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28
Towards Concurrent Type Theory
 INVITED TALK AT TLDI’12
, 2012
"... We review progress in a recent line of research that provides a concurrent computational interpretation of (intuitionistic) linear logic. Propositions are interpreted as session types, sequent proofs as processes in the πcalculus, cut reductions as process reductions, and vice versa. The strong pro ..."
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Cited by 8 (7 self)
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We review progress in a recent line of research that provides a concurrent computational interpretation of (intuitionistic) linear logic. Propositions are interpreted as session types, sequent proofs as processes in the πcalculus, cut reductions as process reductions, and vice versa. The strong prooftheoretic foundation of this type system provides immediate opportunities for uniform generalization, specifically, to embed terms from a functional type theory. The resulting system satisfies the properties of type preservation, progress, and termination, as expected from a language derived via a CurryHoward isomorphism. While very expressive, the language is strictly stratified so that dependent types for functional terms can be enforced during communication, but neither processes nor channels can appear in functional terms. We briefly speculate on how this limitation might be overcome to arrive at a fully dependent concurrent type theory.
Semantics of linear continuationpassing in callbyname
 In Proc. Functional and Logic Programming, Springer Lecture Notes in Comput. Sci
, 2004
"... Abstract. We propose a semantic framework for modelling the linear usage of continuations in typed callbyname programming languages. On the semantic side, we introduce a construction for categories of linear continuations, which gives rise to cartesian closed categories with “linear classical disj ..."
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Cited by 6 (4 self)
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Abstract. We propose a semantic framework for modelling the linear usage of continuations in typed callbyname programming languages. On the semantic side, we introduce a construction for categories of linear continuations, which gives rise to cartesian closed categories with “linear classical disjunctions ” from models of intuitionistic linear logic with sums. On the syntactic side, we give a simply typed callbyname λµcalculus in which the use of names (continuation variables) is restricted to be linear. Its semantic interpretation into a category of linear continuations then amounts to the callbyname continuationpassing style (CPS) transformation into a linear lambda calculus with sum types. We show that our calculus is sound for this CPS semantics, hence for models given by the categories of linear continuations.
A terminating and confluent linear lambda calculus
 PROC. OF 17TH INT. CONFERENCE RTA 2006, VOLUME 4098 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2006
"... We present a rewriting system for the linear lambda calculus corresponding to the {!, ⊸}fragment of intuitionistic linear logic. This rewriting system is shown to be strongly normalizing, and ChurchRosser modulo the trivial commuting conversion. Thus it provides a simple decision method for the eq ..."
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Cited by 6 (0 self)
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We present a rewriting system for the linear lambda calculus corresponding to the {!, ⊸}fragment of intuitionistic linear logic. This rewriting system is shown to be strongly normalizing, and ChurchRosser modulo the trivial commuting conversion. Thus it provides a simple decision method for the equational theory of the linear lambda calculus. As an application we prove the strong normalization of the simply typed computational lambda calculus by giving a reductionpreserving translation into the linear lambda calculus.
Interpreting polymorphic FPC into domain theoretic models of parametric polymorphism
 in: International Colloquium on Automata, Languages and Programming, Proceedings, Vol. 4052 of LNCS, SpringerVerlag
, 2006
"... Abstract. This paper shows how parametric PILLY (Polymorphic Intuitionistic / Linear Lambda calculus with a fixed point combinator Y) can be used as a metalanguage for domain theory, as originally suggested by Plotkin more than a decade ago. Using recent results about solutions to recursive domain e ..."
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Cited by 4 (1 self)
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Abstract. This paper shows how parametric PILLY (Polymorphic Intuitionistic / Linear Lambda calculus with a fixed point combinator Y) can be used as a metalanguage for domain theory, as originally suggested by Plotkin more than a decade ago. Using recent results about solutions to recursive domain equations in parametric models of PILLY, we show how to interpret FPC in these. Of particular interest is a model based on “admissible ” pers over a reflexive domain, the theory of which can be seen as a domain theory for (impredicative) polymorphism. We show how this model gives rise to a parametric and computationally adequate model of PolyFPC, an extension of FPC with impredicative polymorphism. This is the first model of a language with parametric polymorphism, recursive terms and recursive types in a nonlinear setting. 1
Operational semantics and models of linear AbadiPlotkin logic. Manuscript
, 2006
"... Abstract. We present a model of Linear Abadi and Plotkin Logic for parametricity [8] based on the operational semantics of LILY, a polymorphic linear lambda calculus endowed with an operational semantics [3]. We use it to formally prove definability of general recursive types in LILY and to derive r ..."
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Cited by 3 (2 self)
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Abstract. We present a model of Linear Abadi and Plotkin Logic for parametricity [8] based on the operational semantics of LILY, a polymorphic linear lambda calculus endowed with an operational semantics [3]. We use it to formally prove definability of general recursive types in LILY and to derive reasoning principles for the recursive types. 1
Reduction in a linear lambdacalculus with applications to operational semantics
 In RTA
, 2005
"... Abstract. We study betareduction in a linear lambdacalculus derived from Abramsky’s linear combinatory algebras. Reductions are classified depending on whether the redex is in the computationally active part of a term (“surface” reductions) or whether it is suspended within the body of a thunk (“i ..."
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Cited by 2 (0 self)
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Abstract. We study betareduction in a linear lambdacalculus derived from Abramsky’s linear combinatory algebras. Reductions are classified depending on whether the redex is in the computationally active part of a term (“surface” reductions) or whether it is suspended within the body of a thunk (“internal” reductions). If surface reduction is considered on its own then any normalizing term is strongly normalizing. More generally, if a term can be reduced to surface normal form by a combined sequence of surface and internal reductions then every combined reduction sequence from the term contains only finitely many surface reductions. We apply these results to the operational semantics of Lily, a secondorder linear lambdacalculus with recursion, introduced by Bierman, Pitts and Russo, for which we give simple proofs that callbyvalue, callbyname and callbyneed contextual equivalences coincide. 1
Plans, Actions and Dialogues using Linear Logic
, 2008
"... We propose a framework, based on Linear Logic, for finding and executing plans that include dialogue with the aim of simplifying agent design. In particular, we provide a model that allows agents to be robust to unexpected events and failures, and supports significant reuse of agent specifications. ..."
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Cited by 2 (1 self)
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We propose a framework, based on Linear Logic, for finding and executing plans that include dialogue with the aim of simplifying agent design. In particular, we provide a model that allows agents to be robust to unexpected events and failures, and supports significant reuse of agent specifications. Using Linear Logic as the foundational machinery improves upon previous dialogue systems by providing a clear underlying logical model for both planning and execution. The resulting framework has been implemented and several case studies have been considered. Further applications include humancomputer interfaces as well as agent interaction in the semantic web.
2006, ‘Planning as Deductive Synthesis in Intuitionistic Linear Logic
"... Isabelle/HOL of Intuitionistic Linear Logic and consider the support this provides for constructing plans using deductive synthesis of the proof terms. This representation of plans in linear logic provides a concise account of planning with sensing actions, allows the creation and deletion of object ..."
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Cited by 1 (1 self)
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Isabelle/HOL of Intuitionistic Linear Logic and consider the support this provides for constructing plans using deductive synthesis of the proof terms. This representation of plans in linear logic provides a concise account of planning with sensing actions, allows the creation and deletion of objects, and solves the frame problem in an elegant way. Within this setting, we show how planning algorithms are implemented as search strategies within the proof assistant. This allows us to provide a flexible methodology for developing search strategies that is independent of soundness issues. This feature is illustrated in two ways. Firstly, following ideas from logic programming, we show how a significant symmetry in search, caused by context splitting, can be pruned by using a derived inference rule. Secondly, we show how domain specific constraints on synthesis are supported and how they can be used to find contingent or conformant plans. 1
Linear explicit substitutions (extended abstract
 In Proceedings of WESTAPP'98
, 1998
"... Thecalculus [1] adds explicit substitutions to thecalculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper generalises thecalculus to provide a linear calculus of explicit substitutions which analogously descr ..."
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Thecalculus [1] adds explicit substitutions to thecalculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper generalises thecalculus to provide a linear calculus of explicit substitutions which analogously describes the implementation of linear functional programming languages. 1
Verified Planning by Deductive Synthesis in Intuitionistic Linear Logic
"... We describe a new formalisation in Isabelle/HOL of Intuitionistic Linear Logic and consider the support this provides for constructing plans by proving the achievability of given planning goals. The plans so found are provably correct, by construction. This representation of plans in linear logic pr ..."
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Cited by 1 (0 self)
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We describe a new formalisation in Isabelle/HOL of Intuitionistic Linear Logic and consider the support this provides for constructing plans by proving the achievability of given planning goals. The plans so found are provably correct, by construction. This representation of plans in linear logic provides a concise account of planning with sensing actions, allows the creation and deletion of objects, and solves the frame problem in an elegant way. Within this setting, we show how planning algorithms are implemented as search strategies within a theorem proving system. This allows us to provide a flexible methodology for developing search strategies that is independent of soundness issues. This feature is illustrated in two ways. Firstly, following ideas from logic programming, we show how a significant symmetry in search, caused by context splitting, can be pruned by using a derived inference rule. Secondly, we show how domain specific constraints on synthesis are supported and how they can be used to find contingent or conformant plans. We illustrate the approach with example planning scenarios. 1.