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Domain theory for concurrency
, 2003
"... Concurrent computation can be given an abstract mathematical treatment very similar to that provided for sequential computation by domain theory and denotational semantics of Scott and Strachey. ..."
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Concurrent computation can be given an abstract mathematical treatment very similar to that provided for sequential computation by domain theory and denotational semantics of Scott and Strachey.
A formal calculus for informal equality with binding
 In WoLLIC’07: 14th Workshop on Logic, Language, Information and Computation, volume 4576 of LNCS
, 2007
"... Abstract. In informal mathematical usage we often reason using languages with binding. We usually find ourselves placing captureavoidance constraints on where variables can and cannot occur free. We describe a logical derivation system which allows a direct formalisation of such assertions, along w ..."
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Abstract. In informal mathematical usage we often reason using languages with binding. We usually find ourselves placing captureavoidance constraints on where variables can and cannot occur free. We describe a logical derivation system which allows a direct formalisation of such assertions, along with a direct formalisation of their constraints. We base our logic on equality, probably the simplest available judgement form. In spite of this, we can axiomatise systems of logic and computation such as firstorder logic or the lambdacalculus in a very direct and natural way. We investigate the theory of derivations, prove a suitable semantics sound and complete, and discuss existing and future research. 1
2011): Nominal terms and nominal logics: from foundations to metamathematics
 In: Handbook of Philosophical Logic
"... ABSTRACT: Nominal techniques concern the study of names using mathematical semantics. Whereas in much previous work names in abstract syntax were studied, here we will study them in metamathematics. More specifically, we survey the application of nominal techniques to languages for unification, rew ..."
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ABSTRACT: Nominal techniques concern the study of names using mathematical semantics. Whereas in much previous work names in abstract syntax were studied, here we will study them in metamathematics. More specifically, we survey the application of nominal techniques to languages for unification, rewriting, algebra, and firstorder logic. What characterises the languages of this chapter is that they are firstorder in character, and yet they can specify and reason on names. In the languages we develop, it will be fairly straightforward to give firstorder ‘nominal ’ axiomatisations of namerelated things like alphaequivalence, captureavoiding substitution, beta and etaequivalence, firstorder logic with its quantifiers—and as we shall see, also arithmetic. The formal axiomatisations we arrive at will closely resemble ‘natural behaviour’; the specifications we see typically written out in normal mathematical usage. This is possible because of a novel namecarrying semantics in nominal sets, through which our languages will have namepermutations and termformers that can bind as primitive builtin features.
Nominal unification from a higherorder perspective
 In Proceedings of RTA’08
"... Abstract. Nominal Logic is an extension of firstorder logic with equality, namebinding, nameswapping, and freshness of names. Contrarily to higherorder logic, bound variables are treated as atoms, and only free variables are proper unknowns in nominal unification. This allows “variable capture”, ..."
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Abstract. Nominal Logic is an extension of firstorder logic with equality, namebinding, nameswapping, and freshness of names. Contrarily to higherorder logic, bound variables are treated as atoms, and only free variables are proper unknowns in nominal unification. This allows “variable capture”, breaking a fundamental principle of lambdacalculus. Despite this difference, nominal unification can be seen from a higherorder perspective. From this view, we show that nominal unification can be reduced to a particular fragment of higherorder unification problems: higherorder patterns unification. This reduction proves that nominal unification can be decided in quadratic deterministic time. 1
Nominal Algebra and the HSP Theorem
"... Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as firstorder logic, the lambdacalculus, or process calculi. ..."
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Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as firstorder logic, the lambdacalculus, or process calculi. Nominal algebra has a semantics in nominal sets (sets with a finitelysupported permutation action); previous work proved soundness and completeness. The HSP theorem characterises the class of models of an algebraic theory as a class closed under homomorphic images, subalgebras, and products, and is a fundamental result of universal algebra. It is not obvious that nominal algebra should satisfy the HSP theorem: nominal algebra axioms are subject to socalled freshness conditions which give them some flavour of implication; nominal sets have significantly richer structure than the sets semantics traditionally used in universal algebra. The usual method of proof for the HSP theorem does not obviously transfer to the nominal algebra setting. In this paper we give the constructions which show that, after all, a ‘nominal ’ version of the HSP theorem holds for nominal algebra; it corresponds to closure under homomorphic images, subalgebras, products, and an atomsabstraction construction specific to nominalstyle semantics. Keywords: universal algebra, equational logic, nominal algebra, HSP or Birkhoff’s theorem, nominal sets, nominal terms 1
On Universal Algebra over Nominal Sets
"... ... theorem for algebras over nominal sets. We isolate a ‘uniform’ fragment of our equational logic, which corresponds to the nominal logics present in the literature. We give semantically invariant translations of theories for nominal algebra and NEL into ‘uniform’ theories and systematically prove ..."
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... theorem for algebras over nominal sets. We isolate a ‘uniform’ fragment of our equational logic, which corresponds to the nominal logics present in the literature. We give semantically invariant translations of theories for nominal algebra and NEL into ‘uniform’ theories and systematically prove HSP theorems for models of these theories.
Completeness for algebraic theories of local state
"... Abstract. Every algebraic theory gives rise to a monad, and monads allow a metalanguage which is a basic programming language with sideeffects. Equations in the algebraic theory give rise to equations between programs in the metalanguage. An interesting question is this: to what extent can we put ..."
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Abstract. Every algebraic theory gives rise to a monad, and monads allow a metalanguage which is a basic programming language with sideeffects. Equations in the algebraic theory give rise to equations between programs in the metalanguage. An interesting question is this: to what extent can we put equational reasoning for programs into the algebraic theory for the monad? In this paper I focus on local state, where programs can allocate, update and read the store. Plotkin and Power (FoSSaCS’02) have proposed an algebraic theory of local state, and they conjectured that the theory is complete, in the sense that every consistent equation is already derivable. The central contribution of this paper is to confirm this conjecture. To establish the completeness theorem, it is necessary to reformulate the informal theory of Plotkin and Power as an enriched algebraic theory in the sense of Kelly and Power (JPAA, 89:163–179). The new presentation can be read as 14 program assertions about three effects. The completeness theorem for local state is dependent on certain conditions on the type of storable values. When the set of storable values is finite, there is a subtle additional axiom regarding quotient types. 1
Categorical Equational Systems: Algebraic Models and Equational Reasoning
, 2010
"... This dissertation is submitted for the degree of Doctor of PhilosophyDedicated to my parents and my wifeDeclaration This dissertation is the result of my own work done under the guidance of my supervisor, and includes nothing which is the outcome of work done in collaboration except where specifical ..."
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Cited by 7 (2 self)
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This dissertation is submitted for the degree of Doctor of PhilosophyDedicated to my parents and my wifeDeclaration This dissertation is the result of my own work done under the guidance of my supervisor, and includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text. This dissertation is not substantially the same as any that I have submitted or will be submitting for a degree or diploma or other qualification at this or any other University. This dissertation does not exceed the regulation length of 60,000 words, including tables and footnotes. 5
On the construction of free algebras for equational systems
 IN: SPECIAL ISSUE FOR AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP 2007). VOLUME 410 OF THEORETICAL COMPUTER SCIENCE
, 2009
"... The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applica ..."
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The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications.
AN EFFICIENT NOMINAL UNIFICATION ALGORITHM
"... Abstract. Nominal Unification is an extension of firstorder unification where terms can contain binders and unification is performed modulo αequivalence. Here we prove that the existence of nominal unifiers can be decided in quadratic time. First, we linearlyreduce nominal unification problems to ..."
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Abstract. Nominal Unification is an extension of firstorder unification where terms can contain binders and unification is performed modulo αequivalence. Here we prove that the existence of nominal unifiers can be decided in quadratic time. First, we linearlyreduce nominal unification problems to a sequence of freshness and equalities between atoms, modulo a permutation, using ideas as Paterson and Wegman for firstorder unification. Second, we prove that solvability of these reduced problems may be checked in quadratic time. Finally, we point out how using ideas of Brown and Tarjan for unbalanced merging, we could solve these reduced problems more efficiently. 1.