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Nominal rewriting
 Information and Computation
"... Nominal rewriting is based on the observation that if we add support for alphaequivalence to firstorder syntax using the nominalset approach, then systems with binding, including higherorder reduction schemes such as lambdacalculus betareduction, can be smoothly represented. Nominal rewriting ma ..."
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Nominal rewriting is based on the observation that if we add support for alphaequivalence to firstorder syntax using the nominalset approach, then systems with binding, including higherorder reduction schemes such as lambdacalculus betareduction, can be smoothly represented. Nominal rewriting maintains a strict distinction between variables of the objectlanguage (atoms) and of the metalanguage (variables or unknowns). Atoms may be bound by a special abstraction operation, but variables cannot be bound, giving the framework a pronounced firstorder character, since substitution of terms for variables is not captureavoiding. We show how good properties of firstorder rewriting survive the extension, by giving an efficient rewriting algorithm, a critical pair lemma, and a confluence theorem
Nominal Rewriting Systems
 Proceedings of the 6th ACM SIGPLAN symposium on Principles and Practice of Declarative Programming (PPDP 2004), ACM
, 2004
"... We present a generalisation of rstorder rewriting which allows us to deal with terms involving binding operations in an elegant and practical way. We use a nominal approach to binding, in which bound entities are explicitly named (rather than using a nameless syntax such as de Bruijn indices), yet ..."
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Cited by 23 (11 self)
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We present a generalisation of rstorder rewriting which allows us to deal with terms involving binding operations in an elegant and practical way. We use a nominal approach to binding, in which bound entities are explicitly named (rather than using a nameless syntax such as de Bruijn indices), yet we get a rewriting formalism which respects conversion and can be directly implemented. This is achieved by adapting to the rewriting framework the powerful techniques developed by Pitts et al. in the FreshML project. Nominal rewriting can be seen as higherorder rewriting with a rstorder syntax and builtin conversion. We show that standard ( rstorder) rewriting is a particular case of nominal rewriting, and that very expressive higherorder systems such as Klop's Combinatory Reduction Systems can be easily dened as nominal rewriting systems. Finally we study con
uence properties of nominal rewriting.
Universal algebra for termination of higherorder rewriting
 In Proc. RTA ’05
, 2005
"... Abstract. We show that the structures of binding algebras and Σmonoids by Fiore, Plotkin and Turi are sound and complete models of Klop’s Combinatory Reduction Systems (CRSs). These algebraic structures play the same role of universal algebra for term rewriting systems. Restricting the algebraic str ..."
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Cited by 9 (6 self)
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Abstract. We show that the structures of binding algebras and Σmonoids by Fiore, Plotkin and Turi are sound and complete models of Klop’s Combinatory Reduction Systems (CRSs). These algebraic structures play the same role of universal algebra for term rewriting systems. Restricting the algebraic structures to the ones equipped with wellfounded relations, we obtain a complete characterisation of terminating CRSs. We can also naturally extend the characterisation to rewriting on metaterms by using the notion of Σmonoids. 1
Free Σmonoids: A higherorder syntax with metavariables
 In Proc. of APLAS’04, LNCS 3302
, 2004
"... Abstract. The notion of Σmonoids is proposed by Fiore, Plotkin and Turi, to give abstract algebraic model of languages with variable binding and substitutions. In this paper, we give a free construction of Σmonoids. The free Σmonoid over a given presheaf serves a wellstructured term language inv ..."
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Cited by 8 (4 self)
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Abstract. The notion of Σmonoids is proposed by Fiore, Plotkin and Turi, to give abstract algebraic model of languages with variable binding and substitutions. In this paper, we give a free construction of Σmonoids. The free Σmonoid over a given presheaf serves a wellstructured term language involving binding and substitutions. Moreover, the free Σmonoid naturally contains interesting syntactic objects which can be viewed as “metavariables ” and “environments”. We analyse the term language of the free Σmonoid by relating it with several concrete systems, especially the λcalculus extended with contexts. 1
Term Equational Systems and Logics (Extended Abstract)
"... We introduce an abstract general notion of system of equations between terms, called Term Equational System, and develop a sound logical deduction system, called Term Equational Logic, for equational reasoning. Further, we give an analysis of algebraic free constructions that together with an intern ..."
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Cited by 2 (0 self)
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We introduce an abstract general notion of system of equations between terms, called Term Equational System, and develop a sound logical deduction system, called Term Equational Logic, for equational reasoning. Further, we give an analysis of algebraic free constructions that together with an internal completeness result may be used to synthesise complete equational logics. Indeed, as an application, we synthesise a sound and complete nominal equational logic, called Synthetic Nominal Equational Logic, based on the category of Nominal Sets.
Abstract Modularity
"... Modular rewriting seeks criteria under which rewrite systems inherit properties from their smaller subsystems. This divide and conquer methodology is particularly useful for reasoning about large systems where other techniques fail to scale adequately. Research has typically focused on reasoning ab ..."
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Cited by 1 (0 self)
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Modular rewriting seeks criteria under which rewrite systems inherit properties from their smaller subsystems. This divide and conquer methodology is particularly useful for reasoning about large systems where other techniques fail to scale adequately. Research has typically focused on reasoning about the modularity of specific properties for specific ways of combining specific forms of rewriting. This paper is, we believe, the first to ask a much more general question. Namely, what can be said about modularity independently of the specific form of rewriting, combination and property at hand. A priori there is no reason to believe that anything can actually be said about modularity without reference to the specifics of the particular systems etc. However, this paper shows that, quite surprisingly, much can indeed be said.
SecondOrder Equational Logic (Extended Abstract)
"... We provide an extension of universal algebra and its equational logic from first to second order. Conservative extension, soundness, and completeness results are established. ..."
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We provide an extension of universal algebra and its equational logic from first to second order. Conservative extension, soundness, and completeness results are established.
Formal Software Development: From Foundations to Tools
"... This exposé gives an overview of the author’s contributions to the area of formal software development. These range from foundational issues dealing with abstract models of computation to practical engineering issues concerned with tool integration and user interface design. We can distinguish three ..."
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This exposé gives an overview of the author’s contributions to the area of formal software development. These range from foundational issues dealing with abstract models of computation to practical engineering issues concerned with tool integration and user interface design. We can distinguish three lines of work: Firstly, there is foundational work, centred around categorical models of rewriting. A new semantics for rewriting is developed, which abstracts over the concrete term structure while still being able to express key concepts such as variable, layer and substitution. It is based on the concept of a monad, which is wellknown in category theory to model algebraic theories. We generalise this treatment to term rewriting systems, infinitary terms, term graphs, and other forms of rewriting. The semantics finds applications in functional programming, where monads are used to model computational features such as state, exceptions and I/O, and modularity proofs, where
Term Equational Rewrite Systems and Logics
"... We introduce an abstract general notion of system of equations and rewrites between terms, called Term Equational Rewrite System (TERS), and develop a sound logical deduction system, called Term Equational Rewrite Logic (TERL), to reason about equality and rewriting. Further, we give an analysis of ..."
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We introduce an abstract general notion of system of equations and rewrites between terms, called Term Equational Rewrite System (TERS), and develop a sound logical deduction system, called Term Equational Rewrite Logic (TERL), to reason about equality and rewriting. Further, we give an analysis of algebraic free constructions which together with an internal completeness result may be used to synthetise a complete TERL. Indeed, as an application, we derive a sound and complete equational rewrite
General Terms
"... Nominal rewriting extends firstorder rewriting with GabbayPitts abstractors: bound entities are named, matching respects αconversion and can be directly implemented thanks to the use of freshness constraints. In this paper we study two extensions to nominal rewriting. First we introduce a quantif ..."
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Nominal rewriting extends firstorder rewriting with GabbayPitts abstractors: bound entities are named, matching respects αconversion and can be directly implemented thanks to the use of freshness constraints. In this paper we study two extensions to nominal rewriting. First we introduce a quantifier for modelling name generation and restriction. This allows us to model higherorder functions involving local state, and has also applications in concurrency theory. The second extension introduces new constraints in freshness contexts. This allows us to express strategies of reduction and has applications in programming language design and implementation. Finally, we study confluence properties of nominal rewriting and its extensions. N