This paper formalises within first-order logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal Logic, a version of first-order many-sorted logic with equality containing primitives for renaming via name-swapping and for freshness of names, from which a notion of binding can be derived. Its axioms express...

Abstract not found

Abstract not found

A specification language used in the context of an effective theorem prover can provide novel features that enhance precision and expressiveness. In particular, typechecking for the language can exploit the services of the theorem prover. We describe a feature called "predicate subtyping" that uses this capability and illustrate its utility as mechanized in PVS.

this paper is to present such system of sets that may serve as a foundation for denotations of type expressions of a programming language. In this system there is no selfapplication and its existence is consistent with any classical set theory. Still, it is rich enough to contain high-order functions, polymorphism and dependent types. Also recursion operators as well as those needed to build types defined by recursive equations are present. 1 Basic notions

We show that the addition of name induction to the theory EETJ + (LEM -I N ) of explicit elementary types with join yields a theory proof-theoretically equivalent to ID_1.

. Specification languages are best used in environments that provide effective theorem proving. Having such support available, it is feasible to contemplate forms of typechecking that can use the services of a theorem prover. This allows interesting extensions to the type systems provided for specification languages. I describe one such extension called "predicate subtyping" and illustrate its utility as mechanized in PVS. 1 Introduction For programming languages, type systems and their associated typecheckers are intended to ensure the absence of certain undesirable behaviors during program execution [4]. The undesired behaviors generally include untrapped errors such as adding a boolean to an integer, and may (e.g., in Java) encompass security violations. If the language is "type safe," then all programs that can exhibit these undesired behaviors will be rejected during typechecking. Execution is not a primary concern for specification languages, but typechecking can still se...

In this paper we study the relationship between forms of weak induction in theories of operations and numbers. Therefore, we investigate the structure of the natural numbers. Introducing a concept of N-strictness, we give a natural extension of the theory BON which implies the equivalence of operation and N-induction. In addition, we show that in the presence of the non-constructive ¯-operator the above equivalence is provable without this extension. 1 Introduction Applicative theories go back to Feferman's systems of explicit mathematics introduced in [Fef75, Fef79]. They are based on the basic theory of operations and numbers BON which is introduced in [FJ93] as the classic version of Beeson's theory EON (cf. [Bee85]) without induction. Combined with various induction principles, applicative theories provide a natural framework for constructive mathematics and functional programming. If they are strengthened by the so-called non-constructive ¯-operator, a predicatively acceptable ...

We investigate universes axiomatized as sets with natural closure conditions over Frege structures. In the presence of a natural form of induction, we obtain a theory of proof-theoretic strength \Gamma 0 . 1 Introduction Frege structures were introduced by Aczel in [Acz80] as a semantical concept to introduce a notion of sets by means of a partial truth predicate. This approach is closely related to prior work of Scott [Sco75] and was originally developed for questions around Martin-Lof's type theory. In [Bee85, Ch. XVII] Beeson gave a formalization of Frege structures as a truth theory over applicative theories. Applicative theories go back to Feferman's systems of explicit mathematics introduced in [Fef75, Fef79]. These systems provide a logical basis for functional programming. The basic theory for which Frege structures are defined is the basic theory of operations and numbers TON, introduced and studied in [JS95]. It comprises total combinatorial logic and arithmetic. The notion ...

Mahlo universe