Generalised Name Abstraction for Nominal Sets

Clouston, Ranald

doi:10.1007/978-3-642-37075-5_28

Cited by 3 publications

(5 citation statements)

References 18 publications

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“…The separated product gives rise to another symmetric closed monoidal structure on Pm-Nom, with 1 as unit, and the exponential object given by magic wand X − * Y . An explicit characterisation of X − * Y is not needed in the remainder of this paper, but for a complete presentation we briefly recall the description from [23] (see also [19] and [5]). First, define a Pm-action on the set of partial functions f : [23] for a proof and explanation.…”

Section: Separated Productmentioning

confidence: 99%

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Separation and Renaming in Nominal Sets

Moerman¹,

Rot²

2019

Preprint

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Nominal sets provide a foundation for reasoning about names. They are used primarily in syntax with binders, but also, e.g., to model automata over infinite alphabets. In this paper, nominal sets are related to nominal renaming sets, which involve arbitrary substitutions rather than permutations, through a categorical adjunction. In particular, the left adjoint relates the separated product of nominal sets to the Cartesian product of nominal renaming sets. Based on these results, we define the new notion of separated nominal automata. These automata can be exponentially smaller than classical nominal automata, if the semantics is closed under substitutions. * This research has been partially funded by the ERC AdG project 787914 FRAPPANT.

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Section: Separated Productmentioning

confidence: 99%

“…The latter is generalised to [X]Y in [7]. In [5] it is shown that the coincidence [X]Y ∼ = (X − * Y ) only holds under strong assumptions (including that X is single-orbit).…”

Section: Separated Productmentioning

confidence: 99%

Separation and Renaming in Nominal Sets

Moerman¹,

Rot²

2019

Preprint

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“…We use a freshness relation to define conditions involving disjointness of bound variables in such rules as Rule 23 ( Figure 7). Notions of α-equivalence is defined in a general way as in Ranald Clouston's work on Generalised Name Abstraction for Nominal Sets [Clouston 2013]. We explicitly add α-conversion in the elaboration rules for functor application (Rule 29) and for the sequence of declarations (Rule 36).…”

Section: Formalising the Development In Coqmentioning

confidence: 99%

Static interpretation of higher-order modules in Futhark: functional GPU programming in the large

Elsman

Henriksen

Annenkov

et al. 2018

Proc. ACM Program. Lang.

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We present a higher-order module system for the purely functional data-parallel array language Futhark. The module language has the property that it is completely eliminated at compile time, yet it serves as a powerful tool for organizing libraries and complete programs. The presentation includes a static and a dynamic semantics for the language in terms of, respectively, a static type system and a provably terminating elaboration of terms into terms of an underlying target language. The development is formalised in Coq using a novel encoding of semantic objects based on products, sets, and finite maps. The module language features a unified treatment of module type abstraction and core language polymorphism and is rich enough for expressing practical forms of module composition.

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“…The theory of nominal sets based on the idea of permutations of variables and notion of finite support. 2 The theory gives a uniform approach to deal with bound variables and allows for generalisation of binders to various structures [Clo13] We are interested in the particular generalisation where one can bind a set of variables at once. Moreover, since nominal sets offer a uniform approach it can be applied to various structures even if they are quite different from well-studied systems such as the lambda calculus.…”

Section: Variable Binding and Nominal Techniquesmentioning

confidence: 99%

“…Definition 3.4.8 is the usual notion of freshness for a single atom. We can also define a generalised notion of freshness following [Clo13], which will be useful in the context of our formalisation. Definition 3.4.9 (Generalised freshness).…”

Section: Variable Binding and Nominal Techniquesmentioning

confidence: 99%

Adventures in Formalisation: Financial Contracts, Modules, and Two-Level Type Theory

Annenkov

2018

Preprint

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Over the last few decades, software has become essential for the proper functioning of systems in the modern world. Formal verification techniques are slowly being adopted in various industrial application areas, and there is a big demand for research in the theory and practice of formal techniques to achieve a wider acceptance of tools for verification.We present three projects concerned with applications of certified programming techniques and proof assistants in the area of programming language theory and mathematics.The first project is about a certified compilation technique for a domain-specific programming language for financial contracts (the CL language). The code in CL is translated into a simple expression language well-suited for integration with software components implementing Monte Carlo simulation techniques (pricing engines). The compilation procedure is accompanied with formal proofs of correctness carried out in the Coq proof assistant. Moreover, we develop techniques for capturing the dynamic behaviour of contracts with the passage of time. These techniques potentially allow for efficient integration of contract specifications with high-performance pricing engines running on GPGPU hardware.The second project presents a number of techniques that allow for formal reasoning with nested and mutually inductive structures built up from finite maps and sets (also called semantic objects), and at the same time allow for working with binding structures over sets of variables. The techniques, which build on the theory of nominal sets combined with the ability to work with multiple isomorphic representations of finite maps, make it possible to give a formal treatment, in Coq, of a higher-order module system, including the ability to eliminate entirely, at compile time, abstraction barriers introduced by the module system. The development is based on earlier work on static interpretation of modules and provides the foundation for a higher-order module language for Futhark, an optimising compiler targeting data-parallel architectures, such as GPGPUs.The third project is related to homotopy type theory (HoTT), a new branch of mathematics based on a fascinating idea connecting type theory and homotopy theory. HoTT provides us with a new foundation for mathematics allowing for developing machine-checkable proofs in various areas of computer science and mathematics. Along with Vladimir Voevodsky's univalence axiom, HoTT offers a formal treatment of the informal mathematical principle: equivalent structures can be identified. However, in some cases, the notion of weak equality available in HoTT leads to the "infinite coherence" problem when defining internally certain structures (such as a type of n-restricted semi-simplicial types, inverse diagrams and so on). We explain the basic idea of two-level type theory, a version of Martin-Löf type theory with two equality types: the first acts as the usual equality of homotopy type theory, while the second allows us to reason about strict equality. In this system, we ...

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Generalised Name Abstraction for Nominal Sets

Cited by 3 publications

References 18 publications

Separation and Renaming in Nominal Sets

Separation and Renaming in Nominal Sets

Static interpretation of higher-order modules in Futhark: functional GPU programming in the large

Adventures in Formalisation: Financial Contracts, Modules, and Two-Level Type Theory

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