Abstract: We present a logical language which extends the syntax of positive Horn clauses by permitting implications in goals and in the bodies of clauses. The operational meaning of a goal which is an implication is given by the deduction theorem. That is, a goal D ⊃ G is satisfied by a program P if the goal G is satisfied by the larger program P ∪ {D}. If the formula D is the conjunction of a collection of universally quantified clauses, we interpret the goal D ⊃ G as a request to load the code in D prior to attempting G, and then unload that code after G succeeds or fails. This extended use of implication provides a logical explanation of parametric modules, some uses of Prolog’s assert predicate, and certain kinds of abstract datatypes. Both a model-theory and proof-theory are presented for this logical language. We show how to build a possible-worlds (Kripke) model for programs by a fixed point construction and show that the operational meaning of implication mentioned above is sound and complete for intuitionistic, but not classical, logic. 1. Implications as Goals Let A be a syntactic variable which ranges over atomic formulas of first-order logic. Let G range over a class of formulas, called goal formulas, to be specified shortly. We shall assume, however, that this class always contains ⊤ (true) and all atomic formulas. The formulas represented by A and G may contain free variables. Given these two classes, we define definite clauses, denoted by the syntactic variable D, as follows: D: = G ⊃ A | ∀x D | D1 ∧ D2 A program is defined to be a finite set of closed definite clauses. P will be a syntactic variable for programs. A clause of the form ⊤ ⊃ A will often be written as simply A. Let P be a program. Define [P] to be the smallest set of formulas satisfying the following recursive definitions. (i) P ⊆ [P].

Abstract not found

We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We alsoshowhow to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and Spector's Double Negation Shift.

Stephen A. Cook and Bruce M. Kapron Department of Computer Science University of Toronto Toronto, Canada M5S 1A4 1 Introduction Functionals are functions which take natural numbers and other functionals as arguments and return natural numbers as values. The class of "feasible" functionals of finite type was introduced in [6] via the typed lambda calculus, and used to interpret certain formal systems of arithmetic: systems capturing the notion of "feasibly constructive proof" (we equate feasibility with polynomial time computability) . Here we name the functionals of [6] the basic feasible functionals and justify the designation by presenting results which include two programming style characterizations of the class. We also give examples of both feasible and infeasible functionals, and argue that the notion plays a natural role in complexity theory. Type 2 functionals take numbers and ordinary numerical functions as arguments. When these argument functions are 0-1 valued (i.e. sets) ...

This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. We establish the following extraction rule for an extension of GnA # by quantifier--free choice AC--qf and analytical axioms # having the form #x # #y ## sx#z # F0 (including also a `non-- standard' axiom F - which does not hold in the full set--theoretic model but in the strongly majorizable functionals): From a proof GnA # +AC--qf + # # #u 1 , k 0 #v ## tuk#w 0 A0(u, k, v, w) one can extract a uniform bound # such that #u 1 , k 0 #v ## tuk#w # #ukA0 (u, k, v, w) holds in the full set--theoretic type structure. In case n = 2 (resp. n = 3) #uk is a polynomial (resp. an elementary recursive function) in k, u M := #x. max(u0, . . . , ux). In the present paper we show that for n # 2, GnA # +AC--qf+F - proves a generalization of the binary Knig's lemma yielding new conservation results since the conclusion of the above rule can be verified in G max(3,n) A # in this case. In a subsequent paper we will show that many important ine#ective analytical principles and theorems can be proved already in G2A # +AC--qf+# for suitable #. 1

Feferman has proposed a system, FS 0 , as an alternative framework for encoding logics and also for reasoning about those encodings. We have implemented a version of this framework and performed experiments that show that it is practical. Specifically, we describe a formalisation of predicate calculus and the development of an admissible rule that manipulates formulae with bound variables. This application will be of interest to researchers working with frameworks that use mechanisms based on substitution in the lambda calculus to implement variable binding and substitution in the declared logic directly. We suggest that meta-theoretic reasoning, even for a theory using bound variables, is not as difficult as is often supposed, and leads to more powerful ways of reasoning about the encoded theory. x 1 Introduction: why metamathematics? A logical framework is a formal theory that is designed for the purpose of describing other formal theories in a uniform way, and for making the work ...

This is a preprint of a paper that has been submitted to Information and Computation.

this paper we show how to combine the unrestricted countable choice, induction on infinite well-founded trees and restricted classical logic in a constructively given model. For readers faniliar with intuitionistic systems [14], we may succinctly describe the theory we interpret as follows. Expand the extensional version of HA

Presentation of Rewriting and Typing 13 2 Abstract Rewriting Systems 15 2.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 2.2 Abstract Rewriting Systems : : : : : : : : : : : : : : : : : : : : : : : : : : 15 2.3 Morphisms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16 2.4 Properties of Abstract Rewriting Systems : : : : : : : : : : : : : : : : : : : 18 2.5 Strategies : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19 2.6 Criteria : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 2.7 Conclusions and Related Work : : : : : : : : : : : : : : : : : : : : : : : : : 24 3 Topology 27 3.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 3.2 Topology : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28 3.3 Equivalence : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30 3.4 Topological Characte...

) 1 J. M. E. Hyland 2 C.-H. L. Ong 3 University of Cambridge, England Abstract This paper is motivated by the discovery that an appropriate quotient SN 3 of the strongly normalising untyped 3-terms (where 3 is just a formal constant) forms a partial applicative structure with the inherent application operation. The quotient structure satisfies all but one of the axioms of a partial combinatory algebra (pca). We call such partial applicative structures conditionally partial combinatory algebras (c-pca). Remarkably, an arbitrary right-absorptive c-pca gives rise to a tripos provided the underlying intuitionistic predicate logic is given an interpretation in the style of Kreisel's modified realizability, as opposed to the standard Kleenestyle realizability. Starting from an arbitrary right-absorptive c-pca U , the tripos-to-topos construction due to Hyland et al. can then be carried out to build a modified realizability topos TOPm (U ) of non-standard sets equipped with an equali...