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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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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
Stone duality for nominal Boolean algebras with NEW
 In Proceedings of the 4th international conference on algebra and coalgebra in computer science (CALCO 2011), volume 6859 of Lecture Notes in Computer Science
, 2011
"... Abstract. We define Boolean algebras over nominal sets with a functionsymbol Nmirroring the N‘fresh name ’ quantifier. We also define dual notions of nominal topology and Stone space, prove a representation theorem over fields of nominal sets, and extend this to a Stone duality. 1 ..."
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Abstract. We define Boolean algebras over nominal sets with a functionsymbol Nmirroring the N‘fresh name ’ quantifier. We also define dual notions of nominal topology and Stone space, prove a representation theorem over fields of nominal sets, and extend this to a Stone duality. 1
Structural Recursion over Contextual Objects
"... Abstract—A core programming language is presented that allows structural recursion over open LF objects and contexts. The main technical tool is a coverage checking algorithm that also generates valid recursive calls. Termination of callbyvalue reduction is proven using a reducibility semantics. T ..."
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Abstract—A core programming language is presented that allows structural recursion over open LF objects and contexts. The main technical tool is a coverage checking algorithm that also generates valid recursive calls. Termination of callbyvalue reduction is proven using a reducibility semantics. This establishes consistency and allows the implementation of proofs about LF specifications as wellfounded recursive functions using simultaneous pattern matching. I.
Relating Two Semantics of Locally Scoped Names
"... The operational semantics of programming constructs involving locally scoped names typically makes use of stateful dynamic allocation: a set of currentlyused names forms part of the state and upon entering a scope the set is augmented by a new name bound to the scoped identifier. More abstractly, o ..."
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The operational semantics of programming constructs involving locally scoped names typically makes use of stateful dynamic allocation: a set of currentlyused names forms part of the state and upon entering a scope the set is augmented by a new name bound to the scoped identifier. More abstractly, one can see this as a transformation of local scopes by expanding them outward to an implicit toplevel. By contrast, in a neglected paper from 1994, Odersky gave a stateless lambda calculus with locally scoped names whose dynamics contracts scopes inward. The properties of ‘Oderskystyle ’ local names are quite different from dynamically allocated ones and it has not been clear, until now, what is the expressive power of Odersky’s notion. We show that in fact it provides a direct semantics of locally scoped names from which the more familiar dynamic allocation semantics can be obtained by continuationpassing style (CPS) translation. More precisely, we show that there is a CPS translation of typed lambda calculus with dynamically allocated names (the PittsStark νcalculus) into Odersky’s λνcalculus which is computationally adequate with respect to observational equivalence in the two calculi. 1998 ACM Subject Classification F.3.2 operational semantics, F.3.3 functional constructs, F.4.1 lambda calculus and related systems.
Investigations into Algebra and Topology over Nominal Sets
, 2011
"... The last decade has seen a surge of interest in nominal sets and their applications to formal methods for programming languages. This thesis studies two subjects: algebra and duality in the nominal setting. In the first part, we study universal algebra over nominal sets. At the heart of our approach ..."
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The last decade has seen a surge of interest in nominal sets and their applications to formal methods for programming languages. This thesis studies two subjects: algebra and duality in the nominal setting. In the first part, we study universal algebra over nominal sets. At the heart of our approach lies the existence of an adjunction of descent type between nominal sets and a category of manysorted sets. Hence nominal sets are a full reflective subcategory of a manysorted variety. This is presented in Chapter 2. Chapter 3 introduces functors over manysorted varieties that can be presented by operations and equations. These are precisely the functors that preserve sifted colimits. They play a central role in Chapter 4, which shows how one can systematically transfer results of universal algebra from a manysorted variety to nominal sets. However, the equational logic obtained is more expressive than the nominal equational logic of Clouston and Pitts, respectively, the nominal algebra of Gabbay and Mathijssen. A uniform fragment of our logic with the same expressivity
Full Abstraction for Nominal Scott Domains
, 2013
"... We develop a domain theory within nominal sets and present programming language constructs and results that can be gained from this approach. The development is based on the concept of orbitfinite subset, that is, a subset of a nominal sets that is both finitely supported and contained in finitely m ..."
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We develop a domain theory within nominal sets and present programming language constructs and results that can be gained from this approach. The development is based on the concept of orbitfinite subset, that is, a subset of a nominal sets that is both finitely supported and contained in finitely many orbits. This concept appears prominently in the recent research programme of Bojańczyk et al. on automata over infinite languages, and our results establish a connection between their work and a characterisation of topological compactness discovered, in a quite different setting, by Winskel and Turner as part of a nominal domain theory for concurrency. We use this connection to derive a notion of Scott domain within nominal sets. The functionals for existential quantification over names and ‘definite description ’ over names turn out to be compact in the sense appropriate for nominal Scott domains. Adding them, together with parallelor, to a programming language for recursively defined higherorder functions with name abstraction and locally scoped names, we prove a full abstraction result for nominal Scott domains analogous to Plotkin’s classic result about PCF and conventional Scott domains: two program phrases have the same observable operational behaviour in all contexts if and only if they denote equal elements of the nominal Scott domain model. This is the first full abstraction result we know of for higherorder functions with local names that uses a domain theory based on ordinary extensional functions, rather than using the more intensional approach of game semantics.
Instances of computational effects: an algebraic perspective
"... Abstract—We investigate the connections between computational effects, algebraic theories, and monads on functor categories. We develop a syntactic framework with variable binding that allows us to describe equations between programs while taking into account the idea that there may be different ins ..."
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Abstract—We investigate the connections between computational effects, algebraic theories, and monads on functor categories. We develop a syntactic framework with variable binding that allows us to describe equations between programs while taking into account the idea that there may be different instances of a particular computational effect. We use our framework to give a general account of several notions of computation that had previously been analyzed in terms of monads on presheaf categories: the analysis of local store by Plotkin and Power; the analysis of restriction by Pitts; and the analysis of the pi calculus by Stark. I.
MFPS 29 Preliminary Proceedings Normalization by evaluation and algebraic effects
"... We examine the interplay between computational effects and higher types. We do this by presenting a normalization by evaluation algorithm for a language with function types as well as computational effects. We use algebraic theories to treat the computational effects in the normalization algorithm i ..."
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We examine the interplay between computational effects and higher types. We do this by presenting a normalization by evaluation algorithm for a language with function types as well as computational effects. We use algebraic theories to treat the computational effects in the normalization algorithm in a modular way. Our algorithm is presented in terms of an interpretation in a category of presheaves equipped with partial equivalence relations. The normalization algorithm and its correctness proofs are formalized in dependent type theory (Agda). Keywords:
Wellfounded Recursion over Contextual Objects
"... We present a core programming language that supports writing wellfounded structurally recursive functions using simultaneous pattern matching on contextual LF objects and contexts. The main technical tool is a coverage checking algorithm that also generates valid recursive calls. To establish cons ..."
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We present a core programming language that supports writing wellfounded structurally recursive functions using simultaneous pattern matching on contextual LF objects and contexts. The main technical tool is a coverage checking algorithm that also generates valid recursive calls. To establish consistency, we define a callbyvalue smallstep semantics and prove that every welltyped program terminates using a reducibility semantics. Based on the presented methodology we have implemented a totality checker as part of the programming and proof environment Beluga where it can be used to establish that a total Beluga program corresponds to a proof.