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A New Approach to Abstract Syntax Involving Binders
 In 14th Annual Symposium on Logic in Computer Science
, 1999
"... Syntax Involving Binders Murdoch Gabbay Cambridge University DPMMS Cambridge CB2 1SB, UK M.J.Gabbay@cantab.com Andrew Pitts Cambridge University Computer Laboratory Cambridge CB2 3QG, UK ap@cl.cam.ac.uk Abstract The FraenkelMostowski permutation model of set theory with atoms (FMsets) ..."
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Cited by 146 (14 self)
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Syntax Involving Binders Murdoch Gabbay Cambridge University DPMMS Cambridge CB2 1SB, UK M.J.Gabbay@cantab.com Andrew Pitts Cambridge University Computer Laboratory Cambridge CB2 3QG, UK ap@cl.cam.ac.uk Abstract The FraenkelMostowski permutation model of set theory with atoms (FMsets) can serve as the semantic basis of metalogics for specifying and reasoning about formal systems involving name binding, ffconversion, capture avoiding substitution, and so on. We show that in FMset theory one can express statements quantifying over `fresh' names and we use this to give a novel settheoretic interpretation of name abstraction. Inductively defined FMsets involving this nameabstraction set former (together with cartesian product and disjoint union) can correctly encode objectlevel syntax modulo ffconversion. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated n...
Set theory for verification: I. From foundations to functions
 J. Auto. Reas
, 1993
"... A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherord ..."
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Cited by 46 (18 self)
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherorder syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,
Weak Relative PseudoComplements of Closure Operators
 Algebra Universalis
, 1996
"... We define the notion of weak relative pseudocomplement on meet semilattices, and we show that it is strictly weaker than relative pseudocomplementation, but stronger than pseudocomplementation. Our main result is that if a complete lattice L is meetcontinuous, then every closure operator on L a ..."
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Cited by 13 (12 self)
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We define the notion of weak relative pseudocomplement on meet semilattices, and we show that it is strictly weaker than relative pseudocomplementation, but stronger than pseudocomplementation. Our main result is that if a complete lattice L is meetcontinuous, then every closure operator on L admits weak relative pseudocomplements with respect to continuous closure operators on L.
Constructive set theories and their categorytheoretic models
 IN: FROM SETS AND TYPES TO TOPOLOGY AND ANALYSIS
, 2005
"... We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicat ..."
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Cited by 9 (0 self)
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We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories we consider all have sound and complete classes of categorytheoretic models, obtained by axiomatizing the structure of an ambient category of classes together with its subcategory of sets. In certain special cases, the categories of sets have independent characterizations in familiar categorytheoretic terms, and one thereby obtains a rich source of naturally occurring mathematical models for (both predicative and impredicative) constructive set theories.
The Open Calculus of Constructions: An Equational Type Theory with Dependent Types for Programming, Specification, and Interactive Theorem Proving
"... The open calculus of constructions integrates key features of MartinLöf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational t ..."
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Cited by 5 (0 self)
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The open calculus of constructions integrates key features of MartinLöf's type theory, the calculus of constructions, Membership Equational Logic, and Rewriting Logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational theories. We explore the open calculus of constructions as a uniform framework for programming, specification and interactive verification in an equational higherorder style. By having equational logic and rewriting logic as executable sublogics we preserve the advantages of a firstorder semantic and logical framework and especially target applications involving symbolic computation and symbolic execution of nondeterministic and concurrent systems.
Safety Signatures for Firstorder Languages and Their Applications
 In FirstOrder Logic Revisited (Hendricks et all,, eds.), 3758, Logos Verlag
, 2004
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HYPERSOLVER: A Graphical Tool for Commonsense Set Theory
, 1996
"... This paper investigates an alternative set theory (due to Peter Aczel) called Hyperset Theory. Aczel uses a graphical representation for sets and thereby allows the representation of nonwellfounded sets. A program, called HYPERSOLVER, which can solve systems of equations defined in terms of sets ..."
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Cited by 1 (1 self)
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This paper investigates an alternative set theory (due to Peter Aczel) called Hyperset Theory. Aczel uses a graphical representation for sets and thereby allows the representation of nonwellfounded sets. A program, called HYPERSOLVER, which can solve systems of equations defined in terms of sets in the universe of this new theory is presented. This may be a useful tool for commonsense reasoning. 1 INTRODUCTION Set theory has long occupied a unique place in mathematics since it allows various other branches of mathematics to be formally defined within it. The theory has ignited many debates on its nature and a number of different axiomatizations were developed to formalize its underlying `philosophical' principles. Collecting entities into an abstraction for further thought (i.e., set construction) is an important process in mathematics, and this brings in assorted problems [5]. The theory had many groundshaking crises (like the discovery of the Russell's Paradox [6]) throughout it...
Higherorder rewriting via conditional firstorder rewriting in the open calculus of constructions
 Informatik Berichte. Department of Computer Science
"... Abstract. Although higherorder rewrite systems (HRS) seem to have a firstorder flavor, the direct translation into firstorder rewrite systems, using e.g. explicit substitutions, is by no means trivial. In this paper, we explore a twostage approach, by showing how higherorder pattern rewrite sys ..."
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Abstract. Although higherorder rewrite systems (HRS) seem to have a firstorder flavor, the direct translation into firstorder rewrite systems, using e.g. explicit substitutions, is by no means trivial. In this paper, we explore a twostage approach, by showing how higherorder pattern rewrite systems, and in fact a somewhat more general class, can be expressed by conditional firstorder rewriting in the open calculus of constructions (OCC), which itself has been presented and implemented using explicit substitutions. The key feature of OCC that we exploit is that conditions are allowed to contain quantifiers and equations which can be solved using firstorder matching. The way we express HRS works in spite of the fact that structural equality of OCC does not subsume αconversion. Another topic that we touch upon in this paper is the use of higherorder abstract syntax in a classical framework like OCC, because it is often used in connection with higherorder rewriting. 1