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58
Fresh Logic
 Journal of Applied Logic
, 2007
"... Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with resp ..."
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Cited by 219 (28 self)
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Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with respect to metavariables. We present oneandahalfthorder logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of oneandahalfthorder logic derivability, show them equivalent, show that the derivations satisfy cutelimination, and prove correctness of an interpretation of firstorder logic within it. We discuss the technicalities in a wider context as a casestudy for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation
A Proof Theory for Generic Judgments
, 2003
"... this paper, we do this by adding the #quantifier: its role will be to declare variables to be new and of local scope. The syntax of the formula # x.B is like that for the universal and existential quantifiers. Following Church's Simple Theory of Types [Church 1940], formulas are given the ..."
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Cited by 76 (20 self)
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this paper, we do this by adding the #quantifier: its role will be to declare variables to be new and of local scope. The syntax of the formula # x.B is like that for the universal and existential quantifiers. Following Church's Simple Theory of Types [Church 1940], formulas are given the type o, and for all types # not containing o, # is a constant of type (# o) o. The expression # #x.B is ACM Transactions on Computational Logic, Vol. V, No. N, October 2003. 4 usually abbreviated as simply # x.B or as if the type information is either simple to infer or not important
Nominal logic programming
, 2006
"... Nominal logic is an extension of firstorder logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, αequivalence). This article investigates logic programming based on nominal logic. This technique is especial ..."
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Cited by 40 (9 self)
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Nominal logic is an extension of firstorder logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, αequivalence). This article investigates logic programming based on nominal logic. This technique is especially wellsuited for prototyping type systems, proof theories, operational semantics rules, and other formal systems in which bound names are present. In many cases, nominal logic programs are essentially literal translations of “paper” specifications. As such, nominal logic programming provides an executable specification language for prototyping, communicating, and experimenting with formal systems. We describe some typical nominal logic programs, and develop the modeltheoretic, prooftheoretic, and operational semantics of such programs. Besides being of interest for ensuring the correct behavior of implementations, these results provide a rigorous foundation for techniques for analysis and reasoning about nominal logic programs, as we illustrate via two examples.
A Sequent Calculus for Nominal Logic
 IN PROC. LICS’04
, 2004
"... Nominal logic is a theory of names and binding based on the primitive concepts of freshness and swapping, with a selfdual N  (or "new")quantifier, originally presented as a Hilbertstyle axiom system extending firstorder logic. We present a sequent calculus for nominal logic called Fre ..."
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Cited by 30 (16 self)
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Nominal logic is a theory of names and binding based on the primitive concepts of freshness and swapping, with a selfdual N  (or "new")quantifier, originally presented as a Hilbertstyle axiom system extending firstorder logic. We present a sequent calculus for nominal logic called Fresh Logic, or FL, admitting cutelimination. We use FL to provide a prooftheoretic foundation for nominal logic programming and show how to interpret $FOL^{\Delta abla}$, another logic with a selfdual quantifier, within FL.
Induction and coinduction in sequent calculus
 Postproceedings of TYPES 2003, number 3085 in LNCS
, 2003
"... Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and coinduction. These proof principles are based on a proof theoretic (rather than sett ..."
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Cited by 28 (8 self)
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Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and coinduction. These proof principles are based on a proof theoretic (rather than settheoretic) notion of definition [13, 20, 25, 51]. Definitions are akin to (stratified) logic programs, where the left and right rules for defined atoms allow one to view theories as “closed ” or defining fixed points. The use of definitions makes it possible to reason intensionally about syntax, in particular enforcing free equality via unification. We add in a consistent way rules for pre and post fixed points, thus allowing the user to reason inductively and coinductively about properties of computational system making full use of higherorder abstract syntax. Consistency is guaranteed via cutelimination, where we give the first, to our knowledge, cutelimination procedure in the presence of general inductive and coinductive definitions. 1
Focusing on binding and computation
 In IEEE Symposium on Logic in Computer Science
, 2008
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A Simpler Proof Theory for Nominal Logic
 In FOSSACS 2005, number 3441 in LNCS
, 2005
"... Nominal logic is a variant of firstorder logic which provides support for reasoning about bound names in abstract syntax. A key feature of nominal logic is the newquantifier, which quantifies over fresh names (names not appearing in any values considered so far). Previous attempts have been made ..."
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Cited by 28 (11 self)
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Nominal logic is a variant of firstorder logic which provides support for reasoning about bound names in abstract syntax. A key feature of nominal logic is the newquantifier, which quantifies over fresh names (names not appearing in any values considered so far). Previous attempts have been made to develop convenient rules for reasoning with the newquantifier, but we argue that none of these attempts is completely satisfactory. In this paper we develop a new sequent calculus for nominal logic in which the rules for the newquantifier are much simpler than in previous attempts. We also prove several structural and metatheoretic properties, including cutelimination, consistency, and conservativity with respect to Pitts' axiomatization of nominal logic; these proofs are considerably simpler for our system. 1
A Definitional Approach to Primitive Recursion over Higher Order Abstract Syntax
 In Proceedings of the 2003 workshop on Mechanized
, 2003
"... Syntax S. J. Ambler (S.Ambler@mcs.le.ac.uk) R. L. Crole (R.Crole@mcs.le.ac.uk) & A. Momigliano (A.Momigliano@mcs.le.ac.uk) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K. ..."
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Cited by 27 (5 self)
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Syntax S. J. Ambler (S.Ambler@mcs.le.ac.uk) R. L. Crole (R.Crole@mcs.le.ac.uk) & A. Momigliano (A.Momigliano@mcs.le.ac.uk) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K.
Completeness and Herbrand Theorems for Nominal Logic
 Journal of Symbolic Logic
, 2006
"... Nominal logic is a variant of firstorder logic in which abstract syntax with names and binding is formalized in terms of two basic operations: nameswapping and freshness. It relies on two important principles: equivariance (validity is preserved by nameswapping), and fresh name generation (&qu ..."
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Cited by 25 (5 self)
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Nominal logic is a variant of firstorder logic in which abstract syntax with names and binding is formalized in terms of two basic operations: nameswapping and freshness. It relies on two important principles: equivariance (validity is preserved by nameswapping), and fresh name generation ("new" or fresh names can always be chosen).