Relative Expressiveness of Defeasible Logics II
aa r X i v : . [ c s . L O ] F e b Under consideration for publication in Theory and Practice of Logic Programming Relative Expressiveness of Defeasible Logics II
Michael J. Maher
School of Engineering and Information TechnologyUniversity of New South Wales, CanberraACT 2600, AustraliaE-mail: [email protected] 1 January 2003; revised 1 January 2003; accepted 1 January 2003
Abstract (Maher 2012) introduced an approach for relative expressiveness of defeasible logics, and two notions ofrelative expressiveness were investigated. Using the first of these definitions of relative expressiveness, weshow that all the defeasible logics in the DL framework are equally expressive under this formulation ofrelative expressiveness. The second formulation of relative expressiveness is stronger than the first. How-ever, we show that logics incorporating individual defeat are equally expressive as the corresponding logicswith team defeat. Thus the only differences in expressiveness of logics in DL arise from differences in howambiguity is handled. This completes the study of relative expressiveness in DL begun in (Maher 2012). KEYWORDS : defeasible logic, non-monotonic reasoning, relative expressiveness
Introduction
Defeasible logics provide several linguistic features to support the expression of defeasible knowl-edge. There are also a variety of such logics, supporting different intuitions about reasoning ina defeasible setting. The DL framework (Antoniou et al. 2000; Billington et al. 2010) provideslogics that allow ambiguity in the “truth” status of a literal to propagate, and logics that blockambiguity; it has logics that require an individual rule to defeat all competitors, and logics thatallow a “team” of rules to defeat competitors. Given the different inferences supported by the dif-ferent logics, it is interesting to determine whether these logics are equally powerful or whether,perhaps, some are more powerful than the others.In terms of inference strength, (Billington et al. 2010) established the relationship between thedifferent logics of DL . In terms of computational complexity, the logics of DL are equivalent:all have linear complexity (Maher 2001; Billington et al. 2010). Relative expressiveness of thedifferent logics was first investigated in (Maher 2012), which developed a framework, based onsimulation in the presence of additional elements. Two notions of relative expressiveness withinthis framework were investigated: polynomial simulation wrt the addition of facts, and simulationwrt the addition of rules.In this paper we continue this investigation. We will see that all the logics of DL are equallyexpressive, using the first notion of relative expressiveness. Thus we cannot distinguish the logicsbased on this notion. We also establish that individual defeat has equal expressiveness to teamdefeat in the logics of DL wrt addition of rules. This is somewhat surprising, given the apparentgreater sophistication of the team defeat inference rules. Given results in (Maher 2012), thiscompletes the study of relative expressiveness for DL . M.J. Maher
The next two sections summarize the DL framework of defeasible logics and the notions ofrelative expressiveness introduced in (Maher 2012). Then the following two sections togetherprovide the proof that the logics of DL are of equal expressivity (in terms of simulation wrt ad-dition of facts). The first shows the simulation of an ambiguity propagating logic by an ambiguityblocking logic, while the second shows a simulation in the reverse direction. Combined with re-sults of (Maher 2012), this establishes that the logics in DL all have the same expressiveness inthis formulation.The following sections investigate relative expressiveness via the second, stronger formulation.Adapting simulations of (Maher 2012) to the stronger formulation, we establish that individualdefeat has equal expressiveness to team defeat in the logics of DL . Proofs of the results in thispaper are detailed and lengthy. They appear in an appendix. Defeasible Logic
In this section we can only present an outline of the defeasible logics we investigate. Further de-tails can be obtained from (Billington et al. 2010) and the references therein. We address propo-sitional defeasible logics, but the results should extend to a first-order language.A defeasible theory is built from a language Σ of literals (which we assume is closed undernegation) and a language Λ of labels. A defeasible theory D = ( F, R, > ) consists of a set offacts F , a finite set of rules R , each rule with a distinct label from Λ , and an acyclic relation > on Λ called the superiority relation . This syntax is uniform for all the logics considered here. Factsare individual literals expressing indisputable truths. Rules relate a set of literals (the body),via an arrow, to a literal (the head), and are one of three types: a strict rule, with arrow → ; adefeasible rule, with arrow ⇒ ; or a defeater, with arrow ❀ . Strict rules represent inferences thatare unequivocally sound if based on definite knowledge; defeasible rules represent inferencesthat are generally sound. Inferences suggested by a defeasible rule may fail, due to the presencein the theory of other rules. Defeaters do not support inferences, but may impede inferencessuggested by other rules. The superiority relation provides a local priority on rules. Strict ordefeasible rules whose bodies are established defeasibly represent claims for the head of therule to be concluded. The superiority relation contributes to the adjudication of these claimsby an inference rule, leading (possibly) to a conclusion. Given a theory D , the correspondinglanguages are expressed by Σ( D ) and Λ( D ) .Defeasible logics derive conclusions that are outside the syntax of the theories. Conclusionsmay have the form + dq , which denotes that under the inference rule d the literal q can be con-cluded, or − dq , which denotes that the logic can establish that under the inference rule d theliteral q cannot be concluded. The syntactic element d is called a tag. In general, neither con-clusion may be derivable: q cannot be concluded under d , but the logic is unable to establishthat. Tags +∆ and − ∆ represent monotonic provability (and unprovability) where inference isbased on facts, strict rules, and modus ponens. We assume these tags and their inference rulesare present in every defeasible logic. What distinguishes a logic is the inference rule for defea-sible reasoning. The four logics discussed in the Introduction correspond to four different pairsof inference rules, labelled ∂ , δ , ∂ ∗ , and δ ∗ ; they produce conclusions of the form (respectively) + ∂q , − ∂q , + δq , − δq , etc. The inference rules δ and δ ∗ require auxiliary tags and inferencerules, denoted by σ and σ ∗ , respectively. For each of the four principal defeasible tags d , thecorresponding logic is denoted by DL ( d ) .The four principal tags and corresponding inference rules represent different intuitions about elative Expressiveness of Defeasible Logics II ∂ and ∂ ∗ ambiguity is blocked, while in δ and δ ∗ ambiguity is propa-gated; in ∂ and δ rules for a literal act as a team to overcome competing rules, while in ∂ ∗ and δ ∗ a single rule must overcome all competing rules. A more detailed discussion of ambiguity andteam defeat in the DL framework is given in (Billington et al. 2010) and (Maher 2012).The inference rules are presented in the appendix in the form of the definition of a function T D for a given theory D . Given a defeasible theory D , for any set of conclusions E , T D ( E ) denotesthe set of conclusions inferred from E using D and one application of an inference rule. T D is amonotonic function on the complete lattice of sets of conclusions ordered by containment. Theleast fixedpoint of T D is the set of all conclusions that can be drawn from D . We follow standardnotation in that T D ↑ ∅ and T D ↑ ( n + 1) = T D ( T D ↑ n ) .The relative inference strength of the different logics in DL was established in the inclusiontheorem of (Billington et al. 2010). For any tag d , + d ( D ) denotes the set of conclusions of D ofthe form + dq and similarly for − d . Theorem 1 ( Inclusion Theorem (Billington et al. 2010) )Let D be a defeasible theory.(a) +∆( D ) ⊆ + δ ∗ ( D ) ⊆ + δ ( D ) ⊆ + ∂ ( D ) ⊆ + σ ( D ) ⊆ + σ ∗ ( D ) .(b) − σ ∗ ( D ) ⊆ − σ ( D ) ⊆ − ∂ ( D ) ⊆ − δ ( D ) ⊆ − δ ∗ ( D ) ⊆ − ∆( D ) .(c) + δ ∗ ( D ) ⊆ + ∂ ∗ ( D ) ⊆ + σ ∗ ( D ) (d) − σ ∗ ( D ) ⊆ − ∂ ∗ ( D ) ⊆ − δ ∗ ( D ) Parts (a) and (b) are proved in (Billington et al. 2010). Parts (c) and (d) can be established bysimilar methods.
Simulating Defeasible Logics (Maher 2012) introduced a framework for addressing the relative expressiveness of defeasiblelogics. The framework identifies the greater (or equal) expressiveness of L compared to L with the ability to simulate any theory D in a logic L by a theory T ( D ) in the logic L . Simplesimulation was shown not to be sufficiently discriminating, so simulation was required to hold inthe presence of an addition to the theory.The addition of a theory A to a theory D is denoted by D + A . Addition is essentially theunion of the theories, but we require Λ( D ) ∩ Λ( A ) = ∅ , so that the addition of theories preservesthe property that distinct rules have distinct labels. This requirement also has the effect that asuperiority statement in D cannot affect a rule in A , and vice versa. Let D = ( F, R, > ) and A = ( F ′ , R ′ , > ′ ) . Then D + A = ( F ∪ F ′ , R ∪ R ′ , > ∪ > ′ ) . Λ( D + A ) = Λ( D ) ∪ Λ( A ) and Σ( D + A ) = Σ( D ) ∪ Σ( A ) .A simulating theory T ( D ) in general will involve additional literals, rules and labels beyondthose of D . If additions A were permitted to affect these, the notion of simulation would becometrivial, so we restrict additions to have only an indirect effect on T ( D ) , via Σ( D ) . Given a theory D and a possible simulating theory T ( D ) , we say an addition A is modular if Σ( A ) ∩ Σ( T ( D )) ⊆ Σ( D ) , Λ( D ) ∩ Λ( A ) = ∅ , and Λ( T ( D )) ∩ Λ( A ) = ∅ . In general, we will consider a class ofadditions but for any D and T ( D ) only the modular additions in the class will be considered.Since different logics involve different tags, conclusions from theories in different logics can-not be identical. For simulation it suffices that conclusions are equal modulo tags. Given logics L and L , with principal tags d and d , respectively, we say two conclusions α in L and β in L are equal modulo tags if α is + d q and β is + d q or α is − d q and β is − d q . M.J. Maher
Thus we have the following definition of simulation and relative expressiveness. For morediscussion on the motivations for the definitions, see (Maher 2012).
Definition 2
Let C be a class of defeasible theories.We say D in logic L is simulated by D in L with respect to a class C if, for every modularaddition A in C , D + A and D + A have the same conclusions in Σ( D + A ) , modulo tags.We say a logic L can be simulated by a logic L with respect to a class C if every theory in L can be simulated by some theory in L with respect to additions from C .We say L is more (or equal) expressive than L if L can be simulated by L with respect C .Different notions of relative expressiveness arise from different choices for C . There were twoclasses of additions investigated in (Maher 2012): the addition of facts (that is, A has the form ( F, ∅ , ∅ ) ), and the addition of rules (that is, A has the form ( ∅ , R, ∅ ) ). Simulation with respectto addition of rules is stronger than simulation with respect to addition of facts because any factcan equally be expressed as a strict rule with an empty body. We might also consider arbitraryadditions, where A can be any defeasible theory.The main results of (Maher 2012) are that: • DL ( ∂ ) and DL ( ∂ ∗ ) have equal expressiveness, with respect to addition of facts , as do DL ( δ ) and DL ( δ ∗ ) • neither DL ( ∂ ) nor DL ( ∂ ∗ ) is more expressive than DL ( δ ) or DL ( δ ∗ ) , and vice versa,with respect to addition of rules • when arbitrary additions are permitted, of the four defeasible logics under consideration,none is more expressive than any other Blocked Ambiguity Simulates Propagated Ambiguity
We now show that every theory over an ambiguity propagating logic can be simulated by a theoryover the corresponding ambiguity blocking logic. To begin, we show that DL ( ∂ ∗ ) can simulate DL ( δ ∗ ) . Any defeasible theory D is transformed into a new theory. The new theory employs newpropositions strict ( q ) and supp ( q ) , for each literal q , and supp body ( r ) , comp ( r ) , and o ( r ) , foreach rule r . The new theory also introduces labels p d ( r ) , n d ( r, s ) , p s ( r ) , n s ( r, s ) , for each pair r, s of opposing rules in D . These are families of propositions and labels, not predicates, despitethe notation. Definition 3
Let D = ( F, R, > ) be a defeasible theory with language Σ . We define the transformation T of D to T ( D ) = ( F ′ , R ′ , > ′ ) as follows:1. The facts of T ( D ) are the facts of D . That is, F ′ = F .2. Every strict rule of R is included in R ′ .3. For every literal q , R ′ contains str ( q ) : q → strict ( q ) nstr ( q ) : ⇒ ¬ strict ( q ) and the superiority relation contains nstr ( q ) > ′ str ( q ) , for every q . elative Expressiveness of Defeasible Logics II
54. For each literal q in Σ , R ′ contains q ⇒ supp ( q )
5. For each strict or defeasible rule r of the form b , . . . , b n ֒ → r q in R , R ′ contains supp ( b ) , . . . , supp ( b n ) ⇒ supp body ( r ) supp body ( r ) , ¬ o ( r ) ⇒ supp ( q ) and, further, for each rule s = B s ֒ → s ∼ q for ∼ q in R , where s > r , R ′ contains n s ( r, s ) : B s ⇒ o ( r ) p s ( r ) : ⇒ ¬ o ( r ) and the superiority relation contains n s ( s, r ) > ′ p s ( s ) .6. For each strict or defeasible rule r = B r ֒ → r q in R , R ′ contains inf ( r ) : B r , ¬ comp ( r ) , ¬ strict ( ∼ q ) ⇒ q and, further, for each rule s = B s ֒ → s ∼ q for ∼ q in R , where s < r , R ′ contains n d ( r, s ) : supp body ( s ) ⇒ comp ( r ) p d ( r ) : ⇒ ¬ comp ( r ) and the superiority relation contains n d ( r, s ) > ′ p d ( r ) .Parts 1 and 2 of the transformation preserve all the strict inferences from D . Part 3 allows us todistinguish strict conclusions from defeasible conclusions. The structure of these rules – where str ( q ) is strict, nstr ( q ) is defeasible, and nstr ( q ) > str ( q ) – ensures that strict ( q ) is inferreddefeasibly iff q is inferred strictly, and strict ( q ) fails iff strict inference of q fails. A similarstructure of rules was previously used in (Maher 2012) in showing that DL ( ∂ ∗ ) can simulate DL ( ∂ ) wrt addition of facts.We use the proposition supp ( q ) to indicate that the literal q is supported (i.e. + σ ∗ q can beinferred), while the literal q refers to defeasible provability (wrt δ ∗ ). Part 4 ensures that everyliteral that holds defeasibly is also supported. This property is justified by the inclusion theoremof (Billington et al. 2010). Part 5 encodes the inference rules for support (i.e. σ ∗ ). supp body ( r ) indicates that all literals in the body of rule r are supported. The head q of a rule r is supportedif the body of r is supported and r is not overruled (i.e. all rules s that are superior to r fail).The overruling of r is indicated by o ( r ) . The rules n s ( r, s ) and p s ( r ) and the superiority relationensure that ¬ o ( r ) is derived defeasibly iff there is no overruling rule s .Rules inf ( r ) in part 6 encode the inference rules for δ ∗ . q holds defeasibly iff the body of arule r for q holds defeasibly and r has no competing rules (i.e. all rules for ∼ q not inferior to r have a body that fails wrt σ ∗ ). The rules n d ( r, s ) and p d ( r ) and the superiority relation ensurethat ¬ comp ( r ) is derived defeasibly iff there is no competing rule.In this translation, the superiority relation in D is not directly represented by the superiorityrelation in T ( D ) . Instead, the superiority relation in D is used to restrict the instantiation of rulesin the transformation, while the superiority relation in T ( D ) is used to ensure that o ( r ) and ¬ o ( r ) do not both fail, and similarly for comp ( r ) . Example 4
To see the operation of this transformation, consider the following theory D , which demonstratesthe difference between ambiguity propagation and blocking logics. M.J. Maher r : ⇒ p r : ¬ p ⇒ ¬ qr : ⇒ ¬ p r : ⇒ q In DL ( ∂ ∗ ) from D we conclude − ∂ ∗ p and − ∂ ∗ ¬ p , + ∂ ∗ q and − ∂ ∗ ¬ q . In DL ( δ ∗ ) from D weconclude − δ ∗ p and − δ ∗ ¬ p , − δ ∗ q and − δ ∗ ¬ q . We also conclude + σ ∗ p and + σ ∗ ¬ p , + σ ∗ q and + σ ∗ ¬ q . T ( D ) contains the following rules. ⇒ supp body ( r ) supp body ( r ) , ¬ o ( r ) ⇒ supp ( p ) ⇒ supp body ( r ) supp body ( r ) , ¬ o ( r ) ⇒ supp ( ¬ p ) supp ( ¬ p ) ⇒ supp body ( r ) supp body ( r ) , ¬ o ( r ) ⇒ supp ( ¬ q ) ⇒ supp body ( r ) supp body ( r ) , ¬ o ( r ) ⇒ supp ( q ) p ⇒ supp ( p ) p s ( r ) : ⇒ ¬ o ( r ) ¬ p ⇒ supp ( ¬ p ) p s ( r ) : ⇒ ¬ o ( r ) q ⇒ supp ( q ) p s ( r ) : ⇒ ¬ o ( r ) ¬ q ⇒ supp ( ¬ q ) p s ( r ) : ⇒ ¬ o ( r ) inf ( r ) : ¬ comp ( r ) , ¬ strict ( ¬ p ) ⇒ pinf ( r ) : ¬ comp ( r ) , ¬ strict ( p ) ⇒ ¬ pinf ( r ) : ¬ p, ¬ comp ( r ) , ¬ strict ( q ) ⇒ ¬ qinf ( r ) : ¬ comp ( r ) , ¬ strict ( ¬ q ) ⇒ qp d ( r ) : ⇒ ¬ comp ( r ) n d ( r , r ) : supp body ( r ) ⇒ comp ( r ) p d ( r ) : ⇒ ¬ comp ( r ) n d ( r , r ) : supp body ( r ) ⇒ comp ( r ) p d ( r ) : ⇒ ¬ comp ( r ) n d ( r , r ) : supp body ( r ) ⇒ comp ( r ) p d ( r ) : ⇒ ¬ comp ( r ) n d ( r , r ) : supp body ( r ) ⇒ comp ( r ) T ( D ) also contains the following superiority statements. nstr ( p ) > ′ str ( p ) n d ( r , r ) > ′ p d ( r ) nstr ( ¬ p ) > ′ str ( ¬ p ) n d ( r , r ) > ′ p d ( r ) nstr ( q ) > ′ str ( q ) n d ( r , r ) > ′ p d ( r ) nstr ( ¬ q ) > ′ str ( ¬ q ) n d ( r , r ) > ′ p d ( r ) There are some points to highlight in this example. Rules for strict and ¬ strict are omittedfrom the listing above because they are not of interest ( D has no strict rules or facts); we willhave conclusions + ∂ ∗ ¬ strict ( l ) and − ∂ ∗ strict ( l ) , for every literal l . There are no rules n s ( r, s ) in T ( D ) because they only occur when s > r , and the superiority relation in D is empty. Con-sequently, there are no superiority statements of the form n s ( r, s ) > ′ p s ( r ) . It also follows that + ∂ ∗ ¬ o ( r ) is concluded, for each rule r , and hence we can infer + ∂ ∗ supp ( l ) , for each literal l except ¬ q , reflecting the fact that these literals are supported in D , and + ∂ ∗ supp body ( r ) , foreach rule r . We can then infer also + ∂ ∗ supp ( ¬ q ) . It then follows that − ∂ ∗ ¬ comp ( r ) is con-cluded, for each r , using the superiority relation. Then, as a consequence of the rules inf ( r ) , we elative Expressiveness of Defeasible Logics II l fail to be inferred (i.e. we conclude − ∂ ∗ l , for each literal l ). This expressesthe ambiguity propagating behaviour of DL ( δ ∗ ) from within DL ( ∂ ∗ ) . Theorem 5
The ambiguity blocking logics ( DL ( ∂ ) and DL ( ∂ ∗ ) ) can simulate the ambiguity propagatinglogics ( DL ( δ ) and DL ( δ ∗ ) ) with respect to addition of facts. Propagated Ambiguity Simulates Blocked Ambiguity
We now show that every theory over an ambiguity blocking logic can be simulated by a theoryover the corresponding ambiguity propagating logic. To begin, we simulate DL ( ∂ ∗ ) by DL ( δ ∗ ) .Any defeasible theory D is transformed into a new theory T ( D ) . The new theory employs newpropositions strict ( q ) and undefeated ( q ) for each literal q in Σ , and employs labels str ( q ) and nstr ( q ) for each literal q in Σ , and n d ( r, s ) and p d ( r ) for each pair of opposing rules r, s in R . Definition 6
Let D = ( F, R, > ) be a defeasible theory with language Σ . We define the transformation T of D to T ( D ) = ( F ′ , R ′ , > ′ ) as follows:1. The facts of T ( D ) are the facts of D . That is, F ′ = F .2. Every strict rule of R is included in R ′ .3. For every literal q , R ′ contains str ( q ) : q → strict ( q ) nstr ( q ) : ⇒ ¬ strict ( q ) t ( q ) : strict ( q ) ⇒ true ( q ) nt ( q ) : ⇒ ¬ true ( q ) and the superiority relation contains nstr ( q ) > ′ str ( q ) and t ( q ) > ′ nt ( q ) , for every q .4. For each literal q , R ′ contains undefeated ( q ) ⇒ q For each strict or defeasible rule r = B r ֒ → r q for q in R , R ′ contains p d ( r ) : B r , ¬ true ( ∼ q ) ⇒ undefeated ( q ) and, further, for each rule s = B s ֒ → s ∼ q for ∼ q in R , where r > s , R ′ contains n d ( r, s ) : B s ⇒ ¬ undefeated ( q ) and the superiority relation contains n d ( r, s ) > ′ p d ( r ) .Parts 1 and 2 preserve all the strict inferences from D . Part 3 allows us to distinguish strictconclusions from defeasible conclusions. For this transformation – compared to the transforma-tion in the previous section – extra rules t and nt are needed. These rules ensure that δ ∗ and σ ∗ agree on the literals true ( q ) , that is, from T ( D )+ A we conclude + δ ∗ true ( q ) iff we conclude + σ ∗ true ( q ) iff D + A ⊢ +∆ q . (See Lemma 18 in the appendix.) In comparison, we never infer − σ ∗ ¬ strict ( q ) and always infer + σ ∗ ¬ strict ( q ) , independent of D . This also demonstrates a flaw in (Maher 2012). In that paper, the transformation used to simulate DL ( δ ) with DL ( δ ∗ ) fails to use these extra rules, and thus is incorrect. Definition 13 has a corrected transformation. M.J. Maher
Part 4 encodes the inference rules for ∂ ∗ : q holds defeasibly if the body of a rule r for q holdsdefeasibly, ∼ q is not established strictly, and r is not defeated (i.e. all rules not inferior to r havea body that fails wrt ∂ ∗ ). The requirement that r is not defeated is expressed through the use ofrules n d ( r, s ) opposing p d ( r ) for each rule s in D not inferior to r . The rules n d ( r, s ) are superiorto p d ( r ) in T ( D ) , thus ensuring that undefeated ( q ) is inferred iff r is not defeated. Example 7
To see the operation of this transformation, consider (again) the following theory D , whichdemonstrates the difference between ambiguity propagation and blocking logics. r : ⇒ p r : ¬ p ⇒ ¬ qr : ⇒ ¬ p r : ⇒ q In DL ( ∂ ∗ ) from D we conclude − ∂ ∗ p and − ∂ ∗ ¬ p , + ∂ ∗ q and − ∂ ∗ ¬ q . In DL ( δ ∗ ) from D weconclude − δ ∗ p and − δ ∗ ¬ p , − δ ∗ q and − δ ∗ ¬ q . We also conclude + σ ∗ p and + σ ∗ ¬ p , + σ ∗ q and + σ ∗ ¬ q . T ( D ) contains the following rules and superiority relation. n d ( r , r ) : ⇒ ¬ undefeated ( p ) p d ( r ) : ¬ true ( ¬ p ) ⇒ undefeated ( p ) n d ( r , r ) : ⇒ ¬ undefeated ( ¬ p ) p d ( r ) : ¬ true ( p ) ⇒ undefeated ( ¬ p ) n d ( r , r ) : ⇒ ¬ undefeated ( ¬ q ) p d ( r ) : ¬ p, ¬ true ( ¬ q ) ⇒ undefeated ( ¬ q ) n d ( r , r ) : ¬ p ⇒ ¬ undefeated ( q ) p d ( r ) : ¬ true ( q ) ⇒ undefeated ( q ) undefeated ( p ) ⇒ p undefeated ( ¬ p ) ⇒ ¬ p undefeated ( q ) ⇒ q undefeated ( ¬ q ) ⇒ ¬ qn d ( r , r ) > ′ p d ( r ) nstr ( p ) > ′ str ( p ) n d ( r , r ) > ′ p d ( r ) nstr ( ¬ p ) > ′ str ( ¬ p ) n d ( r , r ) > ′ p d ( r ) nstr ( q ) > ′ str ( q ) n d ( r , r ) > ′ p d ( r ) nstr ( ¬ q ) > ′ str ( ¬ q ) The rules concerning strict and true have been omitted. Because there are no facts or strictrules in D we will infer − δ ∗ strict ( s ) , and hence + δ ∗ ¬ true ( s ) and + σ ∗ ¬ true ( s ) for eachliteral s ∈ Σ . However, because of the superiority of n d over p d , we infer − σ ∗ undefeated ( ¬ p ) and − δ ∗ undefeated ( ¬ p ) and hence − σ ∗ ¬ p and − δ ∗ ¬ p (and similarly for p ). Hence, the bodyof p d ( r ) fails, and we infer − δ ∗ ¬ q . Similarly, the body of n d ( r , r ) fails, and hence we infer + δ ∗ q . This reflects the ambiguity blocking behaviour of DL ( ∂ ∗ ) from within the ambiguitypropagating logic DL ( δ ∗ ) .The proof of correctness of this simulation is complicated by the fact that inference rules for δ ∗ and σ ∗ are defined mutually recursively, while the inference rules for ∂ ∗ are directly recursive.This difference in structure makes a direct inductive proof difficult. The problem is resolved bya “tight” simulating transformation that is able to simulate DL ( ∂ ∗ ) (wrt addition of facts) in anyof the DL logics. elative Expressiveness of Defeasible Logics II Theorem 8
For d ∈ { δ, δ ∗ , ∂ } , DL ( d ) can simulate DL ( ∂ ∗ ) with respect to addition of factsCombining Theorems 5 and 8 with results from (Maher 2012), we see that all logics of the DL framework are equally expressive in terms of simulation wrt addition of facts. Simulation of Individual Defeat wrt Addition of Rules
The following definition defining D ′ = ( F ′ , R ′ , < ′ ) from D is repeated from (Maher 2012). Definition 9
We add the following rules1. The facts of D ′ are the facts of D . That is, F ′ = F .2. For each rule r = B ֒ → r q in R , R ′ contains p ( r ) : B ֒ → r h ( r ) s ( r ) : h ( r ) → q and, further, for each rule r ′ = B ′ ֒ → r ′ ∼ q for ∼ q in R , R ′ contains n ( r, r ′ ) : B ′ ֒ → r ′ ¬ h ( r )
3. For every r > r ′ in D , where r and r ′ are rules for opposite literals, D ′ contains p ( r ) > ′ n ( r, r ′ ) and n ( r ′ , r ) > ′ p ( r ′ ) .It was shown in (Maher 2012) that, using this transformation, DL ( ∂ ) simulates DL ( ∂ ∗ ) and DL ( δ ) simulates DL ( δ ∗ ) , wrt addition of facts.On the surface, it might appear that this result extends readily to addition wrt rules: since theadded rules do not participate in the superiority relation of the combined theory, it might beexpected that the difference between team defeat and individual defeat is irrelevant. However,that expectation is misleading. The following example shows that this transformation does not provide a simulation of DL ( ∂ ∗ ) by DL ( ∂ ) wrt addition of rules . Example 10
Let D consist of the rules r : ⇒ pr : ⇒ ¬ p Then T ( D ) consists of the following rules p ( r ) : ⇒ h ( r ) n ( r , r ) : ⇒ ¬ h ( r ) p ( r ) : ⇒ h ( r ) n ( r , r ) : ⇒ ¬ h ( r ) s ( r ) : h ( r ) ⇒ ps ( r ) : h ( r ) ⇒ ¬ p Now, let A be the rule ⇒ p M.J. Maher
Clearly, D + A ⊢ − ∂ ∗ p (and D + A ⊢ − ∂ ∗ ¬ p ), since r cannot be overruled. However, T ( D ) + A ⊢ − ∂h ( r ) , since n ( r , r ) cannot be overruled, and hence s ( r ) fails. This leaves the rule for p in A without competition, and so T ( D ) + A ⊢ + ∂p .A similar but more complex example (given in the appendix) shows the transformation alsodoes not provide a simulation of DL ( δ ∗ ) by DL ( δ ) wrt addition of rules. These problems arisebecause if a rule body succeeds in D for each of q and ∼ q , and the rules are not overruled, thesimulation D ′ has all bodies for q and ∼ q failing. An applicable rule for q (or ∼ q ) in A thus hasa competitor in D , but not in D ′ . In this way D ′ differs from D , and the examples show thataddition of rules can make this difference observable.To avoid these problems, we add extra rules to those in Definition 9. Definition 11
We define T ( D ) as the theory ( F ′ , R ′ , > ′ ) consisting of the facts, rules and superiority statementsfrom D ′ in Definition 9, and the following.4. For each literal q , R ′ contains o ( q ) : one ( q ) ❀ q
5. For each rule r = B r ֒ → r q in R , R ′ contains B r ⇒ one ( q )
6. For every rule r for q , > ′ contains s ( r ) > ′ o ( ∼ q ) .Parts 4 and 5 of this definition introduce an additional rule for each literal ∼ q which, however,is subordinate to the methods to derive q in the original transformation in the sense that a deriva-tion of q in the original transformation will overrule (part 6) a derivation of ∼ q using part 4. Therules in part 4 are defeaters, so they cannot be used to derive any conclusions.The effect of the extended definition on Example 10 is to add the following to the transformedtheory: ⇒ one ( p ) o ( p ) : one ( p ) ❀ p ⇒ one ( ¬ p ) o ( ¬ p ) : one ( ¬ p ) ❀ ¬ ps ( r ) > o ( ¬ p ) s ( r ) > o ( p ) We now have T ( D ) + A ⊢ − ∂p , since the rule o ( ¬ p ) provides a non-failed competitor to therule in A . More generally, we find that, through the extended transformation, team defeat logicscan simulate the corresponding individual defeat logics with respect to addition of rules. Theorem 12
The logic DL ( ∂ ) can simulate DL ( ∂ ∗ ) with respect to addition of rules.The logic DL ( δ ) can simulate DL ( δ ∗ ) with respect to addition of rules. Simulation of Team Defeat wrt Addition of Rules
The same theory D and addition A as in Example 10 demonstrates that the simulation of DL ( ∂ ) by DL ( ∂ ∗ ) wrt addition of facts exhibited in (Maher 2012) does not extend to addition of rules.The transformation below modifies the one of (Maher 2012) by treating strict rules differently elative Expressiveness of Defeasible Logics II q (following Definition 11), andemploying separate defeasible rules to accommodate differences between the δ and σ inferencerules. We use a construction to restrict one class of defeasible rules to use only in simulating σ inference; it is not necessary to restrict the other class because δ ⊆ σ , by the inclusion theorem. Definition 13
We define the transformation T of D to T ( D ) = ( F ′ , R ′ , > ′ ) as follows:1. The facts of T ( D ) are the facts of D . That is, F ′ = F .2. Every strict rule of R is included in R ′ .3. For every literal q , R ′ contains str ( q ) : q → strict ( q ) nstr ( q ) : ⇒ ¬ strict ( q ) t ( q ) : strict ( q ) ⇒ true ( q ) nt ( q ) : ⇒ ¬ true ( q ) and the superiority relation contains nstr ( q ) > ′ str ( q ) and t ( q ) > ′ nt ( q ) , for every q .4. For each ordered pair of opposing rules r i = ( B i ֒ → i ∼ q ) and r j = ( B j ֒ → j q ) in R , where r j is not a defeater, R ′ contains R ij : B i ֒ → i ¬ d ( r i , r j ) R ij : B j ⇒ d ( r i , r j ) R ij : true ( q ) ⇒ d ( r i , r j ) d ( r i , r j ) ⇒ d ( r i ) f ail ( r i ) ⇒ d ( r i ) N F i : B i ⇒ ¬ f ail ( r i ) F i : ⇒ f ail ( r i ) and R ij > ′ R ij iff r j > r i , R ij > ′ R ij for every i and j , and N F i > F i for every i .If there is no strict or defeasible rule r j for q in D then only the last three rules appear in R ′ , foreach i .5. For each literal q , and each strict or defeasible rule r = ( B r ֒ → r q ) in R , R ′ contains B r ⇒ one ( q )
6. For each literal q , R ′ contains s ( q ) : one ( q ) , ¬ true ( ∼ q ) , d ( s ) , . . . , d ( s k ) ⇒ q where s , . . . , s k are the rules for ∼ q
7. For each literal q and for each strict or defeasible rule r for q , R ′ contains supp ( q ) : B r , d σ ( s , r ) , . . . , d σ ( s k , r ) , g, ¬ g ⇒ q where B r is the body of r , s , . . . , s k are the rules for ∼ q , and for every strict or defeasible rule r and opposing rule s , R ′ contains a ( s, r ) : B s ⇒ ¬ d σ ( s, r ) b ( s, r ) : B r ⇒ d σ ( s, r ) The superiority relation contains a ( s, r ) > b ( s, r ) iff s > r . R ′ also contains the rules ⇒ g ⇒ ¬ g M.J. Maher δ ⇐⇒ δ ∗ <> <>∂ ⇐⇒ ∂ ∗ Fig. 1. Relative expressiveness of logics in DL using simulation wrt addition of rules8. For each rule r = B r ֒ → r q in R , R ′ contains B r ⇒ o ( q )
9. For each literal q , R ′ contains o ( q ) : o ( q ) ❀ q and > ′ contains s ( q ) > ′ o ( ∼ q ) .Parts 1–3 allow us to characterize strict conclusions. Part 4 expresses whether a rule is defeatedor not, while part 6 expresses that q can be concluded if there is an applicable strict or defeasiblerule for q , all attempts to strictly derive ∼ q fail finitely, and all opposing rules are defeated.While this expresses properly the inference rules for ∂ and δ , the inference rule for σ omits thecondition on strict derivation of ∼ q and has a slightly different form of defeat. We need part 7 toexpress inference (and defeat) for σ . g and ¬ g are used to restrict the applicability of this rule to σ ∗ ; we have T ( D )+ A ⊢ + σ ∗ g , but T ( D )+ A ⊢ − ∂ ∗ g and T ( D )+ A ⊢ − δ ∗ g (and the same for ¬ g ). Parts 8 and 9 redress the lack of a competitor in the same way as in Definition 11. Theorem 14
The logic DL ( ∂ ∗ ) can simulate DL ( ∂ ) with respect to addition of rules.The logic DL ( δ ∗ ) can simulate DL ( δ ) with respect to addition of rules. Conclusions
We have shown that the logics of the DL framework are equally expressive when relative expres-siveness is formulated as ability to simulate in the presence of additional facts. This involved theintroduction of two new transformations simulating, respectively, a logic that blocks ambiguityand a logic that propagates ambiguity.We also completed the study of relative expressiveness wrt addition of rules. Figure 1 showsthis relation on the logics in DL , where an arrow from d to d expresses that DL ( d ) can besimulated by DL ( d ) with respect to the addition of rules. <> between tags expresses that thetwo corresponding logics have incomparable expressiveness. It is clear that DL breaks into twoclasses of logics of different expressiveness.While the issue of relative expressiveness within the framework DL is now largely resolved,this same approach can be applied to relate these logics to other logics. We can expect the sameresults for the WFDL logics (Maher and Governatori 1999; Maher et al. 2011), because of theirsimilarity to DL , but their relation to the defeasible logics of Nute and Maier (Maier and Nute 2006;Maier and Nute 2010) will be of interest. Even more interesting will be to address other systemsof defeasible reasoning, such as argumentation (Dung 1995; Rahwan and Simari 2009). Acknowledgements:
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Appendix
This appendix contains the inference rules for the logics in DL , proofs of results in the body ofthe paper, and some examples. Theorems, Lemmas, or Examples numbered 1–14 refer to itemsin the body of the paper. Larger numbers refer to items in this appendix. Inference Rules for DL For every inference rule + d there is a closely related inference rule − d allowing to infer thatsome literals q cannot be consequences of D via + d . The relationship between + d and − d isdescribed as the Principle of Strong Negation (Antoniou et al. 2000). These inference rules areplaced adjacently to emphasize this relationship.Some notation in the inference rules requires explanation. Given a literal q , its complement ∼ q is defined as follows: if q is a proposition then ∼ q is ¬ q ; if q has form ¬ p then ∼ q is p . Wesay q and ∼ q (and the rules with these literals in the head) oppose each other. R s ( R sd ) denotesthe set of strict rules (strict or defeasible rules) in R . R [ q ] ( R s [ q ] , etc) denotes the set of rules(respectively, strict rules) of R with head q . Given a rule r , A ( r ) denotes the set of literals in thebody (or antecedent) of r . +∆) +∆ q ∈ T D ( E ) iff either.1) q ∈ F ; or.2) ∃ r ∈ R s [ q ] such that.1) ∀ a ∈ A ( r ) , +∆ a ∈ E − ∆) − ∆ q ∈ T D ( E ) iff.1) q / ∈ F , and.2) ∀ r ∈ R s [ q ] .1) ∃ a ∈ A ( r ) , − ∆ a ∈ E M.J. Maher + ∂ ) + ∂q ∈ T D ( E ) iff either.1) +∆ q ∈ E ; or.2) The following three conditions all hold..1) ∃ r ∈ R sd [ q ] ∀ a ∈ A ( r ) , + ∂a ∈ E , and.2) − ∆ ∼ q ∈ E , and.3) ∀ s ∈ R [ ∼ q ] either.1) ∃ a ∈ A ( s ) , − ∂a ∈ E ; or.2) ∃ t ∈ R sd [ q ] such that.1) ∀ a ∈ A ( t ) , + ∂a ∈ E , and.2) t > s . − ∂ ) − ∂q ∈ T D ( E ) iff.1) − ∆ q ∈ E , and.2) either.1) ∀ r ∈ R sd [ q ] ∃ a ∈ A ( r ) , − ∂a ∈ E ; or.2) +∆ ∼ q ∈ E ; or.3) ∃ s ∈ R [ ∼ q ] such that.1) ∀ a ∈ A ( s ) , + ∂a ∈ E , and.2) ∀ t ∈ R sd [ q ] either.1) ∃ a ∈ A ( t ) , − ∂a ∈ E ; or.2) not ( t > s ) . + ∂ ∗ ) + ∂ ∗ q ∈ T D ( E ) iff either.1) +∆ q ∈ E ; or.2) ∃ r ∈ R sd [ q ] such that.1) ∀ a ∈ A ( r ) , + ∂ ∗ a ∈ E , and.2) − ∆ ∼ q ∈ E , and.3) ∀ s ∈ R [ ∼ q ] either.1) ∃ a ∈ A ( s ) , − ∂ ∗ a ∈ E ; or.2) r > s . − ∂ ∗ ) − ∂ ∗ q ∈ T D ( E ) iff.1) − ∆ q ∈ E , and.2) ∀ r ∈ R sd [ q ] either.1) ∃ a ∈ A ( r ) , − ∂ ∗ a ∈ E ; or.2) +∆ ∼ q ∈ E ; or.3) ∃ s ∈ R [ ∼ q ] such that.1) ∀ a ∈ A ( s ) , + ∂ ∗ a ∈ E , and.2) not ( r > s ) . + δ ) + δq ∈ T D ( E ) iff either.1) +∆ q ∈ E ; or.2) The following three conditions all hold..1) ∃ r ∈ R sd [ q ] ∀ a ∈ A ( r ) , + δa ∈ E , and.2) − ∆ ∼ q ∈ E , and.3) ∀ s ∈ R [ ∼ q ] either.1) ∃ a ∈ A ( s ) , − σa ∈ E ; or.2) ∃ t ∈ R sd [ q ] such that.1) ∀ a ∈ A ( t ) , + δa ∈ E , and.2) t > s . − δ ) − δq ∈ T D ( E ) iff.1) − ∆ q ∈ E , and.2) either.1) ∀ r ∈ R sd [ q ] ∃ a ∈ A ( r ) , − δa ∈ E ; or.2) +∆ ∼ q ∈ E ; or.3) ∃ s ∈ R [ ∼ q ] such that.1) ∀ a ∈ A ( s ) , + σa ∈ E , and.2) ∀ t ∈ R sd [ q ] either.1) ∃ a ∈ A ( t ) , − δa ∈ E ; or.2) not ( t > s ) . + σ ) + σq ∈ T D ( E ) iff either.1) +∆ q ∈ E ; or.2) ∃ r ∈ R sd [ q ] such that.1) ∀ a ∈ A ( r ) , + σa ∈ E , and.2) ∀ s ∈ R [ ∼ q ] either.1) ∃ a ∈ A ( s ) , − δa ∈ E ; or.2) not ( s > r ) . − σ ) − σq ∈ T D ( E ) iff.1) − ∆ q ∈ E , and.2) ∀ r ∈ R sd [ q ] either.1) ∃ a ∈ A ( r ) , − σa ∈ E ; or.2) ∃ s ∈ R [ ∼ q ] such that.1) ∀ a ∈ A ( s ) , + δa ∈ E , and.2) s > r . + δ ∗ ) + δ ∗ q ∈ T D ( E ) iff either.1) +∆ q ∈ E ; or.2) ∃ r ∈ R sd [ q ] such that.1) ∀ a ∈ A ( r ) , + δ ∗ a ∈ E , and.2) − ∆ ∼ q ∈ E , and.3) ∀ s ∈ R [ ∼ q ] either.1) ∃ a ∈ A ( s ) , − σ ∗ a ∈ E ; or.2) r > s . − δ ∗ ) − δ ∗ q ∈ T D ( E ) iff.1) − ∆ q ∈ E , and.2) ∀ r ∈ R sd [ q ] either.1) ∃ a ∈ A ( r ) , − δ ∗ a ∈ E ; or.2) +∆ ∼ q ∈ E ; or.3) ∃ s ∈ R [ ∼ q ] such that.1) ∀ a ∈ A ( s ) , + σ ∗ a ∈ E , and.2) not ( r > s ) . elative Expressiveness of Defeasible Logics II + σ ∗ ) + σ ∗ q ∈ T D ( E ) iff either.1) +∆ q ∈ E ; or.2) ∃ r ∈ R sd [ q ] such that.1) ∀ a ∈ A ( r ) , + σ ∗ a ∈ E , and.2) ∀ s ∈ R [ ∼ q ] either.1) ∃ a ∈ A ( s ) , − δ ∗ a ∈ E ; or.2) not ( s > r ) . − σ ∗ ) − σ ∗ q ∈ T D ( E ) iff.1) − ∆ q ∈ E , and.2) ∀ r ∈ R sd [ q ] either.1) ∃ a ∈ A ( r ) , − σ ∗ a ∈ E ; or.2) ∃ s ∈ R [ ∼ q ] such that.1) ∀ a ∈ A ( s ) , + δ ∗ a ∈ E , and.2) s > r . Proofs of results
We now turn to proofs of results in the body of the paper, and some examples. This part of theappendix has the same structure as the paper itself, to make access easier.All simulation proofs (of DL ( d ) by DL ( d ) , say) have two parts: first we show every conse-quence of D + A in DL ( d ) has a corresponding consequence of T ( D )+ A in DL ( d ) , and thenwe show that every consequence of T ( D )+ A in DL ( d ) in the language of D + A has a corre-sponding consequence of D + A in DL ( d ) . In both cases the proof is by induction on the level n of T ↑ n where T combines the functions in the inference rules for ± d and ± ∆ for D + A in the first part, and combines the functions in the inference rules for ± d and ± ∆ for T ( D )+ A in the second part. The induction hypothesis for the first part is: for k ≤ n , if α ∈ T D + A ↑ n then T ( D )+ A ⊢ α ′ , where α ′ is the counterpart, in DL ( d ) , of α . For the second part it is: for k ≤ n , if α ∈ Σ and α ∈ T T ( D )+ A ↑ n then D + A ⊢ α ′ , where α ′ is the counterpart, in DL ( d ) ,of α . Since T P ↑ ∅ the induction hypothesis is always valid for n = 0 .Throughout this appendix, if r is a rule then B r refers to the body of that rule. For brevity, wewrite + dB , where B is a set of literals, to mean { + dq | q ∈ B } . Blocked Ambiguity Simulates Propagated Ambiguity
The facts and strict rules of D + A and T ( D )+ A are the same, except for rules for strict ( q ) in T ( D )+ A . However strict ( q ) is not used in any other strict rule. Consequently, for any addi-tion A , D + A and T ( D )+ A draw the same strict conclusions in Σ( D + A ) . Furthermore, theseconclusions are reflected in the defeasible conclusions of strict ( q ) . Lemma 15
Let A be any defeasible theory, and let Σ be the language of D + A . Then, for every q ∈ Σ , • D + A ⊢ +∆ q iff T ( D )+ A ⊢ +∆ q iff T ( D )+ A ⊢ + ∂ ∗ strict ( q ) iff T ( D )+ A ⊢ − ∂ ∗ ¬ strict ( q ) • D + A ⊢ − ∆ q iff T ( D )+ A ⊢ − ∆ q iff T ( D )+ A ⊢ − ∂ ∗ strict ( q ) iff T ( D )+ A ⊢ + ∂ ∗ ¬ strict ( q ) Proof
The proof of D + A ⊢ ± ∆ q iff T ( D )+ A ⊢ ± ∆ q is straightforward, by induction on length ofproofs.In the inference rule for + ∂ ∗ strict ( q ) , clause . . must be false, by the structure of the rulesin part 3 of the transformation. Consequently, we infer + ∂ ∗ strict ( q ) iff we infer +∆ strict ( q ) ,which happens iff we infer +∆ q since there is only the one rule for strict ( q ) . Similarly, clause . . of the inference rule for − ∂ ∗ strict ( q ) is true, so we infer − ∂ ∗ strict ( q ) iff we infer − ∆ strict ( q ) ,which happens iff we infer − ∆ q since there is only the one rule for strict ( q ) .6 M.J. Maher
In the inference rule for − ∂ ∗ ¬ strict ( q ) , clause . . is false because the body of nstr ( q ) isempty, and clause . . is false because nstr ( q ) > ′ str ( q ) . Thus we infer − ∂ ∗ ¬ strict ( q ) iff weinfer +∆ strict ( q ) . Finally, in the inference rule for + ∂ ∗ ¬ strict ( q ) , clause . is false, becausethere is no fact or strict rule for ¬ strict ( q ) . and clauses . . and . . are true (the latter because nstr ( q ) > ′ str ( q ) ). Thus, we can infer + ∂ ∗ ¬ strict ( q ) iff we can infer − ∆ strict ( q ) .This lemma establishes that strict provability ( ± ∆ ) from D + A in DL ( δ ∗ ) is captured in DL ( ∂ ∗ ) by the transformation defined above, no matter what the addition A . We now show that DL ( ∂ ∗ ) can simulate the behaviour of both δ ∗ and σ ∗ with respect to addition of facts. Lemma 16
Let D be a defeasible theory, T ( D ) be the transformed defeasible theory as described in Def-inition 3, and let A be a modular set of facts. Let Σ be the language of D + A and let q ∈ Σ .Then • D + A ⊢ + σ ∗ q iff T ( D )+ A ⊢ + ∂ ∗ supp ( q ) • D + A ⊢ − σ ∗ q iff T ( D )+ A ⊢ − ∂ ∗ supp ( q ) • D + A ⊢ + δ ∗ q iff T ( D )+ A ⊢ + ∂ ∗ q • D + A ⊢ − δ ∗ q iff T ( D )+ A ⊢ − ∂ ∗ q Proof
Suppose + σ ∗ q ∈ T D + A ↑ ( n +1) . Then, by the + σ ∗ inference rule, there is a strict or defeasiblerule r in D with head q and body B r such that + σ ∗ B r ⊆ T D + A ↑ n , and for every rule s in D for ∼ q either there is a literal b in the body of s such that − δ ∗ b ∈ T D + A ↑ n or s > r . Hence, by theinduction hypothesis, there is a strict or defeasible rule r in D with head q and body B r such that T ( D )+ A ⊢ + ∂ ∗ supp ( b ) for each b ∈ B r and for every rule s in D for ∼ q either there is a literal b in the body of s such that T ( D )+ A ⊢ − ∂ ∗ b or s > r . Then T ( D )+ A ⊢ + ∂ ∗ supp body ( r ) and for for every rule s in D for ∼ q with s > r T ( D )+ A ⊢ − ∂ ∗ B s , and hence T ( D )+ A ⊢ + ∂ ∗ ¬ o ( r ) . Combining these two conclusions, and given that there is no rule for ¬ supp ( q ) , wehave T ( D )+ A ⊢ + ∂ ∗ supp ( q ) .Suppose + δ ∗ q ∈ T D + A ↑ ( n +1) . Then, by the + δ ∗ inference rule, there is a strict or defeasi-ble rule r in D with head q and body B r such that + δ ∗ B r ⊆ T D + A ↑ n , − ∆ ∼ q ∈ T D + A ↑ n ,and for every rule s in D for ∼ q where r > s , there is a literal b in the body of s such that − σ ∗ b ∈ T D + A ↑ n . Hence, by the induction hypothesis, there is a strict or defeasible rule r in D with head q and body B r such that T ( D )+ A ⊢ + ∂ ∗ B r , T ( D )+ A ⊢ − ∆ ∼ q , andfor every rule s in D for ∼ q where r > s , there is a literal b in the body of s such that T ( D )+ A ⊢ − ∂ ∗ supp ( b ) . By Lemma 15, T ( D )+ A ⊢ − ∂ ∗ ¬ strict ( ∼ q ) . By repeated appli-cation of the − ∂ ∗ inference rule we have T ( D )+ A ⊢ − ∂ ∗ supp body ( s ) for each s , and then T ( D )+ A ⊢ + ∂ ∗ ¬ comp ( r ) . Thus the body of the rule inf ( r ) in T ( D ) holds defeasibly. Onthe other hand, for every rule s for ∼ q in D where r > s there is a literal b in the body of s such that T ( D )+ A ⊢ − ∂ ∗ supp ( b ) so, using the inference rule for − ∂ ∗ and the rule from part4 we must have T ( D )+ A ⊢ − ∂ ∗ b . T ( D )+ A ⊢ + ∂ ∗ B r so, using the rules in part 4 and part 5, T ( D )+ A ⊢ + ∂ ∗ supp body ( r ) . Hence, for the rules for ∼ q where r > s , the rules n d ( s, r ) canbe applied and T ( D )+ A ⊢ − ∂ ∗ ¬ comp ( s ) . Consequently, all rules inf ( s ) for ∼ q fail. From thisfact and the fact that body of rule inf ( r ) is proved defeasibly we conclude T ( D )+ A ⊢ + ∂ ∗ q .Suppose − σ ∗ q ∈ T D + A ↑ ( n +1) . Then, by the − σ ∗ inference rule, − ∆ q ∈ T D + A ↑ n and,for every strict or defeasible rule r in D with head q and body B r , either − σ ∗ b ∈ T D + A ↑ n for elative Expressiveness of Defeasible Logics II b ∈ B r , or there is a rule s in D for ∼ q with body B s such that + δ ∗ B s ⊆ T D + A ↑ n and s > r . Hence, by the induction hypothesis, T ( D )+ A ⊢ − ∆ q and for every strict or defeasiblerule r in D with head q either T ( D )+ A ⊢ − ∂ ∗ supp ( b ) for some b ∈ B r , or there is a rule s in D for ∼ q with s > r and T ( D )+ A ⊢ + ∂ ∗ B s . Hence, either T ( D )+ A ⊢ − ∂ ∗ supp body ( r ) or T ( D )+ A ⊢ − ∂ ∗ ¬ o ( r ) . In either case, we have T ( D )+ A ⊢ − ∂ ∗ supp ( q ) .Suppose − δ ∗ q ∈ T D + A ↑ ( n + 1) . Then, by the − δ ∗ inference rule, − ∆ q ∈ T D + A ↑ n or,for every strict or defeasible rule r in D with head q and body B r , either − δ ∗ b ∈ T D + A ↑ n for some b ∈ B r , +∆ ∼ q ∈ T D + A ↑ n , or there is a rule s in D for ∼ q with body B s suchthat + σ ∗ B s ⊆ T D + A ↑ n and r > s . Hence, by the induction hypothesis, T ( D )+ A ⊢ − ∆ q and for every strict or defeasible rule r in D with head q either (1) T ( D )+ A ⊢ − ∂ ∗ b for some b ∈ B r , (2) T ( D )+ A ⊢ +∆ ∼ q , or (3) there is a rule s in D for ∼ q with r > s and T ( D )+ A ⊢ + ∂ ∗ supp ( b ′ ) for every b ′ ∈ B s . We consider these three cases in turn. In the first case, the rule inf ( r ) fails. In the second case, using part 3, we can conclude T ( D )+ A ⊢ +∆ strict ( ∼ q ) and T ( D )+ A ⊢ − ∂ ∗ ¬ strict ( ∼ q ) , and hence the rule inf ( r ) fails. In the third case, we can conclude T ( D )+ A ⊢ + ∂ ∗ supp body ( s ) and hence, using part 6, T ( D )+ A ⊢ − ∂ ∗ ¬ comp ( r ) . Thus, therule inf ( r ) fails. In each case, the rule inf ( r ) fails. Thus we can derive T ( D )+ A ⊢ − ∂ ∗ q .Suppose + ∂ ∗ supp ( q ) ∈ T T ( D )+ A ↑ ( n +1) . Then, by the + ∂ ∗ inference rule, either + ∂ ∗ q ∈T T ( D )+ A ↑ n , or there is a strict or defeasible rule r in D with head q and body B r such that + ∂ ∗ supp body ( r ) ∈ T T ( D )+ A ↑ n and + ∂ ∗ ¬ o ( r ) ∈ T T ( D )+ A ↑ n . Consequently, + ∂ ∗ supp ( b ) ∈T T ( D )+ A ↑ n , for each b ∈ B r for every rule s in D for ∼ q where s > r , there is b in the body of s such that − ∂ ∗ b ∈ T T ( D )+ A ↑ n . In the first case, by the induction hypothesis, D + A ⊢ + δ ∗ q and then, by the inclusion theorem, D + A ⊢ + σ ∗ q . In the second case, by the induction hypoth-esis, D + A ⊢ + σ ∗ B r for every rule s in D for ∼ q where s > r , there is b in the body of s suchthat D + A ⊢ − δ ∗ b . Applying the inference rule for + σ ∗ , D + A ⊢ + σ ∗ q .Suppose − ∂ ∗ supp ( q ) ∈ T T ( D )+ A ↑ ( n + 1) . Then, by the − ∂ ∗ inference rule, − ∂ ∗ q ∈T T ( D )+ A ↑ n , and for every strict or defeasible rule r in D for q either − ∂ ∗ supp body ( r ) ∈T T ( D )+ A ↑ n or − ∂ ∗ ¬ o ( r ) ∈ T T ( D )+ A ↑ n . In the former case we must have − ∂ ∗ supp ( b ) ∈T T ( D )+ A ↑ n for some b in the body B r of r . In the latter case we must have that for some rule s in D with body B s , s > r and + ∂ ∗ B s ⊆ T T ( D )+ A ↑ n . By the induction hypothesis, we have D + A ⊢ − δ ∗ q (and hence D + A ⊢ − ∆ q ) and, for each r either D + A ⊢ − σ ∗ b for some b ∈ B r or there is an opposing rule s with s > r and D + A ⊢ + δ ∗ B s . Applying the inference rule for − σ ∗ we conclude D + A ⊢ − σ ∗ q .Suppose + ∂ ∗ q ∈ T T ( D )+ A ↑ ( n +1) . Then, by the + ∂ ∗ inference rule, there is a strict or defea-sible rule r in D with head q and body B r such that + ∂ ∗ B r ⊆ T T ( D )+ A ↑ n , + ∂ ∗ ¬ strict ( ∼ q ) ∈T T ( D )+ A ↑ n , and + ∂ ∗ ¬ comp ( r ) ∈ T T ( D )+ A ↑ n . By Lemma 15, D + A ⊢ − ∆ ∼ q . Using thestructure of T ( D ) and the + ∂ ∗ inference rule, for every rule s in D for ∼ q where r > s we musthave − ∂ ∗ supp body ( s ) ∈ T T ( D )+ A ↑ n , and hence − ∂ ∗ supp ( b ) ∈ T T ( D )+ A ↑ n , for some b in the body of s . By the induction hypothesis, D + A ⊢ + δ ∗ B r and for every rule s in D for ∼ q where r > s , there is b in the body of s such that D + A ⊢ − σ ∗ b . Now, applying the + δ ∗ inference rule, we have D + A ⊢ + δ ∗ q .Suppose − ∂ ∗ q ∈ T T ( D )+ A ↑ ( n + 1) . Then, by the − ∂ ∗ inference rule, − ∆ q ∈ T T ( D )+ A ↑ n and, for every strict or defeasible rule r for q in D with body B r , either (1) − ∂ ∗ b ∈ T T ( D )+ A ↑ n for some b ∈ B r , (2) − ∂ ∗ ¬ comp ( r ) ∈ T T ( D )+ A ↑ n , (3) − ∂ ∗ ¬ strict ( ∼ q ) ∈ T T ( D )+ A ↑ n , or,(4) for some rule s for ∼ q in D , + ∂ ∗ B s ⊆ T T ( D )+ A ↑ n , + ∂ ∗ ¬ comp ( s ) ∈ T T ( D )+ A ↑ n , and + ∂ ∗ ¬ strict ( q ) ∈ T T ( D )+ A ↑ n .Hence, using the structure of T ( D ) and Lemma 15, − ∆ q ∈ T T ( D )+ A ↑ n and, for every rule8 M.J. Maher r for q in D with body B r , either (1) − ∂ ∗ b ∈ T T ( D )+ A ↑ n for some b ∈ B r , (2) for some rule s ′ for ∼ q in D we have + ∂ ∗ supp ( b ) ∈ T T ( D )+ A ↑ n for each b ∈ B s ′ , (3) +∆ ∼ q ∈ T T ( D )+ A ↑ n ,or, (4) for some rule s for ∼ q in D , + ∂ ∗ B s ⊆ T T ( D )+ A ↑ n , for every rule r ′ for q , there is b ′ inits body such that − ∂ ∗ supp ( b ′ ) ∈ T T ( D )+ A ↑ n , and − ∆ q ∈ T T ( D )+ A ↑ n .By the induction hypothesis, D + A ⊢ − ∆ q and, for every strict or defeasible rule r for q in D with body B r , either (1) D + A ⊢ − δb for some b ∈ B r , (2) for some rule s for ∼ q in D we have D + A ⊢ + σ ∗ b for each b ∈ B s , (3) D + A ⊢ +∆ ∼ q , or (4) for some rule s for ∼ q in D , D + A ⊢ + δ ∗ B s , for every rule r ′ for q , there is b ′ in its body such that D + A ⊢ − σ ∗ b ′ ,and D + A ⊢ − ∆ q . For each disjunct, applying the inference rule for − δ ∗ , we can conclude D + A ⊢ − δ ∗ q .This result concerns only addition of facts. It was established in (Maher 2012) that it cannotbe extended to addition of rules.Given that the ambiguity blocking logics can simulate each other, as can the ambiguity prop-agating logics (see (Maher 2012)) we have Theorem 17
The ambiguity blocking logics ( DL ( ∂ ) and DL ( ∂ ∗ ) ) can simulate the ambiguity propagatinglogics ( DL ( δ ) and DL ( δ ∗ ) ) with respect to addition of facts.This is Theorem 5 from the body of the paper. Propagated Ambiguity Simulates Blocked Ambiguity
As with the previous simulation, the facts and strict rules of D and T ( D ) are the same, except forrules for strict ( q ) in T ( D ) . Thus, again, for any addition A , D + A and T ( D )+ A draw the samestrict conclusions in Σ( D + A ) . Furthermore, these conclusions are reflected in the defeasibleconclusions of strict ( q ) , true ( q ) and ¬ true ( q ) , and also in support conclusions. Lemma 18
Let D be a defeasible theory, T ( D ) be the transformed defeasible theory as described in Defini-tion 6, and let A be a modular defeasible theory. Let Σ be the language of D + A and let q ∈ Σ .Then • D + A ⊢ +∆ q iff T ( D )+ A ⊢ +∆ q iff T ( D )+ A ⊢ + δ ∗ strict ( q ) iff T ( D )+ A ⊢ + δ ∗ true ( q ) iff T ( D )+ A ⊢ + σ ∗ true ( q ) iff T ( D )+ A ⊢ − δ ∗ ¬ true ( q ) iff T ( D )+ A ⊢ − σ ∗ ¬ true ( q ) • D + A ⊢ − ∆ q iff T ( D )+ A ⊢ − ∆ q iff T ( D )+ A ⊢ − δ ∗ strict ( q ) iff T ( D )+ A ⊢ − δ ∗ true ( q ) iff T ( D )+ A ⊢ − σ ∗ true ( q ) iff T ( D )+ A ⊢ + δ ∗ ¬ true ( q ) iff T ( D )+ A ⊢ + σ ∗ ¬ true ( q ) Proof
The proof of D + A ⊢ ± ∆ q iff T ( D )+ A ⊢ ± ∆ q is straightforward, by induction on length ofproofs.In the inference rule for + δ ∗ strict ( q ) , clause . . must be false, by the structure of the rulesin part 3 of the transformation. Consequently, we infer + δ ∗ strict ( q ) iff we infer +∆ strict ( q ) , elative Expressiveness of Defeasible Logics II +∆ q since there is only the one rule for strict ( q ) . Similarly, clause . . of the inference rule for − δ ∗ strict ( q ) is true, so we infer − δ ∗ strict ( q ) iff we infer − ∆ strict ( q ) ,which happens iff we infer − ∆ q since there is only the one rule for strict ( q ) .Note that − ∆ true ( q ) and − ∆ ¬ true ( q ) are consequences of T ( D )+ A because there are nostrict rules for such literals in T ( D )+ A . Using this fact, the two rules t ( q ) and nt ( q ) and thesuperiority t ( q ) > nt ( q ) , using the inference rule for + δ ∗ , we can infer + δ ∗ ¬ true ( q ) iff we caninfer − σ ∗ strict ( q ) , because . of the inference rule is false, . . and . . are true, and . . . is false. Similarly, using the inference rule for + σ ∗ , we can infer + σ ∗ ¬ true ( q ) iff we can infer − δ ∗ strict ( q ) . Using the inference rules for − δ ∗ and − σ ∗ , we can infer − δ ∗ ¬ true ( q ) iff we caninfer + σ ∗ strict ( q ) , and we can infer − σ ∗ ¬ true ( q ) iff we can infer + δ ∗ strict ( q ) .We need this more detailed characterization of strict consequence, compared to Lemma 15,because both δ ∗ and σ ∗ intermediate conclusions influence δ ∗ conclusions.The next lemma is a key part of the proof. It shows that the structure of T ( D )+ A tightly con-strains the inferences that can be made in the sense that, for the literals of interest, the inferencerules δ ∗ and σ ∗ draw the same conclusions. Lemma 19
Let D be a defeasible theory, T ( D ) be the transformed defeasible theory as described in Defini-tion 6, and let A be a modular set of facts. Let Σ be the language of D + A extended with literalsof the forms undefeated ( p ) , ¬ undefeated ( p ) and ¬ true ( p ) , for p ∈ Σ( D ) .Then, for any q ∈ Σ , • T ( D )+ A ⊢ + δ ∗ q iff T ( D )+ A ⊢ + σ ∗ q • T ( D )+ A ⊢ − δ ∗ q iff T ( D )+ A ⊢ − σ ∗ q Proof
Two parts of the proof follow immediately from the inclusion theorem. These are the forwarddirection of the first statement and the backward direction of the second statement. Furthermore,it is immediate from Lemma 18 that the result holds for literals involving true and for literalsthat are proved strictly. The remaining parts are proved by induction.Recall that T ( D )+ A ⊢ s iff there is an integer n such that s ∈ T T ( D )+ A ↑ n . Note that theresult holds in T T ( D )+ A ↑ , since it is empty. Suppose the result holds for conclusions s with s ∈ T T ( D )+ A ↑ n . We show that it also holds for conclusions in T T ( D )+ A ↑ ( n + 1) .(1) If + σ ∗ q ∈ T T ( D )+ A ↑ ( n + 1) then + σ ∗ undefeated ( q ) ∈ T T ( D )+ A ↑ n , because thereis only one rule for q and it cannot be overruled. Further, if + σ ∗ undefeated ( q ) ∈ T T ( D )+ A ↑ n then for some rule r of D we must have + σ ∗ B r ⊆ T T ( D )+ A ↑ n and + σ ∗ ¬ true ( ∼ q ) ∈T T ( D )+ A ↑ n and, for every rule s in D for ∼ q with r > s , there is p ∈ B s with − δ ∗ p ∈T T ( D )+ A ↑ n , because clause . . . must be false, since n d ( r, s ) > p d ( r ) for every such s .By the induction hypothesis, + δ ∗ B r ⊆ T T ( D )+ A ↑ n , and for each s there is p ∈ B s with − σ ∗ p ∈ T T ( D )+ A ↑ n and, by Lemma 18, + δ ∗ ¬ true ( ∼ q ) and − ∆ ∼ q are consequences of T ( D )+ A . Applying the + δ ∗ inference rule, T ( D )+ A ⊢ + δ ∗ undefeated ( q ) and, applying the − σ ∗ inference rule, T ( D )+ A ⊢ − σ ∗ undefeated ( ∼ q ) since every rule p d ( s ) contains a p with − σ ∗ p ∈ T T ( D )+ A ↑ n . Hence, applying the + δ ∗ inference rule, T ( D )+ A ⊢ + δ ∗ q .If + σ ∗ ¬ undefeated ( q ) ∈ T T ( D )+ A ↑ ( n + 1) then, for some rule s for ∼ q in D , + σ ∗ B s ⊆T T ( D )+ A ↑ n . By the induction hypothesis, + δ ∗ B s ⊆ T T ( D )+ A ↑ n . Applying the + δ ∗ inferencerule, noting that there is no fact or strict rule for undefeated ( q ) and that n d ( r, s ) > p d ( r ) , wehave T ( D )+ A ⊢ + δ ∗ ¬ undefeated ( q ) .0 M.J. Maher (2) If − δ ∗ q ∈ T T ( D )+ A ↑ ( n + 1) then, using the − δ ∗ inference rule and the structure of T ( D )+ A , − ∆ q ∈ T T ( D )+ A ↑ n and either − δ ∗ undefeated ( q ) ∈ T T ( D )+ A ↑ n or +∆ ∼ q ∈T T ( D )+ A ↑ n or + σ ∗ undefeated ( ∼ q ) ∈ T T ( D )+ A ↑ n .If − δ ∗ undefeated ( q ) ∈ T T ( D )+ A ↑ ( n + 1) then either for every rule p d ( r ) , for some p ∈ B r , − δ ∗ p ∈ T T ( D )+ A ↑ n or − δ ∗ ¬ true ( ∼ q ) ∈ T T ( D )+ A ↑ n , or, for some rule n d ( r, s ) , + σ ∗ B s . Bythe induction hypothesis, either for each p d ( r ) there is a p in its body where T ( D )+ A ⊢ − σ ∗ p ,or T ( D )+ A ⊢ + δ ∗ B s for some s . Applying the − σ ∗ , we have T ( D )+ A ⊢ − σ ∗ undefeated ( q ) .If +∆ ∼ q ∈ T T ( D )+ A ↑ n then, by Lemma 18, T ( D )+ A ⊢ − σ ∗ ¬ true ( ∼ q ) . Hence we musthave T ( D )+ A ⊢ − σ ∗ undefeated ( q ) , since ¬ true ( ∼ q ) appears in each rule for undefeated ( q ) .If + σ ∗ undefeated ( ∼ q ) ∈ T T ( D )+ A ↑ n then there is a rule p d ( s ) for undefeated ( ∼ q ) where T ( D )+ A ⊢ + σ ∗ B s and T ( D )+ A ⊢ + σ ∗ ¬ true ( q ) and, for every rule n d ( s, r ) , T ( D )+ A ⊢− δ ∗ B r . By the induction hypothesis, T ( D )+ A ⊢ + δ ∗ B s and, for every rule n d ( s, r ) (wherewe must have s > r in D ), T ( D )+ A ⊢ − δ ∗ B r . Hence, for every r for q in D where r > s we have T ( D )+ A ⊢ − σ ∗ B r . For every other r for q in D there is n d ( r, s ) where T ( D )+ A ⊢ + δ ∗ B s . Hence, applying the − σ ∗ inference rule for undefeated ( q ) , we must have T ( D )+ A ⊢− σ ∗ undefeated ( q ) .Thus, in every case we have T ( D )+ A ⊢ − σ ∗ undefeated ( q ) and consequently T ( D )+ A ⊢− σ ∗ q .If − δ ∗ ¬ undefeated ( q ) ∈ T T ( D )+ A ↑ ( n + 1) then for every rule p d ( r ) , for some p in its body, − δ ∗ p ∈ T T ( D )+ A ↑ n . By the induction hypothesis, for every rule p d ( r ) , for some p in its body, T ( D )+ A ⊢ − σ ∗ p . Applying the − σ ∗ inference rule, T ( D )+ A ⊢ − σ ∗ ¬ undefeated ( q ) .As a consequence of the inclusion theorem and the previous lemma, any inference rule between σ ∗ and δ ∗ (that is, any inference rule except for ∆ and δ ) behaves the same way on Σ -literals in T ( D )+ A . In particular, it applies to ∂ ∗ . Corollary 20
Let Σ be the language of D , Σ ′ be as defined in the previous lemma. Let A be any set of facts.Then if q ∈ Σ ′ • T ( D )+ A ⊢ + δ ∗ q iff T ( D )+ A ⊢ + ∂ ∗ q • T ( D )+ A ⊢ − δ ∗ q iff T ( D )+ A ⊢ − ∂ ∗ q Now we show that the transformation preserves the ∂ ∗ consequences of D + A . Theorem 21
Let D be a defeasible theory, T ( D ) be the transformed defeasible theory as described in Def-inition 6, and let A be a modular set of facts. Let Σ be the language of D + A and let q ∈ Σ .Then • D + A ⊢ + ∂ ∗ q iff T ( D )+ A ⊢ + ∂ ∗ q • D + A ⊢ − ∂ ∗ q iff T ( D )+ A ⊢ − ∂ ∗ q Proof
Suppose + ∂ ∗ q ∈ T D + A ↑ ( n +1) . Then, by the + ∂ ∗ inference rule, either +∆ q ∈ T D + A ↑ n (in which case, we must have T ( D ) + A ⊢ + δ ∗ ) or +∆ q / ∈ T D + A ↑ n and there is a strictor defeasible rule r in D with head q and body B r such that + ∂ ∗ B r ⊆ T D + A ↑ n , − ∆ ∼ q ∈T D + A ↑ n , and for every rule s in D for ∼ q either there is a literal b in the body of s such that elative Expressiveness of Defeasible Logics II − ∂ ∗ b ∈ T D + A ↑ n or r > s . Hence, in the latter case, by the induction hypothesis, there isa strict or defeasible rule r in D + A with head q and body B r such that T ( D )+ A ⊢ + ∂ ∗ B r , T ( D )+ A ⊢ − ∆ ∼ q , and for every rule s in D + A for ∼ q either T ( D )+ A ⊢ − ∂ ∗ B s or r > s .From this statement we derive several facts. (1) By Lemma 18 and the inclusion theorem, T ( D )+ A ⊢ + ∂ ∗ ¬ true ( ∼ q ) . (2) Thus, T ( D )+ A ⊢ + ∂ ∗ ( B r , ¬ true ( ∼ q )) and, for every rule n d ( r, s ) in T ( D ) , T ( D )+ A ⊢ − − ∂ ∗ B s (since rules s where r > s do not give rise to a rule n d ( r, s ) ). Hence, T ( D )+ A ⊢ + ∂ ∗ undefeated ( q ) . (3) Conversely, T ( D )+ A ⊢ − ∂ ∗ undefeated ( ∼ q ) because, for every rule p d ( s ) for undefeated ( ∼ q ) , either T ( D )+ A ⊢ − ∂ ∗ B s or there is a rule n d ( s, r ) superior to p d ( s ) with T ( D )+ A ⊢ + ∂ ∗ B r . Consequently, since the only rule in T ( D ) for q has body undefeated ( q ) (and similarly for ∼ q ), applying the + ∂ ∗ inference rule, we have T ( D )+ A ⊢ + ∂ ∗ q .Suppose − ∂ ∗ q ∈ T D + A ↑ ( n + 1) . Then, by the − ∂ ∗ inference rule, − ∆ q ∈ T D + A ↑ n and,for every strict or defeasible rule r in D with head q and body B r , either − ∂ ∗ b ⊆ T D + A ↑ n for some b ∈ B r , +∆ ∼ q ∈ T D + A ↑ n , or there is a rule s in D for ∼ q with body B s suchthat + ∂B s ⊆ T D + A ↑ n and r > s . Hence, by the induction hypothesis, T ( D )+ A ⊢ − ∆ q and for every strict or defeasible rule r in D with head q either T ( D )+ A ⊢ − ∂ ∗ b for some b ∈ B r , T ( D )+ A ⊢ +∆ ∼ q , or there is a rule s in D for ∼ q where T ( D )+ A ⊢ + ∂ ∗ B s and r > s . Hence, for every rule p d ( r ) in T ( D ) for undefeated ( q ) either T ( D )+ A ⊢ − δ ∗ b for some b ∈ B r , or T ( D )+ A ⊢ − δ ∗ ¬ strict ( ∼ q ) (by Lemma 18), or there is a rule n d ( r, s ) where T ( D )+ A ⊢ + σ ∗ B s . Applying the inference rule for − δ ∗ undefeated ( q ) , we conclude T ( D )+ A ⊢ − δ ∗ undefeated ( q ) and, hence, T ( D )+ A ⊢ − δ ∗ q .Suppose + ∂ ∗ q ∈ T T ( D )+ A ↑ ( n +1) . Then, by the + ∂ ∗ inference rule and using the structureof T ( D ) , either +∆ q ∈ T T ( D )+ A ↑ n , or + ∂ ∗ undefeated ( q ) ∈ T T ( D )+ A ↑ n , − ∆ ∼ q ∈T T ( D )+ A ↑ n , and − ∂ ∗ undefeated ( ∼ q ) ∈ T D + A ↑ n . In the first case we have D + A ⊢ +∆ q and thus D + A ⊢ + ∂ ∗ q , Alternatively, there is a strict or defeasible rule r in D with head q andbody B r such that + ∂ ∗ B r ⊆ T T ( D )+ A ↑ n , + ∂ ∗ ¬ true ( ∼ q ) ∈ T T ( D )+ A ↑ n , and for every rule s in D for ∼ q where r > s there is a literal b in the body B s of s such that − ∂ ∗ b ∈ T T ( D )+ A ↑ n .By the induction hypothesis and Lemma 18, D + A ⊢ + ∂ ∗ B r , D + A ⊢ − ∆ ∼ q , and for everyrule s in D for ∼ q where r > s there is b in the body of s such that D + A ⊢ − ∂b . Applying the + ∂ inference rule, we conclude D + A ⊢ + ∂ ∗ q .Suppose − ∂ ∗ q ∈ T T ( D )+ A ↑ ( n +1) . Then, by the − ∂ ∗ inference rule and using the struc-ture of T ( D ) , − ∆ q ∈ T T ( D )+ A ↑ n and either (1) − ∂ undefeated ( q ) ∈ T T ( D )+ A ↑ n . or(2) +∆ ∼ q ∈ T T ( D )+ A ↑ n , or (3) + ∂ undefeated ( ∼ q ) ∈ T T ( D )+ A ↑ n . By Lemma 18 andCorollary 20 we have D + A ⊢ − ∆ q and D + A ⊢ + ∂ ∗ ¬ true ( q ) .In the first case, for each rule r for q in D either there is a literal p ∈ B r and − ∂p ∈ T T ( D )+ A ↑ n or − ∂ ∗ ¬ true ( ∼ q ) ∈ T T ( D )+ A ↑ n or for some rule s for ∼ q in D where r > s , + ∂ ∗ B s ⊆T T ( D )+ A ↑ n . By the induction hypothesis (and Lemma 18 and Corollary 20), for each rule r for q in D either there is a literal p ∈ B r and D + A ⊢ − ∂ ∗ p , or D + A ⊢ +∆ ∼ q , or for some rule s for ∼ q in D where r > s , D + A ⊢ + ∂ ∗ B s . Applying the − ∂ inference rule, D + A ⊢ − ∂ ∗ q .In the second case, by Lemma 18 and Corollary 20, D + A ⊢ +∆ ∼ q . Consequently, applyingthe − ∂ inference rule, D + A ⊢ − ∂ ∗ q . In the third case, for some rule s for ∼ q in D , + ∂ ∗ B s ⊆T T ( D )+ A ↑ n and, for all rules r for q in D where s > r , for some p ∈ B r , − ∂ ∗ p ∈ T T ( D )+ A ↑ n . By the induction hypothesis, for every rule r for q in D where r > s , for some p ∈ B r , D + A ⊢ − ∂ ∗ p , and D + A ⊢ + ∂ ∗ B s . Applying the − ∂ inference rule, D + A ⊢ − ∂ ∗ q .2 M.J. Maher
Combining Theorem 21 with Lemma 19 and the inclusion theorem, we see that DL ( ∂ ∗ ) canbe simulated by DL ( δ ∗ ) and DL ( δ ) . Theorem 22
For d ∈ { δ, δ ∗ , ∂ } , DL ( d ) can simulate DL ( ∂ ∗ ) with respect to addition of facts Proof D + A ⊢ + ∂ ∗ q iff T ( D )+ A ⊢ + ∂ ∗ q (by Theorem 21) iff T ( D )+ A ⊢ + δ ∗ q (by Corollary 20)iff T ( D )+ A ⊢ + dq (by Lemma 19 and the inclusion theorem). The proof is similar for − ∂ ∗ q .This is Theorem 8 from the body of the paper. Simulation of Individual Defeat wrt Addition of Rules
Example 10 does not apply to DL ( δ ) and DL ( δ ∗ ) . We have T ( D ) + A ⊢ + σh ( r ) and, conse-quently, T ( D ) + A ⊢ − δp , in agreement with D under DL ( δ ∗ ) . The weaker inference strengthof ambiguity propagation masks the distinction that is present for blocked ambiguity reasoning.However, the next example shows that the transformation does not provide a simulation wrt rulesfor the propagating ambiguity logics. Example 23
Let D consist of the rules r : ⇒ pr : ⇒ ¬ pr : ⇒ pr : ⇒ ¬ p with r > r and r > r .Then T ( D ) consists of the following rules p ( r ) : ⇒ h ( r ) n ( r , r ) : ⇒ ¬ h ( r ) n ( r , r ) : ⇒ ¬ h ( r ) p ( r ) : ⇒ h ( r ) n ( r , r ) : ⇒ ¬ h ( r ) n ( r , r ) : ⇒ ¬ h ( r ) p ( r ) : ⇒ h ( r ) n ( r , r ) : ⇒ ¬ h ( r ) n ( r , r ) : ⇒ ¬ h ( r ) p ( r ) : ⇒ h ( r ) n ( r , r ) : ⇒ ¬ h ( r ) n ( r , r ) : ⇒ ¬ h ( r ) s ( r ) : h ( r ) ⇒ ps ( r ) : h ( r ) ⇒ ¬ ps ( r ) : h ( r ) ⇒ ps ( r ) : h ( r ) ⇒ ¬ p with p ( r ) > n ( r , r ) , n ( r , r ) > p ( r ) , p ( r ) > n ( r , r ) , and n ( r , r ) > p ( r ) .Now, let A be the rule ⇒ p elative Expressiveness of Defeasible Logics II D + A ⊢ − δ ∗ p , because for every rule r for p , there is a rule for ¬ p that is not overruledby r ( r does not overrule r , r does not overrule r and A overrules neither).However, considering the transformed theory, T ( D ) + A ⊢ − σh ( r ) , because n ( r , r ) >p ( r ) and, similarly, T ( D ) + A ⊢ − σh ( r ) . Consequently, both rules for ¬ p in T ( D ) + A fail.This leaves the rules for p without competition, and so T ( D ) + A ⊢ + δp , conflicting with thebehaviour of D + A .Following essentially the same argument, this example also applies to DL ( ∂ ∗ ) and DL ( ∂ ) .We show that the transformation defined in Definition 11 (and Definition 9 ) allows the team-defeat logics to simulate their individual-defeat counterparts. We treat the two cases separately,but first we address the effect of the transformation on strict inference. Lemma 24
Consider the transformation T from Definition 11. For any D and A • D + A ⊢ +∆ q iff T ( D )+ A ⊢ +∆ q • D + A ⊢ − ∆ q iff T ( D )+ A ⊢ − ∆ q The proof is a straightforward induction.
Theorem 25
The logic DL ( ∂ ∗ ) can be simulated by DL ( ∂ ) with respect to addition of rules. Proof
We consider the transformation T ( D ) of a defeasible theory D as defined in Definition 11 (andDefinition 9) and show that this transformation provides a simulation of each defeasible theory D in DL ( ∂ ∗ ) from within DL ( ∂ ) .Fix any D and any A that satisfies the language separation condition. Let Σ = Σ( D ) ∪ Σ( A ) .[1] If + ∂ ∗ q ∈ T D + A ↑ ( n +1) then either +∆ q ∈ T D + A ↑ n (in which case T ( D )+ A ⊢ + ∂q )or + ∂ ∗ B r ⊆ T D + A ↑ n , where B r is the body of some strict or defeasible rule r in D + A . Inthe latter case, T D + A ↑ n also contains − ∆ ∼ q and for every rule s for ∼ q in D + A either r > s or − ∂ ∗ p ∈ T D + A ↑ n for some literal p in the body of s . Then, by the induction hypothesis, T ( D )+ A ⊢ + ∂B r , T ( D )+ A ⊢ − ∆ ∼ q and, if r ∈ D , for every rule n ( r, s ) for ¬ h ( r ) in T ( D ) ,either p ( r ) > ′ n ( r, s ) or T ( D )+ A ⊢ − ∂p where p occurs in the body of n ( r, s ) . Thus, using theinference rule for + ∂ , if r ∈ D then T ( D )+ A ⊢ + ∂h ( r ) . If r ∈ A then T ( D )+ A ⊢ + ∂B r so,whether r ∈ D or r ∈ A , there is a rule for q in T ( D )+ A with body B and T ( D )+ A ⊢ + ∂B .Applying the inference rule for − ∂ multiple times, for each strict or defeasible rule s for ∼ q in D we have T ( D )+ A ⊢ − ∂h ( s ) . Furthermore, as noted above, for every rule s ∈ A for ∼ q ,since r > s , − ∂ ∗ p ∈ T D + A ↑ n for some literal p in the body of s . Thus, every rule for ∼ q in T ( D )+ A fails. Now, again applying the inference rule for + ∂ , we have T ( D )+ A ⊢ + ∂q .[2] If − ∂ ∗ q ∈ T D + A ↑ ( n +1) then − ∆ q ∈ T D + A ↑ n and either +∆ ∼ q ∈ T D + A ↑ n (in whichcase T ( D )+ A ⊢ − ∂q ) or, for every strict or defeasible rule r for q in D + A , either − ∂ ∗ p ∈T D + A ↑ n for some p in the body of r or there exists a rule s for ∼ q with body B , + ∂ ∗ B ⊆T D + A ↑ n and r > s . Then, for every strict or defeasible rule p ( r ) in T ( D ) , either − ∂ ∗ p ∈T D + A ↑ n for some p in the body of p ( r ) or there is a rule n ( r, s ) with body B and p ( r ) > ′ n ( r, s ) ,by the structure of T ( D ) . By the induction hypothesis, for every strict or defeasible rule p ( r ) in T ( D ) , either T ( D )+ A ⊢ − ∂p for some p in the body of p ( r ) or there is a rule n ( r, r ′ ) withbody B where T ( D )+ A ⊢ + ∂B and p ( r ) > ′ n ( r, r ′ ) . Since there is only one rule for h ( r ) ,4 M.J. Maher application of the inference rule for − ∂ gives us T ( D )+ A ⊢ − ∂h ( r ) for each strict or defeasiblerule r ∈ D for q . Also by the induction hypothesis, for every strict or defeasible rule r for q in A , T ( D )+ A ⊢ − ∂p for some p in the body of r . Hence T ( D )+ A ⊢ − ∂q .[3] If q ∈ Σ and + ∂q ∈ T T ( D )+ A ↑ ( n +1) then either (1) +∆ q ∈ T T ( D )+ A ↑ n (in which case D + A ⊢ + ∂ ∗ q ), or else (2) + ∂h ( r ) ∈ T T ( D )+ A ↑ n for some strict or defeasible rule r for q in D , or else (3) + ∂B r ⊆ T T ( D )+ A ↑ n for some strict or defeasible rule r for q in A with body B r . In case (3), by the induction hypothesis, D + A ⊢ + ∂B r . In case (2) we must also have thatevery rule for ∼ q in T ( D )+ A fails (except for o ( ∼ q ) , which is overruled); that is, for every rule s for ∼ q in D , − ∂h ( s ) ∈ T T ( D )+ A ↑ n and, for every rule for ∼ q in A with body B , for someliteral p in B − ∂p ∈ T T ( D )+ A ↑ n . In case (3) we must also have that one ( ∼ q ) fails, so thatevery rule for ∼ q in D with body B , for some literal p in B − ∂p ∈ T T ( D )+ A ↑ n . Hence, by theinduction hypothesis, in case (3), for every rule for ∼ q in D + A with body B , for some literal p in B D + A ⊢ − ∂ ∗ p . In both cases (2) and (3), − ∆ ∼ q ∈ T T ( D )+ A ↑ n and hence D + A ⊢ − ∆ ∼ q .Applying the + ∂ ∗ inference rule in case (3), D + A ⊢ + ∂ ∗ q .In case (2), if + ∂h ( r ) ∈ T T ( D )+ A ↑ n then p ( r ) is not a defeater, + ∂B r ⊆ T T ( D )+ A ↑ n where B r is the body of r and for every rule n ( r, s ) with body B ′ either for some literal p in B ′ − ∂p ∈T T ( D )+ A ↑ n or for some rule t for h ( r ) , its body is proved with respect to ∂ and t > s . There isonly one rule for h ( r ) , so this last disjunct reduces to p ( r ) > n ( r, s ) . Using the construction of T ( D ) , r is not a defeater, + ∂B r ⊆ T T ( D )+ A ↑ n where B r is the body of r and for every rule s for ∼ q in D with body B ′ either for some literal p in B ′ , − ∂p ∈ T T ( D )+ A ↑ n or r > s . Furthermore,from the previous paragraph, for every rule for ∼ q in A with body B , for some literal p in B − ∂p ∈ T T ( D )+ A ↑ n . Using the induction hypothesis, D + A ⊢ + ∂ ∗ B r , and for every rule for ∼ q in D + A either for some literal p in the body D + A ⊢ − ∂ ∗ p or r > s . Applying the inferencerule for + ∂ ∗ , we obtain D + A ⊢ + ∂ ∗ q .[4] If q ∈ Σ and − ∂q ∈ T T ( D )+ A ↑ ( n +1) then, using the − ∂ inference rule and the structure of T ( D ) , − ∆ q ∈ T T ( D )+ A ↑ n and either (0) +∆ ∼ q ∈ T T ( D )+ A ↑ n (in which case D + A ⊢ − ∂ ∗ q ),or else (1) − ∂h ( r ) ∈ T T ( D )+ A ↑ n for every rule r for q in D , while for every rule r in A thereis a literal p in the body of r with − ∂p ∈ T T ( D )+ A ↑ n , and − ∂one ( q ) ∈ T T ( D )+ A ↑ n ; or (2) + ∂h ( s ) ∈ T T ( D )+ A ↑ n for some rule s for ∼ q in D ; or (3) + ∂one ( ∼ q ) ∈ T T ( D )+ A ↑ n , in whichcase there is a rule s for ∼ q in D where + ∂B s ⊆ T T ( D )+ A ↑ n , and − ∂h ( r ) ∈ T T ( D )+ A ↑ n forevery rule r for q in D (so that o ( ∼ q ) is not overruled); or (4) there is a rule s for ∼ q in A where + ∂B s ⊆ T T ( D )+ A ↑ n . In any case, using the induction hypothesis, we have D + A ⊢ − ∆ q .In case (1), since − ∂one ( q ) ∈ T T ( D )+ A ↑ n , for every rule r in D for q there is a literal p in B r with − ∂p ∈ T T ( D )+ A ↑ n . Thus all rules for q in D + A fail, and hence D + A ⊢ − ∂ ∗ q . In case (2),we must have, for every rule r in D for q , either s > r (so that p ( s ) > n ( s, r ) or there is a literal p in B r with − ∂p ∈ T T ( D )+ A ↑ n . By the induction hypothesis, we then have D + A ⊢ − ∂ ∗ p , foreach such p . Furthermore, no rule in A can overrule s . Hence, applying the − ∂ ∗ inference rule, D + A ⊢ − ∂ ∗ q .In case (3), since − ∂h ( r ) ∈ T T ( D )+ A ↑ n , either there is a literal p in B r with − ∂p ∈T T ( D )+ A ↑ n or there is a rule s in D for ∼ q with + ∂B s ⊆ T T ( D )+ A ↑ n and r > s (so that p ( r ) > n ( s, r ) ). By the induction hypothesis, for every rule r for q in D either there is a literal p in B r with D + A ⊢ − ∂ ∗ p or there is a rule s in D for ∼ q with D + A ⊢ + ∂ ∗ B s and r > s .Furthermore, from + ∂one ( ∼ q ) we know there is an s in D with (using the induction hypothesis) D + A ⊢ + ∂ ∗ B s , and this s cannot be overruled by any rule r in A . Consequently, applying the − ∂ ∗ inference rule, D + A ⊢ − ∂ ∗ q .In case (4), by the induction hypothesis, we have there is a rule s for ∼ q in A where D + A ⊢ elative Expressiveness of Defeasible Logics II + ∂ ∗ B s and, since s cannot be inferior to any rule, applying the − ∂ ∗ inference rule we have D + A ⊢ − ∂ ∗ q .This concludes the proof that DL ( ∂ ) can simulate DL ( ∂ ∗ ) with respect to addition of rules.We now turn to the corresponding proof for DL ( δ ) and DL ( δ ∗ ) . Theorem 26
The logic DL ( δ ∗ ) can be simulated by DL ( δ ) with respect to addition of rules. Proof
Let A be any set of rules. Let Σ be the language of D + A and let q ∈ Σ . Let T ( D ) be thetransformed defeasible theory as described in Definition 11. Then we claim • D + A ⊢ + σ ∗ q iff T ( D )+ A ⊢ + σq • D + A ⊢ − σ ∗ q iff T ( D )+ A ⊢ − σq • D + A ⊢ + δ ∗ q iff T ( D )+ A ⊢ + δq • D + A ⊢ − δ ∗ q iff T ( D )+ A ⊢ − δq If + δ ∗ q ∈ T D + A ↑ ( n +1) then either +∆ q ∈ T D + A ↑ n (in which case T ( D )+ A ⊢ + δq ) orelse + δ ∗ B r ⊆ T D + A ↑ n , where B r is the body of some strict or defeasible rule r in D + A . Inthe latter case, T D + A ↑ n also contains − ∆ ∼ q and for every rule s for ∼ q in D + A either r > s or − σ ∗ p ∈ T D + A ↑ n for some literal p in the body of s . Then, by the induction hypothesis, T ( D )+ A ⊢ + δB r , T ( D )+ A ⊢ − ∆ ∼ q and, if r and s are in D , for every rule n ( r, s ) for ¬ h ( r ) in T ( D ) , either p ( r ) > ′ n ( r, s ) or T ( D )+ A ⊢ − σp where p occurs in the body of n ( r, s ) and,similarly, the rule p ( s ) for h ( s ) in T ( D ) , either p ( s ) < ′ n ( s, r ) or T ( D )+ A ⊢ − σp where p occurs in the body of p ( s ) . If s is in A then r > s cannot occur (since the rules of A donot participate in the superiority relation) and T ( D )+ A ⊢ − σp where p occurs in the body of s . If r is in A and s is in D then, again, r > s cannot occur and T ( D )+ A ⊢ − σp where p occurs in the body of p ( s ) . Thus, using the inference rules for + δ and − σ , if r is in D then T ( D )+ A ⊢ + δh ( r ) and if s is in D then T ( D )+ A ⊢ − σh ( s ) . Now, applying the inference rulefor + δ , we conclude T ( D )+ A ⊢ + δq .If − δ ∗ q ∈ T D + A ↑ ( n +1) then − ∆ q ∈ T D + A ↑ n (and, hence, T ( D )+ A ⊢ − ∆ q ) and either +∆ ∼ q ∈ T D + A ↑ n (in which case T ( D )+ A ⊢ − δq ) or, for every strict or defeasible rule r for q in D + A , either − δ ∗ p ∈ T D + A ↑ n for some p in the body of r or there exists a rule s for ∼ q with body B s , where + σ ∗ B s ⊆ T D + A ↑ n and r > s . Now, if, for some s for ∼ q in A , + σ ∗ B s ⊆ T D + A ↑ n then, by the induction hypothesis, T ( D )+ A ⊢ + σB s and, applying theinference rule for − δ (and noting that no rule is superior to s ), we have T ( D )+ A ⊢ − δq .Otherwise, for every strict or defeasible rule p ( r ) in T ( D ) , either − δ ∗ p ∈ T D + A ↑ n for some p in the body of p ( r ) or there is a rule n ( r, s ) with body B s and p ( r ) > ′ n ( r, s ) , by the structureof T ( D ) . By the induction hypothesis, for every strict or defeasible rule p ( r ) in T ( D ) , either T ( D )+ A ⊢ − δp for some p in the body of p ( r ) or there is a rule n ( r, s ) with body B s where T ( D )+ A ⊢ + σB s and p ( r ) > ′ n ( r, s ) . In both cases, since there is only one rule for h ( r ) ,application of the inference rule for − δ gives us T ( D )+ A ⊢ − δh ( r ) for each strict or defeasiblerule r for q in D . Hence, no rule s ( r ) can overrule o ( ∼ q ) . Now, if every rule r for q in A has p ∈ B r with − δ ∗ p ∈ T D + A ↑ n then, by the induction hypothesis, T ( D )+ A ⊢ − δp for everysuch rule an hence, by application of the − δ inference rule, T ( D )+ A ⊢ − δq . Otherwise, thereis s for ∼ q in D with T ( D )+ A ⊢ + σB s . By the + σ inference rule T ( D )+ A ⊢ + σone ( ∼ q ) .Consequently, since o ( ∼ q ) cannot be overruled, T ( D )+ A ⊢ − δq .6 M.J. Maher
Hence, in every case, T ( D )+ A ⊢ − δq .If q ∈ Σ and + δq ∈ T T ( D )+ A ↑ ( n +1) then either +∆ q ∈ T T ( D )+ A ↑ n (in which case D + A ⊢ + δ ∗ q ), or + δh ( r ) ∈ T T ( D )+ A ↑ n for some strict or defeasible rule r for q in D .or + δB ⊆ T T ( D )+ A ↑ n for some strict or defeasible rule for q in A with body B . In the lat-ter cases, T T ( D )+ A ↑ n also contains − ∆ ∼ q ; hence D + A ⊢ − ∆ ∼ q . In these cases we mustalso have, for each rule for ∼ q in A , for some p in its body − σp ∈ T T ( D )+ A ↑ n . Hence, bythe induction hypothesis, for each rule for ∼ q in A , for some p in its body D + A ⊢ − σ ∗ p . If + δh ( r ) ∈ T T ( D )+ A ↑ n then p ( r ) is not a defeater, + δB r ⊆ T T ( D )+ A ↑ n where B r is the bodyof r and for every rule n ( r, s ) with body B s either for some literal p in B s − σp ∈ T T ( D )+ A ↑ n or for some rule t for h ( r ) , its body is proved with respect to δ and t > s . There is only onerule for h ( r ) , so this last disjunct reduces to p ( r ) > n ( r, s ) . Using the construction of T ( D ) , r is not a defeater, + δB r ⊆ T T ( D )+ A ↑ n and for every rule s for ∼ q in D either for some literal p in B s , − σp ∈ T T ( D )+ A ↑ n or r > s . Using the induction hypothesis, D + A ⊢ + δ ∗ B r , and forevery rule for ∼ q in D either for some literal p in the body D + A ⊢ − σ ∗ p or r > s . Applyingthe inference rule for + δ ∗ to this statement, and given we have shown that all rules for ∼ q in A fail, we obtain D + A ⊢ + δ ∗ q .If q ∈ Σ and − δq ∈ T T ( D )+ A ↑ ( n +1) then − ∆ q ∈ T T ( D )+ A ↑ n and either (1) +∆ ∼ q ∈T T ( D )+ A ↑ n (in which case D + A ⊢ − δ ∗ q ), or (2) − δh ( r ) ∈ T T ( D )+ A ↑ n for every rule r for q in D and every rule for q in A has a literal p in its body with − δq ∈ T T ( D )+ A ↑ n , or (3)there is a rule s for ∼ q in D where + σh ( s ) ∈ T T ( D )+ A ↑ n , or (4) there is a rule for ∼ q in A with body B and + σB ⊆ T T ( D )+ A ↑ n . (Some conditions are simpler than the inference rule for − δ might suggest because the superiority relation in T ( D )+ A does not involve the rules for q and ∼ q .) Consequently, D + A ⊢ − ∆ q . In the first case, using the induction hypothesis, we have D + A ⊢ − ∆ q and D + A ⊢ +∆ ∼ q ; hence, D + A ⊢ − δ ∗ q . In the second case, for each r , either r is a defeater, or there is a literal p in the body of r such that − δp ∈ T T ( D )+ A ↑ n , or there is a rule s for ∼ q in D (corresponding to rule n ( r, s ) in T ( D ) ) with body B s where + σB s ⊆ T T ( D )+ A ↑ n and r > s . By the induction hypothesis, either r is a defeater, or there is a literal p in the body of r such that D + A ⊢ − δ ∗ p , or there is a rule s for ∼ q in D with body B s where D + A ⊢ + σ ∗ B s and r > s . Similarly, using the induction hypothesis, every rule for q in A has a literal p in itsbody with D + A ⊢ − δ ∗ p . Applying the inference rule for − δ ∗ , we obtain D + A ⊢ − δ ∗ q .In the third case, either +∆ h ( s ) ∈ T T ( D )+ A ↑ n , or + σB ⊆ T T ( D )+ A ↑ n , where B s is thebody of s , and, for every rule r for q in D , either − δp ∈ T T ( D )+ A ↑ n for some literal p in thebody of r or r > s . If +∆ h ( s ) ∈ T T ( D )+ A ↑ n then +∆ B ⊆ T T ( D )+ A ↑ n and s is strict. Usingthe induction hypothesis, D + A ⊢ +∆ B and, hence, D + A ⊢ +∆ ∼ q and, like case (1) above, D + A ⊢ − δ ∗ q . In the other case, by the induction hypothesis, D + A ⊢ + σ ∗ B s and, for everyrule r for q in D , either D + A ⊢ − δ ∗ p for some literal p in the body of r or r > s . Applying theinference rule for − δ ∗ we conclude D + A ⊢ − δ ∗ q .In the fourth case, using the induction hypothesis, there is a rule for ∼ q in A with body B and D + A ⊢ + σ ∗ B . Applying the inference rule for − δ ∗ we conclude D + A ⊢ − δ ∗ q .If + σ ∗ q ∈ T D + A ↑ ( n +1) then either +∆ q ∈ T D + A ↑ n (in which case T ( D )+ A ⊢ + σq ) or + σ ∗ B r ⊆ T D + A ↑ n , where B r is the body of some strict or defeasible rule r in D + A . In thelatter case, for every rule s for ∼ q in D + A either s > r or − δ ∗ p ∈ T D + A ↑ n for some literal p in the body of s . If r ∈ A then r is not inferior to any rule. So, by the induction hypothesis, T ( D )+ A ⊢ + σB r , and, by the + σ inference rule T ( D )+ A ⊢ + σq . If r ∈ D , by the inductionhypothesis, T ( D )+ A ⊢ + σB r , for every rule n ( r, s ) for ¬ h ( r ) in T ( D ) , either n ( r, s ) > ′ p ( r ) or T ( D )+ A ⊢ − δp where p occurs in the body of n ( r, s ) . Thus, using the inference rule for elative Expressiveness of Defeasible Logics II + σ , T ( D )+ A ⊢ + σh ( r ) . Applying the inference rule for − δ multiple times, for each strict ordefeasible rule s for ∼ q in D we have T ( D )+ A ⊢ − δh ( s ) . Now, applying the inference rule for + σ , we have T ( D )+ A ⊢ + σq .If − σ ∗ q ∈ T D + A ↑ ( n +1) then − ∆ q ∈ T D + A ↑ n and for every strict or defeasible rule r for q in D + A , either − σ ∗ p ∈ T D + A ↑ n for some p in the body of r or there exists a rule s for ∼ q in D + A with body B s , + δ ∗ B s ⊆ T D + A ↑ n and s > r . If r ∈ A then − σ ∗ p ∈ T D + A ↑ n for some p in the body of r and hence, by the induction hypothesis, T ( D )+ A ⊢ − σp . If r ∈ D then, forevery strict or defeasible rule p ( r ) in T ( D ) , either − σ ∗ p ∈ T D + A ↑ n for some p in the body of p ( r ) or there is a rule n ( r, s ) with body B s with + δ ∗ B s ⊆ T D + A ↑ n and n ( r, s ) > ′ p ( r ) , by thestructure of T ( D ) . By the induction hypothesis, for every strict or defeasible rule p ( r ) in T ( D ) ,either T ( D )+ A ⊢ − σp for some p in the body of p ( r ) or there is a rule n ( r, s ) with body B s where T ( D )+ A ⊢ + δB and n ( r, s ) > ′ p ( r ) . Application of the inference rule for − σ gives us T ( D )+ A ⊢ − σh ( r ) for each strict or defeasible rule r for q in D . Rules in A for q also fail, asmentioned above. Hence T ( D )+ A ⊢ − σq .If q ∈ Σ and + σq ∈ T T ( D )+ A ↑ ( n +1) then either +∆ q ∈ T T ( D )+ A ↑ n (in which case D + A ⊢ + σ ∗ q ), or + σB ⊆ T T ( D )+ A ↑ n for some strict or defeasible rule for q in A with body B , or + σh ( r ) ∈ T T ( D )+ A ↑ n for some strict or defeasible rule r for q in D . In the second case, bythe induction hypothesis, D + A ⊢ + σ ∗ B and, applying the + σ ∗ inference rule, D + A ⊢ + σ ∗ q .In the third case, if + σh ( r ) ∈ T T ( D )+ A ↑ n then p ( r ) is not a defeater, + σB r ⊆ T T ( D )+ A ↑ n where B r is the body of r and for every rule n ( r, s ) with body B s either for some literal p in B s , − δp ∈ T T ( D )+ A ↑ n or n ( r, s ) > p ( r ) . Using the construction of T ( D ) , r is not a defeater, + σB r ⊆ T T ( D )+ A ↑ n where B r is the body of r and for every rule s for ∼ q with body B s in D either for some literal p in B ′ , − δp ∈ T T ( D )+ A ↑ n or s > r . Using the induction hypothesis, D + A ⊢ + σ ∗ B , and for every rule for ∼ q in D either for some literal p in the body D + A ⊢ − δ ∗ p or s > r . Note also that no rule s for ∼ q in A can be superior to r . Applying the inference rulefor + σ ∗ to this statement, we obtain D + A ⊢ + σ ∗ q .If q ∈ Σ and − σq ∈ T T ( D )+ A ↑ ( n +1) then − ∆ q ∈ T T ( D )+ A ↑ n (and, consequently, D + A ⊢− ∆ q ) and for every strict or defeasible rule r for q in A with body B there is p in B with − σp ∈ T T ( D )+ A ↑ n , and, for every rule r for q in D , − σh ( r ) ∈ T T ( D )+ A ↑ n . From − σh ( r ) either r is a defeater, or there is a literal p in the body of r such that − σp ∈ T T ( D )+ A ↑ n ,or there is a rule s for ∼ q in D (corresponding to rule n ( r, s ) in T ( D ) ) with body B s where + δB s ⊆ T T ( D )+ A ↑ n and s > r . By the induction hypothesis, either r is a defeater, or thereis a literal p in the body of r such that D + A ⊢ − σ ∗ p , or there is a rule s for ∼ q in D withbody B s where D + A ⊢ + δ ∗ B s and s > r . Applying the inference rule for − σ ∗ , we obtain D + A ⊢ − σ ∗ q .Combining Theorems 25 and 26 we have Theorem 12. Simulation of Team Defeat wrt Addition of Rules
The same theory D and addition A as in Example 10 demonstrates that the simulation of DL ( ∂ ) by DL ( ∂ ∗ ) wrt addition of facts exhibited in (Maher 2012) does not extend to addition of rules. Example 27 M.J. Maher
Let D consist of the rules r : ⇒ pr : ⇒ ¬ p and let A be the rule ⇒ p Then D + A ⊢ − ∂p .The transformation presented in (Maher 2012) simulates D wrt addition of facts with the fol-lowing theory D ′ : R : ⇒ ¬ d ( r , r ) R : ⇒ ¬ d ( r , r ) R : ⇒ d ( r , r ) R : ⇒ d ( r , r ) N F : ⇒ ¬ f ail ( r ) N F : ⇒ ¬ f ail ( r ) F : ⇒ f ail ( r ) F : ⇒ f ail ( r ) d ( r , r ) ⇒ d ( r ) d ( r , r ) ⇒ d ( r ) f ail ( r ) ⇒ d ( r ) f ail ( r ) ⇒ d ( r ) ⇒ one ( p ) one ( p ) , d ( r ) ⇒ p ⇒ one ( ¬ p ) one ( ¬ p ) , d ( r ) ⇒ ¬ p with N F > F and N F > F . (Rules R ij have been omitted because there are no strict rulesin D .)Then consequences of D ′ (and D ′ + A ) include − ∂ ∗ d ( r , r ) and − ∂ ∗ f ail ( r ) , and hencealso − ∂ ∗ d ( r ) . Consequently, the only rule for ¬ p in D ′ + A fails and hence, using the rule in A ,we can conclude + ∂ ∗ p .Thus D ′ does not simulate D wrt addition of rules. The weakness of the transformation in theprevious section is also evident here. Lemma 28
Let D be a defeasible theory, T ( D ) be the transformed defeasible theory as described in Defini-tion 13, and let A be a modular defeasible theory. Let Σ be the language of D + A and let q ∈ Σ .Then • D + A ⊢ +∆ q iff T ( D )+ A ⊢ +∆ q iff T ( D )+ A ⊢ + ∂ ∗ strict ( q ) iff T ( D )+ A ⊢ + ∂ ∗ true ( q ) iff T ( D )+ A ⊢ − ∂ ∗ ¬ true ( q ) • D + A ⊢ − ∆ q iff T ( D )+ A ⊢ − ∆ q iff T ( D )+ A ⊢ − ∂ ∗ strict ( q ) iff T ( D )+ A ⊢ − ∂ ∗ true ( q ) iff T ( D )+ A ⊢ + ∂ ∗ ¬ true ( q ) Proof
This result follows immediately from Lemma 18 and the inclusion theorem, since δ ∗ ⊆ ∂ ∗ ⊆ σ ∗ .We say a rule r fails in D if, for some literal p in the body of r , D ⊢ − ∂ ∗ p . Similarly, r failsin T ↑ n if − ∂ ∗ p ∈ T ↑ n for some literal p in the body of r . elative Expressiveness of Defeasible Logics II Theorem 29
The logic DL ( ∂ ) can be simulated by DL ( ∂ ∗ ) with respect to addition of rules. Proof
Let Σ be the language of D + A . Note, that, employing Lemma 18, T ( D )+ A ⊢ + ∂ ∗ ¬ true ( q ) iff T ( D )+ A ⊢ − ∆ q iff D + A ⊢ − ∆ q . Because T ( D )+ A ⊢ − ∂ ∗ g , we can essentially ignore therules supp ( q ) , which are only included for the simulation of DL ( δ ) by DL ( δ ∗ ) .Suppose + ∂q ∈ T D + A ↑ ( n + 1) . Then either +∆ q ∈ T D + A ↑ n (in which case T ( D ) + A ⊢ + ∂ ∗ q ), or − ∆ ∼ q ∈ T D + A ↑ n and there is a non-empty team of strict or defeasible rules for q such that + ∂B r ⊆ T D + A ↑ n for each body B r of each rule r and every rule s for ∼ q either has abody that fails in T D + A ↑ n or s < t for some rule t in the team. t / ∈ A because rules in A do notparticipate in the superiority relation. Then, by the induction hypothesis, T ( D ) + A ⊢ − ∆ ∼ q , T ( D ) + A ⊢ + ∂ ∗ B r for each rule r in the team, and for every rule s for ∼ q either its bodyfails in T ( D ) + A or there is a rule t in the team and t > s . If s ∈ A then its body B s failsin T ( D ) + A . If s ∈ D then either T ( D ) + A ⊢ + ∂ ∗ f ail ( s ) or T ( D ) + A ⊢ + ∂ ∗ d ( s, t ) ; ineither case, T ( D ) + A ⊢ + ∂ ∗ d ( s ) . Considering T ( D ) and the inference rule for + ∂ ∗ , we have T ( D ) + A ⊢ + ∂ ∗ one ( q ) . By Lemma 28, T ( D ) + A ⊢ + ∂ ∗ ¬ true ( ∼ q ) . Hence, the body of s ( q ) is proved. Because > is acyclic, there is a rule in the team for q that is not inferior to any rulein the team for ∼ q . Hence this rule r ′ is not defeated, so d ( r ′ ) fails, and hence the rule s ( ∼ q ) in T ( D ) (from point 6) fails. Hence all rules for ∼ q fail, with the possible exception of o ( ∼ q ) .However s ( q ) > o ( ∼ q ) and hence, applying the + ∂ ∗ inference rule, T ( D ) + A ⊢ + ∂ ∗ q .Suppose − ∂q ∈ T D + A ↑ ( n + 1) . Then − ∆ q ∈ T D + A ↑ n (and hence T ( D ) + A ⊢ − ∆ q )and either (1) +∆ ∼ q ∈ T D + A ↑ n (in which case T ( D ) + A ⊢ − ∂ ∗ q ), or (2) every rule r for q fails, or (3) there is a rule s for ∼ q with body B s such that + ∂B s ⊆ T D + A ↑ n and, for everystrict or defeasible rule t for q , either t fails in T D + A ↑ n , or t > s . In case (2), the rules r in A for q fail and, by the induction hypothesis and the inference rule for − ∂ ∗ , the rules r in A for q fail and, T ( D ) + A ⊢ − ∂ ∗ one ( q ) and hence T ( D ) + A ⊢ − ∂ ∗ q . In case (3), by the inductionhypothesis, there is a rule s for ∼ q with body B s such that T ( D ) + A ⊢ + ∂ ∗ B s and for everystrict or defeasible rule t for q , either t fails in T ( D ) + A , or t > s . If s ∈ A then t > s , for every t , and hence T ( D ) + A ⊢ − ∂ ∗ q . If s ∈ D then T ( D ) + A ⊢ − ∂ ∗ d ( s, t ) (since, via Lemma 28,we also have T ( D ) + A ⊢ − ∂ ∗ true ( q ) ). Using the − ∂ ∗ inference rule, T ( D ) + A ⊢ − ∂ ∗ d ( s ) and hence T ( D ) + A ⊢ − ∂ ∗ q .Suppose q ∈ Σ and + ∂ ∗ q ∈ T T ( D )+ A ↑ ( n + 1) . Then either (1) +∆ q ∈ T T ( D )+ A ↑ n (in which case D + A ⊢ + ∂q ), or − ∆ ∼ q ∈ T T ( D )+ A ↑ n (and hence D + A ⊢ − ∆ ∼ q ) andeither (2) for some r in A for q , B r ⊆ T T ( D )+ A ↑ n , or (3) + ∂ ∗ one ( q ) ∈ T T ( D )+ A ↑ n , + ∂ ∗ ¬ true ( ∼ q ) ∈ T T ( D )+ A ↑ n , and + ∂ ∗ d ( r ) occurs in T T ( D )+ A ↑ n , for each rule r for ∼ q in D . In both cases (2) and (3) we must have, for any rule s for ∼ q in A , for some p in the body B s of s , − ∂ ∗ p ∈ T T ( D )+ A ↑ n . By the induction hypothesis, D + A ⊢ − ∂p for each such p .In case (2), by the induction hypothesis, D + A ⊢ B r . Also, in case (2), the rule o ( ∼ q ) mustfail. Consequently, every rule s for ∼ q in D fails in T T ( D )+ A ↑ n . By the induction hypothesis,every rule s for ∼ q in D fails in D + A . Now, applying the inference rule for + ∂ , D + A ⊢ + ∂q .In case (3) there must be a strict or defeasible rule r for q in D with body B r such that + ∂ ∗ B r ⊆ T T ( D )+ A ↑ n and, using the rules for d ( s ) and d ( s, t ) , for every rule s for ∼ q ,either the body B s of s fails or there is a strict or defeasible rule t for q with body B t such that + ∂ ∗ B t ⊆ T T ( D )+ A ↑ n and t > s . By the induction hypothesis, D + A ⊢ + ∂B r , and, for everyrule s for ∼ q in D , either D + A ⊢ − ∂B s or there is a strict or defeasible rule t for q with body0 M.J. Maher B t such that D + A ⊢ + ∂B t and t > s . As noted above, for any rule s for ∼ q in A , B s fails in D + A . Hence, by the inference rule for + ∂ , D + A ⊢ + ∂q .If q ∈ Σ and − ∂ ∗ q ∈ T T ( D )+ A ↑ ( n + 1) then, using the inference rule for − ∂ ∗ and thestructure of T ( D ) , − ∆ q ∈ T T ( D )+ A ↑ n and either (a) +∆ ∼ q ∈ T T ( D )+ A ↑ n (in which case D + A ⊢ − ∂ ∗ q ), or (b) − ∂ ∗ one ( q ) ∈ T T ( D )+ A ↑ n , or (c) − ∂ ∗ d ( s ) ∈ T T ( D )+ A ↑ n for somerule s for ∼ q in D , or (d) + ∂ ∗ one ( ∼ q ) ∈ T T ( D )+ A ↑ n and + ∂ ∗ d ( r ) ∈ T T ( D )+ A ↑ n for everyrule r for q in D , or (e) for some s for ∼ q in A , + ∂ ∗ B s ⊆ T T ( D )+ A ↑ n .If (b) − ∂ ∗ one ( q ) ∈ T T ( D )+ A ↑ n then for every strict or defeasible rule for q in D fails.Applying the induction hypothesis and the inference rule for − ∂ , we have D + A ⊢ − ∂q . If c) − ∂ ∗ d ( s ) ∈ T T ( D )+ A ↑ n for some rule s for ∼ q in D , then there is no strict or defeasible rule r for q that defeats s . If (d) then there is a rule s for ∼ q with body B s such that + ∂ ∗ B s ⊆T T ( D )+ A ↑ n and every rule r for q in D is defeated by a strict or defeasible rule for ∼ q . Inboth cases (c) and (d), applying the induction hypothesis and the inference rule for − ∂ , we have D + A ⊢ − ∂q . If (e) then, by the induction hypothesis, D + A ⊢ + ∂B s and hence, applying theinference rule for − ∂ , D + A ⊢ − ∂q . Theorem 30
The logic DL ( δ ) can be simulated by DL ( δ ∗ ) with respect to addition of rules. Proof
Let Σ be the language of D + A . Note that, for any q ∈ Σ( D ) , T ( D )+ A ⊢ + σ ∗ ¬ true ( q ) and, employing Lemma 18, T ( D )+ A ⊢ + δ ∗ ¬ true ( q ) iff T ( D )+ A ⊢ − ∆ q iff D + A ⊢ − ∆ q .Also note that T ( D )+ A ⊢ − δ ∗ g , T ( D )+ A ⊢ + σ ∗ g , and T ( D )+ A ⊢ + σ ∗ ¬ g , where g is theproposition used in part 7 of Definition 13.Suppose + δq ∈ T D + A ↑ ( n + 1) . Then either +∆ q ∈ T D + A ↑ n , or − ∆ ∼ q ∈ T D + A ↑ n and there is a non-empty team of strict or defeasible rules for q such that + δB r ⊆ T D + A ↑ n foreach body B r of each rule r and every rule s for ∼ q either has a body that fails in T D + A ↑ n or s < t for some rule t in the team. Then, by the induction hypothesis, either T ( D ) + A ⊢ +∆ q (in which case T ( D ) + A ⊢ + δ ∗ q ), or T ( D ) + A ⊢ − ∆ ∼ q , T ( D ) + A ⊢ + δ ∗ B r for eachrule r in the team, and for every rule s for ∼ q with body B s either T ( D ) + A ⊢ − σB s or thereis a rule t in the team and t > s . If s ∈ A then T ( D ) + A ⊢ − σB s . If s ∈ D then either T ( D ) + A ⊢ + δ ∗ f ail ( s ) or T ( D ) + A ⊢ + δ ∗ d ( s, t ) ; in either case, T ( D ) + A ⊢ + δ ∗ d ( s ) .Considering T ( D ) , and the inference rule for + δ ∗ we have T ( D ) + A ⊢ + δ ∗ one ( q ) . By Lemma18, T ( D ) + A ⊢ + delta ∗ ¬ true ( ∼ q ) . Hence, the body of s ( q ) is proved. Because > is acyclic, there is a rule in theteam for q that is not inferior to any rule in the team for ∼ q . Hence this rule r ′ is not defeated,so d ( r ′ ) fails, and hence the rule for ∼ q in T ( D ) from point 6 fails. Similarly, d σ ( r ′ , s ) fails,and hence the rules for ∼ q in T ( D ) from point 7 fail. Hence all rules for ∼ q fail, with thepossible exception of o ( ∼ q ) . However s ( q ) > o ( ∼ q ) and hence, applying the + δ ∗ inferencerule, T ( D ) + A ⊢ + δ ∗ q .Suppose − δq ∈ T D + A ↑ ( n + 1) . Then − ∆ q ∈ T D + A ↑ n (and hence T ( D ) + A ⊢ − ∆ q )and either (1) +∆ ∼ q ∈ T D + A ↑ n (in which case T ( D ) + A ⊢ − δ ∗ q ), or (2) every rule r for q contains a body literal p with − δp ∈ T D + A ↑ n , or (3) there is a rule s for ∼ q with body B s suchthat + σB s ⊆ T D + A ↑ n and, for every strict or defeasible rule t for q , either t fails in T D + A ↑ n ,or t > s . In case (2), the rules r in A for q fail and, by the induction hypothesis and the inference elative Expressiveness of Defeasible Logics II − δ ∗ , the rules r in A for q fail in T ( D ) + A , so T ( D ) + A ⊢ − δ ∗ one ( q ) and hence T ( D ) + A ⊢ − δ ∗ q . In case (3), by the induction hypothesis, there is a rule s for ∼ q with body B s such that T ( D ) + A ⊢ + σ ∗ B s and for every strict or defeasible rule t for q , either t fails in T ( D ) + A , or t > s . If s ∈ A then t > s , for every t , and hence T ( D ) + A ⊢ − δ ∗ q . If s ∈ D then T ( D ) + A ⊢ − δ ∗ d ( s, t ) (since, via Lemma 18, we also have T ( D ) + A ⊢ − δ ∗ true ( q ) ).Using the − δ ∗ inference rule, T ( D ) + A ⊢ − δ ∗ d ( s ) . The bodies of rules from point 7 of thetransformation also fail (wrt δ ∗ ), because of the presence of g . Hence T ( D ) + A ⊢ − δ ∗ q .Suppose q ∈ Σ and + δ ∗ q ∈ T T ( D )+ A ↑ ( n + 1) . Then either (1) +∆ q ∈ T T ( D )+ A ↑ n (inwhich case D + A ⊢ + δq ), or else − ∆ ∼ q ∈ T T ( D )+ A ↑ n and either (2) there is a strict ordefeasible rule r for q in A where + δ ∗ B r ⊆ T T ( D )+ A ↑ n and for all rules for ∼ q in T ( D )+ A ,the body of the rule contains a literal p with − σ ∗ p ∈ T T ( D )+ A ↑ n , or (3) each of + δ ∗ one ( q ) , + δ ∗ ¬ true ( ∼ q ) , and + δ ∗ d ( s ) occurs in T T ( D )+ A ↑ n , for each rule s for ∼ q in D .Hence, in case (3), there is a strict or defeasible rule r for q with body B r such that + δ ∗ B r ⊆T T ( D )+ A ↑ n and, for every rule s for ∼ q , either − σ ∗ p ∈ T T ( D )+ A ↑ n , for some p in the body B s of s , or there exists t in D for q with + δ ∗ B t ⊆ T T ( D )+ A ↑ n and t > s . By the inductionhypothesis, D + A ⊢ − ∆ ∼ q , D + A ⊢ + δB r , and, for every rule s for ∼ q , either D + A ⊢ − σB s or D + A ⊢ + δB t and t > s . By the inference rule for + δ , D + A ⊢ + δq .In case (2), using the structure of T ( D ) , for the rules supp ( ∼ q ) , originating from some rule s for ∼ q in D , either for some p in B s , − σ ∗ p ∈ T T ( D )+ A ↑ n or, for some t , − σ ∗ d σ ( t, s ) ∈T T ( D )+ A ↑ n (and, hence, + δ ∗ B t ⊆ T T ( D )+ A ↑ n and t > s ). Now, by the induction hypothesis, D + A ⊢ + δB r ; D + A ⊢ − ∆ ∼ q ; for all rules for ∼ q in A , the body of the rule contains a literal p with D + A ⊢ − σ ∗ p ; and for all rules for ∼ q in D , either the body of the rule contains a literal p with D + A ⊢ − σ ∗ p or there is a rule t for q in D with t > s and D + A ⊢ + δB t . Applyingthe inference rule for + δ , D + A ⊢ + δq .If q ∈ Σ and − δ ∗ q ∈ T T ( D )+ A ↑ ( n + 1) then, using the inference rule for − δ ∗ and thestructure of T ( D )+ A , − ∆ q ∈ T T ( D )+ A ↑ n (and, hence, D + A ⊢ − ∆ q ) and either +∆ ∼ q ∈T T ( D )+ A ↑ n (in which case D + A ⊢ − δq ), or else for every rule r for q in A , there is a literal p in B r such that − δ ∗ p ∈ T T ( D )+ A ↑ n and either (1) − δ ∗ ¬ true ( ∼ q ) ∈ T T ( D )+ A ↑ n (in whichcase D + A ⊢ +∆ ∼ q and hence D + A ⊢ − δq ), or (2) − δ ∗ one ( q ) ∈ T T ( D )+ A ↑ n , or (3) − δ ∗ d ( s ) ∈ T T ( D )+ A ↑ n for some rule s for ∼ q in D . Or (4) + σ ∗ one ( ∼ q ) ∈ T T ( D )+ A ↑ n and + σ ∗ d ( r ) ∈ T T ( D )+ A ↑ n for every rule r for q in D , or (5) there is a rule s for ∼ q in A and + σ ∗ B s ⊆ T T ( D )+ A ↑ n . Or (6) the body of a rule supp ( ∼ q, s ) is supported for some rule s for ∼ q (that is, + σ ∗ B s ⊆ T T ( D )+ A ↑ n and, for each rule r for q , + σ ∗ d σ ( r, s ) ∈ T T ( D )+ A ↑ n ).For (2) and (3), by the induction hypothesis, for every rule r for q in A , there is a literal p in B r such that D + A ⊢ − δp . If (2) − δ ∗ one ( q ) ∈ T T ( D )+ A ↑ n then every strict or defeasiblerule for q in D fails. Applying the induction hypothesis and the inference rule for − δ , we have D + A ⊢ − δq . If (3) − δ ∗ d ( s ) ∈ T T ( D )+ A ↑ n for some rule s for ∼ q in D , then + σ ∗ B s ⊆T T ( D )+ A ↑ n , where B s is the body of s , and for every strict or defeasible rule r for q with body B either − δ ∗ B r ∈ T T ( D )+ A ↑ n or + σ ∗ B s ⊆ T T ( D )+ A ↑ n , where B s is the body of s , and r > s . Applying the induction hypothesis, D + A ⊢ + σB s and, for every r for q , D + A ⊢ − δB r or D + A ⊢ + σB s and r > s . Hence, by the inference rule for − δ , D + A ⊢ − δq .If (4) then there is a rule s for ∼ q with body B s such that + σ ∗ B s ⊆ T T ( D )+ A ↑ n and forevery rule r for q in D either there is a literal p in the body of r such that − δ ∗ p ∈ T T ( D )+ A ↑ n or there is a rule s ′ for ∼ q with body B ′ such that + σ ∗ B ′ ⊆ T T ( D )+ A ↑ n and s ′ > r . Applyingthe induction hypothesis, for every rule r for q in D either there is a literal p in the body of r M.J. Maher such that D + A ⊢ − δp or there is a rule s ′ for ∼ q with body B ′ such that D + A ⊢ + σB ′ and s ′ > r . t follows, by the inference rule for − δ , that D + A ⊢ − δq .If (5) then, by the induction hypothesis, D + A ⊢ + σB s and, since s is not inferior to any rule,the inference rule for − δ gives us D + A ⊢ − δq .In case (6), since + σ ∗ d σ ( r, s ) ∈ T T ( D )+ A ↑ n , we must have + σ ∗ B s ⊆ T T ( D )+ A ↑ n andeither there is a literal p in B r such that − δ ∗ p ∈ T T ( D )+ A ↑ n or r > s . By the inductionhypothesis, D + A ⊢ + σ ∗ B s and, for every rule r for q in D either there is a literal p in B r suchthat D + A ⊢ − δ ∗ p or r > s . By the − δ inference rule, D + A ⊢ − δq .Suppose + σq ∈ T D + A ↑ ( n + 1) . Then either +∆ q ∈ T D + A ↑ n , or there is a strict ordefeasible rule r for q such that + σB r ⊆ T D + A ↑ n where B r is the body of r and every rule s for ∼ q has a body with a literal p such that − δp ∈ T D + A ↑ n or s > r . Then, by the inductionhypothesis, either T ( D )+ A ⊢ +∆ q (in which case T ( D )+ A ⊢ + σ ∗ q ), or T ( D )+ A ⊢ + σ ∗ B r ,and every rule s for ∼ q has a body with a literal p such that − δ ∗ p ∈ T D + A ↑ n or s > r . If r ∈ A then r is not inferior to any rule and, by the inference rule for + σ ∗ , T ( D ) + A ⊢ + σ ∗ q .If r ∈ D then, by the + σ ∗ inference rule, T ( D ) + A ⊢ + σ ∗ B r . Furthermore, again by the + σ ∗ inference rule, for every s for ∼ q in D , T ( D ) + A ⊢ + σ ∗ d σ ( s, r ) , since a ( s, r ) > b ( s, r ) iff s > r . Note that there is no superiority relation between the rules in T ( D ) for q and ∼ q . Hence,applying the inference rule for + σ ∗ , T ( D ) + A ⊢ + σ ∗ q .Suppose − σq ∈ T D + A ↑ ( n + 1) . Then − ∆ q ∈ T D + A ↑ n and either every rule r for q contains a body literal p and − σp ∈ T D + A ↑ n , or there is a rule s for ∼ q with body B s suchthat + δB s ⊆ T D + A ↑ n and s > r . (Note that, for r ∈ A , only the first possibility can apply.)Then, by the induction hypothesis, T ( D ) + A ⊢ − ∆ q and either every rule r for q contains abody literal p such that T ( D ) + A ⊢ − σ ∗ p , or there is a rule s for ∼ q with body B s such that T ( D ) + A ⊢ + δ ∗ B s and s > r . (In particular, every rule for q in A contains a body literal p with T ( D ) + A ⊢ − σ ∗ p .) If all rules for q in D fall in the former case, we have − σ ∗ one ( q ) ,and all rules supp ( q ) fail. Otherwise, there is an s that is not inferior to any rule for q and hence T ( D ) + A ⊢ − σ ∗ d ( s, r ) and T ( D ) + A ⊢ − σ ∗ d ( s ) . Similarly, T ( D ) + A ⊢ − σ ∗ d σ ( s, r ) . Ineither case, all rules for q fail, and hence T ( D ) + A ⊢ − σ ∗ q .Suppose q ∈ Σ and + σ ∗ q ∈ T T ( D )+ A ↑ ( n + 1) . Then either (1) +∆ q ∈ T T ( D )+ A ↑ n (inwhich case D + A ⊢ + σq ), or else either (2) for some rule r for q in A , + σ ∗ B r ⊆ T T ( D )+ A ↑ n ,or (3) + σ ∗ one ( q ) ∈ T T ( D )+ A ↑ n , and + σ ∗ d ( s ) occurs in T T ( D )+ A ↑ n , for each rule s for ∼ q in D , or (4) for some strict or defeasible rule r for q in D , + σ ∗ B r ⊆ T T ( D )+ A ↑ n and + σ ∗ d σ ( s, r ) occurs in T T ( D )+ A ↑ n , for each rule s for ∼ q in D .In case (2), by the induction hypothesis, D + A ⊢ + σB r and hence, by the inference rule for σ , D + A ⊢ + σq .In case (3), there is a strict or defeasible rule r for q with body B r such that + σ ∗ B r ⊆T T ( D )+ A ↑ n and, for every rule s for ∼ q in D , either − δ ∗ p ∈ T T ( D )+ A ↑ n , for some p in thebody B s of s or there is a rule t for q with body B t such that + σ ∗ B t ∈ T T ( D )+ A ↑ n . and s > t .By the induction hypothesis, D + A ⊢ + σB r , and, for every rule s for ∼ q , either D + A ⊢ − δB s or there is a rule t for q with body B t such that D + A ⊢ + σ ∗ B t and s > t . Because ¿ is acyclic,there is a rule t for q such that D + A ⊢ + σB t and, for every rule s for ∼ q either D + A ⊢ − δB s or s > t . By the inference rule for + σ , D + A ⊢ + σq .In case (4), there is a strict or defeasible rule r for q with body B r such that + σ ∗ B r ⊆T T ( D )+ A ↑ n and, for each rule s for ∼ q in D , either − δ ∗ p ∈ T T ( D )+ A ↑ n , for some p in thebody B s of s , or s > r . By the induction hypothesis, D + A ⊢ + σB r , and, for each s , either D + A ⊢ − δp or s > r . By the + σ inference rule, D + A ⊢ + σq . elative Expressiveness of Defeasible Logics II q ∈ Σ and − σ ∗ q ∈ T T ( D )+ A ↑ ( n + 1) then, using the inference rule for − σ ∗ and thestructure of T ( D ) , − ∆ q ∈ T T ( D )+ A ↑ n (and hence D + A ⊢ − ∆ q ), and either (1) +∆ ∼ q ∈T T ( D )+ A ↑ n (in which case D + A ⊢ − σ ∗ q ), or (2) for each strict or defeasible rule r for q in A , there is a literal p in B r such that − σ ∗ p ∈ T T ( D )+ A ↑ n and for each strict or defeasible rule r for q in D , either there is a literal p in B r such that − σ ∗ p ∈ T T ( D )+ A ↑ n , or there is a rule s for ∼ q in D such that + δ ∗ B s ⊆ T T ( D )+ A ↑ n and s > r . By the induction hypothesis, in case(2), D + A ⊢ − σB r for the rules r in A and, for rules r in D , either D + A ⊢ − σB r or thereis a rule s for ∼ q in D such that D + A ⊢ + δ ∗ B s and s > r . Applying the − σ inference rule, D + A ⊢ − σqσq