Well-Founded Semantics for Defeasible Logic

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Well-Founded Semantics for Defeasible Logic FREDERICK MAIER∗ Florida Institute for Human and Machine Cognition 40 S Alcaniz St Pensacola, FL 32502

DONALD NUTE Institute for Artificial Intelligence The University of Georgia Athens, GA 30605

Abstract Fixpoint semantics are provided for ambiguity blocking and propagating variants of Nute’s defeasible logic. The semantics are based upon the well-founded semantics for logic programs. It is shown that the logics are sound with respect to their counterpart semantics and complete for locally finite theories. Unlike some other nonmonotonic reasoning formalisms such as Reiter’s default logic, the two defeasible logics are directly skeptical and so reject floating conclusions. For defeasible theories with transitive priorities on defeasible rules, the logics are shown to satisfy versions of Cut and Cautious Monotony. For theories with either conflict sets closed under strict rules or strict rules closed under transposition, a form of Consistency Preservation is shown to hold. The differences between the two logics and other variants of defeasible logic—specifically those presented by Billington, Antoniou, Governatori, and Maher—are discussed. KEYWORDS: defeasible logic, logic programming, well-founded semantics, stable model semantics, ambiguity blocking and propagation.

1 Introduction An argument is deductively valid if it is impossible for its premises to be true and its conclusion to be false. The notion of deductive validity is central to first-order logic and standard modal logics. Logical systems intended to capture deductive validity must be monotonic: if the truth of some set of premises S guarantees the truth of a proposition p, then the truth of any set of premises T that contains S must also guarantee the truth of p. But a proposition may be true even though a particular agent is not justified in believing it, and an agent may be justified in believing a proposition even though it is false. If we allow that there can be ∗ A completed version of this work appears as: Frederick Maier and Donald Nute. Well-founded semantics for defeasible logic. Synthese, 176(2): 243-274, 2010. Available at www.springer.com. http://dx.doi.org/10.1007/s11229-009-9492-1

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Frederick Maier and Donald Nute

justified false beliefs and also that it should at least sometimes be possible to discover that a belief we thought justified was indeed false, then we must conclude that a logic of epistemic justification must be nonmonotonic. For suppose that some epistemic state E justifies belief in a false proposition p and that I represents some information that when discovered by an agent in epistemic state E would allow the agent to recognize that p is false. Then the epistemic state produced by learning I in E would not justify belief in p. It is in this sense that epistemic justification is not monotonic. Members of the Artificial Intelligence community have long recognized, at least implicitly, that the logic of epistemic justification must be nonmonotonic. Or at least they have recognized that modeling anything like human reasoning requires a system that is nonmonotonic, and this amounts to the same thing. The development of systems of nonmonotonic reasoning among the AI community is at least in part motivated by the recognition that agents may be justified in accepting p at some point in time yet justified in rejecting p at a later point in time, not because they have discovered that some of their original evidence for p was false but because they have learned new, different information that rebuts or undercuts p. There have been other reasons for developing nonmonotonic formalisms, but the nonmonotonicity of epistemic justification is the primary motivation for the systems developed in this paper. We should like an approach to nonmonotonic reasoning to provide both a semantics that explains the meanings of the epistemic states the system describes and a constructive proof theory for deriving which propositions are justified by a particular epistemic state. Most approaches have provided one or the other of these but not both. The defeasible logics described in this paper were originally developed proof theoretically, but until now no completely satisfactory semantics was available. Other systems such as inheritance networks (Horty et al. 1990), default logic (Reiter 1980), autoepistemic logic (Moore 1985), and various semantics for logic programming such as the stable model semantics (Gelfond and Lifschitz 1988) have focused on the semantics of nonmonotonic theories and not on producing constructive proof methods. In this paper, we will do several things. We will describe two versions of defeasible logic. We will discuss how they resemble and differ from other nonmonotonic formalisms, and how they differ from the variants of defeasible logic discussed elsewhere, specifically those widely published by David Billington, Grigoris Antoniou, and others (Antoniou et al. 2000a; 2000b; 2001; 2006; Billington 1993; Governatori et al. 2004; Maher and Governatori 1999; Maher et al. 2001). We will also present semantics for these two logics. The semantics presented here are versions of the well-founded semantics developed for logic programs (Van Gelder et al. 1991). We will show that our logics are sound and for (locally) finite theories complete with respect to their semantics. We will also show that these logics are not always complete where infinite theories are involved, and we will discuss why this should be the case.

Well-Founded Semantics for Defeasible Logic

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2 Defeasible logic As is usually the case for most authors writing on nonmonotonic reasoning, we will present propositional versions of the logics. The language of defeasible logic is made up of literals and rules. A literal is either an atom (a ground atomic formula of first-order logic) or the negation of an atom. A rule is a triple made up of sets of literals and one of the connectors →, ⇒, or . Rules are often written using variables, but we will treat these as standing for the sets of their ground instances. Definition 1 If S is a finite set of literals and p is a literal, then S → p is a strict rule, S ⇒ p a defeasible rule, and S p an undercutting defeater (or simply a defeater). In each of these cases, S is the body of the rule and p is the head of the rule. Where r is either S → p, S ⇒ p, or S p, we will let body(r ) = S and head (r ) = p. We may read S → p as “If S , then definitely p,” S ⇒ p as “If S , then defeasibly (normally, apparently, evidently) p”, and S p as “If S, then maybe p. Strict rules with empty bodies are facts and defeasible rules with empty bodies are called presumptions. ∅ ⇒ p is read “Presumably, p.” Strict and defeasible rules are applied in defeasible logic in the usual way, that is, if we can derive the body of the rule, then we also derive the head of the rule. But a central notion of defeasible logic is that a defeasible rule may be defeated by other rules. For example, consider the two rules “If Squeaky has fur, then defeasibly Squeaky does not fly” and “If Squeaky has wings, then defeasibly Squeaky flies”. We cannot, on pain of contradiction, detach the heads of both of these rules even if we know that Squeaky has both fur and wings. So to detach the conclusion “Squeaky doesn’t fly” when we know that Squeaky has fur, we need to check to see whether the body of the second rule, “Squeaky has wings”, is derivable. To apply one rule, we need to show that the other rule cannot be applied. Thus, we will develop a consequence relation |∼ and a “non-consequence” relation ∼|. Where T is a defeasible theory (defined below) and p a literal, T |∼ p indicates that p is defeasibly derivable from T and T ∼| p indicates that it can be demonstrated that p is not defeasibly derivable from T . Thus, defeasible logic is a system of proofs and refutations. The role of defeaters is solely to defeat other arguments that might otherwise establish a literal. An undercutting defeater can prevent the application of a defeasible rule when the head of the undercutting defeater and the head of the defeasible rule are incompatible. For example, the defeater ¬ has-intact-flight-feathers ¬flies might be used to prevent a proof of flies from bird ⇒ flies, but it cannot be used to directly prove ¬flies. A different notion of an undercutting defeater was developed independently by John Pollock (1974). Each of Pollock’s undercutting defeaters undercuts a specific rule that is named in the head of the defeater. Our language does not include expressions that refer to other expressions in the language. However, defeasible logic still allows control over which defeasible rules may actually be undercut by a defeater with a conflicting head. If we do not want a defeasible rule to be undercut by a defeater, then we can specify that the defeasible rule takes

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priority or has precedence over the defeater. For example, we could give bat ⇒ flies priority over the defeater ¬ has-intact-flight-feathers ¬flies. Rules are collected together to form defeasible theories. Defeasible theories also include what are called conflict sets, which are sets of literals that cannot consistently hold. A collection of rules conflict if their heads form a conflict set. Theories also specify the precedence (i.e., priority, preference) relation over rules described above. Different versions of defeasible logic specify different constraints on each of these components and determine how they are used to prove and refute literals. For a given theory D, we use At(D) to refer to the set of atoms appearing in the rules and conflict sets of D, and Lit(D) to refer to these atoms and their negations. Definition 2 A defeasible theory D is a triple ⟨R, C , ≺⟩, where R is a countable set of rules, C a countable set of finite sets of literals in Lit(D) such that for any literal p ∈ At(D), {p, ¬p} ∈ C , and ≺ an acyclic binary relation over the non-strict rules in R. The set C is the set of conflict sets. A set of the form {p, ¬p} is called a minimal conflict set, and we say that C is minimal in D (and write CMIN ) if it contains only minimal conflict sets. We say that conflict sets (and by extension, defeasible theories) are closed under strict rules if, for all c ∈ C , if A → p is a rule and p ∈ c, then {A ∪ (c − {p})} ∈ C . It is not a necessary condition that a defeasible theory be closed under strict rules, and indeed it is computationally expensive to have them closed, but it is often a necessary prerequisite to drawing reasonable conclusions from a theory. The nonminimal sets obtained by closing the theory under the strict rules are called extended conflict sets. For a given theory, we use Rs , Rd , and Ru to denote the strict, defeasible, and defeater rules of R, respectively. The sets Rs [p], Rd [p], and Ru [p] refer to those rules with head p. C [p] denotes the set of conflict sets containing p. 2.1 Direct skepticism All the versions of defeasible logic that have been created to date are directly skeptical formalisms: In order to derive a literal p there must be some rule for it with a body that is also derived. In other words, the justification for a literal should itself be justified. This can be contrasted with many other formalisms such as default logic (Reiter 1980) and the stable model/answer-set semantics for logic programs (Gelfond and Lifschitz 1988; 1991), where consequences are defined indirectly as the intersection of extensions (with each system defining extensions in its own way). These formalisms allow what Makinson and Schlechta (1991) call floating conclusions, which are consequences having no support common to every extension. Directly skeptical formalisms generally have no mechanism to allow floating conclusions. Consider Ginsberg’s extended Nixon Diamond, cast here as defeasible theory. Example 1 D = ⟨R, CMIN , ∅⟩, where R is

Well-Founded Semantics for Defeasible Logic 1. 2. 4. 6. 8.

∅ → Nixon {Nixon} → Republican {Quaker } ⇒ Dove {Hawk } → ¬Dove {Hawk } → Extremist

3. 5. 7. 9.

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{Nixon} → Quaker {Republican} ⇒ Hawk {Dove} → ¬Hawk {Dove} → Extremist

If we momentarily disregard defeaters and priorities among rules, then extensions for defeasible theories can be defined in a straightforward way, similar to using the Gelfond-Lifschitz transformation of (Gelfond and Lifschitz 1988). Definition 3 A set S ∈ Lit(D) is an extension of D = ⟨Rs ∪ Rd , C , ∅⟩ iff S = Cn(D S ), where 1. D S = Rs ∪ {r |r ∈ Rd & ∀c ∈ C [head (r )](c − head (r ) * S )}, and 2. Cn(D S ) is the closure of the rules of D S . The definition yields two extensions for the example, E1 and E2 . E1 : {Nixon, Republican, Quaker , Dove, ¬Hawk , Extremist} E2 : {Nixon, Republican, Quaker , ¬Dove, Hawk , Extremist} Since Extremist is in each extension, it is a consequence of the theory under the indirectly skeptical view. However, since no rule for Extremist has a body that appears in both extensions, Extremist is a floating conclusion. No version of defeasible logic derives it. The majority of researchers accept floating conclusions as reasonable and view direct skepticism’s inability to derive floating conclusions as a defect. Makinson and Schlechta (1991) call the directly skeptical approach “just wrong”. The issue appears in the context of semantics for logic programs, where it is sometimes said that the well-founded semantics is overly weak (drawing fewer conclusions than is reasonable). Brewka (1996) writes that the WFS “provides a very poor approximation to the answer–set semantics.” The suggestion, apparently, is that the standard to which the WFS should be compared is the indirectly skeptical answer-set semantics. Acceptance of floating conclusions is not universal, however. Defeasible logic rejects them, as do other proponents of direct skepticism such as Horty (2002) and Antonelli (2005). Horty’s position, expressed in the case of the Nixon example, is that the two possibilities undermine each other. It is possible that this individual would adopt an extreme position, as either a dove or a hawk. But it seems equally reasonable to imagine that such an individual, rather than being pulled to one extreme or the other, would combine elements of both views in a more balanced, measured position falling toward the center of the political spectrum (Horty 2002).

Prakken (2002) opposes this and writes that reliance on intuition pumps of the above sort can be a very dangerous affair, and that intuitions in many other examples support floating conclusions. He also writes that in at least some of Horty’s examples there is implicit information which, when made explicit, would prevent one from drawing the floating conclusion. One may achieve this in the present example by adding the below rules, which state that Quakers who are also Republicans

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usually are neither hawks nor doves. The modified theory has a third extension, one in which Dove, Hawk , and Extremist do not appear. 10. {Republican, Quaker } ⇒ ¬Dove 11. {Republican, Quaker } ⇒ ¬Hawk We agree that one may be misled by poor examples or faulty intuitions, but notice that opponents on both sides of this issue depend on examples and intuitions to support their positions. We have not found in the literature disciplined arguments either for or against floating conclusions that do not rely on these intuition pumps. Furthermore, whichever position one takes, one must then attempt to explain away apparent counterexamples. If we restrict ourselves to the formal structure of a given example, all we know is that there is no support for the floating conclusion that is free of conflict. Defeasible logic takes the position that in that case, one should not consider a statement justified. Direct skepticism does have one formal advantage: it clearly allows us to create a constructive proof procedure for nonmonotonic reasoning. By their nature, floating conclusions cannot be directly supported by the most obvious sorts of direct proofs. A strategy for overcoming this might be some sort of reasoning by cases. But we have not seen a constructive proof procedure that captures the indirectly skeptical approach. 2.2 Ambiguity Defeasible logic also differs from many other nonmonotonic reasoning formalisms in the way it has traditionally handled ambiguity. Consider the defeasible theory below. Example 2 D = ⟨R, CMIN , ∅⟩, R is 1. ∅ ⇒ p

2. ∅ ⇒ ¬p

3. {p} ⇒ ¬q

4. ∅ ⇒ q

The first two rules are vacuously supported and they also conflict. We cannot detach their heads on pain of contradiction. Furthermore, there is no mechanism for choosing between p and ¬p. The two literals are said to be ambiguous, and there is some debate in the literature regarding the proper handling of ambiguity. It is clear that neither p nor ¬p should be considered justified, that is, neither D |∼ p nor D |∼ ¬p should hold. One possible course of action is to consider both p and ¬p as refuted, which effectively blocks or localizes the ambiguity to just those literals. This is the course taken by Horty in (Horty et al. 1990) and by most forms of defeasible logic. If one does this, then since p is refuted, all support for ¬q vanishes, and only q is left with any support. Indeed, under the ambiguity blocking view, q is proved while ¬q is refuted. Neither literal is ambiguous. Another course of action is simply to refrain from concluding D ∼| p and D ∼| ¬p. Since the status of ¬q depends upon resolving the status of p, the ambiguity of p is effectively propagated to ¬q. Adopting ambiguity propagation yields in a sense a more extreme form of skepticism, in that fewer conclusions can be drawn. In the


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example, p might hold (there is conflicting information about it and no way to resolve the conflict), and if it does hold, then there would be evidence for both q and ¬q, and so q and ¬q would be ambiguous. In the ambiguity propagating view, the safe course of action is to conclude nothing at all about p. Most authors prefer propagating ambiguity to blocking it. In 1987 Pollock wrote that defeasible logic was the only formalism he could think of that blocks ambiguity (the major papers on inheritance networks appeared at roughly this time)(Pollock 1987). The issue of ambiguity is discussed in detail in (Stein 1990). Since it is ambiguity propagation that seems to have the greatest support, we will consider some reasons for adopting ambiguity blocking instead. We will do this by considering two kinds of reasons that have been or could be given for not inferring q in the example above. One reason is that in an example like 2, we have equally good evidence for and against p. So p is ambiguous, and it’s ambiguity infects q. However, if we delete the first two rules in example 2, we still have equally good evidence for and against p, namely, none. The claim is that p is ambiguous if we have evidence both for and against it and we can’t resolve the situation, but p is not ambiguous if we have no evidence for or against it and we can’t resolve the situation. In either case, we cannot tell whether we should accept p or accept ¬p. There is certainly a technical difference in the two cases, but the proponent of blocking ambiguity cannot see that the technical difference is a difference that should matter. The second reason is that the proponent of propagating ambiguity may point out that there are models for example 2 that contain q and models that do not. That is certainly true if by “models” we mean default extensions. And this does allow us to embrace ambiguity propagation and floating conclusions at the same time. But we will see later in this paper that this theory, and all defeasible theories, have a unique well-founded model. We will also see that, depending on how we define well-foundedness, this model may either block or propagate ambiguity although of necessity there can be no floating conclusions. So we cannot decide between blocking or propagating ambiguity on this basis since it depends on what kind of model we decide to use for our theories. In the next section we will describe two almost identical defeasible logics. The first logic blocks ambiguity, but by modifying its structure slightly an ambiguity propagating logic is produced. Proponents of propagating ambiguity have sometimes rejected defeasible logic because the most familiar versions block ambiguity, but at best that is just a reason for rejecting the ambiguity blocking versions. Section 4 presents semantics for both ambiguity blocking and ambiguity propagating versions of defeasible logic. 3 Proof systems The basic idea behind the proof system of each defeasible logic is that a literal p can be derived from a defeasible theory just in case p is the head of some strict or undefeated defeasible rule in the theory and all of the literals in the body of the rule are also derivable. The systems we will discuss are Nute’s ambiguity blocking logic

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NDL (Nute 1999; 2003; Donnelly 1999) and its ambiguity propagating counterpart ADL (Maier and Nute 2006). Proofs form argument trees with nodes labeled with tagged literals. For a given node n, we will use child (n) to refer to the set of its children, and label (n) to refer to the label of n. Earlier systems of defeasible logic began to appear in the late 1980’s (Nute, 1986; 1987; 1994; Covington et al. 1997). David Billington in (Billington 1993) presents a quantified version of one of Nute’s logics and shows it to be cumulative. Billington, together with Grigoris Antoniou, Michael Maher, Guido Governatori, and others, would later go on to publish dozens of papers on this logic and its many offshoots, studying their formal properties, developing implementations, and relating them to alterative formalisms (Antoniou et al. 2000a; 2000b; 2001; 2006; Governatori et al. 2004; Maher et al. 1999; 2001). In these variants of defeasible logic, proofs are typically linear sequences of tagged literals. Definition 4 Let D be a defeasible theory. A defeasible argument tree for D is a finite tree τ such that every node of τ is labeled with one of +p or −p, where p is any literal in Lit(D). If τ is a defeasible argument tree for D and n is a node in τ , then τ is a positive node iff n is labeled +p, and n is a negative node iff n is labeled −p. Definition 5 Let A be a set of literals, and n a node of a defeasible argument tree τ . 1. A succeeds at n iff for all q ∈ A, there is a child of n labeled +q. 2. A fails at n iff there is a q ∈ A and a child of n labeled −q. A tree over D with root +p indicates that p is defeasibly derivable from D; a tree over D with root −p indicates that p is defeasibly refuted. The different defeasible logics specify different conditions that nodes in the tree must satisfy in order for the tree as a whole to constitute a proof in the logic. The proof conditions for NDL are described below. NDL is the first defeasible logic to incorporate a cycle check, thereby weeding out circular arguments. Maher and Governatori (1999) present a well-founded semantics for the variant of defeasible logic appearing in (Billington 1993). The semantics—since it is based on a notion of well-foundedness—naturally detects circular arguments; however, since proofs are linear sequences of literals, the logic of (Billington 1993) cannot. NDL is also the first to introduce conflict sets to deal with indirect incompatibilities between rules. Definition 6 An argument tree τ over defeasible theory D is an NDL-proof for D iff for each node n of τ , one of the following obtains. 1. label (n) = +p and either a. there is an r ∈ Rs [p] such that body(r ) succeeds at n, or b. there is an r ∈ Rd [p] such that i. body(r ) succeeds at n, and


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ii. for all c ∈ C [p] there is a q ∈ c − {p} such that for all s ∈ R[q], either body(s) fails at n or else s ≺ r . 2. label (n) = −p and a. for all r ∈ Rs [p], body(r ) fails at n, and b. for all r ∈ Rd [p], either i. body(r ) fails at n, or ii. there is a c ∈ C [p] such that for all q ∈ c − {p}, there is a s ∈ R[q] such that body(s) succeeds at n and s ⊀ r . 3. label (n) = −p and n has an ancestor m in τ with label (m) = −p, and all nodes between n and m are negative. The third condition is called failure-by-looping. Since conclusions cannot be established by circular arguments, failure-by-looping can help to show that a literal cannot be derived from a defeasible theory. Earlier versions of defeasible logic lacked this feature, and its absence helps these logics to have low complexity (e.g. the logic discussed in (Maher et al. 2001) is shown to have linear complexity). However, it also entails that the logics fail to draw reasonable conclusions in some cases. Example 3 D = ⟨R, CMIN , ∅⟩, R is 1. ∅ → mammal 3. {bat} ⇒ furry 5. {bat} ⇒ flies

2. {furry, has wings} ⇒ bat 4. {bat} ⇒ has wings 6. {mammal } ⇒ ¬flies

In earlier versions of defeasible logic, although we could easily see that there was no way to show D |∼ bat, we could not demonstrate this in the proof theory. That is, we could not show D ∼| bat (bat would be undetermined, neither proven nor refuted.) Consequently, neither could we show D |∼ ¬flies. Failure-by-looping provides a mechanism for showing D ∼|NDL bat, which then allows us to show D |∼NDL ¬flies. The requirement in failure-by-looping that the nodes between n and m are all negative is needed to ensure that literals are not simultaneously provable and refutable. For instance, in the below theory, if the requirement is omitted, then p can be both derived and refuted. The sequence +p, −q, +p, −q (interpreted as a linear tree) shows D |∼ p, while −p, +q, −p shows D ∼| p. Example 4 D = ⟨R, CMIN , ∅⟩, R is 1. ∅ ⇒ p

2. ∅ ⇒ q

3. {q} ⇒ ¬p

4. {p} ⇒ ¬q

The NDL-consequences of Example 1 are D |∼NDL Nixon, D |∼NDL Republican, and D |∼NDL Quaker , but also D ∼|NDL Dove, D ∼|NDL Hawk , and D ∼|NDL Extremist. That Extremist, Dove and Hawk are all refuted indicates that NDL both blocks ambiguity and rejects floating conclusions. In Example 2, we have D ∼|NDL p, D ∼|NDL ¬p, D ∼|NDL ¬q, and D |∼NDL q. As was noted before, most

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researchers feel that ambiguity propagation, which in Example 2 would prevent concluding q, is intuitively more reasonable. Brewka (2001) quickly dismisses an earlier version of defeasible logic precisely because it is ambiguity blocking. It turns out, however, that a very minor modification to the proof system of NDL produces an ambiguity propagating defeasible logic—ADL. The modification creating ADL affects only part 2.b.ii of Definition 6. It specifies that p is defeated only if every defeasible rule in support of p fails or else is defeated by a satisfied strict rule or a satisfied defeasible rule of higher precedence for each element q ∈ c − {p} (for some c ∈ C [p]). In NDL, a rule whose precedence was either higher or incomparable could be used. The modified proof theory is shown below. Definition 7 An argument tree τ for D is an ADL-proof for D iff each node n of τ satisfies conditions 1 or 3 for NDL, or else satisfies condition 2 below: 2. label (n) = −p and a. for all r ∈ Rs [p], body(r ) fails at n, and b. for all r ∈ Rd [p], either i. body(r ) fails at n, or ii. there is a c ∈ C [p] such that for all q ∈ c − {p}, there is a s ∈ R[q] such that body(s) succeeds at n and s is strict or else r ≺ s. Apart from this modification, the proof system is left unchanged. It can be readily seen that every proof in ADL is a proof in NDL. Definition 8 Let D be a defeasible theory, L either NDL or ADL, and τ an L-proof for D. 1. τ is an L-proof of p in D iff τ is an L-proof for D, p ∈ Lit(D), and the root node of τ is labeled +p. 2. τ is an L-refutation of p in D iff τ is an L-proof for D, p ∈ Lit(D), and the root node of τ is labeled −p. 3. D |∼L p iff there is an L-proof of p in D. 4. D ∼|L p iff there is an L-refutation of p in D. 5. If S ⊆ Lit(D), then D |∼L S iff for all p ∈ S , D |∼L p. 6. If S ⊆ Lit(D), then D ∼|L S iff there is a p ∈ S such that D ∼|L p. Proposition 1 For all defeasible theories D, 1) if D |∼ADL p, then D |∼NDL p, and 2) if D ∼|ADL p, then D ∼|NDL p. Proof Seen by examination of definitions for proof in NDL and ADL. In Example 2, since the rules for p and ¬p are incomparable, neither D ∼|ADL p nor D ∼|ADL ¬p can be shown (nor can D |∼ADL p or D |∼ADL ¬p.) Because D ∼|ADL p cannot be shown, D ∼|ADL ¬q cannot be shown, and so neither can D |∼ADL q. Each literal is underdetermined in ADL.


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4 Well-founded semantics for ADL and NDL The consequences of defeasible theories have traditionally been specified proof theoretically. An adequate semantic counterpart to NDL (or ADL) has not existed until now. The two semantics described here are based on the well-founded semantics, a fixpoint semantics originally specified for normal logic programs (Van Gelder et al. 1991). The proof systems for NDL and ADL are sound relative to their counterpart semantics, and while completeness does not hold in general, NDL and ADL are complete for the class of locally finite theories, defined below. We note that a previous fixpoint semantics for NDL was described by Sam Donnelly (1999). However, while NDL is sound relative to this semantics, it is not complete even for finite theories. This is not surprising, since the semantics in (Donnelly 1999) is indirectly skeptical and allows floating conclusions. As this is so, for some finite theories the logic cannot derive all the consequences according to the semantics.

4.1 Unfounded sets and immediate consequences Definition 9 An interpretation I of a defeasible theory D is an ordered pair I = ⟨T, U⟩, where T ⊆ Lit(D) and U ⊆ Lit(D). An interpretation I is coherent iff T ∩ U = ∅. When speaking of multiple interpretations, we will sometimes use integer or ordinal subscripts, as in In = ⟨Tn , Un ⟩. Interpretations can be ordered according to the amount of information they provide about literals: I1 ⊑ I2 if and only if both T1 ⊆ T2 and U1 ⊆ U2 . This is often called the knowledge or Fitting ordering. The semantics here is based upon the notion of an unfounded set of literals. Intuitively, these are collections of literals for which no external support exists. The only way to prove an unfounded literal is to use literals that are themselves unfounded. This idea is formalized below. Definition 10 A set S ⊆ Lit(D) is unfounded in NDL with respect to D and an interpretation I = ⟨T, U⟩ iff for all literals p ∈ S : 1. For every r ∈ Rs [p], body(r ) ∩ (U ∪ S ) ̸= ∅. 2. For every r ∈ Rd [p], (a) body(r ) ∩ (U ∪ S ) ̸= ∅, or (b) there is a c ∈ C [p] such that for each q ∈ c − {p} there is a rule s ∈ R[q] such that i) body(s) ⊆ T and ii) s ⊀ r . Definition 11 (Unfounded Sets in ADL) The definition of unfounded set in ADL is exactly the same as for NDL, save that condition 2b.ii is replaced with the following requirement: r ≺ s or s is strict.

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Example 5 D = ⟨R, CMIN , ∅⟩, R is 1. ∅ ⇒ p

2. ∅ ⇒ ¬p

3. {p} ⇒ r

4. {q} ⇒ q

In Example 5, the sets {p, ¬p}, {q}, and {p, ¬p, q, r } are all unfounded sets under NDL with respect to D and ⟨∅, ∅⟩. However, only {q} constitutes an unfounded set under ADL. Since neither rule 1 nor 2 is preferred to the other, {p, ¬p} is not unfounded under ADL, and since p is not unfounded, neither is r . Unfounded sets are closed under union. Since this is so, then for a given defeasible theory D and interpretation I, there exists a greatest unfounded set wrt D and I. We define this below as an operator UD , which produces sets of literals from interpretations. Definition 12 For defeasible theory D and interpretation I, UD (I) =

∪ {S | S is an unfounded set wrt to D and I}

Lemma 1 If S is a set of sets unfounded in NDL (ADL) wrt to defeasible theory D and ∪ interpretation I = ⟨T, U⟩, then S is unfounded in NDL (ADL) wrt to D and I. Proof ∪ Suppose p ∈ S . Let Sj ∈ S such that p ∈ Sj . If r ∈ Rs [p], then since Sj is ∪ ∪ unfounded, body(r ) ∩ (Sj ∪ U) ̸= ∅. But since Sj ⊆ S , body(r ) ∩ ( S ∪ U) ̸= ∅. ∪ Similarly, if r ∈ Rd [p], then either body(r ) ∩ ( S ∪ U) ̸= ∅ or there is a c ∈ C [p] such that for all u ∈ c − {p}, there is some rule s such that body(s) ⊆ T and s ⊀ r ∪ (for ADL, r ≺ s or s is strict). Generalizing on p, it follows that S is unfounded in NDL (ADL) with respect to D and I. The operator UD can be viewed as producing the set of literals that are unfounded with respect to an interpretation. The immediate consequence operator TD , familiar from logic programming and defined below for defeasible theories, generates the set of literals that are well-founded with respect to that interpretation. Definition 13 If D is a defeasible theory and I = ⟨U, T⟩ an interpretation, then a rule r ∈ RD is a witness of provability for p wrt D and I if one of the below conditions applies. 1. r ∈ Rs [p] and body(r ) ⊆ T. 2. r ∈ Rd [p] and body(r ) ⊆ T, and for each conflict set c ∈ C [p], there exists a q ∈ c − {p} such that for all s ∈ R[q], s ≺ r or body(s) ∩ U ̸= ∅.


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Definition 14 If D is a defeasible theory and I an interpretation, then the immediate consequences of D wrt I, written TD (I) is the set TD (I) = {p| there exists a witness of provability for p wrt D and I}. Observe that TD (I) is defined identically for both NDL and ADL, whereas UD (I) is (implicitly) defined differently for each (because each has a slightly different notion of unfounded set). Regardless of the definition used, however, TD (I) and UD (I) are monotone. The proof of this proceeds in large part identically for both ADL and NDL (and when needed, the differences are indicated). Lemma 2 For any defeasible theory D and interpretations I1 and I2 , if I1 ⊑ I2 , then 1. TD (I1 ) ⊆ TD (I2 ), and 2. UD (I1 ) ⊆ UD (I2 ). Proof Suppose I1 ⊑ I2 and let p ∈ TD (I1 ). Then either there exists an r ∈ Rs [p] such that body(r ) ⊆ T1 , or there exists a rule r ∈ Rd [p] such that body(r ) ⊆ T1 and for each conflict set c ∈ C [p], there exists a q ∈ c − {p} such that for all s ∈ R[q], s ≺ r or body(s) ∩ U1 ̸= ∅. Since I1 ⊑ I2 , T1 ⊆ T2 and U1 ⊆ U2 . By substitution, we have r ∈ Rs [p], and body(r ) ⊆ T2 , or r ∈ Rd [p] and body(r ) ⊆ T2 and for each conflict set c ∈ C [p], there exists a q ∈ c − {p} such that for all s ∈ R[q], s ≺ r or body(s) ∩ U2 ̸= ∅. It can be seen that p ∈ TD (I2 ). Generalizing on p, TD (I1 ) ⊆ TD (I2 ). Suppose now that p ∈ UD (I1 ). Since UD (I1 ) is the greatest unfounded set in NDL (ADL) wrt D and I1 , 1. For every r ∈ Rs [p], body(r ) ∩ (U1 ∪ UD (I1 )) ̸= ∅. 2. For every r ∈ Rd [p], (a) body(r ) ∩ (U1 ∪ UD (I1 )) ̸= ∅, or (b) There is a c ∈ C [p] such that for each q ∈ c − {p} there is a rule s ∈ R[q] such that body(s) ⊆ T1 and s ⊀ r (for ADL, r ≺ s or s is strict ). Since T1 ⊆ T2 and U1 ⊆ U2 , we can substitute T2 for T1 and U2 for U1 . Generalizing on p, we see that UD (I1 ) is unfounded in NDL (ADL) wrt D and I2 . So by the definition of UD , UD (I1 ) ⊆ UD (I2 ). Both TD (I) and UD (I) produce sets of literals, and these can be composed to form a monotone operator WD (I) which maps interpretations to interpretations. WD is itself used to define a unique sequence of interpretations. Definition 15 WD (I) = ⟨TD (I), UD (I)⟩

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Definition 16 If D is a defeasible theory, then (ID ) = (ID,0 , ID,1 , . . .) is the sequence of interpretations such that ID,0 = ⟨∅, ∅⟩ ID,α+1 = WD (ID,α ) (for successor ordinals α + 1) ID,α = lub({ID,β |β < α}) (where α is a limit ordinal). It is clear from the definition of (ID ) and the monotonicity of TD and UD that this sequence is itself monotonically increasing. Importantly, the sequence is also coherent, in the sense that for any literal p and interpretation ⟨TD,λ , UD,λ ⟩ in this sequence, p cannot both be in TD,λ and UD,λ . The proof of this is similar to one for logic programs appearing in (Van Gelder et al. 1991). Note that here, it does not matter at all whether unfounded sets are defined according to NDL or ADL. Lemma 3 If D is a defeasible theory, then for all λ ≥ 0, TD,λ ∩ UD,λ = ∅. Proof For λ = 0, the lemma holds. Suppose it holds for all α < λ and let λ be a successor ordinal. Let A be some set of literals such that A ∩ Tλ ̸= ∅. Since A ∩ Tλ ̸= ∅, there must be some κ ≤ λ such that Iκ is the earliest in the sequence (I0 , I1 , . . .) for which Tκ ∩ A ̸= ∅. For all ι < κ, Tι ∩ A = ∅. Let p be a literal such that p ∈ A and p ∈ Tκ . Then by definition of immediate consequence there must be a witness of provability for p wrt D and Iι for some for some ι < κ. As this is so, either definition 13.1 or 13.2 holds. Suppose it’s 13.1. Then there is an r ∈ Rs [p] such that body(r ) ⊆ Tι . Since body(r ) ⊆ Tι and (ID ) is monotone increasing, body(r ) ⊆ Tλ−1 . By the inductive hypothesis, body(r ) ∩ Uλ−1 = ∅. By choice of A, A ∩ Tι = ∅, and so body(r ) ∩ A = ∅. Since p ∈ A and there exists an r ∈ Rs [p] such that body(r ) ∩ (Uλ−1 ∪ A) = ∅, A fails the definition of unfounded set with respect to D and Iλ−1 . Suppose it’s 13.2. Then there is an r ∈ Rd [p] such that body(r ) ⊆ Tι for some ι < κ and for each conflict set c ∈ C [p], there exists a q ∈ c − {p} such that for all s ∈ R[q], s ≺ r or body(s) ∩ Uι ̸= ∅. By choice of A, A ∩ Tι = ∅. So, body(r ) ∩ A = ∅. Since body(r ) ⊆ Tι and (ID ) is monotone, body(r ) ⊆ Tλ−1 . Since body(r ) ⊆ Tλ−1 , then by the inductive hypothesis body(r ) ∩ Uλ−1 = ∅. Also, if body(s) ∩ Uι ̸= ∅, then body(s) ∩ Uλ−1 ̸= ∅. If this is so, then by the inductive hypothesis, body(s) * Tλ−1 . Generalizing on s, there exists a p ∈ A and a rule r ∈ Rd [p] such that body(r ) ∩ (Uλ−1 ∪ A) = ∅ and for all conflict sets c ∈ C [p] there is a q for which for all rules s ∈ R[q], s ≺ r or else body(s) * Tλ−1 . A again violates the definition of unfounded set with respect to D and Iλ−1 . Generalizing on A, no set intersecting Tλ is unfounded with respect to D and Iλ−1 . Conversely, no set unfounded wrt D and Iλ−1 intersects Tλ . In particular, Uλ —the greatest unfounded set with respect to D and Iλ−1 —does not intersect Tλ . We conclude that Tλ ∩ Uλ = ∅. If λ is a limit ordinal, then if Tλ ∩ Uλ ̸= ∅, for some p we have p ∈ Tλ and


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p ∈ Uλ . By definition of Iλ , there must be a least successor ordinal κ < λ such that p ∈ Tκ and p ∈ Uκ . This contradicts the assumption that the hypothesis holds for all α < λ, and so for all p ∈ Tλ we have p ∈ / Uλ . 4.2 The well-founded model Definition 9 does not insist that interpretations are coherent. This allows the set of interpretations to form a complete lattice under the ⊑ relation. Since WD is monotone, then by the Knaster-Tarski Theorem (Tarski 1955), least and greatest fixpoints of WD exist. We take the least fixpoint to be the well–founded model of a defeasible theory. Definition 17 The well-founded model wfm(D) of defeasible theory D is the least fixpoint of WD . We will sometimes write the well-founded model of D as ID,WF to emphasize that it is an interpretation, and often we will eliminate the theory in the subscript (writing IWF ) when the theory is not in doubt. The “consequences” of a theory are defined based upon the well-founded model. Definition 18 Let D be a defeasible theory, L one of NDL or ADL, and IWF = ⟨TWF , UWF ⟩ D’s well-founded model under L. Define the expression D |≈L p to mean p ∈ TWF under L and D ≈|L p to mean p ∈ UWF under L. Reviewing Example 5, we find that the well-founded model according to NDL is ⟨ ∅, {p, ¬p, q, ¬q, r ,¬r }⟩. All literals are ultimately unfounded. Since there are no rules for either ¬r or ¬q, both of these are unfounded. q is unfounded because it depends directly on itself. In NDL, since the supported rules for p and ¬p conflict, they are taken as unfounded. Under ADL, the model is ⟨∅, {¬r , q, ¬q}⟩. The literals p and ¬p are undetermined in ADL, and this indeterminacy is propagated to r . 4.3 Soundness and partial completeness The proof systems of NDL and ADL are sound wrt to their counterpart semantics. As the proofs showing this for NDL and ADL are similar, we emphasize NDL below, stating in parentheses where the conditions for ADL differ. Theorem 1 (Soundness) If D is a defeasible theory and L one of NDL or ADL, then D |∼L p implies D |≈L p, and D ∼|L p, implies D ≈|L p. Proof If D |∼L p or D ∼|L p, then there is a proof tree τ showing this. We induct on the depth of τ . (Base case) Suppose τ is just a single node n labeled +δp or −δp. We consider each case separately.

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(Case 1) Suppose that n is labeled +δp. Then either 6.1.a or 6.1.b of the definition of proof obtains. If it’s 6.1.a, since n has no children, there must be a rule r ∈ Rs [p] such that body(r ) = ∅. By definitions 13 (witness of provability) and 14 (immediate consequence), p ∈ TD (I0 ). If 6.1.b obtains, then there is some rule r ∈ Rd [p] that succeeds at n. Since n has no children, body(r ) = ∅. Let c ∈ C [p]. Since definition 6.1.b is satisfied, then there is some q ∈ c − {p} such that for every rule s ∈ R[q], body(s) fails at n or else s ≺ r . Since τ consists of a single node, body(s) cannot fail and so s ≺ r . Generalizing on c, each conflict set in c ∈ C [p] contains a q ̸= p such that for every rule s ∈ R[q], s ≺ r . By definitions 13 and 14, p ∈ TD (I0 ). (Case 2) Suppose that n is labeled −δp. Since τ consists of only a single node, failure-by-looping cannot apply and there can be no strict rules with head p. Therefore 6.2.b must obtain. Let r ∈ Rd [p]. Since node n has no children, condition 6.2.b.ii must hold. Let c ∈ C [p] and q ∈ c − {p}, and let s ∈ R[q] such that body(s) succeeds and s ⊀ r (for ADL, this restriction is strengthened to r ≺ s or s strict). Such an s must exist since 6.2.b.ii holds. Since τ has only a single node, body(s) = ∅ and so body(s) ⊆ T0 . Generalizing on q, for every q ∈ c − {p}, there exists a rule s ∈ R[q] such that body(s) ∈ T0 and s ⊀ r (again, for ADL, r ≺ s or s is strict). Generalizing on r , every rule for p satisfies definition 10.2.b (for ADL, 11.2.b), and so the unit set {p} is an unfounded set wrt to D and I0 . (Induction) Suppose for all integers j ≤ k , if τ has depth j and its root is labeled +δp (−δp), then D |≈L p (D ≈|L p). Suppose τ has depth k + 1. (Case 1) Suppose the root n of τ is labeled +δp. Then 6.1.a or 6.1.b again holds. If 6.1.a holds, then there is a strict rule r ∈ Rs [p] such that body(r ) succeeds at n. For all q ∈ body(r ), there is a child of m labeled +δq. Each such q is the root of a valid argument tree of maximum depth k , and so by the inductive hypothesis, D |≈L body(r ). Thus, there is a least ordinal α such that body(r ) ⊆ Tα . By definition of TD , p ∈ TD (Iα ). If 6.1.b holds, then there is a defeasible rule r ∈ Rd [p] such that body(r ) succeeds at n. As before, D |≈L body(r ). Let c ∈ C [p]. Since 6.1.b holds, there is a u ∈ c−{p} such that for all rules s ∈ R[u], either body(s) fails at n or else s ≺ r . Suppose body(s) fails at n. By definition of failure, there exists a child m of n labeled −δv , where v ∈ body(s). Node m is thus the head of a valid proof tree of depth ≤ k , and so by the inductive hypothesis, D ≈|L body(s). Generalizing on s and then c, for every c ∈ C [p], there is a u ∈ c − {p} such that if s ∈ R[u], then D ≈|L body(s), or else s ≺ r . Let β be the least ordinal such that for rule r and each rule s, body(r ) ⊆ Tβ and body(s)∩Uβ ̸= ∅ or s ≺ r . By definition of immediate consequence, p ∈ TD (Iβ ) = Tβ+1 . (Case 2) Suppose the root of τ is labeled −δp. Any branch of a proof tree involving failure-by-looping need not extend beyond the topmost node where definition 6.3 (failure-by-looping) applies. As this is so, the tree can be trimmed to that point, and so 6.3 only applies to the leaves of the tree. We may assume without loss of generality that τ is of this form. Define N to be the set of nodes of τ labeled with −δu for any u, and S to be


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the set of the u’s. Let n be any node in N . Then n is labeled −δq for some q ∈ S . Node n is either a leaf or an internal node. We treat each case separately. (Case 2.a) If n is an internal node, then 6.2 obtains. Suppose r ∈ Rsd [q]. If r ∈ Rs [q], then body(r ) fails at n. By definition of failure, n has a child m labeled −δv , where v ∈ body(r ). By definition of N and S , m ∈ N and v ∈ S . Let r ∈ Rd [q]. Since 6.2 holds at n, either (i) the body(r ) fails at n and so there is a a ∈ body(r ) and a child m of n such that m is labeled −δa and m ∈ N (and so a ∈ S ), or (ii) there is a conflict set c ∈ C [q] such that for any u ∈ c − {q}, there is an s ∈ R[u] such that s ⊀ r (for ADL, r ≺ s or s is strict) and for each v ∈ body(s), there is a subtree of n with root labeled +δv that constitutes a defeasible proof. Each such subtree has depth ≤ k and so by the inductive hypothesis D |≈L body(s). Thus for each r ∈ Rsd [q], either body(r ) ∩ S ̸= ∅, or else r ∈ Rd [q] and there is a c ∈ C [q] such that for any u ∈ c − {q}, there is a s ∈ R[u] such that s ⊀ r (for ADL, r ≺ s or s is strict) and D |≈L body(s). (Case 2.b) Suppose that n is a leaf node. Then either 6.2 or 6.3 obtains. If 6.2 obtains, then as was shown in the base case, rs [q] = ∅ and for each r ∈ Rd [q] there is a c ∈ C [q] such that for every u ∈ c − {q}, there exists a rule s ∈ R[u] such that body(s) ⊆ T0 and s ⊀ r (for ADL, s is strict or else r ≺ s). If 6.3 obtains, then there is a non-leaf node labeled −δq, and we have shown there that (1) for each r ∈ Rs [q], body(r ) ∩ S ̸= ∅, and (2) for each r ∈ Rd [q] either body(r ) ∩ S ̸= ∅, or else there is a conflict set c ∈ C [q] such that for all u ∈ c − {q} there is a rule s ∈ R[u] such that D |≈L body(s) and s ⊀ r (for ADL, r ≺ s or s is strict). Generalizing on n, let λ be the least ordinal such that for any rule s above, body(s) ⊆ Tλ . Given the above 2 cases, by definition S is unfounded with respect to D and Iλ . Since p ∈ S , D ≈|L p. Example 6 Let D = ⟨R, CMIN , ∅⟩, where R is the following infinite set of rules. ∅⇒p {q0 } ⇒ ¬p {qn+1 } ⇒ qn (for each n ∈ N) Since proofs must be of finite depth, the above example shows that completeness does not hold in general. From the standpoint of the two fixpoint semantics, the set {¬p, q0 , q1 , q2 , . . .} is clearly unfounded wrt to D and I0 . That is, {¬p, q0 , q1 , q2 , . . .} ⊆ U1 . As this is so, the rule ∅ ⇒ p is allowed to fire, and so p ∈ T2 . The well-founded model of this example under both semantics is ⟨{p}, {¬p, q0 , ¬q0 , q1 , ¬q1 , . . .}⟩· From the proof-theoretical standpoint, however, since the refutation of ¬p involves a branch of infinite depth, neither NDL and ADL are capable of showing D |∼ p. Though completeness cannot be claimed to hold in general, it does hold for the class of locally finite defeasible theories, defined below (the given definition grew out of a discussion with David Billington). Basically, locally finite theories are those where every literal has a finite dependency graph. The precise notion

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of dependency is given in Definition 19. Since every finite theory is necessarily locally finite, completeness holds for finite theories as a special case. More generally, we can say a literal with a finite dependency graph is well-founded (alternatively, unfounded) according to one of the semantics if and only if it is provable (refutable) according to the corresponding proof system. Definition 19 Let D be a defeasible theory. If p ∈ Lit(D), then DepD (p) is the smallest set such that 1. p ∈ DepD (p), and 2. for each q ∈ DepD (p), if c ∈ C [q], then c ⊆ DepD (p), and 3. for each q ∈ DepD (p), if r ∈ R[q], then body(r ) ⊆ DepD (p). Definition 20 Let D be a defeasible theory. For each p ∈ Lit(D), p is locally finite in D iff DepD (p) is finite; and D is locally finite iff for each p ∈ Lit(D), p is locally finite in D. As is clear from the definition, if p is locally finite in D, then for all q ∈ DepD (p), q is locally finite in D. In Example 6, DepD (p) = {p, ¬p}∪{qn |n ∈ N}∪{¬qn |n ∈ N} and so p is not locally finite. Recall that though p is well-founded, no proof tree for +p exists in either NDL or ADL. The concept of a literal being locally finite in a theory does not precisely capture the notion of provability and refutability in the proof systems. There are literals that are not locally finite in a given theory but which nevertheless can be proven (refuted) using a finite proof tree. This is due to the way priorities are used in the proof tree: e.g., inferior rules in some cases are not explicitly mentioned in the proof tree. Nevertheless, if a literal is locally finite, then the semantics and the proof systems are in agreement. Theorem 2 (Completeness for locally finite literals) If D is a defeasible theory, p is a locally finite literal in D, and L is one of ADL or NDL, then D |≈L p implies D |∼L p and D ≈|L p implies D ∼|L p. Proof We will prove by induction on the sequence (ID ) that for any λ ≥ 0, if p ∈ Tλ (p ∈ Uλ ) then D |∼L p (D ∼|L p). I0 is empty and so the hypothesis trivially holds for λ = 0. Suppose the hypothesis holds for all λ < α. Since each literal first appears well-founded or unfounded in (ID ) at some interpretation indexed by a successor ordinal, we may assume without loss of generality that α is a successor ordinal. (Case 1) Suppose p ∈ Tα . Then there is some rule r ∈ Rsd [p] such that body(r ) ⊆ Tα−1 . If r ∈ Rs [p], then by the inductive hypothesis D |∼L a for each a ∈ body(r ), and so for each there exists a defeasible proof tree with root labeled +δa. We may append these proofs to a node labeled +δp to form a proof showing D |∼L p. Since body(r ) is finite, the proof tree is finite. If r ∈ Rd [p], then as before for each a in body(r ), D |∼L a. Also, for all c ∈ C [p] there is a q ∈ c − {p} such that for all rules s ∈ R[q], s ≺ r or else there is a


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v ∈ body(s) such that v ∈ Uα−1 . By the inductive hypothesis, D ∼|L v . Since p is locally finite in D, DepD (p) is finite, and so the number of v ’s is finite. Adding a tree for each v for each conflict set c ∈ C [p] as well as adding trees for each a ∈ body(r ) to a root labeled +δp forms a proof tree that satisfies definition 6.1 (For ADL, it’s def. 7.1). And so D |∼L p. (Case 2) Suppose p ∈ Uα . By definition of (ID ), Uα is an unfounded set wrt D and Iα−1 . Let S = Uα ∩ DepD (p). Since DepD (p) is finite, S is finite. Let τ0 be the tree consisting of a single unmarked node labeled −δp. From τ0 , we construct a series of trees. Given a tree τi , we form a new tree τi+1 by picking any unmarked node x ∈ τi labeled −δq for some q in S . Note that q ∈ DepD (p). Since q ∈ Uα and Uα is an unfounded set wrt D and Iα−1 , for each rule r ∈ Rsd [q], there is literal a ∈ body(r ) such that either (a) a ∈ Uα−1 or (b) a ∈ Uα or (c) r ∈ Rd [q] and there exists a conflict set c ∈ C [q] such that for all u ∈ c − {q}, there is a rule s such that body(s) ⊆ Tα−1 and s ⊀ r (for ADL, s is strict or r ≺ s). We consider each case in turn. Observe that by definition of DepD (p), each literal a must be in S , as must each body(s) referenced above. (2.a) a ∈ Uα−1 . By inductive hypothesis D ∼|L a. We may append a proof tree τa with root −δa to node x and mark each node of τa . (2.b) a ∈ Uα . If x does not already have a child labeled −δa, then append to x a node y labeled −δa. If y satisfies condition 2 or 3 in Definition 6 (For ADL, 7), then mark y. Otherwise, leave y unmarked. (2.c) Since for all u ∈ c − {q}, there is a rule s such that body(s) ⊆ Tα−1 , then by the inductive hypothesis for each v ∈ body(s) a proof tree τv exists with root +δv . We may append these to node x and mark all nodes of τv . After applying one of the cases 2.a–2.c for each rule r ∈ Rsd [q], examine the resulting tree to see if there is an unmarked non-leaf node z in the tree such that all the children of z are marked. If such a node z is found, mark it. Repeat this procedure until there are no more unmarked nodes in the tree all of whose children are marked. The resulting tree is τi+1 . ∪∞ Let τ = i=0 τi . Suppose x is a marked node in τ . If x was added to τ using case 2.a, then x occurs within a subtree of τ that is a proof tree. So x must satisfy one of the conditions in Definition 6 (7). Similarly, if x was added using case 2.c, then x is part of a valid proof tree and so satisfies the proof conditions. If x was added to τ and marked according to case 2.b, then x is a leaf node in τ and x satisfies condition 2 or 3 of Definition 6(7). Otherwise, x is a non-leaf node in τ , x was added to τ using condition 2.b, and x was marked because all of its children were marked. Looking at the cases used to add the children of x to τ (we add a child for each rule for q), we see that x must satisfy condition 2 in Definition 6(7). So if τ is finite and if every node in τ is marked, then τ is a proof tree. Since cases 2.a–2.c append to node x other nodes labeled with a literal directly from S or else from a proof tree whose root is labeled with a literal from S , and since both of these are finite structures, it must be the case that the branching

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factor of τ is finite. So if τ is infinite, then τ must have an infinitely long branch. Consider such a branch. Every node in this branch (other than the top node) must have been added using case 2.b since all the other branches add proof trees which are finite. Since it was added by 2.b, q ∈ S . So, every node in the branch must be labeled −δq for some literal q ∈ S . Furthermore, no node in the branch satisfies condition 3 in Definition 6(7) since if it did, it would have been marked when it was added to τ and it would therefore have no children. But since S is finite, there must be some literal q such that two different nodes in our infinite branch are labeled −δq. But then one of these two nodes does satisfy condition 3 of Definition 6(7), which is a contradiction. Therefore, τ is not infinite. Since τ is not infinite, we can let j be a non-negative integer such that τ = τj . Suppose τj has an unmarked node. Since a node must be marked if all its children are marked, τj must have an unmarked leaf node x . This node must have been added by case 2.b of our construction, and so we can let q be a literal such that x is labeled −δq, and q ∈ Uα . Since x is not marked, it satisfies neither condition 2 nor 3 of Definition 6(7). If there is no rule r ∈ Rsd [q], then x satisfies condition 2 of Definition 6(7). So there is a rule r ∈ Rsd [q], and one of the cases 2.a–2.c applies to x . So there must be some m > j such x has a child node in τm . Then x is not a leaf node in τm and x is not a leaf node in τ , a contradiction. Therefore, every node in τ satisfies some condition in Definition 6(7) and τ is a proof tree.

5 Properties of Defeasible Logics 5.1 Specificity The so-called Tweety Triangle is probably the most familiar example in the literature on nonmonotonic reasoning. Example 7 D = ⟨R, CMIN , ∅⟩, R is 1. 2. 3. 4. 5.

∅ → bird (Tweety) ∅ → penguin(Tweety) {bird (Tweety)} ⇒ flies(Tweety) {penguin(Tweety)} ⇒ ¬flies(Tweety) {penguin(Tweety)} → bird (Tweety)

Neither fies(Tweety) nor ¬flies(Tweety) are consequences in either NDL or ADL. But a common intuition is that we should be able to show ¬flies(Tweety). The reason given to support this is that because of r5 , penguin(Tweety) is more specific or, in the context of the theory, gives more information than bird (Tweety). Of course, we can get the desired result by adding r3 ≺ r4 to the precedence relation. Example 8 D = ⟨R, CMIN , ∅⟩, R is

Well-Founded Semantics for Defeasible Logic 1. 2. 3. 4. 5.

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∅ → university-student(Joe) ∅ → adult(Joe) {adult(Joe)} ⇒ self-supporting(Joe) { university-student(Joe) } ⇒ ¬ self-supporting(Joe) { university-student(Joe) } ⇒ adult(Joe)

This example has exactly the same form as the Tweety Triangle except that in this case r5 is a defeasible rule. Yet our intuition is that we should be able to show |∼ ¬ self-supporting(Joe) even though in both NDL and ADL we can’t show either |∼ self-supporting(Joe) or |∼ ¬ self-supporting(Joe). This shows that specificity can be based on defeasible rules as well as on strict rules. Example 9 D = ⟨R, CMIN , ∅⟩, R is 1. 2. 3. 4. 5. 6. 7. 8.

∅ → university-student(Joe) ∅ → adult(Joe) ∅ → self-supporting(Joe) {university-student(Joe)} ⇒ ¬ self-supporting(Joe) {adult(Joe)} ⇒ self-supporting(Joe) {university-student(Joe)} ⇒ adult(Joe) {self-supporting(Joe)} ⇒ works-full-time(Joe) {university-student(Joe)} ⇒ ¬works-full-time(Joe)

In this final example, because Joe is a self-supporting university student, we have evidence both that he does and that he does not have a full-time job. But neither r7 nor r8 is more specific than the other since, although Joe is self-supporting, university students normally are not self-supporting (using specificity.) So we should get ̸|∼ ¬ works-full-time(Joe) and ̸|∼ works-full-time(Joe), as we do in both NDL and ADL. In the first two examples, if we delete all the facts and presumptions in D and add back in the facts in body(r4 ), then from the resulting theory we can derive body(r5 ). But we cannot derive body(r4 ) from the theory we get when we remove all the facts and presumptions from D and add back in the facts in body(r3 ). And if we perform these operations on the theory in the third example using r7 and r8 , we can derive the body of neither rule from the other. This suggests the following formalization of specificity. Definition 21 Let D be a defeasible theory and let A ⊆ Lit(D). 1. RD /A = {r : r ∈ RD and body(r ) ̸= ∅} ∪ {∅ → p : p ∈ A}. 2. Lit(RD /A) = {p : p ∈ Lit(D) and there is r ∈ RD /A such that p ∈ body(r ) or p = head (r )}. 3. D/A = ⟨RD /A, CD /Lit(RD /A), ≺D /Lit(RD /A)⟩. Definition 22 Let L be either NDL or ADL, D be a defeasible theory, r1 ∈ RD,du and r2 ∈ RD,du . Then r1 is more L-specific than r2 with respect to D iff D/body(r1 ) |∼L body(r2 ) and D/body(r2 ) ∼|L body(r1 ).

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Definition 23 A defeasible theory D satisfies specificity for NDL (ADL) iff for any r1 ∈ RD,du and r2 ∈ RD,du , if r1 is more NDL-specific (ADL-specific) than r2 , then r2 ≺D r1 . → An earlier defeasible logic D< defined in (Nute 1992) lacked precedence relations but built specificity directly into the proof procedure. We can do something similar with NDL and ADL by doing proofs for the appropriate related defeasible theories to show that one rule takes precedence over the other in the target specificitysatisfying theory.

5.2 Cut and Cautious Monotony One method of evaluating nonmonotonic formalisms is to examine the abstract properties of their consequence relations—see, for instance (Makinson 1994). Gabbay (1985) proposes that a nonmonotonic formalism counts as a nonmonotonic logic if and only if its consequence relation |∼ satisfies the below three properties. His proposal has been widely accepted. 1. Reflexivity: If p ∈ Γ, then Γ |∼ p. 2. Cut: If Γ |∼ p and Γ ∪ {p} |∼ q, then Γ |∼ q. 3. Cautious Monotony: If Γ |∼ p and Γ |∼ q, then Γ ∪ {p} |∼ q. Cautious Monotony allows lemmatization: if p is a consequence of a given theory, then we may add it to the theory without changing the consequences. Cut is its converse: we can remove the things that are consequences of the remainder without affecting the results. That is, we can safely remove the redundant statements. Together they constitute Cumulativity; they yield a logic which is in some sense stable and well-behaved. NDL and ADL both satisfy Reflexivity under one reasonable construal of the condition. Technically, only rules and not literals can belong to a defeasible theory, and only literals and not rules can be derived from a defeasible theory. But whenever a fact ∅ → p is in D, we have D |∼NDL p and D |∼ADL p. Neither logic in general satisfies reasonable versions of Cut and Cautious Monotony. Nevertheless, they do satisfy these properties provided that ≺ is transitive. Observe, however, that since a defeasible theory consists of a set of rules and the consequences of the theory are literals, if we are to add a consequence to the theory, we must add it as either a strict or a defeasible rule (more specifically, as a fact or a presumption). Provided ≺ is transitive, it does not matter which sort of rule is added. The more interesting and stronger case is where we add the fact, and the theorems below pertain to that case. Definition 24 Let D be a defeasible theory and let A ⊆ Lit(D). Then D ⊕ A = ⟨RD ∪ {∅ → p : p ∈ A}, CD , ≺D ⟩


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Lemma 4 Let L be one of NDL or ADL and D a defeasible theory such that ≺D is transitive. Let E = D ⊕ {p}. For all κ ≥ 0, if p ∈ TD,κ and q ∈ TD,κ , then q ∈ TE ,WF . For all κ ≥ 0, if p ∈ TD,κ and q ∈ UD,κ , then q ∈ UE ,WF . Proof Since ID,0 = ⟨∅, ∅⟩, the hypothesis is trivially satisfied for κ = 0. Suppose it holds for all 0 < κ < λ and that p ∈ TD,λ . We may assume wlog that λ is a successor ordinal. (Case 1) Suppose that q ∈ TD,λ . If q = p, then q is a fact of E and so clearly q ∈ TE ,WF . We’ll show that q ̸= p also implies q ∈ TE ,WF . If q ̸= p, then there exists a rule rq ∈ RD,sd [q] such that body(rq ) ⊆ TD,λ−1 . Since RD [q] ⊆ RE [q], rq ∈ RE [q]. By the inductive hypothesis, body(rq ) ⊆ TE ,WF . If rq is strict, then clearly q ∈ TE ,WF by definition of TE and IE ,WF . Suppose that rq is defeasible. Then for each conflict set c ∈ C [q] the below proposition α holds: There exists a w ∈ c − {q} such that for all rw ∈ RD [w ], either rw ≺D rq or body(rw ) ∩ UD,λ−1 ̸= ∅·

(α)

Let c1 ∈ C [q]. Considering α above for c1 , either w = p or else w ̸= p. Assume that w = p. Given α above and that p ∈ TD,λ , any witness of p’s provability in D must be a defeasible rule (otherwise rw ⊀ rq and body(rw ) ∩ UD,WF = ∅ would obtain, violating α). Let rp be a witness of provability for p in D relative to Iλ . Then body(rp ) ∩ UD,λ−1 = ∅. Given α, it must be the case that rp ≺D rq . Since ≺D is acyclic, rq ⊀ rp . Since rp is a witness of provability for p in D (relative to Iλ ), there is a u ∈ c1 − {p} such that for every rule ru ∈ RD [u], either body(ru ) ∩ UD,λ−1 ̸= ∅ or ru ≺D rp . Since rq is a witness of provability for q in D and rq ⊀ rp , then u ̸= q. Thus for each ru ∈ RD [u], either body(ru ) ∩ UD,λ−1 ̸= ∅ or ru ≺D rq . Since u ̸= p, it is the case that RD [u] = RE [u], and so (using the inductive hypothesis) we have u ∈ c1 − {q} and for each ru ∈ RE [u], either body(ru ) ∩ UE ,WF ̸= ∅ or ru ≺D rq . Observe that if w ̸= p, then RD [w ] = RE [w ], and so by the inductive hypothesis and α above, we have w ∈ c1 − {q} and for all rw ∈ RE [w ], either rw ≺D rq or body(rw ) ∩ UE ,WF ̸= ∅. Generalizing on c1 , for each conflict set c ∈ CE [q], there is a w ∈ c − {q} such that for each s ∈ RE [w ], body(s) ∩ UE ,WF ̸= ∅ or else s ≺D rq . Since body(rq ) ⊆ TE ,WF , then by definition of TE and IE ,WF , q ∈ TE ,WF . (Case 2) Suppose q ∈ UD,λ , and let a be any literal such that a ∈ UD,λ . Since Iλ is coherent and p ∈ TD,λ , a ̸= p and so RD [a] = RE [a]. Let ra be any rule in RD,sd [a]. If ra is strict, then by definition of unfounded sets in NDL and ADL, it must be that body(ra ) ∩ (UD,λ ∪ UD,λ−1 ) ̸= ∅. By the inductive hypothesis, body(ra ) ∩ (UD,λ ∪ UE ,WF ) ̸= ∅. Suppose that ra is defeasible. Then either body(ra ) ∩ (UD,λ ∪ UD,λ−1 ) ̸= ∅, or else there exists a c ∈ CD [a] such that for each w ∈ c − {a} there is a rule rw ∈ RD [w ] such that body(rw ) ⊆ TD,λ−1 and rw ⊀ ra (for ADL, ra ≺D rw or rw is strict). If the former, then by the inductive hypothesis body(ra ) ∩ (UD,λ ∪ UE ,WF ) ̸= ∅. Suppose it’s the latter. Since RD ⊆ RE and CD = CE , it follows that c ∈ CE [a] and for

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each w ∈ c − {a} there is an rw ∈ RE [w ] such that (by the inductive hypothesis) body(rw ) ⊆ TD,λ−1 ⊆ TE ,WF and rw ⊀ ra (for ADL, ra ≺D rw or rw is strict). Generalizing on ra , for each r ∈ RE ,s [a], body(r ) ∩ (UD,λ ∪ UE ,WF ) ̸= ∅, and for each r ∈ RE ,d [a] either body(r ) ∩ (UD,λ ∪ UE ,WF ) ̸= ∅, or else there exists a conflict set c ∈ CE [a] such that for each w ∈ c − {a} there is a rule rw ∈ RE [w ] such that body(rw ) ⊆ TE ,WF and rw ⊀ r (for ADL, r ≺D rw or rw is strict). Generalizing on a, it can be seen that UD,λ satisfies the definition of unfounded set wrt E and IE ,WF (according to NDL and ADL, respectively), and so UD,λ ⊆ UE (UE ,WF ) (hence UD,λ ⊆ UE ,WF ). Since q ∈ UD,λ , it follows that q ∈ UE ,WF . Theorem 3 (Cautious Monotony) Let L be one of NDL or ADL and D a defeasible theory such that ≺D is transitive. If D |≈L p and D |≈L q, then D ⊕ {p} |≈L q. If D |≈L p and D ≈|L q, then D ⊕ {p} ≈|L q. Proof Let E = D ⊕ p, and suppose D |≈L p and D |≈L q (alternatively D ≈|L q). There then is a least λ ≥ 0 such that p ∈ TD,λ and q ∈ TD,λ (q ∈ UD,λ ). By Lemma 4 q ∈ TE ,WF and so E |≈L q (q ∈ UE ,WF and so E ≈|L q). Lemma 5 Let L be one of NDL or ADL and D a defeasible theory such that ≺D is transitive and D |≈L p. Let E = D ⊕ {p}. For all κ ≥ 0, if p ∈ TE ,κ and q ∈ TE ,κ , then q ∈ TD,WF . If p ∈ TE ,κ and q ∈ UE ,κ , then q ∈ UD,WF . Proof Since IE ,0 = ⟨∅, ∅⟩, the above claim is trivially satisfied for κ = 0. Suppose it holds for all 0 < κ < λ. We may assume wlog that λ is a successor ordinal. Since ∅ → p is a rule of E , clearly p ∈ TE ,λ . (Case 1) Suppose q ∈ TE ,λ . Since by assumption D |≈L p, if q = p, then q ∈ TD,WF . Suppose q ̸= p. There then exists a rule r ∈ RE [q] such that body(r ) ⊆ TE ,λ−1 . Since q ̸= p, r ∈ RD [q] and by the inductive hypothesis body(r ) ⊆ TD,WF . If r is strict, then clearly q ∈ TD,WF by definition of TD and ID,WF . So suppose r is defeasible. As before, body(r ) ⊆ TD,WF . Let c ∈ CE [q]. There then exists a w ∈ c − {q} such that for all s ∈ RE [w ], body(s) ∩ UE ,λ−1 ̸= ∅ or s ≺D r . Observe that the rule t = ∅ → p is a rule of RE and that body(t) ∩ UE ,λ−1 = ∅ and t ⊀ r . Thus p ̸= w , and so RE [w ] = RD [w ]. By the inductive hypothesis for all s ∈ RD [w ], body(s) ∩ UD,WF ̸= ∅ or else s ≺D r . Note that CE = CD . Generalizing on c, we have r ∈ RD,ds and body(r ) ⊆ TD,WF , and either r is strict or for each c ∈ CD [q] there exists a w ∈ c − {q} such that for all s ∈ RD [w ], body(s) ∩ UD,WF ̸= ∅ or else s ≺D r . By definition of TD and ID,WF , q ∈ TD,WF . (Case 2) Suppose q ∈ UE ,λ , and let a be any literal such that a ∈ UE ,λ . By coherence, a ̸= p and so RE [a] = RD [a]. Since a ∈ UE ,λ , then for all ra ∈ RE ,sd [a], either


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A· body(ra ) ∩ (UE ,λ ∪ UE ,λ−1 ) ̸= ∅ (and so by inductive hypothesis, body(ra ) ∩ (UE ,λ ∪ UD,WF ) ̸= ∅), or else B · ra ∈ RE ,d [a] and there is a c ∈ CE [a] such that for each w ∈ c − {a} there is a rule sw ∈ RE [w ] where body(sw ) ⊆ TE ,λ−1 (and so by hypothesis body(sw ) ⊆ TD,WF ) and sw ⊀ ra (for ADL, sw is strict or ra ≺D sw ). Suppose that body(ra ) ∩ (UE ,λ ∪ UD,WF ) = ∅. Then B must hold. For each w above, if w ̸= p, then RE [w ] = RD [w ]. If w = p, then since D |≈L p, there exists a rule rp ∈ RD [p] such that body(rp ) ⊆ TD,WF . If rp is strict, then rp ⊀ ra . Suppose rp is defeasible. Then for some u ∈ c − {p}, for each ru ∈ RD [u] either body(ru ) ∩ UD,WF ̸= ∅ or ru ≺D rp . Either u = a or u ̸= a. If u = a, then either body(ra ) ∩ UD,WF ̸= ∅ or ra ≺D rp ; observe that if it’s the latter, then also rp ⊀ ra . If u ̸= a, then from B above, it must be that for some su ∈ RD [u], body(su ) ⊆ TD,WF and su ⊀ ra (for ADL, su is strict or ra ≺D su ). Since ID,WF is coherent, it must also be that su ≺D rp . In the case of NDL, since su ⊀ ra and su ≺ rp , we have rp ⊀ ra . For ADL, since su ≺ rp , it must be that ra ≺D su , and so ra ≺D rp . Generalizing on ra , for each strict or defeasible ra ∈ RD [a] either body(ra ) ∩ (UE ,λ ∪UD,WF ) ̸= ∅, or else ra is defeasible and there exists a conflict set c ∈ CD [a] such that for each w ∈ c −{a} there is a rule s ∈ RD [w ] such that body(s) ⊆ TD,WF and either body(ra ) ∩ UD,WF ̸= ∅ or s ⊀ ra (for ADL, s is strict or ra ≺D s). Generalizing on a, UE ,λ satisfies the definition of an unfounded set wrt D and ID,WF (in NDL and ADL, respectively), and so UE ,λ ⊆ UD (ID,WF ). However UD,WF = UD (ID,WF ). Since q ∈ UE ,λ , it follows that q ∈ UD,WF . Theorem 4 (Cut) Let L be one of NDL or ADL and D a defeasible theory such that ≺D is transitive. If D |≈L p and D ⊕ {p} |≈L q, then D |≈L q. If D |≈L p and D ⊕ {p} ≈|L q, then D ≈|L q. Proof Let E = D ⊕ p. Since ∅ → p is a rule of E , clearly E |≈L p. Suppose E |≈L q (alternatively, E ≈|L q). Then there exists a least λ ≥ 0 such that p ∈ TE ,λ and q ∈ TE ,λ (q ∈ UE ,λ ). By Lemma 5, q ∈ TD,WF (q ∈ UD,WF ). The examples below illustrate why, when ≺ is not transitive, ADL and NDL fail Cautious Monotony and NDL fails Cut. Though we omit a discussion of it here, the examples also serve as counterexamples even if D ⊕ {p} is used to indicate that p is added as a presumption. Furthermore, ADL fails Cut as well, provided D ⊕ {p} means p is added as a fact. Example 10 D = ⟨R, C , ≺⟩, where C is closed under strict rules, ≺ = {3 ≺ 2, 2 ≺ 1}, and R is 1. ∅ ⇒ q 5. ∅ → s

2. {s} ⇒ p 6. ∅ → t

3. {} ⇒ r 7. {r } → ¬t

4. {q, r } → ¬p

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The relevant extended conflict sets of the above theory are {p, q, r }, and {r , t}. Observe that though we have 3 ≺ 2 and 2 ≺ 1, we do not have 3 ≺ 1. Under both NDL and ADL, since rule 1 is supported and rule 2 is inferior to 1, we can derive q. r is refuted because the strict rule 6 defeats rule 3 (the only rule for r ). Since r is refuted, rule 4 fails; furthermore, rule 3 is inferior to rule 2, and so p is defeasibly provable in both NDL and ADL. Since p is derivable from the theory anyway, suppose that we add it to the theory as a fact, i.e. the strict rule ∅ → p. Reconsidering the literal q, we see that though rule 1 is still supported, there is now no literal v ∈ {p, q, r } − {q} such that all rules for v fail or are inferior to rule 1. And so q cannot be derived from D ⊕ {p} in either NDL or ADL. This shows that both NDL and ADL fail to satisfy Cautious Monotony. Example 11 D = ⟨R, C , ≺⟩, where C is closed under strict rules, ≺ = {1 ≺ 3, 2 ≺ 1, 6 ≺ 5} and R is 1. ∅ ⇒ p 5. {r } ⇒ t

2. ∅ ⇒ q 6. ∅ ⇒ ¬t

3. ∅ ⇒ r

4. {q, r } → ¬p

Example 11 shows that NDL fails Cut. The relevant extended conflict set is {p, q, r }. In this example, q is refuted under NDL because its only rule (2) is not superior to either 1 or 3. Given this, rule 4 fails. Since rule 4 fails and rule 2 is inferior to rule 1, we can derive p. Since rule 1 is inferior to 3, we can derive r . Since 5 is then supported and superior to 6, we conclude t. The literal ¬t is refuted. However, adding ∅ → p changes the consequences. In that modified theory, r is refuted. This causes rule 5 to fail, which allows ¬t to be derivable. And so we have D |∼ p and D ⊕ {p} |∼ ¬t, but D ̸|∼ ¬t. 5.3 Consistency Preservation A central issue in nonmonotonic reasoning is deciding what to do when we have evidence supporting inconsistent conclusions. The goal is to resolve such conflicts whenever possible rather than to draw absurd results. That being the case, we do not want our nonmonotonic inferences to produce any new inconsistencies. If whatever underlying monotonic theory we may have is consistent, then the complete theory including whatever defaults or defeasible rules we adopt should remain consistent. When this is the case, we will say that a theory preserves consistency under the logic we are using. Example 12 D = ⟨R, CMIN , ∅⟩, R is 1. ∅ ⇒ p

2. ∅ ⇒ q

3. {p} → r

4. {q} → ¬r

Unfortunately, the above example shows that defeasible theories do not preserve consistency in general. Since conflict sets are here minimal, the two presumptions do not conflict, and so we get D |∼L p and D |∼L q. But then we must apply the


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strict rules, and we get D |∼L r and D |∼L ¬r . If we discard the two presumptions, however, we get D ∼|L a for each a ∈ {p, q, r , ¬r }, and so the strict part of the theory is consistent. Notice, however, that if we close the conflict set of D under strict rules, we get a different result. {p, ¬r }, {q, r }, and {p, q} are extended conflict sets. In this extended theory D → , the two presumptions conflict and we get D → ∼|NDL a for each a ∈ {p, q, r , ¬r }. For ADL, all literals are ambiguous. It turns out that there are two ways to guarantee that a defeasible theory preserves consistency for both NDL and ADL. One is to close the conflict sets under strict rules. The other is to add all transpositions of strict rules to the theory. Theorems 5 and 6, respectively, state each form of preservation. The transposition-based version is similar to a result reported in (Caminada 2006) for what the author calls semi-normal extended logic programs. There are other good reasons why we might want to close the strict rules of a theory under transpositions, but this method of preservation does not solve the problem of our example, at least not for NDL. If we want to block ambiguity, the correct result is to refute both p and q. Adding the transposition of strict rules, while it preserves consistency, also prevents us from doing this. To achieve the results we want, we need to close the conflict set under strict rules. Theorem 5 (Consistency Preservation I ) Let D = ⟨R, C , ≺⟩ be a defeasible theory with C closed under strict rules and L one of NDL or ADL. If c ∈ C and D |≈L c, then c ⊆ Cl (Rs ). Proof Let IWF = ⟨TWF , UWF ⟩ be the well-founded model of D according to L. We will induct on the sequence I0 , I1 , . . . , IWF , showing that for all α ≥ 0, if c ∈ C and c ⊆ Tα , then c ⊆ Cl (Rs ). This obviously holds for α = 0. Suppose it holds for all α < β and suppose that c ∈ C and c ⊆ Tβ , and for all γ < β, c * Tγ . We can assume wlog that β is a successor ordinal. Since c ⊆ Tβ , for each p ∈ c, there is a rule r ∈ Rsd [p] such that r is a witness of provability wrt D and Iβ−1 . Thus, for each r , body(r ) ⊆ Tβ−1 . Let S be any minimal set of such r ’s such that for each p ∈ c there exists a witness for p in S . If S contains a nonempty subset A consisting entirely of defeasible rules, then since each rule r ∈ S is a witness of provability relative to D and Iβ−1 , it follows by Definition 13 that for each s ∈ A, there exists a s ′ ∈ A such that s ′ ≺ s. However, this is impossible since ≺ is acyclic and S is finite, and so S must consist entirely of strict rules. Let X = {q|r ∈ S and q ∈ body(r )}. X may be viewed as the set of literals obtained by replacing some p in c with body(r ), where r ∈ S and head (r ) = p. Since conflict sets are closed under strict rules and S consists entirely of strict rules, X itself is a conflict set. However, since for each r ∈ S , body(r ) ⊆ Tβ−1 , it follows by inductive hypothesis that X ⊆ Cl (Rs ). Since each r ∈ S is strict and body(r ) ⊆ X , and c = {head (r )|r ∈ S }, it follows that c ⊆ Cl (Rs ). For the other form of consistency preservation, we will show that if a set X and its element-wise negation ¬X =def {¬p|p ∈ X } are both well-founded relative to a particular theory, then the strict rules alone of that theory are sufficient to derive X . A form of consistency preservation immediately follows.

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Definition 25 For any literal p and set of literals A, if r = A → p, then trans(r ) =def {A/q ∪ {¬p} → ¬q|q ∈ A}. A defeasible theory D is closed under strict transpositions iff for all r ∈ RD,s , trans(r ) ⊆ RD,s . Lemma 6 Let D be a defeasible theory with Rs closed under strict transpositions and let L be NDL or ADL. If X is a set of literals such that and D |≈L X and D |≈L ¬X , then X ∈ Cl (Rs ). Proof Let IWF = ⟨TWF , UWF ⟩ be the well-founded model of D according to L. We will induct on the sequence I0 , I1 , . . . , IWF , showing that for all α ≥ 0, if for any finite set X such that X ⊆ Tα and ¬X ⊆ TWF , then X ⊆ Cl (Rs ). The hypothesis obviously holds for α = 0. Suppose it holds for all α < β and that β is the least ordinal such that X ⊆ Tβ . We can assume wlog that β is a successor ordinal. Since X ⊆ Tβ , for each p ∈ X , there is a rule r ∈ Rsd [p] such that r is a witness of provability of p wrt D and Iβ−1 . And so for each r , body(r ) ⊆ Tβ−1 . Let S be any minimal set of witnesses for X (one witness for each literal of X ), and suppose that S contains a nonempty subset A of defeasible rules. If s ∈ A, then since s is a witness of provability for head (s) relative to Iβ−1 , it follows by coherence that for no ordinal γ does body(s) ∩ Uγ ̸= ∅ hold. However, since ¬X ⊆ TWF , it follows that there is some s ′ ∈ Rsd [¬head (s)] such that body(s ′ ) ⊆ Tα for some interpretation Iα ⊑ IWF . As this is so, body(s ′ ) ∩ Uβ−1 = ∅. Since s is a witness of provability for head (s) relative to Iβ−1 , it must be the case that s ′ ≺ s. However, since s ′ is a witness of provability relative to Iα and body(s) ∩ Uα = ∅, it must be the case that s ≺ s ′ . Since ≺ is acyclic, this is a contradiction, and so S must consist entirely of strict rules. Since ¬X ⊆ TWF , for any r ∈ S , ¬head (r ) ⊆ TWF . Since body(r ) ⊆ Tβ−1 and (I) is monotone, we have (body(r ) ∪ ¬head (r )) ⊆ TWF . Since for each t ∈ trans(r ), t = ((body(r )/q) ∪ {¬head (r )}) → ¬q for some q ∈ body(r ), it follows that body(t) ⊆ TWF and consequently ¬q ⊆ TWF . Since for each literal p ∈ body(r ) there is a rule in trans(r ) with head ¬p, it follows that ¬body(r ) ⊆ TWF . Let Y = {q|r ∈ S and q ∈ body(r )}. Since each r ∈ S is a witness of provability wrt D and Iβ−1 , it follows that Y ⊆ Tβ−1 . However, for each r ∈ S , it is also the case that ¬body(r ) ⊆ TWF , and so ¬Y ⊆ TWF . By inductive hypothesis, Y ⊆ Cl (Rs ). This implies that for each r ∈ S , body(r ) ⊆ Cl (Rs ) and so head (r ) ∈ Cl (Rs ). Since X = {head (r )|r ∈ S }, X ⊆ Cl (Rs ). Theorem 6 (Consistency Preservation II ) Let D be a defeasible theory with Rs closed under strict transpositions and let L be NDL or ADL. For any p ∈ At(D), if D |≈L {p, ¬p}, then {p, ¬p} ⊆ Cl (Rs ).


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Proof The theorem follows directly from Lemma 6. Observe that Theorem 6 is in a sense weaker than Theorem 5, in that it pertains to explicitly contradictory pairs of literals such as {p, ¬p}. The guarantee does not extend to non-minimal conflict sets such as {small , medium, large}. It is Theorem 5 that provides this wider guarantee. With that in mind, when we define standard defeasible theories below, we insist that conflict sets are closed under strict rules.

6 Related Work The semantics for ADL and NDL are based upon the well-founded semantics for logic programs (Van Gelder et al. 1991), but unlike the simple WFS for logic programs, ADL and NDL incorporate priorities among rules. Attempts to extend the WFS to incorporate logic programs with prioritized rules have been made, notably by Brewka (1996) and by Schaub and Wang (2002). Both formalisms correspond to the WFS in the case of unprioritized programs and so, like defeasible logic, neither one guarantees consistency preservation for arbitrary sets of rules. Both differ from defeasible logic in that nonmonotonicity in logic programming is achieved through the use of default negation. Both the approach of (Brewka 1996) and of (Schaub and Wang 2002) are ambiguity propagating and so under any reasonable translation will yield quite different results than NDL. It is not known at this point the extent to which they differ from ADL (although they will differ). Both are based upon an alternating fixpoint procedure and so incorporate a forward looking “consistency check.” The use of extended conflict sets in defeasible logic can be viewed as an attempt to avoid a check of this sort. In an infinite theory, it may not be possible to enumerate the consequences of having a rule fire, and so we cannot even recognize when it’s “safe” to apply a defeasible rule. In a sense, defeasible logic assumes that the consistency check has been done beforehand; the conflicts are simply part of the theory. It is debatable, however, whether explicitly specifying conflict is better than simply truncating, due to computational constraints, the consistency check that is inherent in the alternating fixpoint procedure. The specific defeasible logics we have presented here have a lineage going back over two decades. In 1986, Nute published an early version of defeasible logic, LDR or Logic for Defeasible Reasoning (Nute 1986). This system does not include precedence relations, conflict sets, or failure-by-looping, and the system was “semistrict”, meaning that a strict rule r can be defeated by other strict rules when the derivation of head (r ) depends on the use of defeasible rules. LDR is ambiguity blocking and satisfies specificity. A successor of LDR which first appeared in 1987 includes precedence relations and specificity (Nute 1987; Billington et al. 1990; Nute et al. 1989), and a family of defeasible logics, all of which are ambiguity blocking and none of which include conflict sets or failure-by-looping appeared in → (Nute 1992). One of the latter, D→ < , was the immediate predecessor of NDL. D< is ambiguity blocking, uses a precedence relation, satisfies specificity, and incorporates preemption or team defeat, but it does not use conflict sets or failure-by-looping.

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Of the logics studied and developed by Billington, Antoniou, Governatori, Maher and others, they appear mostly based on the logic of (Billington 1993). That logic, → which we call BDL, modifies D→ < . Neither D< nor BDL have a mechanism for detecting circular reasoning equivalent to failure-by-looping. As was stated earlier, however, Maher and Governatori (1999) produced a well-founded semantics for it which does incorporate a cycle-check. The proof system is presumably sound but not complete relative to this semantics. Normal forms for BDL—in which priorities and defeaters are removed—are given in (Antoniou et al. 2001). An argumentation semantics is defined for variants of BDL in (Governatori et al. 2004). A detailed presentation of how to transform defeasible theories (of BDL and its variants) into logic programs is given in (Antoniou et al. 2006). These earlier and parallel versions of defeasible logic distinguish between strict and defeasible derivations. This is achieved by placing additional tags on literals in derivations. E.g., in BDL, the expression +∆p in a derivation indicates that p is derivable using only Rs (and −∆p indicates that p cannot be so derived), while +δp means that p is derivable using all of R (and −δp means p cannot be so derived). A derivation in BDL is a linear sequence of tagged literals that satisfies certain constraints. This distinction between strict and defeasible derivations provides a mechanism to handle conflict. If the body of a strict is only defeasibly derivable, then the rule is treated as a defeasible rule. Thus, if there is a conflicting rule whose body is also derivable, then the consequent of the strict rule is not derived. This prevents the logic from concluding contradictions except for those due to strict rules alone. Since strict rules serve this dual function, BDL permits the priority relation to range over strict rules (D→ < does not). Example 13 D = ⟨R, CMIN , ∅⟩, R is 1. ∅ ⇒ married

2. {married } → ¬bachelor

3. ∅ ⇒ bachelor

The dual function of strict rules is a simple and computationally efficient method of handling conflict. In cases such as Example 13 above, however, it will not yield intuitively acceptable results. In BDL and D→ < , married and bachelor do not conflict, and so married is defeasibly derivable. Since the body of rule 2 is only defeasibly derivable, rule 2 is considered defeasible. Since rules 2 and 3 conflict and are defeasible, both bachelor and ¬bachelor are defeasibly refuted. In contrast, if conflict sets are closed under strict rules, then NDL and ADL hold that married and bachelor conflict and refrain from deriving either. Antoniou calls the approach taken in ADL and NDL the “purist view” (Antoniou 2006). In the below passage, he defends the dual treatment of strict rules in BDL (note that the literals in square braces are our own and not his): If p [married ] is definitely known, then q [¬bachelor ] is also definitely derived. Otherwise, if p [married ] is defeasibly known, then usually q [¬bachelor ] is true. In our example, p [married ] is not definitely known, so we do not jump automatically to the conclusion q [¬bachelor ] once p [married ] is (defeasibly) proven, but must consider counterarguments in favor of ¬q [bachelor ](Antoniou 2006).


31

We have replaced the simple letters (e.g., p) with more suggestive literals (such as married ), but we do not think this is harmful, and it makes it clear that the defense is not entirely satisfying. If married is defeasibly known then it should be the case that ¬bachelor is also defeasibly known, simply because all evidence for married is evidence for ¬bachelor , and all evidence for bachelor is evidence against married . We feel that the most reasonable choice in the above case is to treat strict rules as strict. How else is one to specify a linguistic convention or an analytic truth? Extended conflict sets were introduced in NDL and ADL to handle cases like this. We allow the evidence of the strict rules to affect our derivations in cases like this when we require that our conflict sets should be closed under the strict rules of a defeasible theory. D→ < and BDL are ambiguity blocking in the same sense as NDL: ambiguous literals are taken as refuted. In order to incorporate ambiguity propagating behavior into BDL, (Antoniou et al., 2000a; 2000b; 2006; Governatori et al. 2004) discuss the use of the tags +Σ and −Σ to indicate when a literal has a supporting argument and when no such argument exists. By supporting argument we mean that the literal would be derivable in the absence of all conflict by simple forward chaining). It is these arguments that are used to construct Makinson and Schlechta’s so-called “zombie paths”. This method of introducing ambiguity propagation is quite distinct from that used in ADL (which is more akin to the manner in which it is achieved in the original well-founded semantics).

7 Conclusions We have defined both an ambiguity blocking and an ambiguity propagating version of defeasible logic and we have developed a well-founded semantics for each. With this choice of logics, one’s position on whether to block or propagate ambiguity can guide one’s choice of which logic to use, but it cannot provide a reason for rejecting defeasible logic as a viable approach to nonmonotonic reasoning. Since every defeasible theory has a unique well-founded model whether we choose NDL or ADL, there can be no such thing as a floating conclusion on this approach. This will not satisfy those who support the floating conclusions that arise when using default extensions or stable models or another approach that generates multiple interpretations for defeasible theories. They will claim that the examples that give rise to floating conclusions provide decisive evidence for these other approaches. Our own view is that these examples are simply not so conclusive as the authors believe. We have also presented significant theorems about these logics and their semantics, notably soundness and partial completeness, coherence, consistency preservation, cut and cautious monotony. Some of the properties of defeasible logic, including completeness, depend on specific properties of the defeasible theories we consider. We can collect these properties in the definition of what we will call a standard defeasible theory. Definition 26

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A defeasible theory D is standard iff D is locally finite, D satisfies specificity, CD is closed under the strict rules in RD , and ≺D is transitive. As we have seen, both NDL and ADL are sound with respect to the corresponding well-founded semantics for all defeasible theories, and are complete with respect to the corresponding well-founded semantics for standard defeasible theories. From these two results and our other semantical results, we get the following syntactical theorems. Theorem 7 (Coherence) If D is a defeasible theory and L is either NDL or ADL, there is no p ∈ Lit(D) such that both D |∼L p and D ∼|L p. Theorem 8 (Cumulativity) Let D be a defeasible theory and L one of NDL or ADL. 1. If ∅ → p ∈ RD , then D |∼L p; 2. If D |∼L p, then (a) D |∼L q iff D ⊕ {p} |∼L q, and (b) D ∼|L q iff D ⊕ {p} ∼|L q. Theorem 9 (Consistency Preservation) Let D be a defeasible theory and L one of NDL or ADL. If S ∈ CD and D |∼L S , then S ∈ Cl (RD,s ). Theories must be locally finite for our completeness result. Conflict sets must be closed under strict rules to guarantee consistency preservation, and the precedence relation must be transitive to ensure cumulativity. Specificity is not needed for any of these results, but it is needed to get proper results in simple cases like the Tweety Triangle. References Antonelli, G.A. 2005. Grounded consequence for defeasible logic. Cambridge University Press. Antoniou, G. 2006. Defeasible reasoning: A discussion of some intuitions. International Journal of Intelligent Systems 21(6), 545–558. Antoniou, G., Billington, D., Governatori, G., and Maher, M. J. 2000a.A flexible framework for defeasible logics. In Proceedings of the 8th Workshop on Non-Monotonic Reasoning. Antoniou, G., Billington, D., Governatori, G., Maher, M. J., and Rock, A. 2000b. A family of defeasible logics and its implementation. In Proceedings of the 14th European Conference on Artificial Intelligence (ECAI 2000). IOS Press, 459–463. Antoniou, G., Billington, D., Governatori, G., and Maher, M. J. 2001. Representation results for defeasible logic. ACM Transactions on Computational Logic 2(2), 255-287. Antoniou, G., Billington, D., Governatori, G., and Maher, M. J. 2006. Embedding defeasible logic into logic programming. Theory and Practice of Logic Programming 6(6), 703–735.


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Billington, D. 1993. Defeasible logic is stable. Journal of Logic and Computation 3(4), 379–400. Billington, D., Coster, K. D., and Nute, D. 1990. A modular translation from defeasible nets to defeasible logics. Journal of Experimental & Theoretical Artificial Intelligence 2(2), 151–177. Brewka, G. 1996. Well-founded semantics for extended logic programs with dynamic preferences. Journal of Artificial Intelligence Research 4, 19–36. Brewka, G. 2001. On the relationship between defeasible logic and well-founded semantics. In Proceedings of the 6th International Conference on Logic Programming and Nonmonotonic Reasoning (LPNMR 2001). Springer, LNCS 2173, 121–132. Caminada, M. 2006. Well-founded semantics for semi-normal extended logic programs. Paper presented at the 11th International Workshop on Non–Monotonic Reasoning (NMR 2006). May 30–June 1, 2006, Lake District, UK. Covington, M.A., Nute, D., & Vellino, A. 1997. Prolog programming in depth. Upper Saddle River, New Jersey: Prentice-Hall. Donnelly, S. 1999. Semantics, Soundness, and Incompleteness for a Defeasible Logic. Master’s thesis, The University of Georgia. Gabbay, D. 1985. Theoretical foundations for non-monotonic reasoning in expert systems. In Apt, K. (ed), Logics and models of concurrent systems. New York: Springer-Verlag New York, 439–457. Gelfond, M. and Lifschitz, V. 1988. The stable model semantics for logic programming. In Proceedings of the 5th International Conference on Logic Programming (ICLP 1988). MIT Press, 1070–1080. Gelfond, M. and Lifschitz, V. 1991. Classical negation in logic programs and disjunctive databases. New Generation Computing 9(3/4), 365–385. Governatori, G., Maher, M. J., Antoniou, G., and Billington, D. 2004. Argumentation semantics for defeasible logic. Journal of Logic and Computation. 14(5), 675–702. Horty, J. F. 2002. Skepticism and floating conclusions. Artificial Intelligence 135(1-2), 55–72. Horty, J. F., Thomason, R. H., and Touretzky, D. S. 1990. A skeptical theory of inheritance in nonmonotonic semantic networks. Artificial Intelligence 42(2-3), 311–348. Maher, M. J. and Governatori, G. 1999. A semantic decomposition of defeasible logics. In Proceedings of the 16th national conference on Artificial intelligence (AAAI 1999). AAAI/MIT Press, 299–305. Maher, M. J., Rock, A., Antoniou, G., Billington, D., and Miller, T. 2001. Efficient defeasible reasoning systems. International Journal on Artificial Intelligence Tools 10(4), 483–501. Maier, F. and Nute, D. 2006. Ambiguity propagating defeasible logic and the wellfounded semantics. In Proceedings of the 10th European Conference on Logics in Artificial Intelligence (JELIA 2006). Springer, LNAI 4160, 306–318. Makinson, D. 1994. General patterns in nonmonotonic reasoning. In D. Gabbay & C. Hogger (Eds.), Handbook of Logic for Artificial Intelligence and Logic Programming, Vol. III. Oxford: Oxford University Press, 35–110. Makinson, D., Schlechta, K. 1991. Floating conclusions and zombie paths: two deep difficulties in the ‘directly skeptical’ approach to inheritance nets. Artificial Intelligence, 48(2), 199–209. Moore, R.C. 1985. Semantical considerations on nonmonotonic logic. Artificial Intelligence, 25(1), 75–94.

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Nute, D. 1986. LDR: A logic for defeasible resasoning. ACMC Research Report 01-0013. The University of Georgia. Nute, D. 1987. Defeasible reasoning. In Proceedings of the 20th Hawaii International Conference on System Science, 470–477. Nute, D. 1992. Basic defeasible logic. In L.F. del Cerro, M. Penttonen (Eds.) Intensional Logics for Programming. Oxford University Press, 125–154. Nute, D. 1994. Defeasible logic. In D. Gabbay and C. Hogger (Eds.), Handbook of Logic for Artificial Intelligence and Logic Programming, Vol. III. Oxford University Press, 353–395. Nute, D. 1999. Norms, priorities, and defeasibility. In P. McNamara and H. Prakken (Eds.), Norms, Logics and Information Systems. IOS Press, 201–218. Nute, D. 2003. Defeasible logic: Theory, implementation, and applications. In Proceedings of the 14th International Conference on Applications of Prolog (INAP 2001). Springer, LNCS 2543, 151–169. Nute, D., Billington, D., & Coster, K.D. 1989. Defeasible logic and inheritance hierarchies with exceptions. In Proceedings of the T¨ ubingen Workshop on Semantic Nets and Nonmonotonic Reasoning, Vol. I (pp. 69–82). University of T¨ ubingen. Pollock, J.L. 1974. Knowledge and Justification. Princeton University Press. Pollock, J.L. 1987. Defeasible reasoning. Cognitive Science, 11(4), 481–518. Prakken, H. 2002. Intuitions and the modelling of defeasible reasoning: some case studies. In S. Benferhat, E. Giunchiglia (Eds.), Proceedings of the 9th International Workshop on Non-Monotonic Reasoning (NMR 2002), 91–99. Reiter, R. 1980. A logic for default reasoning. Artificial Intelligence 13(1–2), 81–132. Schaub, T. and Wang, K. 2002. Preferred well-founded semantics for logic programming by alternating fixpoints. Preliminary report. In S. Benferhat, E. Giunchiglia (Eds.), Proceedings of the 9th International Workshop on Non-Monotonic Reasoning (NMR 2002), 238–246. Stein, L.A. 1990. Resolving Ambiguity in Nonmonotonic Reasoning. Dissertation, Brown University. Tarski, A. 1955. A lattice theoretic fixpoint theorem and its application. Pacific Journal of Mathematics 5(2), 285–309. Van Gelder, A., Ross, K. A., and Schlipf, J. 1991. The well-founded semantics for general logic programs. Journal of the ACM 38(3), 619-49.

Well-Founded Semantics for Defeasible Logic

Well-Founded Semantics for Defeasible Logic

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