A Functional Approach to Non-Monotonic Logic

s Axiom sets and their extensions are viewed as functions from the set of formulas in the language, to a set of four truth-values t,f, s for undefined, and k for contradiction. Such functions form a lattice with 'contains less information' as the partial order and 'combination of several sources of knowledge' as the l.u.b. operation We demonstrate the relevance of this approach by giving concise proofs for some previously known results about normal default rules. For non-monotonic rules in general (not only normal default rules) we define a stronger version of the minimality requirement on consistent fixpoints, and prove that it is sufficient for the existence of a derivation of the fixpoint. 1 . I n t r o d u c t i o n a n d overv iew. Non-monotonic logic may be studied either in terms of non-monotonic inference rules (Reiter 1980, Goodwin 1984) or in terms of non-monotonic operators in the language such as the Unless operator (Sandewall 1972, McDermott and Doyle 1980). In this paper we pursue the former approach. The concept of fixpoints b central to the study of non-monotonic logic: for a given set v of propositions and a given set of rules, we are looking for an extension Le. a set s' of propositions which contains v as a subset, and which is a fixpoint of the set of rules. Fixpoints are also used in the denotational semantics approach to the theory of programming languages (Scott 1970; see also e.g. Manna 1974, Stoy 1977, Blikle 1981). There, the recursive definition of a function is viewed as a functional, i.e. an operator on partial functions, and the function b viewed as the fixpoint of the same functional In t hb paper we propose that the functional approach that is taken in denotational semantics, can be adapted and serve conveniently for the study of non-monotonic logic. This b attractive since logical inference b often viewed as a high-level form of computation, and since computational inference often needs to be non-monotonic. The power of t hb approach b demonstrated through simple proofs of some of Reiter's (1980) results for normal default theories. Other results in the paper apply to non-monotonic rules in general. Although several fixpoints may exbt in the monotonie case, the criterium of being • minimal fixpoint (i.e. all other fixpoints are 'larger) is a sufficient one, and there b only one minimal fixpoint, which b then the lea$t fixpoint. Other, larger fixpoints contain spurious information which b not warranted by the given facts and This research wee eponeored by the Swedish Boerd of Technical Development. inference rules. In the case of non-monotonic rules, there is of course in general no single least fixpoint, and the criterium of fixpoints being minimal is not sufficient: there may be minimal fixpoints which have the given set of propositions as a subset, but which still can not be reached or approached (in the sense of a l imit) by any derivation using the given set of rules. In this paper we define a concept of approachable fixpoint, which is stronger than the concept of minimal fixpoint, and which is proven to be a sufficient condition for the existence of a derivation that reaches or approaches the fixpoint. The following formal machinery is used. We start from two domains, a domain L whose elements are called formulae and a domain J of truth-values. V is the domain of valuations i.e. continuous functions from L to J. A set of axioms is seen as a valuation that maps some formulas (the axioms) to t (for true) and "a l l " other formulas to u (for undefined). (Exception is made for the top element of the domain L) . Derivation of theorems is done by proceeding from the init ial valuation to others where some formulas change value from a to t or / (for false). A set of inference rules corresponds therefore to a binary relation on valuations, i.e. a subset of V x V, which we shall call a deduction. A derivation using a deduction F is a sequence of valuations,