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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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Cited by 23 (6 self)
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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 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 20 (10 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
Simple nominal type theory
"... Abstract. Nominal logic is an extension of firstorder logic with features useful for reasoning about abstract syntax with bound names. For computational applications such as programming and formal reasoning, it is desirable to develop constructive type theories for nominal logic which extend standa ..."
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Cited by 8 (1 self)
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Abstract. Nominal logic is an extension of firstorder logic with features useful for reasoning about abstract syntax with bound names. For computational applications such as programming and formal reasoning, it is desirable to develop constructive type theories for nominal logic which extend standard type theories for propositional, first or higherorder logic. This has proven difficult, largely because of complex interactions between nominal logic’s nameabstraction operation and ordinary functional abstraction. This difficulty already arises in the case of propositional logic and simple type theory. In this paper we show how this difficulty can be overcome, and present a simple nominal type theory which enjoys properties such as type soundness and strong normalization, and which can be soundly interpreted using existing nominal set models of nominal logic. We also sketch how recursion combinators for languages with binding structure can be provided. This is an important first step towards understanding the constructive content of nominal logic and incorporating it into existing logics and type theories. 1
Structural Recursion with Locally Scoped Names
"... This paper introduces a new recursion principle for inductively defined data modulo αequivalence of bound names that makes use of Oderskystyle local names when recursing over bound names. It is formulated in simply typed λcalculus extended with names that can be restricted to a lexical scope, tes ..."
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Cited by 6 (2 self)
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This paper introduces a new recursion principle for inductively defined data modulo αequivalence of bound names that makes use of Oderskystyle local names when recursing over bound names. It is formulated in simply typed λcalculus extended with names that can be restricted to a lexical scope, tested for equality, explicitly swapped and abstracted. The new recursion principle is motivated by the nominal sets notion of “αstructural recursion”, whose use of names and associated freshness sideconditions in recursive definitions formalizes common practice with binders. The new calculus has a simple interpretation in nominal sets equipped with name restriction operations. It is shown to adequately represent αstructural recursion while avoiding the need to verify freshness sideconditions in definitions and computations. The paper is a revised and expanded version of (Pitts, 2010). 1
Toward a General Theory of Names, Binding and Scope
, 2005
"... Highlevel formalisms for reasoning about names and binding such as de Bruijn indices, various flavors of higherorder abstract syntax, the Theory of Contexts, and nominal abstract syntax address only one relatively restrictive form of scoping: namely, unary lexical scoping, in which the scope of a ..."
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Cited by 4 (0 self)
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Highlevel formalisms for reasoning about names and binding such as de Bruijn indices, various flavors of higherorder abstract syntax, the Theory of Contexts, and nominal abstract syntax address only one relatively restrictive form of scoping: namely, unary lexical scoping, in which the scope of a (single) bound name is a subtree of the abstract syntax tree (possibly with other subtrees removed due to shadowing). Many languages exhibit binding or renaming structure that does not fit this mold. Examples include binding transitions in the #calculus; unique identifiers in contexts, memory heaps, and XML documents; declaration scoping in modules and namespaces; anonymous identifiers in automata, type schemes, and Horn clauses; and pattern matching and mutual recursion constructs in functional languages. In these cases, it appears necessary to either rearrange the abstract syntax so that lexical scoping can be used, or revert to firstorder techniques. The purpose
Presheaf models for the πcalculus
 In Proc. CTCS’97, volume 1290 of LNCS
, 1997
"... Abstract. The finite πcalculus has an explicit settheoretic functorcategory model that is known to be fully abstract for strong late bisimulation congruence. We characterize this as the initial free algebra for an appropriate set of operations and equations in the enriched Lawvere theories of Plo ..."
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Abstract. The finite πcalculus has an explicit settheoretic functorcategory model that is known to be fully abstract for strong late bisimulation congruence. We characterize this as the initial free algebra for an appropriate set of operations and equations in the enriched Lawvere theories of Plotkin and Power. Thus we obtain a novel algebraic description for models of the πcalculus, and validate an existing construction as the universal such model. The algebraic operations are intuitive, covering name creation, communication of names over channels, and nondeterminism; the equations then combine these features in a modular fashion. We work in an enriched setting, over a “possible worlds ” category of sets indexed by available names. This expands significantly on the classical notion of algebraic theories, and in particular allows us to use nonstandard arities that vary as processes evolve. Based on our algebraic theory we describe a category of models for the πcalculus, and show that they all preserve bisimulation congruence. We develop a direct construction of free models in this category; and generalise previous results to prove that all freealgebra models are fully abstract. 1
Oneandahalfth order terms: CurryHoward and incomplete derivations
"... Abstract. The CurryHoward correspondence connects Natural Deduction derivation with the lambdacalculus. Predicates are types, derivations are terms. This supports reasoning from assumptions to conclusions, but we may want to reason ‘backwards ’ from the desired conclusion towards the assumptions. ..."
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Abstract. The CurryHoward correspondence connects Natural Deduction derivation with the lambdacalculus. Predicates are types, derivations are terms. This supports reasoning from assumptions to conclusions, but we may want to reason ‘backwards ’ from the desired conclusion towards the assumptions. At intermediate stages we may have an ‘incomplete derivation’, with ‘holes’. This is natural in informal practice; the challenge is to formalise it. To this end we use a oneandahalfth order technique based on nominal terms, with two levels of variable. Predicates are types, derivations are terms — and the two levels of variable are respectively the assumptions and the ‘holes ’ of an incomplete derivation. 1
Dependent Types for a Nominal Logical Framework
, 2012
"... We present a logical framework based on the nominal approach to representing syntax with binders. First we extend nominal terms, which have a builtin nameabstraction operator and a firstorder notion of substitution for variables, with a captureavoiding substitution operator for names. We then bu ..."
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We present a logical framework based on the nominal approach to representing syntax with binders. First we extend nominal terms, which have a builtin nameabstraction operator and a firstorder notion of substitution for variables, with a captureavoiding substitution operator for names. We then build a dependent type system for this extended syntax