PDF Comparing Abductive Theories

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Comparing Abductive Theories

Katsumi Inoue

¹

and Chiaki Sakama

²

Abstract. This paper introduces two methods for comparing explanation power of different abductive theories. One is comparing explainability for observations, and the other is comparing explana- tion contents for observations. Those two measures are represented by generality relations over abductive theories. The generality relations are naturally related to the notion of abductive equivalence introduced by Inoue and Sakama. We also analyze the computational complexity of these relations.

1 Introduction

Abduction has been used in many applications of AI including diag- nosis, design, updates, and discovery. Abduction is incorporated in problem-solving and programming technologies as abductive logic programming [11]. In the process of building knowledge bases, we need to update an abductive theory in accordance with situation change and discovery of surprising facts. For example, to refine an incomplete description, one may need to add more details to a part of the current theory. Such a refinement is expected to ensure that the revised theory is more powerful in abductive reasoning than the previous one. Then, it is important to evaluate abductive theories by comparing abductive power of each theory in such processes.

In predicate logic, comparison of information contents between theories is done by comparing their logical consequences. For example, given two first-order theoriesT1andT2,T1is considered more informative thanT2ifT2 |=ψimpliesT1 |=ψfor any formulaψ, i.e.,T1 |= T2. In this case, it is also saidT1 is more general than T2[13, 14]. On the other hand,T1andT2are equally informative if T1 |=T2andT2|=T1, that is, ifT1andT2are logically equivalent (T1 ≡ T2). Recently, Inoue and Sakama considered the generality conditions for answer set programming (ASP) [9] and for Reiter’s default logic [10]. These generality/equivalence relations compare monotonic/nonmonotonic theories in terms of deduction.

The topic of our interest in this paper is how to compare abductive theories. That is, we seek conditions under which an abductive theory has more explanation power than another abductive theory. As far as the authors know, no answer to this question is given in the literature of abduction. To understand the problem, suppose that an abductive theoryA1is defined to be stronger than another abductive theoryA2. This might imply that there is a formula which can be explained in the former but cannot be in the latter. Then, we would expect thatA1 has more background knowledge thanA2orA1has more hypotheses thanA2. However, the situation is not so simple because addition of background knowledge may violate the consistency of some combination of hypotheses. Hence, relationships between

1National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan. email:[email protected]

2 Wakayama University, Sakaedani, Wakayama 640-8510, Japan. email:

[email protected]

amounts of background theories and hypotheses need to be analyzed in depth to compare abductive theories precisely.

In this paper, we consider two logical frameworks for abduction, first-order abduction and abductive logic programming (ALP). Then, we introduce two methods for comparing explanation power of different abductive theories, which were originally introduced by Inoue and Sakama [8] to identify equivalence of two abductive theories.

The first one is aimed at comparing explainability for observations in different theories, while the second one is aimed at comparing ex- planation contents for observations. Those two comparison measures are represented by generality relations over abductive theories. More- over, the generality relations can naturally be related to the notion of abductive equivalence in [8]. Note that the proposed techniques for first-order abduction can also be applied to comparing frameworks for explanatory induction in inductive logic programming.

The rest of this paper is organized as follows. Section 2 introduces two generality relations for comparing abductive first-order theories.

Section 3 applies the similar techniques to ALP. Section 4 relates the abductive generality relations to abductive equivalence. Section 5 discusses the complexity issues. Section 6 gives concluding remarks.

2 Generality Relations in First-order Abduction

In this section, we consider abductive theories represented in first- order logic, which have often been used in abduction in AI, e.g., [17].

In this setting, abductive theories are compared by two measures.

Definition 1 Suppose thatBandHare sets of first-order formulas, whereBrepresents background knowledge andHis a set of (can- didate) hypotheses. We call a pair(B, H)a (first-order) abductive theory. Given a formulaO as an observation, a setEof formulas belonging toH³is an explanation ofO in(B, H)ifB∪E |=O andB∪Eis consistent. We say thatOis explainable in(B, H)if it has an explanation in(B, H).

2.1 Comparing Explainability

We first consider a measure for comparing explainability between abductive theories.

Definition 2 An abductive theory A1 = (B1, H1) is more (or equally) explainable than an abductive theoryA2= (B2, H2), written asA1 ≥A2, if every observation explainable inA2is also explainable inA1.

3In this paper we do not specify howHis constructed. For example, when hypotheses contain variables, we could just assume that the setHis closed under instantiation. In another case, we could specify the language ofH with a bias and then define that any formula which is constructed fromH and satisfies the bias belongs toH. This latter treatment enables us to deal with comparing theories for inductive logic programming (ILP) [14] within the same logical framework as abduction. In any case, we simply denote as E⊆HwhenEis a set of formulas belonging toH.

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Example 1 Consider three abductive theories A1 = (B1, H1), A2= (B2, H2)andA3= (B3, H3), where

B1 = {sprinkler was on⊃grass is wet}, H1 = {sprinkler was on, rained last night}, B2 = B1∪ {rained last night⊃grass is wet}, H2 = H1∪ { ¬(sprinkler was on ⊃grass is wet)}, B3 = B2∪ {grass is wet⊃shoes are wet}, H3 = H1∪ { ¬(sprinkler was on ⊃shoes are wet)}. Then,A3 ≥A2 ≥A1holds. In fact, every observation explainable inAiis explainable inAi+1fori= 1,2. Notice thatA1≥A2also holds becauserained last nightcan be explained by itself in both A1 andA2. By contrast,shoes are wet is explainable inA3, but is not in eitherA1 orA2, i.e.,A2 6≥A3. Note that each additional hypothesis inHj\H1forj= 2,3has no effect in explaining any formula as it cannot be added toBjwithout violating the consistency.

We provide a necessary and sufficient condition for the explainable generality relation. In the following,T h(Σ)denotes the set of logical consequences of a setΣof first-order formulas.

Definition 3 An extension of an abductive theoryA = (B, H)is T h(B∪S)whereS is a maximal set of formulas belonging toH such thatB∪Sis consistent. The set of all extensions ofAis denoted asExt(A).

Lemma 1 ([17]) LetObe a (possibly infinite) set of formulas. There is an explanation that explains every formula inOin(B, H)iff there is an extensionXof(B, H)such thatO⊆X.

Theorem 2 LetA1 = (B1, H1)andA2 = (B2, H2)be abductive theories. Then,A1 ≥A2holds iff for any extensionX2ofA2, there is an extensionX1ofA1such thatX2⊆X1.

Proof: (⇐) By Lemma 1, if an observationOis explainable inA2, there isX2 ∈Ext(A2)such thatO∈X2. For any suchX2, there isX1 ∈ Ext(A1)such thatX2 ⊆ X1. Then,O ∈ X1 andO is explainable in(B1, H1)by Lemma 1. Hence,A1≥A2.

(⇒) Assume that there isX2∈Ext(A2)such thatX26⊆X1for anyX1 ∈ Ext(A1). Pick a formulaψⁱfor eachX1i ∈ Ext(A1) such thatψⁱ∈(X2\X1i) (6=∅), and letObe the set ofψⁱ’s from everyX1i. Then,O ⊆ X2 butO 6⊆X1 for anyX1 ∈ Ext(A1).

By Lemma 1,V

F∈OFis explainable inA2but is not explainable in

A1. Hence,A16≥A2. 2

There are several classes of abductive theories in which we can see explainable generality holds under some simple conditions.

Proposition 3 (Assumption-freeness) Suppose two abductive the- ories(B1,L)and(B2,L), whereLis the set of all literals in the underlying language. Then,(B1,L)≥(B2,L)iffB2|=B1. Proof: Any extension of an abductive theory (Bi,L) is logically equivalent to a (complete) model ofBi. By Theorem 2,(B1,L) ≥ (B2,L)iff, for any modelM ofB2, there is a modelNofB1such thatM ⊆N. Because bothMandNare complete,M ⊆Nimplies M =N. Hence, any model ofB2is a model ofB1. 2 Proposition 4 (Semi-monotonicity) Suppose that (B, H1) and (B, H2) are two abductive theories with the same background knowledge. IfH1⊇H2, then(B, H1)≥(B, H2).

Proof: For any abductive theory (B, H), we can associate a prerequisite-free normal default theory∆ = (DH, B), whereDH = {^:h_h |h∈H}. Then there is a 1-1 correspondence between the extensions of∆(in the sense of Reiter [18]) andExt((B, H))[17, Theorem 4.1]. By the semi-monotonicity of normal default theories [18, Theorem 3.2],H1 ⊇ H2implies that, for any extensionF of

∆2 = (DH₂, B), there is an extensionEof∆1 = (DH₁, B)such thatF ⊆E. By Theorem 2, the result holds. 2 For abductive theoriesA1= (B1, H)andA2= (B2, H)with the same hypotheses,B1|=B2implies neitherA1≥A2norA2≥A1. This explains the name of semi-monotonicity in Proposition 4.

Example 2 Suppose the abductive theoriesA= (B, H)andA⁰ = (B⁰, H)whereB={a∧b⊃p},B⁰=B∪ {¬b}, andH={a, b}.

Then,A⁰ 6≥Abecausephas the explanation{a, b}inAbut is not explainable inA⁰. On the other hand,A 6≥A⁰because¬bhas the explanation∅inA⁰but is not explainable inA.

2.2 Comparing Explanations

We next provide a second measure for comparing abductive theories.

This time we compare explanation contents.

Definition 4 An abductive theory A1 = (B1, H1) is more (or equally) explanatory than an abductive theoryA2= (B2, H2), written asA1DA2, if, for any observationO, every explanation ofOin A2is also an explanation ofOinA1.

Example 3 For three abductive theories in Example 1,A3DA2 D A1holds. AlthoughA1 ≥A2holds, we see thatA1 4A2because {rained last night}is an explanation ofgrass is wet inA2but is not inA1.

It is easy to see that the relationDis stronger than the relation≥, that is,A1DA2impliesA1≥A2. Now we show the necessary and sufficient condition for explanatory generality.

Theorem 5 LetA1 = (B1, H1)andA2 = (B2, H2)be abductive theories. Then,A1 DA2holds iffB1 |= B2 andH1 ⊇ H2 hold, whereHi={E⊆H_i|B_i∪Eis consistent}fori= 1,2.

Proof: Note that any explanationEof an observationOin(Bi, Hi) satisfies that (1)B_i∪E|=Oand (2)E∈ Hi.

(⇒) SupposeA1 D A2. Then for any formulaOand any setE of formulas,B2∪E |=OandE ∈ H² implyB1∪E |=Oand E ∈ H1. By the fact thatB2∪E |=OimpliesB1∪E |= Ofor anyO, we haveB2∪E|=B1∪Efor anyE∈ H²∩ H¹. Then, B2 |= B1 holds whenE = ∅. By the fact thatE ∈ H2 implies E∈ H1, we also haveH2⊆ H1. Hence, the result holds. 2 Corollary 6 LetA1 = (B1, H1)andA2= (B2, H2)be abductive theories. Then,A1DA2holds iffB1|=B2andA1≥A2hold.

Proof: The setHiin Theorem 5 contains every subsetEofHisuch thatBi∪E is consistent.Hi can be characterized byExt(Ai)as each consistent theory is a subset of some extension. Then, it can be proved that H¹ ⊇ H² iff for anyX2 ∈ Ext(A2), there is X1 ∈Ext(A1)such thatX2⊆X1. Hence, the result follows from

Theorem 2. 2

Corollary 7 IfH1⊇H2, then(B, H1)D(B, H2)holds.

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3 Generality Relations in Abductive Logic Programming

In this section, we turn our attention to generality relations in abduc- tive logic programming (ALP) [11]. The most significant difference between abduction in first-order logic and ALP is that ALP allows the nonmonotonic negation-as-failure operatornotin a background program. When the background programPis nonmonotonic, the fact thatP∪Eis consistent for some setEof hypotheses does not neces- sarily imply thatP∪E⁰is consistent forE⁰⊂E. Hence comparing abductive power in ALP should be checked in a more naive manner upon each subset of hypotheses.

Definition 5 An abductive (logic) program is a pairhP,Γiwhere

• Pis a (logic) program, which is a set of rules of the form:

L1;· · ·;L_k;not L_k+1;· · ·;not L_l

←Ll+1, . . . , Lm, not Lm+1, . . . , not Ln (1) where eachLiis a literal(n≥m≥l≥k≥0), andnotrepre- sents negation as failure (NAF). The symbol;represents disjunction. The left-hand side of the rule is the head, and the right-hand side is the body. A program containing variables is a shorthand of its ground instantiation.

• Γis a set of literals, called abducibles. Any instance of an abducible is also an abducible.

Logic programs mentioned above belong to the class of gen- eral extended disjunctive programs (GEDPs) [6]. If any rule of the form (1) in a programPdoes not containnotin its head, i.e.,k=l, P is called an extended disjunctive program (EDP) [4]. Moreover, if the head of any rule in an EDP P contains no disjunction, i.e., k=l≤1,Pis called an extended logic program (ELP). A semantics of a logic program is given by the answer set semantics [4, 6].

We denote the set of all ground literals in the language of a program asLit. For a programP, the set of answer sets ofPis denoted asAS(P). WhenPis an EDP,AS(P)is an antichain in2^Lit, that is, for any two answer setsS1, S2∈AS(P),S1 ⊆S2impliesS1=S2

[4], but this is not the case for a GEDP. A semantics for ALP is given by extending answer sets of the background program with addition of abducibles. Such an extended answer set is called a belief set, which has also been called a generalized stable model [11].

Definition 6 LetA=hP,Γibe an abductive program, andE⊆Γ.

A belief set ofA(with respect toE) is a consistent answer set of the logic programP ∪E. The set of all belief sets ofAis denoted as BS(A). A setS ∈BS(A)is often denoted asSEwhenSis a belief set with respect toE.

Definition 7 LetA = hP,Γibe an abductive program, andGa conjunction of ground literals called an observation. We will often identify a conjunctionGwith the set of literals inG. A setE ⊆Γ is an explanation ofGinAif every ground literal inGis true in a belief set ofAwith respect toE.⁴WhenGhas an explanation inA, Gis explainable inA.

Note that restrictions in ALP can be removed so that not only literals but rules can be allowed as abducibles and that observations can contain NAF formulas as well as literals. As in the case of first-order abduction, two generality relations are defined for ALP as follows.

4This definition provides credulous explanations. Alternatively, skeptical ex- planations are defined asE⊆Γsuch thatGis true in every belief set of Awith respect toE.

Definition 8 LetA1=hP1,Γ1iandA2 =hP2,Γ2ibe abductive programs, andGan observation.A1is more (or equally) explainable thanA2, written asA1 ≥A2, if every observation explainable inA2

is also explainable inA1. On the other hand,A1is more (or equally) explanatory thanA2, written asA1DA2, if, for any observationG, every explanation ofGinA2is also an explanation ofGinA1. Example 4 LetA1 = hP1,Γiand A2 = hP2,Γibe abductive programs, whereP1={p←a, a←b},P2={p←a, p←b}, andΓ ={a, b}. Then,A1≥A2andA2≥A1, whileA1 DA2but A24A1. In fact,{b}is an explanation ofainA1, but is not inA2.

The following results hold for two generality relations.

Theorem 8 LetA1 =hP1,Γ1iandA2 =hP2,Γ2ibe abductive programs. Then,A1≥A2holds iff for any belief setS2ofA2, there is a belief setS1ofA1such thatS2⊆S1.

Proof: (⇐) IfGis explainable inA2, there isS2 ∈BS(A2)such thatG⊆S2. For any suchS2, there isS1∈BS(A1)such thatS2⊆ S1. Then,G⊆S1andGis explainable inA1. Hence,A1≥A2.

(⇒) Assume that there isS2 ∈BS(A2)such thatS2 6⊆S1 for anyS1 ∈BS(A1). For eachS1i ∈BS(A1), pick a literalLⁱsuch thatLⁱ∈(S2\S1i) (6=∅), and letGbe the set ofLⁱ’s from every S1i. Then,G⊆S2butG6⊆S1for anyS1∈BS(A1). That is,G is explainable inA2but is not inA1, i.e.,A16≥A2. 2 Theorem 9 LetA1 =hP1,Γ1iandA2 =hP2,Γ2ibe abductive programs. Then,A1DA2holds iff for anyE ⊆Γ2and anySE ∈ BS(A2), there isT_E∈BS(A1)such thatE⊆Γ1andS_E⊆T_E. Proof: (⇒) SupposeA1DA2. Then, for any observationGand any E⊆Γ2, the fact thatG⊆SEfor someSE∈BS(A2)implies that G⊆T_Efor someT_E∈BS(A1). Thus,S_E⊆T_E.

(⇐) SupposeSE ∈ BS(A2)for anyE ⊆Γ2 implies the exis- tence ofT_E∈BS(A1)withE⊆Γ1such thatS_E⊆T_E. Then, for any observationG,G⊆SEimpliesG⊆TE. That is, ifGhas an explanationEinA2,Ghas the same explanationEinA1. 2 Theorem 8 and Theorem 9 might look similar, but the condition of the latter is finer-grained than that of the former. In fact, as in the case of first-order abduction,A1DA2impliesA1≥A2.

4 Connection to Abductive Equivalence

In this section, we consider the relationship between the generality relations in abduction proposed in this paper and the equivalence relations in abduction proposed in the literature. Inoue and Sakama [8] study different types of equivalence relations in abduction: explainable/explanatory equivalence of abductive theories under both first-order abduction and ALP. Pearce et al. [16] characterize a part of these problems in the context of equilibrium logic. In the following, an abductive frameworkAmeans either a first-order abductive theoryA= (B, H)or an abductive logic programA=hP,Γi.

Definition 9 ([8]) LetA1andA2be abductive frameworks.

1. A1andA2are explainably equivalent if, for any observationO,⁵ Ois explainable inA1iffOis explainable inA2.

2. A1andA2are explanatorily equivalent if, for any observationO, Eis an explanation ofOinA1iffEis an explanation ofOinA2.

5This definition of explainable equivalence for ALP is not exactly the same as that in [8, Definition 4.3]. In [8] an observation is a single ground literal, while we allow a conjunction of ground literals as an observation.

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Explainable equivalence requires that two abductive frameworks have the same explainability for any observation. Explainable equivalence may reflect a situation that two programs have different knowledge to derive the same goals. On the other hand, explanatory equivalence assures that two abductive frameworks have the same explanation contents for any observation. Explanatory equivalence is stronger than explainable equivalence: if two abductive frameworks are explanatorily equivalent then they are explainably equivalent.

By Definitions 2, 4, 8, and 9, it is obvious that all generality relations defined in this paper are “anti-symmetric”⁶ in the sense that two abductive frameworks are explainably/explanatorily equivalent iff one is both more (or equally) and less (or equally) explainable/explanatory than another at the same time.

Proposition 10 LetA1andA2be abductive frameworks.

1. A1andA2are explainably equivalent iffA1≥A2andA2≥A1. 2. A1andA2are explanatorily equivalent iffA1DA2andA2DA1. With this correspondence and results in previous sections, we can derive either new characterizations of abductive equivalence or new (and simple) proofs of previously presented results. For first-order abduction, the following results can be verified with new proofs.

Proposition 11 Two first-order abductive theoriesA1 andA2 are explainably equivalent iffExt(A1) =Ext(A2)holds.

Proposition 12 For first-order abductive theoriesA1 = (B1, H1) andA2= (B2, H2), the following four statements are equivalent.

1. A1andA2are explanatorily equivalent.

2. A1andA2are explainably equivalent andB1 ≡B2. 3. B1≡B2andH1 =H2.

4. B1≡B2andH1⁰ =H2⁰, where

H_i⁰={h∈Hi|Bi∪ {h} is consistent}fori= 1,2.

For ALP, the next results can be newly obtained. In the following, for any setX, letmax(X) ={x∈X | ¬∃y∈X. x⊂y}.

Theorem 13 Let A1 = hP1,Γ1i and A2 = hP2,Γ2i be ab- ductive programs. Then,A1 andA2 are explainably equivalent iff max(BS(A1)) =max(BS(A2)).

Proof: (⇒) By Theorem 8,A1 ≥ A2 implies that, for anyS2 ∈ max(BS(A2))there existsS1 ∈BS(A1)such thatS2 ⊆S1, and then there existsS1⁰ ∈max(BS(A1))such thatS1 ⊆S⁰1. ByA2≥ A1, there existsS2⁰ ∈BS(A2)such thatS1⁰ ⊆S2⁰, and then there existsS⁰⁰2 ∈ max(BS(A2))such thatS2⁰ ⊆ S2⁰⁰. ThenS2 ⊆ S2⁰⁰

holds, but because both belong tomax(BS(A2)),S2 =S2⁰⁰holds.

Hence,S2(=S⁰1)also belongs tomax(BS(A1)), and thus the result holds. (⇐) can be proved by tracing the above proof backward. 2 Theorem 14 LetA1=hP1,Γ1iandA2=hP2,Γ2ibe abductive programs.A1andA2are explanatorily equivalent iffC¹=C²holds andmax(AS(P1∪E)) =max(AS(P2∪E))for anyE ∈ Ci, whereCi={E⊆Γi|P_i∪E is consistent}fori= 1,2.

Proof: (⇒) Suppose that A1 and A2 are explanatorily equivalent.

By Theorem 9,A1 D A2 implies that, for anyE ⊆ Γ2 and any S_E ∈ BS(A2), there is T_E ∈ BS(A1)such thatE ⊆ Γ1 and SE⊆TE. Then, for anyE⊆Γ2and anyS∈max(AS(P2∪E)), E ⊆Γ1and there isT ∈AS(P1∪E)such thatS⊆T, and then

6The relations≥andDare also preorders, i.e., reflexive and transitive, for both first-order abduction and ALP.

there isT⁰∈max(AS(P1∪E))such thatT ⊆T⁰. ByA2DA1, there isS⁰ ∈ AS(P2∪E)such thatT⁰ ⊆ S⁰, and then there is S⁰⁰ ∈max(AS(P2∪E))such thatS⁰ ⊆S⁰⁰. ThenS ⊆S⁰⁰holds and both belong tomax(AS(P2∪E)), which implyS = T⁰ = S⁰⁰, and thusS ∈max(AS(P1∪E)). Hence, (1) ifE ⊆ Γ2 and P2∪Eis consistent thenE ⊆ Γ1 andP1∪E is consistent, and (2)max(AS(P2∪E)) ⊆ max(AS(P1∪E))for anyE ⊆ Γ2. Similarly, (3) ifE⊆Γ1andP1∪Eis consistent thenE⊆Γ2and P2∪Eis consitent, and (4)max(AS(P1∪E))⊆max(AS(P2∪ E))for anyE⊆Γ1. By (1) and (3),C1=C2holds. By (2) and (4), max(AS(P1∪E)) =max(AS(P2∪E))holds for anyE ⊆Γ1

and for anyE⊆Γ2. Hence, the result follows.

(⇐) can be proved in a similar way. 2

Two logic programsP1andP2are strongly equivalent with respect to a rule setRifAS(P1∪R) =AS(P2∪R)for any logic program R⊆ R[7]. This equivalence notion is a restricted version of strong equivalence [12], and is called relative strong equivalence [7].⁷The next result was originally shown in [8]⁸ and then was discussed in [16] for EDPs. Now it can be simply proved by the antichain prop- erty ofAS(P)for any EDPP.

Corollary 15 LetA1 =hP1,ΓiandA2 =hP2,Γibe abductive programs with the same hypotheses such that bothP1 andP2 are EDPs. Also, letP_i⁰=Pi∪{ ←L,¬L|L∈Lit}fori= 1,2. Then, A1 andA2are explanatorily equivalent iffP1⁰andP2⁰are strongly equivalent with respect toΓ.

5 Complexity Results

We show that the computational complexity of deciding generality between abductive theories becomes more complex in general than that of abductive equivalence presented in [8].

Theorem 16 LetA1 andA2 be two propositional abductive theo- ries. Deciding ifA1≥A2isΠ^P3-complete.

Proof: LetA1 = (B1, H1)andA2 = (B2, H2). We here identify Ext(Ai)with the extensions of the prerequisite-free normal default theory(DHi, Bi)fori= 1,2as in the proof of Proposition 4. For any subsetS⊆H2, checking ifE=T h(B2∪S)is an extension of A2is coNP-complete [19]. IfE ∈Ext(A2)then deciding if there does not existF ∈Ext(A1)such thatE ⊆F can be determined by checking if the formulaV

B2∧V

Sbelongs to some extension ofA1, which isΣ^P2-complete [5]. Thus, we can chooseS ⊆ H2

in nondeterministic polynomial time with aΣ^P2-oracle to decide if A1 6≥A2holds. Hence, the original problem is the complement of this, and belongs toΠ^P3. We omit the proof ofΠ^P3-hardness because

of the space limitation. 2

Theorem 17 LetA1 andA2 be two propositional abductive theo- ries. Deciding ifA1DA2isΠ^P3-complete.

Proof: Follows from Corollary 6 and Theorem 16. 2

7This definition is due to [7], and is slightly different from the notion of rel- ativized equivalence in [20, 16]. In [20],P₁andP₂are defined as strongly equivalent relative to a literal setUiffAS(P₁∪R) =AS(P₂∪R)for any setRof rules that are constructed using literals inU.

8The condition of EDPs was missing in [8, Theorem 4.4]. In fact, only Theo- rem 14 holds for GEDPs. Moreover, to characterize inconsistent programs in ALP, an EDP having the answer setLitshould be translated to an EDP without an answer set in Corollary 15.

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Theorem 18 LetA1=hP1,Γ1iandA2=hP2,Γ2ibe abductive programs. Deciding ifA1≥A2is (i)Π^P2-complete whenP1andP2

are ELPs, and is (ii)Π^P3-complete whenP1andP2are GEDPs.

Proof: A computation problem in GEDPs reduces in polynomial time to the corresponding problem in EDPs [6], so we here consider the cases that eachPiis either an ELP or an EDP.

(Membership) For any guessS⊆Lit, deciding ifS∈BS(A2)is NP-complete for an ELPP2(resp.Σ^P2-complere for an EDPP2) [2].

For such anS, deciding if there does not existT ∈ BS(A1)such thatS ⊆T can be determined by credulous reasoning that contains S, which is NP-complete for an ELPP1(resp.Σ^P2-complere for an EDPP1) [2]. Hence, by Theorem 8,A1 6≥A2can be nondeterministically solvable with two calls to an NP-oracle (resp. aΣ^P2-oracle).

Therefore, the complement is inΠ^P2 (resp.Π^P3).

(Hardness) We prove for the ELP case. LetΦ = ∀X∃Y.φbe a closed QBF, whereφ= Wn

j=1Cj is a DNF formula, that is,Cj is a conjunction of literals. LetA1 = hP1,Γ1iandA2 =hP2,Γ2i be abductive programs such thatP1 = {g←Cj | 1 ≤ j ≤ n}, Γ1=X∪ ¬X∪Y∪ ¬Y,P2 ={g← }, andΓ2=X∪ ¬X, where

¬X ={¬x|x∈X}and¬Y ={¬y|y∈Y}. Note that bothP1

andP2are ELPs. We prove that:A1≥A2⇔Φis valid.

(⇒) SupposeA1 ≥A2. By Theorem 8, for anyS ∈ BS(A2), there isT ∈BS(A1)such thatS ⊆T. In particular, for anyIX ⊆ X, there is a belief setS∈BS(A2)with respect toIX∪¬(X\IX), and henceIX∪¬(X\IX)⊆Tfor someT ∈BS(A1). Sinceg∈S, gmust be inTtoo. Then, someC_j(1≤j≤n)must be true under IX∪ ¬(X\IX)andIY∪ ¬(Y\IY)for someIY ⊆Y. Hence,φis true under such an interpretation. SinceI_Xwas arbitrary,Φis valid.

(⇐) SupposeΦis valid. Then for anyIX ⊆X,φis true under I_X∪ ¬(X\I_X)andI_Y ∪ ¬(Y\I_Y)for someI_Y ⊆Y. Then some Cjis true under this interpretation, and hencegholds. It is easy to see for anyS ∈BS(A2)that there isT ∈BS(A1)such thatS ⊆T. By Theorem 8,A1≥A2holds.

For the EDP case, we can apply a transformation of a QBF

∀X∃Y∀Z.φinto a disjunctive program, which is analogous to the one presented in [1, Theorem 3.1] and [2, Lemma 2]. 2 Theorem 19 LetA1=hP1,Γ1iandA2=hP2,Γ2ibe abductive programs. Deciding ifA1DA2is (i)Π^P2-complete whenP1andP2

are ELPs, and is (ii)Π^P3-complete whenP1andP2are GEDPs.

Proof: Like Theorem 18, we can assume that eachP_iis either an ELP or an EDP. For any guessS ⊆Lit, deciding ifSE ∈BS(A2)for someE ⊆Γ2 is NP-complete for an ELPP2(resp.Σ^P2-complere for an EDPP2) [2]. For any suchE, deciding if AS(P1∪E) 6=

∅is NP-complete for an ELPP2 (resp.Σ^P2-complere for an EDP P2) [1]. ForSE, deciding if there does not exist T ∈ AS(P1 ∪ E)such thatSE ⊆ T can be determined by credulous reasoning that containsS_E, which is NP-complete for an ELPP1 (resp.Σ^P2- complere for an EDPP1) [2]. Hence, by Theorem 9,A14A2can be nondeterministically solvable with three calls to an NP-oracle (resp.

aΣ^P2-oracle). Therefore, the complement is inΠ^P2 (resp.Π^P3). The hardness can be shown in the same way as in Theorem 18. 2

6 Discussion

The relation≥introduced in this paper can be represented by generality relations defined by Inoue and Sakama [9, 10]. We briefly sketch the relationships here. For first-order abductive theoriesA1= (B1, H1)andA2= (B2, H2), by identifyingExt(Ai)with the extensions of the prerequisite-free normal default theory(DHi, B_i)for

i= 1,2, we can prove thatA1 ≥A2iffA1 |=^[_dt A2, where|=^[_dt is a Hoare order defined on the class of default theories [10]. On the other hand, for abductive logic programsA1 = hP1,Γ1iand A2=hP2,Γ2i, letP_i⁰(i= 1,2) be the GEDP defined by

P_i⁰=P_i∪ {l;not l← | l∈Γi}.

Then,BS(Ai) = AS(P_i⁰)holds [6]. With this result, we can see thatA1≥A2iffP1⁰|=^[_lpP2⁰, where|=^[_lpis a Hoare order defined on the class of GEDPs (originally defined on the class of EDPs in [9]).

Besides work on generality relations in ASP [9], a general correspondence framework has been proposed in [3, 15] to compare logic programs. This framework is defined to compare equivalence and inclusion between the semantics of logic programs instead of generality, but the notions of projection and contexts are also introduced to enable a variety of equivalence comparison. Incorporating these notions into our generality framework is a topic of future work.

REFERENCES

[1] T. Eiter and G. Gottlob. On the computational cost of disjunctive logic programs: propositional case. Annals of Mathematics and Artificial In- telligence, 15:289–323, 1995.

[2] T. Eiter, G. Gottlob and N. Leone. Abduction from logic programs:

semantics and complexity. Theoretical Computer Science, 189:129–

177, 1997.

[3] T. Eiter, H. Tompits and S. Woltran. On solution correspondences in answer-set programming. In: Proc. IJCAI-05, pp. 97–102, 2005.

[4] M. Gelfond and V. Lifschitz. Classical negation in logic programs and disjunctive databases. New Generation Computing, 9:365–385, 1991.

[5] G. Gottlob. Complexity results for nonmonotonic logics. J. Logic and Computation, 2:397–425, 1992.

[6] K. Inoue and C. Sakama. Negation as failure in the head. J. Logic Programming 35, pp. 39–78, 1998.

[7] K. Inoue and C. Sakama. Equivalence of logic programs under updates.

In: Proc. 9th European Conference on Logics in Artificial Intelligence, LNAI 3229, pp. 174–186, Springer, 2004.

[8] K. Inoue and C. Sakama. Equivalence in abductive logic. In: Proc.

IJCAI-05, 2005, pp. 472–477.

[9] K. Inoue and C. Sakama. Generality relations in answer set programming. In: Proc. 22nd International Conference on Logic Programming, LNCS 4079, pp. 211–225, Springer, 2006.

[10] K. Inoue and C. Sakama. Generality and equivalence relations in default logic. In: Proc. 22nd Conference on Artificial Intelligence (AAAI- 07), pp. 434–439, 2007.

[11] A. Kakas, R. Kowalski and F. Toni. The role of abduction in logic programming. In: D. Gabbay, C. Hogger and J. Robinson, editors, Hand- book of Logic in Artificial Intelligence and Logic Programming, Vol. 5, pp. 235–324, Oxford University Press, 1998.

[12] V. Lifschitz, D. Pearce and A. Valverde. Strongly equivalent logic programs. ACM Transactions on Computational Logic, 2:526–541, 2001.

[13] T. Niblett. A study of generalization in logic programs. In: Proc. 3rd European Working Sessions on Learning, pp. 131–138, Pitman, 1988.

[14] S.-H. Nienhuys-Cheng and R. De Wolf. Foundations of Inductive Logic Programming. LNAI 1228, Springer, 1997.

[15] J. Oetsch, H. Tompits and S. Woltran. Facts do not cease to exist because they are ignored: relativised uniform equivalence with answer-set projection. In: Proc. 22nd Conference on Artificial Intelligence (AAAI- 07), pp. 458–464, 2007.

[16] D. Pearce, H. Tompits and S. Woltran. Relativised equivalence in equilibrium logic and its applications to prediction and explanation: prelim- inary report. In: Proc. LPNMR’07 Workshop on Correspondence and Equivalence for Nonmonotonic Theories, pp. 37–48, 2007.

[17] D. Poole. A logical framework for default reasoning. Artificial Intelli- gence, 36:27–47, 1988.

[18] R. Reiter. A logic for default Reasoning. Artificial Intelligence, 13:81–

132, 1980.

[19] R. Rosati. Model checking for nonmonotonic logics: algorithm and complexity. In: Proc. IJCAI-99, pp. 76–81, 1999.

[20] S. Woltran. Characterizations for relativized notions of equivalence in answer set programming. In: Proc. 9th European Conference on Logics in Artificial Intelligence, LNAI 3229, pages 161–173, Springer, 2004.