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Minimal Consequence in Sentential Logic CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector J. LOGIC PROGRAMMING 1990:9:19-31 19 MINIMAL CONSEQUENCE IN SENTENTIAL LOGIC MARY-ANGELA PAPALASKARI AND SCOTT WEINSTEIN D We define minimal consequence in sentential logic and present a number of results of a model-theoretic and recursion-theoretic character about this newly introduced nonmonotonic consequence relation. We show that the minimal consequence relation is not compact and is lT: and not 2;. We also connect this relation to questions about the completion of theories by “negation as failure”. We give a complete characterization of the class of theories in sentential logic which can be consistently completed by “nega- tion as failure” using the newly introduced notion of a subconditional theory. We show that the class of theories consistently completable by negation as failure is II! and not 2;. 1. INTRODUCTION The simplicity of sentential logic makes for a transparent exposition of many interesting aspects of the model theory and formalization of minimal consequence. On the other hand, sentential logic already contains enough complexity so that many issues concerning important model-theoretic features of more interesting languages are raised. Significant aspects of minimal consequence can thus be articulated and studied in a clear and natural way, which illuminates their develop- ment for more sophisticated languages. Moreover, certain properties particular to sentential logic, e.g., that there is a decision procedure for validity, will reflect on aspects of minimal consequence. As we will see, the minimal consequences of a consistent theory are always consistent. The minimal consequences of a theory are defined in terms of truth in all minimal models of the theory and, as such, extend the set of semantic consequences Address correspondence to Dr. S. Weinstein, Department of Philosophy, 305 Logan Hall, University of Pennsylvania, Philadelphia, PA 19104-6385. Received June 1987; accepted July 1988. THE JOURNAL OF LOGIC PROGRAMMING OElsevier Science Publishing Co., Inc., 1990 655 Avenue of the Americas, New York, NY 10010 0743-1066/90/$3.50 20 MARY-ANGELA PAPALASKARI AND SCOm WEINSTEIN of the theory by some consistent set of sentences. In the case where an incomplete theory has a unique minimal model, its minimal consequences will be complete. We begin by fixing notation, in Section 2. Section 3 gives the definitions of minimal model and minimal consequence, following those with examples that illustrate some of the key features of these notions. Section 4 contains an exposition of the central model-theoretic properties of minimal consequence, namely minimal satisfiability of satisfiable theories and noncompactness. In Section 5 we undertake a study of some interesting fragments: here we define subconditional theories and compare them with conditional (or Horn) theories; since this is of great computa- tional import with regard to the consistent application of negation as failure, we offer a syntactic characterization of the class of subconditional theories and show that they are the largest class of theories that remain consistent under the applica- tion of negation as failure. Section 6 deals with complexity-theoretic aspects of minimal consequence and subconditional theories; we show that the minimal consequence relation is II! and not 2: and that even in the case of theories with unique minimal models it is A\ and neither r.e. or co-r.e., while the question of determining whether a theory has a unique minimal model is also II! and not 2:. The notion of minimal consequence studied here in the context of sentential logic is related to the notion of circumscription which arises in the study of nonmono- tonic reasoning in first-order logic; see [4] for results on the complexity of minimal consequence and minimal satisfiability in first-order logic. 2. NOTATION A language 9 of sentential logic is a set of sentence letters and sentences built up from them using the usual sentential connectives. Given a set S of sentence letters, we can define a language for sentential logic as the smallest set _.Ysuch that S c 9; if + ~9, then (7+) ~9; if @ET and #EP’, then $J A 4 ~9. We will use the term atom to refer to sentence letters and the term basic formula to refer to atoms and their negations. For convenience, as is usual, the symbols “ V “, “ -+ ” and “ c) ” are introduced as abbreviations. Note that the language 9 is uniquely determined by the set S, and is of the same cardinality as S (if S is infinite), so from now on a language will be given as its set of atoms. The symbols p, q, r, s (possibly subscripted) will be used to denote distinct atoms of S. For a countable language, the atoms will typically be denoted by pi, p2,. , although in some cases, for reasons of clarity of exposition, a number of additional symbols will be used, again with the assumption that distinct symbols denote distinct atoms of the language. Boldfaced versions of these symbols ( p, q, r, s) will be used where it is necessary to have a symbol ranging over the atoms of the language. A structure for a language 9 is defined as a subset of S, so the set of structures for 9’ is of cardinality 2s; thus, the set of structures of any countably infinite language has the cardinal number of the continuum.’ We will assume that the reader is familiar with the definitions of a model of a sentence or a theory, satisfaction, and semantic consequence, as well as various ‘We use the symbol w to denote the set of finite ordinals, i.e., natural numbers. Hence o is the least infinite ordinal and the smallest infinite cardinal, since we identify cardinals with initial ordinals. MINIMAL CONSEQUENCE IN SENTENTIAL LOGIC 21 results concerning these (see, e.g., [l, Section 1.21). By a theory is meant any set of well-formed formulas of the language (not necessarily closed under semantic conse- quence). We will use the symbols A, JV (possibly subscripted) for structures, and uppercase Greek letters for theories. For a consequence relation I=~ , let Cn,(I) = {$I 1r !=,$}. We say that I=~ is monotonic iff VI’VA(r c A * Cn,(r) c Cn,(A)). Note that the classical consequence relation, t= , is monotonic. 3. MINIMAL CONSEQUENCE IN SENTENTIAL LOGIC We begin by defining the notions of minimal model and minimal consequence. Dejkition 3. I. A I=,,, r (A is a minimal model of r) iff _Mt= I and VM( JV~ I “4%.X). DeJinition 3.2. r !=,,, C#B(+ is a minimal consequence of r) iff V.M(dl=,,, I Ml= $). Thus, a model A of a theory is minimal if the theory has no models that are proper submodels of M, and a sentence (p is a minimal consequence of a theory I (or I minimally entails +) if + is true in all minimal models of I’. Note that the symbol “ t=,,, ” is used both as a relation between models and theories and as a relation between theories and sentences, but no confusion should arise, since the meaning will always be clear from the context. The following examples illustrate some interesting features of minimal consequence. In each of them it is assumed that I is a theory in a countable language. Example 3.1. Let I = 8. The set of models of I’ is 9(S), and so B is the unique minimal model for I. Thus I i=,,, 7p1 A . - A -p,, for pi E S, 1 I i s n. The theory ru{p,A --. A p, } has a unique minimal model also, namely { pi 11 I i I n},so rU{plA --- ~p,}~~,p,~ --- A,P,. We immediately see from this simple example that the relation I=~ is nonmono- tonic. Example 3.2. Let I’= { pzi VP,,+~ Ii E w}. Every model of I must contain either pzi or pzi+,, or both, for each i E o. The minimal models of l? will be the ones that contain exactly one of pzi or p2i+l. So the set of minimal models of I is of cardinality 2”, i.e., the cardinal number of the continuum. These examples depict two extreme cases: Cn,(I’) in Example 3.1 is complete, whereas Cn,( I) in Example 3.2 has continuum many models. The following example illustrates yet a different aspect of minimal consequence, namely a theory I’ such that Cn,( I’) = Cn( I). Example 3.3. Let r = { pi V pi 1i, j E w, i Zj}. Then I has sentences of the form ~PO+Plr~Po-,P2r..., and 7pi+po,Tpi+p1 ,..., 7po+pj ,..., for j#i, so in any model .M of I, if pi 44, then for all j # i, pi E AY. Thus the models of I are 22 MARY-ANGELA PAPALASKARI AND SCOTT WEINSTEIN those where at most one element is missing, and the minimal models of P are those where exactly one element is missing. The importance of this example lies in the fact that the set of structures 8 = { _Mil pi e.Mi, pj EJ#~, i #j} cannot be characterized as the set of models of any theory, as can be shown by a straightforward application of the compactness theorem. Hence, there is no set of sentences Z such that the models of Z are exactly the minimal models of l?. Although there are sets of structures that cannot be characterized as the models of any theory, but that can be characterized as the set of minimal models of some theory, there are also sets of structures which cannot be characterized as the minimal models of any theory.
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