Signed Formulas and Annotated Logics

Transcription

1 Signed Formulas and Annotated Logics James J. Lu Neil V. Murray Erik Rosenthal Bucknell University, PA. cs. bucknell. edu SUNY at Albany, NY. albany. edu University of New Haven, CT. brodsky %nhu. yale. edu Abstract Multiple-valued logics (hwl s) have received increasing attention from researchers in order to model certain epistemological concepts such as inconsistency. Signed formulas and annotated logics are two approaches that some authors have used to analyze MVL s. In this paper we ezplore the relationship between signed formulas and annotated logics. A special case of the signed resolution rule is shown to be equivalent to - and thus to unify - the two inference rules, resolution and reduction, of annotated logic, raising the possibility of an SLD style resolution rule for annotated logic programs. Keywords: Multiple-valued Logic, Signed Formula, Resolution, Annotated Logic 1 Int ro duct ion Multiple-valued logics have received increasing attention from researchers in order to model certain epistemological concepts such as inconsistency. One approach to these logics has been signed formulas, investigated independently by Hahnle [4] and by Murray and Rosenthal [12, 131. Other authors (for example, Blair and Subrahmanian [2], Kifer and Lozinski [5, 61, Kifer and Subrahmanian [7, 81, Lu, Henschen, Subrahmanian, and da Costa [Ill) have studied annotated logics. The class of signed formulas, which places no restriction on the domain of truth values, is much wider than the class of annotated logics. A special case of the signed resolution rule is shown to be equivalent to -and thus to unify - the two inference rules, resolution and reduction, of annotated logic. This research was supported in part by the National Science Foundation under grants CCR and CCR The logic of signed formulas is defined in the next section, and annotated logics are described in Section 3. Section 3.3 examines the restriction of signed resolution that is equivalent to the two inference rules of annotated logics. The possibility of an SLD-style resolution rule for annotated logic programs is explored in the last section. We have omitted proofs of theorems as well as a description of experimental results comparing regular signed resolution with annotated signed resolution due to space considerations. 2 Signed Formulas To simplify the presentation, we will restrict attention to propositional logics. However, this is without loss of generality since the main results lift to the first order case. 2.1 Multiple-valued Logics We assume a language A consisting of logical formulas built in the usual way from a set d of atoms, a set of connectives, and a set of logical constants. For the sake of completeness, we precisely define a formula in A as follows: 1. Atoms are formulas. 2. If 0 is an n-ary connective and Fl, Fz,..., Fn are formulas, then so is O(F1, Fz,..., Fn). Associated with A is a finite set A of truth values. An interpretation for A is a function from d to A; i.e., an assignment of truth values to every atom in A. A connective 0 of arity n denotes a function 0 : An -+ A. Interpretations are extended in the usual way to mappings from formulas to A. Alternatively, a formula F of A can be thought of as denoting a mapping from interpretations to A XB IEEE

2 We use the term sign for any subset of A, and overload it by also using it for any expression that denotes a subset of A. We define a signed formula to be an expression of the form S : F, where S is a sign and F is a formula in A. We are interested in signed formulas because they represent queries of the form, "Are there interpretations under which F evaluates to a truth value in S?" In a refutational theorem proving setting for classical logic, the query is typically {true) : F, where the formula F is the negation of a goal or conclusion, conjoined with some axioms and hyoptheses; the answer hoped for is no. To answer arbitrary queries, we map formulas in A to formulas in a classical propositional logic A,. We call A, the language of signed formulas and define it as follows: The atoms are signed formulas and the connectives are (classical) conjunction and disjunction. We emphasize that a signed formula S : F is an atom in A,, regardless of the size or complexity of F, and thus has no component parts in the language A,. The set of truth values is of course {true, false}. 2.2 A-Consistent interpretations An arbitrary interpretation for A, may make an assignment of true or false to any signed formula (i.e., to any atom) in the usual way. Our goal is to focus attention only on those interpretations that relate to the sign in a signed formula. We thus define a A- consistent interpretation I, for h, to be an interpretation for which there exists an interpretation I for A such that for each atom S : F, S : F is true under I, if and only if the formula F in A is mapped into S by I. Intuitively, A-consistent means an assignment of true to all signed formulas whose signs are simultaneously achievable via some interpretation over the original language. If F1 and Fz are formulas in A,, we write F1 +A Fz if whenever I, is a A-consistent interpretation and I,(F1) = true, then &(Fa) = true. The following lemma is immediate since each interpretation in A maps a formula to exactly one element of A. Lemma 1 Let I, be a A-consistent interpretation, let A be an atom and F a formula in A, and let SI and S2 be signs. Then: 1. Is(O : F) = false; 2. I,(A : F) = true; 3. S1 C Sz if and only if S1 : F b~ Sz : F for all formulas F in A; 4. There is exactly one b E A such that IS({b} : A) = true. Although we feel that the focus on A-consistent interpretations is intuitive, it can also be motivated by the following technical observation: The formulas that we are considering in A, do not have any occurrences of negation; only A and V appear as classical interpreted symbols. Whether or not such formulas are satisfiable with respect to A-consistent interpretations, they are trivially satisfiable with respect to arbitrary interpretations. 2.3 A-atomic formulas Many classical inference rules begin with links (complementary pairs of literals). Such rules typically deal only with formulas in which all negatives are at the atomic level. Similarly, the inference techniques that we wish to develop here require that signs be at the "atomic level." To that end, we call a formula A- atomic if it has the property that whenever S : A is an atom in the formula, then A is an atom in A; we call it elementary if whenever S : A is an atom, S is a singleton. Remark 1 There is a natural one-to-one correspondence between interpretations over A and A-consistent interpretations over A, as follows: Given an interpretation I over A, define the interpretation I, over A, by I,((b} : A) = true iff 6 = I(A). Since connectives in A are functions, I, uniquely extends to an interpretation over all atoms of A,. Conversely, by part 4 of Lemma 1, the value of a A-consistent interpretation on elementary atoms determines a unique interpretation over A. 2.4 Signed Resolution In this section we describe a sound and complete resolution procedure for propositional clausal signed logic called signed resolution; for a more complete description see [12]. We present only the binary version of signed resolution (i.e., without the factoring operation introduced in [13]). For propositional signed clauses, binary signed resolution with basic factoring, defined in Subsection 3.2, is complete. Definition (Signed Resolution) Suppose C1 and Cz are the clauses C1: Si : A V El Cz: Sa : A V Ez. Then a signed-resolvent of C1 and C2 on the signed atoms SI : A and S2 : A is the signed clause (Si n Sz) : A V E1 V Ez. The signed atom (SI n Sz) : A is called the remnant 49

3 of the resolution. We say that C1 and C2 are signresolvable on the signed atoms SI : A and S2 : A. It has been pointed out in [13] that the clauses on which signed resolution acts on need not be A-atomic, that they may be arbitrary atoms of Ad. However, this generality plays no role in the study carried out here, and for the remainder of this paper we restrict attention to A-atomic formulas in clause form. Remark 2 Since no A-consistent interpretation may satisfy the signed atom 0 : A, we may assume no signed clauses contain signed atoms with the empty set as the sign. Therefore, if an application of signed resolution produces an impossible remnant, we may simply drop the remnant from the resolvent. The notions of deduction, refutation, and proof are defined in the usual way. 2.5 Regular Signed Clauses In general, when searching for a signed refutation of an arbitrary set of signed clauses, the best one can do is to apply signed resolution repeatedly until the empty clause is obtained. However, suppose we are concerned only with finding proofs of clauses whose signs satisfy some condition C. An interesting question then arises as to what, if any, benefits can be derived from knowledge of this property C. We study this question by examining in detail one particular property C: regular signs. As we shall see, restricting signed resolution to regular signs is essentially equivalent to the two inference rules of annotated logics. To accomplish this, we assume that the set of signs A is not simply an unordered set of objects but instead forms a complete lattice under some ordering 5; the greatest and least elements of A are denoted T and I, respectively. Definition Suppose (P; S} is a partially ordered set and Q C P. Then 1 Q = {y E Pl(3z E Q)z 5 y}. The set Q is the smallest upset containing Q (see [3]). If Q is a singleton set {a}, then we simply write t x. Definition A subset Q of P is regular if for some z E P, Q =t z or Q = (t z) (the set complement of 1 z). In the former case we call Q positive and in the latter case negative. A signed formula is regular if every sign that occurs in it is regular. By Remark 2, we may assume that no regular signed formulas have a sign of the form (t z), where z = I since in this case (t z) = 0. Regular formulas correspond to a class of logics, discussed in the next section, that have recently been investigated rather extensively. In particular, these logics have been shown to provide a reasonable foundation for reasoning systems that need to deal with uncertainty and inconsistency. Moreover, Hiihnle [4] has considered a class of multiple-valued logics - he calls them regular logics - that are closely related to regular formulas when the lattice of truth values is linear. 3 Annotated Logics It turns out that the logic As restricted to regular signed formulas correspond exactly to annotated logics as studied in [ll]. The utility of annotated logics have been considered extensively in the literature [2, 5, 6, 7, 81. We formalize the relationship in this section. 3.1 Annotated Logics PI We briefly recapitulate the basics of propositional annotated logics PI. Definition If A is an atom, and p E A, then 1. A : p is an annotated atom. 2. An annotated literal is either an annotated atom or - A, where A is an annotated atom. We shall consider only annotated formulas that are constructed from annotated literals (positive or negative), and the connectives A and V. An interpretation is a mapping from the set of atoms to A, and we use Z to denote the set of all interpretations for PI. Definition (Satisfaction) An interpretation I is said to satisfy 1. the annotated atom A : p iff I(A) 2 p. 2. the annotated literal - ( A : p) iff I does not satisfy A : p. 3. the disjunction F1 V F2 if it satisfies F1 or F2. 4. the conjunction F1 A Fa if it satisfies F1 and Fa. The relationship between regular signed formulas and the annotated logic PI is made precise in Theorem 1. We define a mapping tr from annotated literals to regular signed atoms as follows: If A : p is a positive annotated literal, then tr(a : p) =t 1.1 : A, and if - (A : p) is a negative annotated literal, then tr(- (A : p)) = (1 p) : A. The mapping may be extended to any formula as follows: If F and G are annotated formulas, then tr(f A G) = tr(f) Atr(G), and tr(f V G) = tr(f) V tr(g). 50

4 Observe that tr is a bijection between annotated formulas and regular signed formulas. Theorem 1 Suppose F is an annotated formula and I is an interpretation. Then I satisfies F iff I, (the corresponding interpretation over A,) satisfies tr(f). The above results are straightforward and simply confirm the intuition that annotated logics are special instances of signed logics. In the next section we describe the relationship between the resolution procedures for annotated logics and the one for signed formulas. 3.2 P-resolution vs. Signed Resolution A sound and complete resolution proof procedure was introduced for clausal annotated logics in [5]. An extension to the procedure was studied in [ll]. The procedure, called p-resolution, contains two inference rules: annotated resolution and reduction. We show that, under the mapping tr, these two procedures correspond to disjoint instances of signed resolution. Definition (Annotated Resolution) We say that two annotated literals L1 and L2 are complementary if they have the respective forms A : p and - ( A : p), where p 2 p. Given the annotated clauses c1: (L1 v 01) c2: (L2 v 0 2 ) where L1 and L2 are complementary, then the annotated resolvent of C1 and C2 on the annotated literals L1 and La is D1 V D2. We say that either C1 resolves with C2, or C2 resolves with C1, with the understanding that the annotation of the positive annotated literal that is resolved upon is greater than or equal to the annotation of the negative annotated literal resolved upon. C1 and C2 are also said to be annotated resolvable on the annotated literals L1 and Lz. We use U and n to denote least upper bound and greatest lower bound, respectively. Definition (Reduction) Given the annotated clauses C1: A: pi V El Cz: A:p2 V E2 where p1 and p2 are incomparable, then the annotated clause A : u {p~, p2} V El V E2 is called a reductant of C1 and C2. In addition, C1 and C2 are said to be reducible on the annotated literals A : p1 and A : p2. Lemma 2 Suppose L1 and L2 are the complementary annotated literals A : p and - (A : p), respectively. If SI : A = tr(l1) and S2 : A = tr(l2), then slns2 = 0. Corollary Suppose C1 and C, are annotated clauses that are annotated-resolvable on the annotated literals L1 and La. Then tr(c1) and tr(c2) are signresolvable with the impossible remnant on the signed atoms tr(l1) and tr(l2). Lemma 3 Suppose L1 and L2 are the annotated literals A : pi and A : p2, respectively. Let SI : A = tr(l1) and S2 : A = tr(l2). Then slns2 =I p, where CL = U hp2). Corollary Suppose C1 and C2 are annotated clauses that are reducible on the annotated literals L1 and La. Then tr(c1) and tr(c2) are sign-resolvable with the remnant non-trivial on the signed atoms tr(l1) and tr(l2). The last two lemmas demonstrate that, in effect annotated resolution and reduction correspond to, respectively, the case when signed resolution produces an impossible remnant and the case when signed resolution produces a non-trivial remnant. We may therefore regard signed resolution as unifying annotated resolution and reduction. We extend the function tr to apply to a sequence of clauses in the obvious way: Given a sequence of annotated clauses V = Cl,..., C,, tr(d) denotes the sequence of signed clauses tr(cl),..., tr(c,). The next lemma is immediate from the previous two. Lemma 4 Let F be a set of annotated clauses. Then V is a p-deduction of F iff tr(v) is a signed deduction of tr(f). It follows immediately that 2) is a p-proof (i.e. a p- refutation) of the set F iff tr(v) is a signed proof of tr(f). A consequence is that signed resolution, along with basic factoring, is refutation complete for first order annotated clauses. This follows by the soundness and completeness result for p-resolution for annotated logics [ll]. By basic factoring we mean that whenever a signed clause contains two signed atoms S : A1 and S : A2 and A1 and A2 are unifiable via mgu 8, then we merge them into the atom S : Ale. The reason this is called basic factoring is because the signs of the atoms being merged must be identical sets of truth values. This differs from the more complex factoring defined in[l3]. Theorem 2 Suppose F is an unsatisfiable set of annotated clauses. Then there is a signed refutation of {tr(c)lc E F}. 51

5 This also provides the first completeness result for signed resolution in the first order case. Up to now, signed resolution has been shown to be refutation complete only for propositional signed clauses. Corollary Signed resolution with basic factoring is refutation complete for unsatisfiable sets of first order regular signed clauses. 3.3 Regular Signed Resolution The results of the previous section give us another way of implementing a proof procedure for annotated logics. In this section we futher characterize the class of signed deductions that correspond to p-deductions. The next theorem shows that a minimal set of regular signs whose intersection is empty will contain exactly one negative sign. Theorem 3 Suppose S1,..., S, are regular signs whose intersection is empty, and suppose that no subset of {SI,..., s,} has an empty intersection. Then for some j, 15 j 5 n, S, = (T xj) and Si =t xi, for ifjandx1,..., x,ea. We say a literal L1 E C1 supports a literal L E C if L = L1 and C = C1 or if C is the result of C1 resolving with another clause on the literal L1, and L is the remnant. In addition, if L1 E C1 supports L2 E Cz, and L2 supports L3 E C,, then L1 also supports L3. Equivalently we say that L3 is supported by L1. In a deduction D, the clauses C1 and C2 indirectly resolve upon the literals L1 E C1 and La E C2 if there is a resolution of two clauses Kl V D1 and K2 V D2 in on K1 and K2, and K; is supported by Li, for i = 1,2. Note that when C1 and C2 (directly) resolve, they also indirectly resolve. A simple syntactic condition that guarantees no two literals of the forms (T 2) : A and (1 y) : A would ever indirectly resolve with one another is to require that resolvents contain only regular signs. This is stated in the next theorem. Definition A signed deduction from a set of signed clauses is said to be regular if it does not contain an direct resolution of two literals of the form (T z) : A and (t y) : A, and every resolvent is a regular signed clause. Theorem 4 Suppose D is a regular proof of a set S. Then 2) does not contain an indirect resolution on two literals with negative signs. Theorem 5 A deduction D is a p-deduction iff tr(d) is a regular signed deduction. Corollary 1 (Completeness of Regular Signed Resolution) Suppose F is an unsatisfiable set of regular signed clauses. Then there is a regular signed proof of F. From the Corollary to Theorem 2, Corollary 1 may be lifted to the first order case. Corollary 2 Suppose F is an unsatisfiable set of first order annotated clauses and 2) is a p-refutation of F. Then tr( D) is a regular signed proof of tr(f). Unrestricted signed resolution allows resolution between any two clauses of the form (T 2) : A V C and (T y) : A V D. In light of Theorem 3, irrelevant deductions that would be avoided by presolution may be performed by signed resolution. 4 Annotated Logic Programs An Annotated Logic Program, abbreviated ALP, is a set of annotated clauses each having exactly one positive annotated literal. Typically, an annotated clause in an ALP is written in the form of an implication A c B1 A...A B,, where A is the positive annotated literal in the clause, while - B1,..., - B,, n >_ 0 are all the negative annotated literals in the clause. The symbol A is said to be the head of the clause, while the conjunction B1 A...A B, is called the body of the clause. A query is a headless clause. The most popular technique for answering queries in Logic Programs is based on SLD-resolution. There are several advantages to using an SLD-style proof procedure. First, it has a close resemblance to the execution behavior of conventional programming languages (e.g. Pascal). This feature makes Prolog, and logic programming in general, more accessible to programmers who are familiar with the imperative style of programming. Secondly, as noted in [8], using SLDresolution, the choice of clauses that need to be considered at each deduction step is restricted to the current goal and the program clauses. Unfortunately, finding an SLD-style proof procedure has proven elusive for ALPS [7, 81. Approximate versions have been developed[8, 141, but none are completely satisfactory. The difficulty in finding an SLD-proof procedure lies in the need to compute reductants. This is illustrated in the next example. Example Consider the ALP P written over FOUR f t) p: t t; p: and the query c p : T. It is easy to see P p : T. However, annotated resolution alone does not lead to a refutation. It is necessary to compute the reductant p : T t from the two program clauses. This resolves 52

6 using annotated resolution with the query to yield a refutation. In terms of resolution theorem proving, an SLDproof procedure does not exist for ALPS because of the incompatibility of p-resolution with the linear restriction strategy [9, lo]. Kifer and Subrahmanian attempted to circumvent this difficulty by first compiling an ALP P to its closure, the set of all possible reductants that can be generated from the clauses in P. Utilizing the closure, they were able to define an SLD-style proof procedure using a simplified version of p-resolution. On the other hand, since signed resolution consists of a single rule of inference, it is amenable to a linear restriction. The completeness of such a restriction on signed resolution in the ground case can be proved by the usual technique (see [l]) of induction on the number of excess literals. It follows thus signed resolution provides an attractive basis for defining an SLDresolution query answering procedure for Annotated Logic Programs. This would eliminate the difficulty faced by p-resolution of having to compute with nonprogram clauses. Acknowledgements We benefited from discussions with Reiner Hahnle. References [l] R. Anderson and W.W. Bledsoe, A Linear Format for Resolution with Merging and a New Technique for Establishing Completeness, in: J.ACM, 17(3) (1970), [7] M. Kifer and V.S.Subrahmanian, On the Expressive Power of Annotated Lo ics, in: Proceedings of the North American Con&ence on Logic Programming, MIT Press (1989), [8] M. Kifer and V.S.Subrahmanian, Theory of Generalized Annotated Logic Programmin and its Applications, in: the Journal of Logic Programmzng, 12, (1992), [9] D. W. Loveland, A Linear Format for Resolution, in: PTOC. IRIA Symp. on Automatic Demonstration, Lecture Notes in Mathematics, Springer, (1970), [lo] D. W. Loveland, A Unifying View of some Linear Herbrand Procedures, in: J. ACM, 19 (1972), [ll] J. J. Lu, L.J. Henschen, V.S. Subrahmanian and N.C.A. da Costa, Reasoning in Paraconsistent Logics, in: Automated Reasoning: Essays in Honor of Woody Bledsoe (R. Boyer ed.), Kluwer Academic (1991), [12] N. V. Murray and E. Rosenthal, Resolution and Path Dissolution in Multiple-valued Logics, in: Proceedings of the International Symposium on Methodologies for Intelligent Systems, Springer v.542 (1991), [13] N. V. Murray and E. Rosenthal, Signed Formulas: A Classical Approach to Multiple-valued Logics, TR91-12 (1991), SUNY at Albany. Presented at the 1992 summer meeting of the ASL. [14] V.S. Subrahmanian, Paraconsistent Disjunctive Databases, in: Theoretical Computer Science, 93 (1992), [2] H.A. Blair and V.S. Subrahmanian, Paraconsistent Logic Programming, in: Theoretical Computer Science, 68 (1989), [3] B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press (1990). [4] R. Hahnle, Uniform Notation Tableau Rules for Multiple-valued Lo@s, in: Proceedin s of the International Symposzum on Multiple- +slued Logic, (1991), [5] M. Kifer and E. Lozinskii, RI: A Logic for Reasoning with Inconsistency, in: IEEE Symposium on Logic in Computer Science, (1989), [6] M. Kifer and E. Lozinskii, A Logic for Reasoning with Inconsistency, in: Journal of Automated Reasoning, to appear. 53