that the language L DLCA is strictly more expressive than its nominal-free fragment L nf DLCA . Our third conjec-ture is that we can find a polysize reduction of the satisfiability problem for L nf DLCA to the satisfiability problem of boolean modal logic with a bounded number of modalparameters which is known to be EXPTIME-complete [36]. We leave the proof of theprevious three conjectures to future work. We leave to future work (i) the proof of theprevious three conjectures, and (ii) the development of tableau-based automated rea-soning procedures for the language L DLCA and for its fragments L sa , ap DLCA , L if , cf DLCA and L nf DLCA which can be used for programming artificial agents endowed with cognitiveattitudes.
Well-foundedness
Future work will also be devoted to study a variant of our logicDLCA under the assumption of converse well-foundedness for the relation (cid:22) i , P andwell-foundedness for the relation (cid:22) i , D . As emphasized in Section 3, these propertiesare required to make agents’ beliefs and desires consistent, namely, to guarantee thatthe formulas ¬ ( B i ϕ ∧ B i ¬ ϕ ) , ¬ B i ⊥ , ¬ ( D i ϕ ∧ D i ¬ ϕ ) and ¬ D i (cid:62) become valid. We willdefine the logic DLCA wf to be the extension of the logic DLCA of Definition 13 by thefollowing two axioms: (cid:104)≡ i (cid:105) ψ → (cid:104)≡ i (cid:105) ( ψ ∧ [ ≺ i , P ] ¬ ψ ) ( CWF (cid:22) i , P ) (cid:104)≡ i (cid:105) ψ → (cid:104)≡ i (cid:105) ( ψ ∧ [ (cid:31) i , D ] ¬ ψ ) ( WF (cid:22) i , D ) The main idea of the polynomial embedding is to exploit the iteration construct ∗ of PDL for the transla-tion tr of the cognitive programs, by stipulating that tr ( ≡ i ) = ( any i ∪− any i ) ∗ , tr ( (cid:22) i , P ) = P ∗ i , tr ( (cid:22) i , D ) = D ∗ i ,and homomorphic otherwise, where A i is agent i ’s set of atomic programs (or actions), A = (cid:83) i ∈ Agt A i is theset of PDL atomic programs, any i = (cid:83) a i ∈ A i and, finally, P ∗ i and D ∗ i are special atomic programs in A i . GL ) axiom from provabilitylogic [14]. Our conjecture is that the logic DLCA wf so defined is sound and completefor the class of multi-agent cognitive models (MCMs) whose relations (cid:22) i , D and (cid:22) i , P are, respectively, well-founded and conversely well-founded. Ceteris paribus preference
We also plan to study a ceteris paribus notion of dyadicpreference in the sense of Von Wright [55], which has been recently formalized in amodal logic setting by van Benthem et al. [48]. According to Von Wright, for an agentto have a preference of ϕ over ψ , she should prefer a situation in which ϕ is true toa situation in which ψ is true, all other things being equal . Our aim is to show thatthe DLCA framework is expressive enough to capture both the static and the dynamicaspects of this notion of ceteris paribus preference.
Acknowledgments
This work was supported by the ANR project CoPains (“Cognitive Planning in Persua-sive Multimodal Communication”). Support from the ANR-3IA Artificial and NaturalIntelligence Toulouse Institute is also gratefully acknowledged.
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