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13
Nominal Logic: A First Order Theory of Names and Binding
 Information and Computation
, 2001
"... This paper formalises within firstorder 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 L ..."
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Cited by 161 (15 self)
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This paper formalises within firstorder 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 firstorder manysorted logic with equality containing primitives for renaming via nameswapping and for freshness of names, from which a notion of binding can be derived. Its axioms express...
Analyzing Proofs in Analysis
 LOGIC: FROM FOUNDATIONS TO APPLICATIONS. EUROPEAN LOGIC COLLOQUIUM (KEELE
, 1993
"... ..."
Subtypes for Specifications: Predicate Subtyping in PVS
 IEEE Transactions on Software Engineering
, 1998
"... 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 ..."
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Cited by 31 (4 self)
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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.
Towards a SetTheoretic Type Theory
, 1988
"... 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 highorder function ..."
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Cited by 2 (1 self)
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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 highorder 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
A Theory of Explicit Mathematics Equivalent to ID_1
"... We show that the addition of name induction to the theory EETJ + (LEM I N ) of explicit elementary types with join yields a theory prooftheoretically equivalent to ID_1. ..."
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Cited by 2 (2 self)
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We show that the addition of name induction to the theory EETJ + (LEM I N ) of explicit elementary types with join yields a theory prooftheoretically equivalent to ID_1.
Subtypes for Specification
 IEEE Transactions on Software Engineering
, 1997
"... . 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 sp ..."
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Cited by 2 (0 self)
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. 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...
Natural Numbers and Forms of Weak Induction in Applicative Theories
, 1995
"... 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 Nstrictness, we give a natural extension of the theory BON which implies the equivalence of operati ..."
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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 Nstrictness, we give a natural extension of the theory BON which implies the equivalence of operation and Ninduction. In addition, we show that in the presence of the nonconstructive ¯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 socalled nonconstructive ¯operator, a predicatively acceptable ...
Universes over Frege Structures
 Annals of Pure and Applied Logic
, 1996
"... 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 prooftheoretic strength \Gamma 0 . 1 Introduction Frege structures were introduced by Aczel in [Acz80] as a semantical concept t ..."
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Cited by 1 (0 self)
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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 prooftheoretic 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 MartinLof'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 ...
Sets, Complements and Boundaries
"... The relations among a set, its complement, and its boundary are examined constructively. A crucial tool is a theorem that allows the construction of a point where a segment comes close to the boundary of a set in a Banach space. Brouwerian examples show that many of the results are the best possible ..."
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Cited by 1 (1 self)
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The relations among a set, its complement, and its boundary are examined constructively. A crucial tool is a theorem that allows the construction of a point where a segment comes close to the boundary of a set in a Banach space. Brouwerian examples show that many of the results are the best possible.