Second-order predicate

Abstract: In this paper, we present a framework for the semantics and the computation of aggregates in the context of logic programming. In our study, an aggregate can be an arbitrary interpreted second order predicate or function. We define extensions of the Kripke-Kleene, the well-founded and the stable semantics for aggregate programs. The semantics is based on the concept of a three-valued immediate consequence operator of an aggregate program. Such an operator approximates the standard two-valued immediate consequence operator of the program, and induces a unique Kripke-Kleene model, a unique well-founded model and a collection of stable models. We study different ways of defining such operators and thus obtain a framework of semantics, offering different trade-offs between precision and tractability. In particular, we investigate conditions on the operator that guarantee that the computation of the three types of semantics remains on the same level as for logic programs without aggregates. Other results show that, in practice, even efficient three-valued immediate consequence operators which are very low in the precision hierarchy, still provide optimal precision.

Abstract: An algorithm is presented which eliminates second–order quantifiers over predicate variables in formulae of type ∃P1, . . . , Pnψ where ψ is an arbitrary formula of first–order predicate logic. The resulting formula is equivalent to the original formula – if the algorithm terminates. The algorithm can for example be applied to do interpolation, to eliminate the second–order quantifiers in circumscription, to compute the correlations between structures and power structures, to compute semantic properties corresponding to Hilbert axioms in non classical logics and to compute model theoretic semantics for new logics. An earlier version of the paper has been published in [GO92b].

Abstract: Takeuti [3] showed that the consistency of analysis (i.e. second order number theory) is finitistically implied by the Hauptsatz for second order logic» i.e. by the proposition that every theorem of this system is derivable without cut. We will prove that, conversely, the Hauptsatz for this system follows from a certain generalization of the consistency of analysis; namely from: I. Every countable set of relations among natural numbers is included in an o)-model. An co-model is a collection of relations among natural numbers which is closed under the second order comprehension axiom. Henkin [l ] has shown that a second order formula is derivable with the cut rule if and only if it is valid in all (countable) co-models. When the given set of relations consists only of the successor relation, I asserts the consistency of analysis. The formalism for second order predicate logic which we will use is obtained from the system of predicate logic of finite order given in Schutte [2] by dropping all expressions and bound variables of types other than 0 (individuals), 1 (propositions) and (0, 0, • • • , 0) (relations among individuals). Thus, expressions of type 0 are built up from constants and free variables of type 0 using function constants. The expressions of type (0, • • • , 0) are constants, free variables and expressions Xx? • • • x£4(x?, • • • , x£), where A (a°u • • • , a£) is a wff (expression of type 1). The logical symbols other than X are —*, v and V. The notation and terminology of [2] will be assumed. In particular, the notions of strict derivation and partial valuation will be the same as in [2], except that they refer to the second order logic and not the full system of [2], and that we require of a partial valuation that whenever VxA(x) is true (t), then so is A{a) for some free

Abstract: Plotkin and Abadi have proposed a syntactic system for parametricity based on a second order predicate logic. This paper shows three theorems about that system. The first is consistency of the system, which is proved by the method of relativization. The second is that polyadic parametricities of recursive types are equivalent to each other. The third is that the theory of parametricity for recursive types is self-realizable. As a corollary of the third theorem, the theory of parametricity for recursive types satisfies the term extraction property.