Some Simplified Forms of Reasoning with Distance-Based Entailments

Transcription

1 Soe Siplified Fors of Reasoning with Distance-Based Entailents Ofer Arieli 1 and Anna Zaansky 2 1 Departent of Coputer Science, The Acadeic College of Tel-Aviv, Israel. oarieli@ta.ac.il 2 Departent of Coputer Science, Tel-Aviv University, Israel. annaz@post.tau.ac.il Abstract. Distance seantics is a robust way of handling dynaically evolving and possibly contradictory inforation. In this paper we show that in any cases distance-based entailents can be coputerized in a general and odular way. We consider two different approaches for reasoning with distance seantics, apply the on soe coon cases, and show their relation to other known probles. 1 Introduction Distance seantics has a proinent role in reflecting the rationality behind the principle of inial change. This is a priary otif in different areas, such as belief revision, database integration, and decision aking in the context of social choice theory. While there is no consensus about the exact nature of this seantics and the properties that is should satisfy, soe particular distance-based approaches have been extensively used in those areas and are ore coon in practice. As shown in [1, 2], any of these distance-based seantics have siilar representations in ters of entailent relations, so it is not surprising that siilar coputational fors ay be used for providing reasoning platfors in those cases. The goal of this paper is therefore to consider soe of the coputational aspects behind these approaches, that is: to identify soe general principles for distance coputations and apply the on soe specific, nevertheless coon test cases. For this, we consider the following two reasoning paradigs: Deductive systes. This traditional approach to autoated reasoning should be taken with care in our context, as any classically valid rules do not hold when distance seantics is involved. Consider, for instance, an inference syste for ajority votes, in which ψ follows fro Γ if there are ore forulas in Γ iplying ψ than those iplying ψ. As it is shown below, this consideration can be used for a distance seantics. Yet, it is evident that this syste is neither reflexive nor onotonic (p follows fro {p} but not fro {p, p}). Moreover, it is not even closed under logical equivalence as, e.g., {p, p} and {p, p, p} have different conclusions (which also invalidates Supported by the Israel Science Foundation, grant No

2 the contraction rule in this case). This exaple indicates that even sound systes for distance seantics, based on structural rules, are hard to get. In the sequel we shall define a sound and coplete syste for one case (in reasoning with unifor distances), and consider soe useful sound systes for soe other distance seantics. For the latter, we shall consider situations in which autoated solvers for known probles ay be incorporated for reaching copleteness. Set coputations. The other approach for reasoning with distance seantics is based on coputations of a inial nonepty intersection of sets of interpretations. The eleents of each set are equally distant fro the forulas that they evaluate, and the inial nonepty intersection of those sets deterines the valuations that are closest to the preises. This idea resebles that of Grove [5], who defines revision in ters of set intersections. We introduce iterative processes for coputing those intersections for different distance-based settings, and show the correspondence between this proble and siilar probles in the context of constraint prograing. In the sequel we consider a general fraework for reasoning with distance seantics and concentrate on three coon cases: in reasoning, reasoning by voting, and reasoning by suations of distances. Each of these reasoning strategies is augented with different distances (etrics), and algoriths for coputing entailents induced by these settings are provided. It is shown that in soe cases distance seantics is reducible to other well-known probles (e.g., a variation of SAT), and so off-the-shelf solvers for those probles ay be useful for distance-based reasoning as well. 2 Distance-based Seantics 2.1 Preliinaries We fix a propositional language L with a finite set Atos = {p 1,..., p } of atoic forulas. A finite ultiset of forulas in L is called a theory. For a theory Γ, we denote by Atos(Γ ) the set of atoic forulas that occur in Γ. The set of valuations for L is Λ = { p 1 : a 1,..., p n : a a 1,..., a {t, f}}. The set of odels of a forula ψ is a subset of Λ, defined as follows: od(p i ) = { p 1 : a 1,..., p i : t,..., p n : a a 1,..., a i 1, a i+1,..., a {t, f}}, od( ψ) = Λ \ od(ψ), od(ψ ϕ) = od(ψ) od(ϕ), od(ψ ϕ) = od(ψ) od(ϕ). For a theory Γ = {ψ 1,..., ψ n } we define od(γ ) = od(ψ 1 )... od(ψ n ). For defining distance-based entailents we recall the definitions in [1, 2]. Definition 1. A pseudo-distance on U is a total function d : U U N so that for all ν, µ U d(ν, µ) = d(µ, ν) (syetry), and d(ν, µ) = 0 iff ν = µ (identity preservation). A distance (etric) d on U is a pseudo-distance that satisfies the triangular inequality: for all ν, µ, σ U, d(ν, σ) d(ν, µ) + d(µ, σ).

3 Exaple 1. It is easy to verify that the following functions are distances on the space Λ of two-valued valuations on Atos: The drastic distance: d U (ν, µ) = 0 if ν = µ and d U (ν, µ) = 1 otherwise. The Haing distance: d H (ν, µ) = {p Atos ν(p) µ(p)}. Definition 2. A nueric aggregation function is a total function f whose arguent is a ultiset of real nubers and whose values are real nubers, such that: (i) f is non-decreasing in the value of its arguent, (ii) f({x 1,..., x n }) = 0 iff x 1 = x 2 =... x n = 0, and (iii) f({x}) = x for every x R. Aggregation functions are, e.g., suation, average, iu, and so forth. Notation 1 Given a finite set S and a (pseudo) distance d, denote: d S = {d(s 1, s 2 ) s 1, s 2 S}. Definition 3. Given a theory Γ = {ψ 1,..., ψ n }, a valuation ν Λ, a pseudodistance d, and an aggregation function f, define: { in{d(ν, µ) µ od(ψi )} if od(ψ i ), d(ν, ψ i ) = 1 + d Λ otherwise. δ d,f (ν, Γ ) = f({d(ν, ψ 1 ),..., d(ν, ψ n )}). Note 1. In the two extree degenerate cases, when ψ is either a tautology or a contradiction, all the valuations are equally distant fro ψ. In the other cases, the valuations that are closest to ψ are its odels and their distance to ψ is zero. This also iplies that δ d,f (ν, Γ ) = 0 iff ν od(γ ) (see [2]). The next definition captures the intuition that the relevant interpretations of a theory Γ are those that are δ d,f -closest to Γ (see also [9]). Definition 4. The ost plausible valuations of Γ (with respect to a pseudo distance d and an aggregation function f) are defined as follows: { { ν Λ µ Λ δd,f (ν, Γ ) δ d,f (µ, Γ ) } if Γ, d,f (Γ ) = Λ Distance-based entailents are now defined as follows: otherwise. Definition 5. For a pseudo distance d and an aggregation function f, define Γ = d,f ψ if d,f (Γ ) od(ψ). 3 Exaple 2. Let Γ = {p, p, q}. As q is not related to the contradiction in Γ, there is no intuitive justification for concluding q fro Γ. This, however, is not possible in classical logic, as Γ is not consistent. In our case, on the other hand, we have that Γ = du,σ q while Γ = du,σ q, Γ = du,σ p and Γ = du,σ p. Siilar results are obtained for = dh,σ. 3 I.e., conclusions should hold in all the ost plausible valuations of the preises.

4 2.2 Coputing distance-based entailents Proposition 1. Denote by = the classical (two-valued) entailent. For every pseudo distance d and aggregation function f, (a) If Γ is satisfiable, then Γ = d,f ψ iff Γ = ψ. (b) For every Γ there is a ψ such that Γ = d,f ψ. Proof (outline). Part (a) follows fro the fact that d(ν, ψ) = 0 iff ν od(ψ) and δ d,f (ν, Γ ) = 0 iff ν od(γ ). Thus, Γ is satisfiable iff d,f (Γ ) = od(γ ) (see also [2]). Part (b) follows fro the fact that for every Γ, d,f (Γ ) (as Λ is finite, there are always valuations that are inially δ d,f -distant fro Γ ). Taken together, the two ites of Proposition 1 iply that every entailent relation induced by our fraework coincides with the classical entailent with respect to consistent preises, while (unlike classical logic) it is not trivial with respect to inconsistent theories. Thus one cannot hope for better coplexity results than those for the classical propositional logic, as for consistent preises the entailent proble is conp-coplete. On the other hand, it is clear fro Definition 5 and the fact that Λ is finite, that for coputable distances and aggregation functions, distance-based reasoning for finite propositional languages is in EXP, i.e., it is decidable with (at ost) exponential coplexity. The purpose of this work is, therefore, to consider soe useful distance-based settings for which there are soe practical ways of coputing entailents. In particular, we consider soe cases in which distance-based reasoning is reducible to the question of satisfiability, and so off-the-shelf SAT-solvers ay be incorporated for autoated coputations of distance-based consequences. For this, we first need the following definitions: Definition 6. Let d be a pseudo distance. Define a function R d : L N 2 Λ by R d (ψ, i) = {µ ν od(ψ) d(µ, ν) i}. Also, let R 0 d (ψ) = R d(ψ, 0) and R i d (ψ) = R d(ψ, i) \ R d (ψ, i 1) for any i N +. Note 2. For any ψ, the sequence R d (ψ, i) is non-decreasing in i (with respect to set inclusion). Also, for every i N, ν R i d (ψ) iff d(ν, ψ) = i. Thus, for a satisfiable ψ, R 0 d (ψ) = od(ψ) = d,f (ψ), and R d (ψ, k) = Λ for k = d Λ. If ψ is not satisfiable, then R d (ψ, i) = R i d (ψ) = for every i. Lea 1. If all the forulas in a theory Γ = {ψ 1,..., ψ n } are satisfiable, then there is soe 0 k d Λ, such that 1 i n R d(ψ i, k). Proof. By Note 2, for every 1 i n there is a k i d Λ such that for every j k i, R d (ψ i, j) = Λ. Let k = {k i 1 i n}. Then R d (ψ, k) = Λ for all 1 i n, and so 1 i n R d(ψ i, j) = Λ. Definition 7. A pseudo distance d is called inductively representable, if there is a coputable function G : 2 Λ 2 Λ such that for every forula ψ and every i N, R d (ψ, i) = G(R d (ψ, i 1)). G is called an inductive representation of d. Exaple 3. As Propositions 6 and 9 below show, both d U and d H are inductively representable.

5 3 MinMax Reasoning In this section we study distance-based reasoning by in- ethods, that is: iniization of ial distances. This kind of reasoning ay be viewed as a skeptical approach, since it iniizes worst cases (ial distances). Distance entailents of this type are induced by the aggregation function. 3.1 Inductively representable distances Min- distance-based reasoning is characterized as follows: 4 Proposition 2. For any pseudo distance d and theory Γ = {ψ 1,..., ψ n }, a) If there is a non-satisfiable eleent in Γ, then d, (Γ ) = Λ. b) If all the eleents in Γ are satisfiable, then d, (Γ ) = 1 i n R d(ψ i, ), where is the inial nuber such that 1 i n R d(ψ i, ) is not epty. Corollary 1. For every pseudo distance d and a theory Γ of satisfiable forulas, d, (Γ ) = 1 i n R d(ψ i, ), where = in{δ d, (ν, Γ ) ν Λ}. In case that d is inductively representable by G, the results above induce the iterative procedure in Figure 1, for coputing d, (Γ ): MPV(G, {ψ 1,..., ψ n }) /* Most Plausible Valuations of {ψ 1,..., ψ n} w.r.t. d and */ /* G an inductive representation of d */ for i {1,..., n}: X i od(ψ i ) if X j is epty for soe j {1,..., n}, return Λ while (X 1... X n) is nonepty: for i {1,..., n}: X i G(X i ) return (X 1... X n ) Fig. 1. Coputing the ost plausible valuations of {ψ 1,..., ψ n} w.r.t. d and Proposition 3. If d is inductively representable by G, then for every theory Γ, MPV(G, Γ ) terinates after at ost d Λ iterations and coputes d, (Γ ). Proof. It is easy to see that in the i-th iteration it holds that X j = R d (ψ j, i) for every 1 j n. Hence, by Lea 1, the condition in the loop is satisfied after at ost d Λ iterations, and so the procedure always terinates. Also, by Proposition 2, the procedure returns d, (Γ ). Note 3. It holds that d, (Γ ) = d, (Γ \ {ϕ}) for every tautology ϕ Γ. Thus, in the coputations above, tautological forulas ay be disregarded. 4 Due to a lack of space soe proofs are oitted.

6 3.2 Unifor distances In this section we consider unifor distances, a generalization of the drastic distance (see Exaple 1). Definition 8. A distance d on Λ is called unifor, if there is soe k d > 0, s.t.: { 0 if ν = µ, d(ν, µ) = otherwise. Proposition 4. Let d be a unifor distance and Γ = {ψ 1,..., ψ n }. Then: { od(γ ) if od(γ ), d, (Γ ) = Λ otherwise. k d Proof. If od(γ ), then d,f (Γ ) = od(γ ) for every d and f (see [2]). Otherwise, there is at least on eleent in Γ that is not satisfiable, and so, for every valuation µ Λ, δ d, (µ, Γ ) = {d(µ, ψ 1 ),..., d(µ, ψ n )} = k d. It follows, then, that in this case d, (Γ ) = Λ. Proposition 4 iplies that for describing reasoning with unifor distances under the in strategy, it is enough to exaine the drastic distance (and so henceforth we focus only on d U ): Corollary 2. For any unifor distances d 1, d 2 and any theory Γ, d1,(γ ) = d2,(γ ). Thus, for every forula ψ, Γ = d1, ψ iff Γ = d2, ψ. Another consequence of Proposition 4 is that = du, is strongly paraconsistent: Only tautological forulas follow fro inconsistent theories: Corollary 3. If Γ is inconsistent, then Γ = du, ψ iff ψ is a tautology. Proof. Suppose that Γ is not consistent. By Proposition 4, du,(γ ) = Λ, and so Γ = du, ψ iff du,(γ ) od(ψ), iff od(ψ) = Λ, iff ψ is a tautology. By Proposition 1 and Corollary 3 we conclude that reasoning with unifor distances under the in strategy has a soewhat crude nature : either the set of preises is classically consistent, in which case the set of conclusions coincides with that of the classical entailent, or, in case of contradictory preises, only tautologies are entailed. It follows that in this case questions of satisfiability and logical entailent are reducible to siilar probles in standard propositional logic, and distance considerations do not cause further coputational coplications. Moreover, standard SAT-solvers and theore provers ay be incorporated for ipleenting this kind of reasoning. Next we provide two ethods for coputerized reasoning in this case. One is based on the procedure MPV defined in the previous section, and the other is based on deduction systes. Proposition 5. If ψ is satisfiable, then R du (ψ, 0) = od(ψ) and R du (ψ, i) = Λ for every i > 0.

7 Proposition 6. The function G U : 2 Λ 2 Λ defined by G U (V ) = Λ for all V Λ, is an inductive representation of d U. Proof. Iediate fro Proposition 5. Corollary 4. For every theory Γ, the procedure MPV(G U, Γ ) terinates after at ost du Λ iterations and coputes du,(γ ). Proof. By Propositions 3 and 6. Another way of coputing consequences of the entailent relation = du, is by the deduction syste S u, defined in Figure 2. This syste anipulates expressions of the for Γ : V, where Γ is a theory and V Λ. Axios: : Λ (A 0 ) {ψ} : od(ψ) if od(ψ) (A 1 ) {ψ} : Λ if od(ψ) = (A 2 ) Inference Rules: Γ 1 : V 1 Γ 2 : V 2 Γ 1 Γ 2 : V 1 V 2 if od(γ 1 Γ 2) (I 1) Γ 1 : V 1 Γ 2 : V 2 Γ 1 Γ 2 : Λ if od(γ 1 Γ 2 ) = (I 2 ) Fig. 2. The syste S u Definition 9. For a theory Γ and a set V Λ, denote by S u Γ : V that Γ : V is provable in S u, and by Γ S u ψ that S u Γ : V for soe V od(ψ). Exaple 4. Let Γ = {p, q, p q}. We show that Γ S u p p. Indeed, p : { p : t, q : t, p : t, q : f } q : { p : t, q : t, p : f, q : t } p q : {p : f, q : f} p, q : { p : t, q : t } (I p, q, p q : Λ 2 ) (I 1 ) Thus, Γ S u ψ iff od(ψ) = Λ. In particular, Γ S u p p. Proposition 7 (soundness and copleteness). Γ = du, ψ iff Γ S u ψ. Proof (outline). The ain observation is that for every theory Γ, S u Γ : V iff V = du,(γ ). This iediately iplies the proposition, since if Γ = du, ψ then du,(γ ) od(ψ). By the observation above, Γ : d, (Γ ) is provable in S u, and so Γ S u ψ. Conversely, if Γ S u ψ, then there is soe V od(ψ) such that Γ : V is provable in S u. By the ain observation again, V = du,, thus du,(γ ) od(ψ), and so Γ = du, ψ.

8 3.3 Haing distances Next, we exaine in- reasoning with the Haing distance d H (see Exaple 1). First, we consider soe iportant cases in which reasoning with Haing distances coincides with reasoning with unifor distances. Definition 10. Denote: K n i = i j=1 ( j n Definition 11. A forula ψ is i-validated for i N +, if aong the 2 Atos(ψ) valuations on Atos(ψ), at ost K Atos(ψ) i valuations do not satisfy ψ. Note 4. An i-validated forula is also j-validated, for 1 i j Atos(ψ). Exaple 5. Any tautology is 1-validated (thus it is i-validated for any i). Also, every literal (i.e., an atoic forula or its negation) is 1-validated. Moreover, as a disjunction of literals is either a tautology or is falsified by only one valuation, Lea 2. Every clause is 1-validated. Proposition 8. If Γ consists of 1-validated forulas, then for every aggregation function f and every forula ψ, Γ = dh,f ψ iff Γ = du,f ψ. Proof (outline). It is sufficient to show that for any 1-validated forula ψ and any µ Λ, d H (µ, ψ) = d U (µ, ψ). Indeed, if µ od(ψ), d H (µ, ψ) = d U (µ, ψ) = 0. Otherwise, µ od(ψ), and so d H (µ, ψ) = d U (µ, ψ) = 1. This follows fro the fact that if ψ is i-validated, then for every µ Λ, d H (µ, ψ) i. Corollary 5. If Γ is a set of clauses, then Γ = dh,f ψ iff Γ = du,f ψ for every aggregation f and forula ψ, Γ = dh, ψ iff Γ S u ψ. Proof. The first ite follows fro Lea 2 and Proposition 8; the second ite follows fro the first ite and Proposition 7. For 1-validated preises we therefore have a sound and coplete proof syste. The reainder of this section deals with the other cases. Definition 12. For µ Λ, denote by Diff(µ, i) the set of valuations differing fro µ in exactly i atos. The next result is an analogue, for Haing distances, of Proposition 6. Proposition 9. The function G H : 2 Λ 2 Λ, defined for every V Λ by G H (V ) = V µ V Diff(µ, 1), is an inductive representation of d H. ). Proof. Straightforward fro the definition of R dh. Proposition 10. MPV(G H, Γ ) terinates after no ore than dh Λ iterations and returns dh,(γ ). If Γ consists of i-validated forulas, MPV(G H, Γ ) terinates after at ost i iterations. Proof. The first part follows fro Propositions 3 and 9. As for every µ Λ d H (µ, ψ) i when ψ is i-validated, we have that R dh (ψ, i) = Λ for all ψ Γ. In the notations of Figure 1, then, after i iterations X 1... X n = Λ, so MPV ust terinate by the i-th iteration.

9 4 Reasoning by Voting Definition 13. Given a ultiset D = {d 1,..., d n }, denote the nuber of zeros in D by Zero(D). A k -voting function vote, where k < N, is defined as k follows: 0 if Zero(D) = n, 1 vote k (D) = 2 if k n Zero(D) < n, 1 otherwise. In what follows, we shall assue that the arguent of vote k is a ultiset of eleents in {0, 1} (e.g., a ultiset of drastic distances). In this case, it is easy to verify that vote k and requires a quoru of at least k is an aggregation function. Intuitively, vote k siulates a pole of the votes to deterine iplications of inconsistent theories. Indeed, if Γ is not consistent and there are valuations that satisfy at least k of the eleents of Γ, then d U,vote k (Γ ) contains all such valuations. Otherwise, du,vote k (Γ ) = Λ. Note that for every k 1 2 the function vote acts as a ajority function for inconsistent theories. Definition 14. Let Γ = {ψ 1,..., ψ n }. Let Sub k (Γ ) be the set of all subsets of Γ of size k n, and denote od (Γ ) = k H Sub k (Γ ) od(h). Proposition 11. For every theory Γ, od(γ ) if od(γ ), du,vote k (Γ ) = od k (Γ ) otherwise, if od k (Γ ), Λ otherwise. By Proposition 11, votedu (Γ ) is coputable as follows:, k Vote k ({ψ 1,..., ψ n }) /* Most plausible valuations of {ψ 1,..., ψ n } w.r.t. d U and vote k */ for i {1,..., n}: X i od(ψ i ) Y if (X 1... X n ) is nonepty, return (X 1... X n ) T for every subset I of {1,..., n} of size k n : Y Y j I X j if Y is nonepty return Y else return Λ Fig. 3. Coputing the ost plausible valuations of {ψ 1,..., ψ n} w.r.t. d U and vote k Proposition 12. Vote k (Γ ) always terinates and returns du,vote k (Γ ). Proof. Iediately follows fro Proposition 11.

10 5 Suation of Distances Suation of distances is probably the ost coon approach for distancebased reasoning. In this section we consider this kind of reasoning. Again, we first consider the general case and then concentrate on ore specific distances. 5.1 Arbitrary pseudo distances Consider the syste S Σ in Figure 4. Again, S Σ anipulates expressions of the for Γ : V, where Γ is a theory and V Λ. We denote by Γ SΣ ψ that Γ : V is provable in S Σ (i.e., SΣ Γ :V ) for soe V od(ψ). Axios: : Λ (A 0) {ψ} : od(ψ) if od(ψ) (A 1) {ψ} : Λ if od(ψ) = (A 2) Inference Rule: Γ 1 : V 1 Γ 2 : V 2 Γ 1 Γ 2 : V 1 V 2 if V 1 V 2 (J 1 ) Fig. 4. The syste S Σ Proposition 13 (soundness). For every d, if Γ SΣ ψ then Γ = d,σ ψ. Proof (outline). If Γ SΣ ψ then SΣ Γ : V for soe V od(ψ). By induction on the length of the proof of Γ : V in S Σ one shows that this iplies that V = d,σ (Γ ), and so d,σ (Γ ) od(ψ). Thus, Γ = d,σ ψ. For another way of reasoning with suation of distances we note that, as in the case of and the voting function, it is possible to identify distancesuation conclusions by a set-theoretical condition. Definition 15. Let d be an inductively representable pseudo distance. Denote: Ω i 1,...,i n d ({ψ 1,..., ψ n }) = n k=1 Ri k d (ψ i ). Proposition 14. For an inductively representable pseudo distance d and a theory Γ = {ψ 1,..., ψ n }, let be the inial nuber s.t. Ω i1,...,in d ({ψ 1,..., ψ n }) is not epty for soe sequence i 1,..., i n in which Σk=1 n i k =. Then d,σ (Γ ) = j j n = Ωj 1,...,j n d (Γ ). Proposition 14 indicates that reasoning with suation of distances is a constraint prograing proble: given a theory Γ = {ψ 1,..., ψ n }, the goal is to iniize the value of Σj=i n i j for which the intersection R i 1 d (ψ 1 )... R i n d (ψ n ) is not epty. Hence, CLP-solvers ay be used here for checking entailents. Note 5. For any pseudo distance d it holds that d,σ (Γ ) = d,σ (Γ \{ϕ}) whenever ϕ is a tautology or a contradiction. Thus, tautologies and contradictions have a degenerate role in the coputations above (cf. Note 3).

11 5.2 Unifor distances As we show below, suation of unifor distances is closely related to the -SAT proble. 5 Definition 16. Let SAT(Γ ) be the set of all the satisfiable ultisets in Γ and SAT(Γ ) the set of the ially satisfiable eleents in SAT(Γ ) (that is, SAT(Γ ) consists of all Υ SAT(Γ ) such that Υ Υ for every Υ SAT(Γ )). Denote: od(sat(γ )) = {µ Λ µ od(υ ) for soe Υ SAT(Γ )}. Note 6. Clearly, SAT(Γ ) is not epty whenever Γ has a satisfiable eleent. Also, all the eleents in SAT(Γ ) have the sae size. Proposition 15. Let d be a unifor distance. Then: { {ν od(υ ) Υ SAT(Γ )} if SAT(Γ ), d,σ (Γ ) = Λ otherwise. By Proposition 15 we conclude the following: Entailents w.r.t. = d,σ ay be coputed by -SAT solving techniques and by incorporating off-the shelf -SAT solvers (see, e.g., [3, 6, 7, 11]). As in the case of, unifor distances behave siilarly w.r.t. suation: Corollary 6. For any two unifor distances d 1, d 2 and a theory Γ, d1,σ(γ ) is the sae as d2,σ(γ ), and so Γ = d1,σ ψ iff Γ = d2,σ ψ. By the second ite above, we ay concentrate on the drastic distance d U. Now, the syste S Σ defined in Figure 4 is not coplete for = du,σ, as its inference rule does not cover all the inter-relations aong the preises. By Proposition 15, for a coplete syste one ay add the following rule: Γ 1 : V 1 Γ 2 : V 2 Γ 1 Γ 2 : od(sat(γ 1 Γ 2 )) if V 1 V 2 = (J 2 ) Obviously, (J 2 ) is not an inference rule in the usual sense, as its conclusion is not affected by V 1 and V 2. As such, this rule is not very useful. Yet, the cobination of (J 1 ) and (J 2 ) ay be helpful, e.g., in the context of belief revision, as: a) if the condition of (J 1 ) is satisfied, the ost plausible valuations of the revised theory should not be recoputed, and b) if the condition of (J 1 ) is not et, (J 2 ) indicates the auxiliary source of coputations, naely: revision can be deterined by -SAT calculations. Definition 17. Denote by S u Σ the syste S Σ together with (J 2 ). Proposition 16 (soundness and copleteness). Γ S u Σ ψ iff Γ = du,σ ψ. 5 The original forulation of -SAT is about finding a valuation that satisfies a ial set of clauses in a set Γ (see, e.g., [10] for soe coplexity results and [8] for related approxiation ethods). By the -SAT proble in our context we ean an extended version of the proble, according to which one has to find all the valuations that satisfy a ial set of forulas fro a ultiset Γ.

12 5.3 Haing distances Suation of Haing distances is very coon in the context of belief revision and database integration. Yet, the deductive systes developed so far for this seantics are liited to a very narrow fragent of propositional languages. One exaple is the logic MF, introduced in [4], in which the preises are sets of literals. In this case, reasoning with = dh,σ reduces to counting ajority votes: Fact 1 Let Γ be a ultiset of literals. Then Γ = dh,σ ψ iff ψ is in the transitive closure of Maj(Γ ), where Maj(Γ ) consists of the literals in Γ whose nuber of appearances in Γ is strictly bigger than the appearances in Γ of their negations. Based on this fact, the odal logic MF in [4] assues that the set Γ of preises consists only of literals, and represents the fact that a literal l appears i ties in Γ by the odal operator BΓ i. Then, l follows fro Γ (l is believed; B Γ l) if it holds that BΓ i l Bj Γ l for soe i > j 0. The following result suggests an alternative way for autoated reasoning with suation of Haing distances, in ore general contexts: Proposition 17. For every theory Γ and a forula ψ we have that Γ = dh,σ ψ if Γ SΣ ψ. If Γ is a set of clauses, then Γ = dh,σ ψ iff Γ S u Σ ψ. Proof. The first part of the proposition is a particular case of Proposition 13; The second part follows fro Propositions 8 and 16. Proposition 17 provides a very first step toward autoated reasoning with Haing distances. More general techniques are a subject for future work. References 1. O. Arieli. Coonsense reasoning by distance seantics. In Proc. TARK 07, pages 33 41, O. Arieli. Distance-based paraconsistent logics. International Journal of Approxiate Reasoning, Accepted. 3. M. L. Bonet, J. Levy, and F. Manyá. Resolution for Max-SAT. Artificial Intelligence, 171(8 9): , L. Cholvy and C. Garion. A logic to reason on contradictory beliefs with a ajority approach. In IJCAI 01 Workshop on Inconsistency in Data and Knowledge, A. Grove. Two odellings for theory change. J. Phil. Logic, 17: , F. Heras and J. Larrosa. New inference rules for efficient Max-SAT solving. In Proc. AAAI 06, pages AAAI Press, S. Joy, J. Mitchell, and B. Borchers. A branch-and-cut algorith for -sat and weighted -sat. Satisfiability Proble: Theory and Applications, 35, H. Karloff and U. Zwick. A 7/8-approxiation algorith for MAX 3SAT? In Proc. FOCS 97. IEEE Press, S. Konieczny and R. Pino Pérez. Merging inforation under constraints: a logical fraework. Journal of Logic and Coputation, 12(5): , C. Papadiitriou. Coputational coplexity. Addison-Wesley, USA, Z Xing and W. Zhang. axsolver: An efficient exact algorith for (weighted) iu satisfiability. Artificial Intelligence, 164(1 2):47 80, 2005.