From: AAAI-99 Proceedings. Copyright © 1999, AAAI (www.aaai.org). All rights reserved.

A Semantic Decomposition of Defeasible

M.J. Maher and G. Governatori School of Computing and Information Technology, Griffith University Nathan, QLD 4111, Australia {mjm,guido}@cit.gu.edu.au

Abstract In the third section we present the metaprogram that en- codes the basic behavior of the Defeasible syntactic We investigate defeasible logics using a technique which de- constructs. We outline Kunen’s and the well- composes the semantics of such logics into two parts: a spec- ification of the structure of defeasible reasoning and a seman- founded programs, and show that the tics for the meta- in which the specification is writ- composition of these semantics with the metaprogram pro- ten. We show that Nute’s Defeasible Logic corresponds to duces, respectively, Defeasible Logic and Well-Founded De- Kunen’s semantics, and develop a defeasible logic from the feasible Logic. We also show, using the metaprogram, that well-founded semantics of Van Gelder, Ross and Schlipf. We explicit failure operators can be added to defeasible logics also obtain a new defeasible logic which extends an existing in a conservative way. Finally, we present some future work language by modifying the specification of Defeasible Logic. and conclusions. Thus our approach is productive in analysing, comparing and designing defeasible logics. Defeasible Logic Introduction Outline of Defeasible Logic A rule r consists of its antecedent A(r) (written on the left; In this paper we start from Nute’s Defeasible Logic (Nute, A(r) may be omitted if it is the empty ) which is a finite 1987; Nute 1994). This logic has an expressive , a set of literals, an arrow, and its consequent C(r) which is a strongly skeptical semantics and a tractable computational literal. In writing rules we omit set notation for antecedents. behavior. Our interest, in this paper, is to decompose De- There are three kinds of rules: Strict rules are denoted by feasible Logic into parts, for the analysis of the logic, and → also to reassemble it with different parts to create new and A p, and are interpreted in the classical sense: when- different logics. ever the premises are indisputable (e.g. facts) then so is the conclusion. An example of a strict rule is “Emus are birds”. We show that one component of Defeasible Logic is ( ) → ( ) Kunen’s semantics for logic programs (Kunen, 1987). As Written formally: emu X bird X . from facts and strict rules only is called definite inference. a consequence of this link, inference in Defeasi- ⇒ ble Logic (where arbitrary are allowed) is Defeasible rules are denoted by A p, and can be de- feated by contrary evidence. An example of such a rule is computable, and inference in propositional Defeasible Logic ( ) ⇒ ( ) is polynomial. bird X flies X , which reads as follows: “Birds typi- cally fly”. The technique that we use – meta-programming – allows ; us to provide several different semantics to the syntactic ele- Defeaters are denoted by A p and are used to prevent ments of Defeasible Logic without violating the underlying some conclusions. In other words, they are used to defeat some defeasible rules by producing evidence to the contrary. intuitive of the syntax. Thus we can create sev- ( ) ; ¬ ( ) eral different defeasible logics, all adhering to the defeasible An example is the rule heavy X flies X ,which structure underlying Defeasible Logic. (Of course, the com- reads as follows: “If an animal is heavy then it may not be putational complexity of such logics varies with the seman- able to fly”. The main point is that the information that an tics.) In particular, we show that a defeasible logic devel- animal is heavy is not sufficient evidence to conclude that it oped using unfounded sets corresponds exactly to the use of doesn’t fly. It is only evidence that the animal may not be the well-founded semantics of logic programs (Van Gelder able to fly. et al., 1991). A superiority relation on R is a relation > on R (that is, The paper is organized as follows. The next section intro- the transitive closure of > is irreflexive). When r1 >r2, duces Defeasible Logic and its . We establish then r1 is called superior to r2,andr2 inferior to r1.This a bottom-up characterization of the consequences of a de- expresses that r1 may override r2. For example, given the feasible theory that serves as our semantics for Defeasible defeasible rules Logic. We also define Well-Founded Defeasible Logic and r : bird(X) ⇒ flies(X) show that it is coherent and consistent. r0 : brokenW ing(X) ⇒¬flies(X) which contradict one another, no conclusive decision can be +∂:IfP (i +1)=+∂q then either made about whether a bird with a broken wing can fly. But (1) +∆q ∈ P (1..i) or 0 if we introduce a superiority relation > with r >r,thenwe (2) (2.1) ∃r ∈ Rsd[q]∀a ∈ A(r):+∂a ∈ P (1..i) and can indeed conclude that it cannot fly. (2.2) −∆ ∼q ∈ P (1..i) and A defeasible theory consists of a set of facts, a set of rules, (2.3) ∀s ∈ R[∼q] either and a superiority relation. A conclusion of a defeasible the- (2.3.1) ∃a ∈ A(s):−∂a ∈ P (1..i) or ory D is a tagged literal and can have one of the following (2.3.2) ∃t ∈ Rsd[q] such that four forms: ∀a ∈ A(t):+∂a ∈ P (1..i) and t>s • +∆q, which is intended to mean that q is definitely prov- able in D. Let us illustrate this definition. To show that q is provable •−∆ defeasibly we have two choices: (1) We show that q is al- q, which is intended to mean that we have proved that ready definitely provable; or (2) we need to argue using the q is not definitely provable in D. defeasible part of D as well. In particular, we require that • +∂q, which is intended to mean that q is defeasibly prov- there must be a strict or defeasible rule with head q which able in D. can be applied (2.1). But now we need to consider possi- •−∂q which is intended to mean that we have proved that ble “counterattacks”, that is, reasoning chains in support of q is not defeasibly provable in D. ∼ q. To be more specific: to prove q defeasibly we must show that ∼ q is not definitely provable (2.2). Also (2.3) Definite provability involves only strict rules and facts. we must consider the set of all rules which are not known to ∼ Proof Theory be inapplicable and which have head q (note that here we consider defeaters, too, whereas they could not be used to In this presentation we use the formulation of Defeasible support the conclusion q; this is in line with the motivation Logic given in (Billington 1993). A defeasible theory D of defeaters given above). Essentially each such rule s at- ( ) is a triple F, R, > where F is a set of literals (called facts), tacks the conclusion q.Forq to be provable, each such rule R a finite set of rules, and > a superiority relation on R.In s must be counterattacked by a rule t with head q with the expressing the proof theory we consider only propositional following properties: (i) t must be applicable at this point, rules. Rules such as the previous examples are interpreted and (ii) t must be stronger than (i.e. superior to) s. Thus as the set of their variable-free instances. each attack on the conclusion q must be counterattacked by Given a set R of rules, we denote the set of all strict rules a stronger rule. in R by Rs, the set of strict and defeasible rules in R by The definition of the proof theory of defeasible logic is Rsd, the set of defeasible rules in R by Rd, and the set of completed by the condition −∂. It is nothing more than a defeaters in R by Rdf t . R[q] denotes the set of rules in R strong of the condition +∂. with consequent q. In the following ∼p denotes the comple- ment of p,thatis,∼p is ¬p if p is an atom, and ∼p is q if p is ¬q. −∂:IfP (i +1)=−∂q then Provability is defined below. It is based on the of (1) −∆q ∈ P (1..i) and a derivation (or proof)inD =(F, R, >). A derivation is a (2) (2.1) ∀r ∈ Rsd[q] ∃a ∈ A(r):−∂a ∈ P (1..i) or finite sequence P =(P (1),...P(n)) of tagged literals sat- (2.2) +∆ ∼q ∈ P (1..i) or isfying the following conditions (P (1..i) denotes the initial (2.3) ∃s ∈ R[∼q] such that part of the sequence P of length i): (2.3.1) ∀a ∈ A(s):+∂a ∈ P (1..i) and (2.3.2) ∀t ∈ Rsd[q] either +∆:IfP (i +1)=+∆q then either ∃a ∈ A(t):−∂a ∈ P (1..i) or t >s6 q ∈ F or ∃r ∈ Rs[q] ∀a ∈ A(r):+∆a ∈ P (1..i) To prove that q is not defeasibly provable, we must first That means, to prove +∆q we need to establish a proof establish that it is not definitely provable. Then we must for q using facts and strict rules only. This is a deduction establish that it cannot be proven using the defeasible part in the classical sense – no proofs for the negation of q need of the theory. There are three possibilities to achieve this: to be considered (in to defeasible provability below, either we have established that none of the (strict and de- where opposing chains of reasoning must be taken into ac- feasible) rules with head q can be applied (2.1); or ∼ q is count, too). definitely provable (2.2); or there must be an applicable rule −∆:IfP (i +1)=−∆q then s with head ∼q such that no applicable rule t with head q is q 6∈ F and superior to s. ∀r ∈ Rs[q] ∃a ∈ A(r):−∆a ∈ P (1..i) The elements of a derivation are called lines of the deriva- To prove −∆q,i.e.thatq is not definitely provable, q must tion. We say that a tagged literal L is provable in D = not be a fact. In addition, we need to establish that every (F, R, >), denoted D ` L, iff there is a derivation in D strict rule with head q is known to be inapplicable. Thus for such that L is a line of P . every such rule r there must be at least one antecedent a for Under some assumptions, the logic and conditions con- which we have established that a is not definitely provable cerning defeasible provability can be simplified (Antoniou (−∆a). et al., 1998) but we do not need those assumptions here. A Bottom-Up Characterization of Defeasible Logic Logic in this wider sense. We will write D ` +∆q to ex- ∈ +∆ The proof theory provides the basis for a top-down press q F , and similarly with other conclusions. (backward-chaining) implementation of the logic. However, The analysis of the proof theory of Defeasible Logic in there are advantages to a bottom-up (forward-chaining) im- (Maher et al., 1998) was based on a 4-tuple defined purely plementation. Furthermore, a bottom-up definition of the in terms of the (top-down) proof theory. By 1, that logic provides a bridge to the logics we will define later. For 4-tuple is precisely F . (+∆ −∆ + − ) +∆ ∩ these reasons we now provide a bottom-up definition of De- An , , ∂, ∂ is coherent if −∆=∅ + ∩− = ∅ feasible Logic. and ∂ ∂ . An extension is consistent if whenever p ∈ +∂ and ∼ p ∈ +∂,forsomep,thenalso D TD We associate with an operator which works on 4- ∈ +∆ ∼ ∈ +∆ tuples of sets of literals. We call such 4-tuples an extension. p and p . Intuitively, coherence says that no literal is simultaneously provable and unprovable. Con- 0 0 0 0 TD(+∆, −∆, +∂,−∂)=(+∆, −∆ , +∂ , −∂ ) where sistency says that a literal and its negation can both be de- feasibly provable only when it and its negation are definitely 0 +∆ = F ∪{q |∃r ∈ Rs[q] A(r) ⊆ +∆} provable; hence defeasible inference does not introduce in- consistency. A logic is coherent (consistent) if the meaning 0 −∆ = −∆ ∪ ({q |∀r ∈ Rs[q] A(r) ∩−∆ 6= ∅} − F ) of each theory of the logic, when expressed as an extension, is coherent (consistent). 0 +∂ =+∆ ∪{q |∃r ∈ Rsd[q] A(r) ⊆ +∂, The following result was shown in (Billington, 1993). ∼q ∈−∆, and 2 Defeasible Logic is coherent and consistent. ∀s ∈ R[∼q] either ( ) ∩− 6= ∅ A s ∂ ,or Well-Founded Defeasible Logic ∃t ∈ R[q] such that A(t) ⊆ +∂ and t>s} It follows from the above definitions that defeasible theories such as 0 : ⇒ −∂ = {q ∈−∆ |∀r ∈ Rsd[q] A(r) ∩−∂ 6= ∅,or r p p ∼q ∈ +∆,or conclude neither +∂p nor −∂p. In some contexts it is desir- ∃s ∈ R[∼q] such that A(s) ⊆ +∂ and able for a logic to recognize such “loops” and to conclude ∀t ∈ R[q] either −∂p. Building on the bottom-up definition of the previ- A(t) ∩−∂ 6= ∅,or ous subsection, and inspired by the work of (Van Gelder et t >s6 } al., 1991), we define a well-founded defeasible logic which The set of extensions forms a complete lattice under the draws such conclusions. pointwise containment ordering1, with ⊥ =(∅, ∅, ∅, ∅) as The central definition required is that of an unfounded set. its least . The least upper bound operation is the Since defeasible logic involves both definite and defeasible pointwise union, which is represented by ∪. It can be shown inference, we need two definitions. A set S of literals is that TD is monotonic and the Kleene sequence from ⊥ is unfounded with respect to an extension E and definite infer- ∆ increasing. Thus the limit F =(+∆F , −∆F , +∂F , −∂F ) ence (or -unfounded) if: For every literal s in S,andfor → of all finite elements in the sequence exists, and TD has a every strict rule B s either least fixpoint L =(+∆L, −∆L, +∂L, −∂L).WhenD is a • B ∩−∆E 6= ∅,or finite propositional defeasible theory F = L. • B ∩ S 6= ∅ The extension F captures exactly the described in the proof theory. This definition is very similar to the definition of un- founded set in (Van Gelder et al., 1991). The main differ- Theorem 1 Let D be a finite propositional defeasible the- ences are that the basic elements of S are literals (and nega- ory and q a literal. tion is classical negation) and “negation as failure” is not • D ` +∆q iff q ∈ +∆F present in the bodies of rules. • D `−∆q iff q ∈−∆F The corresponding definition for defeasible inference is more complex, since there are more factors that influence • D ` +∂q iff q ∈ +∂F defeasible inference. Nevertheless, the basic idea is the • `− ∈− D ∂q iff q ∂F same. The restriction of Theorem 1 to finite propositional theo- We use ,→ to denote that the arrow of a rule is not spec- ries derives from the formulation of the proof theory; proofs ified. That is, B,→ s refers to a rule that might be strict, are guaranteed to be finite under this restriction. How- defeasible, or a defeater. ever, the bottom-up semantics and the following work do AsetS of literals is unfounded with respect to an exten- not need this restriction, and so apply to predicate defeasi- sion E and defeasible inference (or ∂-unfounded)if:For ble logic rules that represent infinitely many propositional every literal s in S, and for every strict or defeasible rule rules. Indeed, for the remainder of this paper we will take r1 : A(r1) ,→ s in D either the bottom-up semantics F as representative of Defeasible • A(r1) ∩−∂E 6= ∅,or 1 (a1,a2,a3,a4) ≤ (b1,b2,b3,b4) iff ai ⊆ bi for i =1, 2, 3, 4. • A(r1) ∩ S 6= ∅,or • there is a rule r2 : A(r2) ,→∼ s in D such that A(r2) ⊆ • defeater(Name, Head, Body),and + : ( ) → ∂E and for every rule r3 A r3 , s in D either • sup(Rule1,Rule2), ( ) ∩− 6= ∅ – A r3 ∂E ,or M consists of the following clauses. We first introduce the – r3 >r6 2. predicates defining classes of rules, namely ∆ Clearly the classes of -unfounded and ∂-unfounded sets supportive rule(Name, Head, Body):- are both closed under unions. Hence there is a greatest ∆- strict( ) ∆ Name, Head, Body . unfounded set wrt E (denoted by UD (E)), and a greatest ∂ supportive rule(Name, Head, Body):- ∂-unfounded set wrt E (denoted by UD(E)). Let UD(E)= ∆ ∂ defeasible(Name, Head, Body). (∅,UD (E), ∅,UD(E)). We define WD(E)=TD(E) ∪UD(E). rule(Name, Head, Body):- Let Iα be the elements of the (possibly transfinite) Kleene supportive rule(Name, Head, Body). sequence starting from ⊥ =(∅, ∅, ∅, ∅). {Iα | α ≥ 0} is an rule(Name, Head, Body):- increasing sequence and thus has a limit. defeater(Name, Head, Body). Let WF =(+∆WF, −∆WF, +∂WF, −∂WF) be the limit of this sequence. Then WF defines the conclusions We introduce now the clauses defining the predicates cor- of Well-Founded Defeasible Logic. If a literal q ∈ +∆WF responding to +∆, −∆, +∂,and−∂. These clauses spec- we write D `WF +∆q, and similarly with the three other ify the structure of defeasible reasoning in Defeasible Logic. sets in WF. Arguably they convey the conceptual simplicity of Defeasi- We can verify that the resulting logic is sensible in the ble Logic more clearly than does the proof theory. following sense. c1 definitely(X):- Proposition 3 Well-Founded Defeasible Logic is coherent fact(X). and consistent. c2 definitely(X):- To illustrate the definitions, consider the following Well- strict(R, X, [Y1,...,Yn]), Founded Defeasible Logic theory. definitely(Y1),...,definitely(Yn). r1 : b ⇒ a c3 not definitely(X):- r2 : ¬c ⇒ a not definitely(X). r3 : d ⇒ a defeasibly( ) r4 : a ⇒¬c c4 X :- definitely( ) r5 : true ⇒ d X . r6 : true ⇒¬d c5 defeasibly(X):- − ¬ − not definitely(∼ X), With respect to the extension E where ∂ a, ∂b and supportive rule( [ ]) −∂c hold, an unfounded set is U = {a, ¬c, d, ¬d}. Well- R, X, Y1,...,Yn , − − ¬ − defeasibly(Y1),...,defeasibly(Yn), Founded Defeasible Logic will conclude ∂a, ∂ c, ∂d not overruled( ) and −∂¬d, whereas conventional Defeasible Logic will con- S, R, X . clude only −∂d and −∂¬d. c6 overruled(S, R, X):- sup(S, R), Decomposition of Defeasible Logics rule(S, ∼ X, [U1,...,Un]), defeasibly( ) defeasibly( ) In this section we show how a defeasible logic can be de- U1 ,..., Un , not defeated( ∼ ) composed into a metaprogram specifying the structure of T,S, X . defeasible reasoning, and a semantics for the meta-language c7 defeated(T,S,∼ X):- (logic programming). We first introduce the metaprogram, sup(T,S), then the two semantics that, when composed with the supportive rule(T,X,[V1,...,Vn]), metaprogram, produce the two logics defined previously. Fi- defeasibly(V1),...,defeasibly(Vn). nally we discuss an example where the metaprogram is mod- not defeasibly( ) ified. c8 X :- not defeasibly(X). The Defeasible Logic Metaprogram The first three clauses address definite provability, while In this section we introduce a metaprogram M in a logic the remainder address defeasible provability. The clauses programming form that expresses the essence of the defea- specify if and how a rule in Defeasible Logic can be over- sible reasoning embedded in the proof theory. The metapro- ridden by another, and which rules can be used to defeat gram assumes that the following predicates, which are used an over-riding rule, among other aspects of the structure of to represent a defeasible theory, are defined. defeasible reasoning. • fact(Head), We have permitted ourselves some syntactic flexibility in presenting the metaprogram. However, there is no technical • strict( ) Name, Head, Body , difficulty in using conventional logic programming syntax • defeasible(Name, Head, Body), to represent this program. This metaprogram is similar to – though briefer and more We use P|=K α to denote that the Kunen semantics for intelligible than – the meta-interpreter d-Prolog for Defeasi- the program P supports α. ble Logic defined in (Covington et al., 1997). The d-Prolog We can now relate the bottom-up characterization of a de- meta-interpreter was designed for execution by Prolog, with feasible theory D with the Kunen semantics for the corre- the Prolog implementation of negation-as-failure. It con- sponding program D. tains many complications due to this intended use. D Given a defeasible theory D =(F, R, >), the correspond- Theorem 4 Let D be a defeasible theory and denote its ing program D is obtained from M by adding facts accord- metaprogram counterpart. ing to the following guidelines: For each literal p, 1. fact(p). for each p ∈ F ; 1. D ` +∆p iff D|=K definitely(p); `−∆ D|= not definitely( ) 2. strict(ri,p,[q1,...,qn]). 2. D p iff K p ; for each rule ri : q1,...,qn → p ∈ R; 3. D ` +∂p iff D|=K defeasibly(p); 3. defeasible(ri,p,[q1,...,qn]). 4. D `−∂p iff D|=K not defeasibly(p); : ⇒ ∈ for each rule ri q1,...,qn p R; Thus Kunen’s semantics of D characterizes the conse- 4. defeater(ri,p,[q1,...,qn]). quences of Defeasible Logic. Defeasible Logic can be de- for each rule ri : q1,...,qn ; p ∈ R; composed into M and Kunen’s semantics. 5. sup(ri,rj). This has a further interesting implication: The conse- for each pair of rules such that ri >rj . quences of predicate Defeasible Logic are computable. That is, if we permit predicates and uninterpreted function sym- Kunen Semantics bols of arbitrary arity, then the four sets of consequences are recursively enumerable. This follows from the fact Kunen’s semantics (Kunen, 1987) is a 3-valued semantics that Kunen’s semantics is recursively enumerable (Kunen, for logic programs. A partial is a mapping t f 1989). It contrasts with most nonmonotonic logics, in which from ground atoms to one of three truth values: (true), the consequence relation is not computable. (false), and u (unknown). This mapping can be extended to For propositional Defeasible Logic, we can use the rela- all formulas using Kleene’s 3-valued logic. tionship with Kunen’s semantics to establish a polynomial Kleene’s truth tables can be summarized as follows. If φ bound on the cost of computing consequences. In unpub- is a boolean combination of the atoms t, f,andu, its truth lished work we have a more precise bound. value is t iff all the possible ways of putting in t or f for the various occurrences of u lead to a value t being computed in ordinary 2-valued logic: φ gets the value f iff ¬φ gets the Well-Founded Semantics value f,andφ gets the value u otherwise. These truth values The presentation of the well-founded semantics in this sec- can be extended in the obvious way to predicate logic, think- tion is based on (Van Gelder et al., 1991). ing of the quantifiers as infinite disjunction or conjunction. The notion of unfounded sets is the cornerstone of well- The Kunen semantics of a program P is obtained from a founded semantics. These sets provide the basis to derive sequence {In} of partial interpretations, defined as follows. negative conclusions in the well-founded semantics. 1. I0(α)=u for every atom α Definition 5 Given a program P, its Herbrand base H, and 2. In+1(α)=t iff for some clause a partial interpretation I,asetA ⊆ H is an unfounded set with respect to I iff each atom α ∈ A satisfies the following β:-φ condition: For each instantiated rule R of P whose head is in the program, α = βσ for some ground substitution σ α, (at least) one of the following holds: such that 1. Some subgoal of the body is false in I. ( )=t In φσ . 2. Some positive subgoal of the body occurs in A 3. In+1(α)=f iff for all the clauses The greatest unfounded set of P with respect to I (UP (I)) β:-φ is the union of all the unfounded sets with respect to I. ( ) ( ) in the program, and all ground substitution σ,ifα = βσ, Definition 6 The transformations TP I , UP I , and ( ) then WP I are defined as follows: ( )=f In φσ . • α ∈ TP iff there is some instantiated rule R of P such 4. In+1(α)=u otherwise. that α is the head of R, and each subgoal of R is true in I. We shall say that the Kunen semantics of P supports α iff • UP (I) is the greatest unfounded set with respect to I. there is an interpretation In, for some finite n, such that • = ∪¬ ( ) ¬ ( ) In(α)=t. This semantics has an equivalent characteri- WP TP UP I ,where UP I denotes the set ob- zation in terms of 3-valued . We refer tained from UP (I) by taking the complement of each atom the reader to (Kunen, 1987) for more details. in UP (I). We are now able to introduce the notion of well-founded The meaning of such expressions can be given by appro- semantics. The well-founded semantics of a program P is priately modifying clauses c2,c5,c6 and c7 of the metapro- represented by the least fixpoint of WP . gram. If an element Yi of the body of a rule has the We write P|=WF α to mean that α receives the value t form fail∂ Zi then the body of c5 (say) should contain in the well-founded semantics of P. not defeasibly(Zi) in place of defeasibly(Yi).There- The following theorem establishes a correspondence be- sulting metaprogram is more general in that it defines a more tween Well-Founded Defeasible Logic and the well-founded expressive language but it has no different effect on theories semantics of D. that do not use the failure operators. Defeasible logic rules that do not involve explicit failure retain the same interpre- D Theorem 7 Let D be a defeasible theory and denote its tation that they had before. metaprogram counterpart. The resulting language, when we use the Kunen seman- For each ground literal p tics, generalizes both Defeasible Logic and Courteous Logic 1. D `WF +∆p iff D|=WF definitely(p); Programs (Grosof, 1997). Indeed, it was already shown in ` −∆ D|= not definitely( ) (Antoniou et al., 1998) that Courteous Logic Programs can 2. D WF p iff WF p ; be expressed by Defeasible Logic theories, by encoding uses 3. D `WF +∂p iff D|=WF defeasibly(p); of the fail operator with defeasible rules. With the explicit 4. D `WF −∂p iff D|=WF not defeasibly(p); failure operators we can express fail directly. Thus Cour- teous Logic Programs are essentially a syntactic of Thus Well-Founded Defeasible Logic can be decomposed (with the same semantics as) the extended Defeasible Logic. into M and the well-founded semantics. As a result, the consequences of a propositional Well- Future Work Founded Defeasible Logic theory can be computed in time polynomial in the size of the theory, using the fact that the This work opens up several variations of Defeasible Logic, well-founded semantics of propositional logic programs can in addition to the ones we have presented. The many dif- be computed in polynomial time (Van Gelder et al., 1991). ferent semantics for negation in logic programs have cor- However, predicate Well-Founded Defeasible Logic is not responding different semantics for the defeasible logic syn- computable, in contrast with predicate Defeasible Logic, tax. For example, the composition of stable model semantics which is computable. (Gelfond and Lifschitz, 1988) with the metaprogram might Through the relationship between Kunen’s semantics and produce a credulous version of defeasible logic. the well-founded semantics of logic programs we can es- Equally, we can vary the fundamental defeasible structure tablish the relationship between Defeasible Logic and Well- by modifying the metaprogram. Such changes will generally Founded Defeasible Logic. The latter is an extension of De- not alter the computational complexity, since it is the seman- feasible Logic in the sense that it respects the conclusions tics of the meta-language which has the dominant effect on that are drawn by Defeasible Logic but generally draws more complexity. conclusions from the same syntactic theory. The results also open up alternative implementations of defeasible logics. One possibility is to execute the metapro- Theorem 8 Let D be a defeasible theory. For every conclu- gram and data in a logic programming system with the ap- sion C, propriate semantics. The connection between Defeasible if D ` C then D `WF C Logic and Kunen’s semantics suggests an implementation incorporating constructive negation (Stuckey, 1995). Defeasible Logic with Explicit Failure Although the meaning of conclusions −∂p and −∆p is ex- Conclusion pressed in terms of failure-to-prove, there is not a way to We have provided a semantic decomposition of defeasible express directly within the above defeasible logics that a logics into two parts: a metaprogram which specifies the literal should fail to be proved; tagged literals are not per- fundamental defeasible structure of the logic (e.g. when a mitted in rules. In contrast, most logic programming-based rule can be defeated by another), and a semantics which de- formalisms employ “negation as failure” to express directly termines how the meta-language is interpreted. that a literal should fail. We showed that Nute’s Defeasible Logic is character- The characterization of defeasible logics by a metapro- ized by Kunen’s semantics and that Well-Founded Defea- gram and a semantics for logic programs provides a way sible Logic is characterized by the well-founded semantics. for these logics to be extended with explicit failure without We also briefly developed a variant of Defeasible Logic with modifying their underlying semantics (i.e. a conservative explicit failure by modifying the metaprogram. Thus differ- extension). We introduce two operators on literals: fail∆ q ent defeasible logics can be obtained by varying either the which expresses that it should be proved that the literal q metaprogram or the semantics of the meta-language. cannot be proven definitely, and fail∂ q which expresses Decomposition is a useful tool for the analysis and com- that it should be proved that q cannot be proven defeasibly. parison of logics. 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