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40
A new approach to abstract syntax with variable binding
 Formal Aspects of Computing
, 2002
"... Abstract. The permutation model of set theory with atoms (FMsets), devised by Fraenkel and Mostowski in the 1930s, supports notions of ‘nameabstraction ’ and ‘fresh name ’ that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variablebinding op ..."
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Cited by 207 (44 self)
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Abstract. The permutation model of set theory with atoms (FMsets), devised by Fraenkel and Mostowski in the 1930s, supports notions of ‘nameabstraction ’ and ‘fresh name ’ that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variablebinding operations. Inductively defined FMsets involving the nameabstraction set former (together with Cartesian product and disjoint union) can correctly encode syntax modulo renaming of bound variables. 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 notion of structural recursion for defining syntaxmanipulating functions (such as capture avoiding substitution, set of free variables, etc.) and a notion of proof by structural induction, both of which remain pleasingly close to informal practice in computer science. 1.
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...
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 145 (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...
Injective positively ordered monoids
 I, J. Pure Appl. Algebra
, 1992
"... We define in this paper a certain notion of completeness for a wide class of commutative (pre)ordered monoids (from now on P.O.M.’s). This class seems to be the natural context for studying structures like measurable function spaces, equidecomposability types of spaces, partially ordered abelian gro ..."
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Cited by 17 (10 self)
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We define in this paper a certain notion of completeness for a wide class of commutative (pre)ordered monoids (from now on P.O.M.’s). This class seems to be the natural context for studying structures like measurable function spaces, equidecomposability types of spaces, partially ordered abelian groups and cardinal algebras. Then, we can prove that roughly speaking, spaces of measures with values in complete P.O.M.’s are complete P.O.M.’s. Furthermore, this notion of completeness yields us an ‘arithmetical ’ characterization of injective P.O.M.’s.
Some connections between set theory and computer science
 Proceedings of the third Kurt Godel Colloquium on Computational Logic and Proof Theory, Lecture Notes in Computer Science
, 1993
"... Abstract. Methods originating in theoreticar computer science for shoiving that certain decision probrems are Npcomplete have also been used to show that certain compactness theorems are equivalent in ZF set theory to the Boolean Prime ldear rheorem (BpI). conversely, there is some evidence that se ..."
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Cited by 12 (4 self)
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Abstract. Methods originating in theoreticar computer science for shoiving that certain decision probrems are Npcomplete have also been used to show that certain compactness theorems are equivalent in ZF set theory to the Boolean Prime ldear rheorem (BpI). conversely, there is some evidence that set theoretic methods connnected with research on BPI may prove useful in computer science. we survey what is known and then look at some new exampres and exprore the underlying reasons for the successful application of quite similar methods to soive"different problems. 1
The Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."
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Cited by 8 (2 self)
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This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.
Adding a total order to ACL2
 In Third International Workshop on the ACL2 Theorem Prover and its Applications (ACL22002
, 2002
"... Abstract. We show that adding a total order to ACL2, via new axioms, allows for simpler and more elegant definitions of functions and libraries of theorems. We motivate the need for a total order with a simple example and explain how a total order can be used to simplify existing libraries of theore ..."
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Cited by 7 (2 self)
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Abstract. We show that adding a total order to ACL2, via new axioms, allows for simpler and more elegant definitions of functions and libraries of theorems. We motivate the need for a total order with a simple example and explain how a total order can be used to simplify existing libraries of theorems (i.e., ACL2 books) on finite set theory and records. These ideas have been incorporated into ACL2 Version 2.6, which includes axioms positing a total order on the ACL2 universe. 1 Introduction ACL2 [7, 6, 8] is a logic of total functions. One particularly pleasant consequence is that many properties of functions can be stated as unconditional rewrite rules. For example, we can prove (equal ( * y ( * x z)) ( * x ( * y z))) without having to establish that x, y, and z are numbers. Such unconditional rewrite rules lead to simpler libraries of theorems, which in turn improve the ability of ACL2 to reduce large terms automatically and efficiently. Unfortunately, it is problematic to exploit fully the totality of functions in ACL2 Version 2.5. One is often forced to use rewrite rules with hypotheses because of the lack of a definable total order on the ACL2 universe.
Choice principles in constructive and classical set theories
 POHLERS (EDS.): PROCEEDINGS OF THE LOGIC COLLOQUIUM 2002
, 2002
"... The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models ..."
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Cited by 4 (3 self)
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The objective of this paper is to assay several forms of the axiom of choice that have been deemed constructive. In addition to their deductive relationships, the paper will be concerned with metamathematical properties effected by these choice principles and also with some of their classical models.
Wehrung: The HahnBanach Theorem implies the existence of a nonLebesgue measurable set
 Fund. Math
, 1991
"... Few methods are known to construct non Lebesguemeasurable sets of reals: most standard ones start from a wellordering of R, orfrom the existence of a nontrivial ultrafilter over ω, and thus need the axiom of choice AC or at least the Boolean Prime Ideal theorem BPI (see [5]). In this paper we pre ..."
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Cited by 4 (2 self)
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Few methods are known to construct non Lebesguemeasurable sets of reals: most standard ones start from a wellordering of R, orfrom the existence of a nontrivial ultrafilter over ω, and thus need the axiom of choice AC or at least the Boolean Prime Ideal theorem BPI (see [5]). In this paper we present a new way for proving the existence
An infinite color analogue of Rado’s theorem
, 2005
"... Let R be a subring of the complex numbers and a be a cardinal. A system of linear homogeneous equations with coefficients in R is called aregular over R if for every acoloring of R there is a monochromatic solution to that system in distinct variables. Rado in 1943 classified those systems of line ..."
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Cited by 3 (0 self)
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Let R be a subring of the complex numbers and a be a cardinal. A system of linear homogeneous equations with coefficients in R is called aregular over R if for every acoloring of R there is a monochromatic solution to that system in distinct variables. Rado in 1943 classified those systems of linear homogeneous equations that are aregular over R for all positive integers a. For every infinite cardinal a, we classify those systems of linear homogeneous equations that are aregular over R. As a corollary, for every positive integer s, we have 2 ℵ0> ℵs if and only if the equation x0 + sx1 = x2 + · · · + xs+2 is ℵ0regular over R. The case s = 1 is due to Erdős and Kakutani.