Resolving Inductive Definitions with Binders in Higher-Order Typed Functional Programming

Lakin, Matthew R.; Pitts, Andrew M.

doi:10.1007/978-3-642-00590-9_4

Cited by 6 publications

(21 citation statements)

References 24 publications

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“…In [19] we claimed that αML correctly implements names and binding modulo α-equivalence. To back up this claim we must show a correctness result for αML akin to the following theorem for the FreshML language, which was proved in [30] and [27].…”

Section: Lam (Lam (Var X))mentioning

confidence: 97%

“…We have practical experience of the NPNAS approach in the context of the functional logic programming language αML [19]. This is a call-by-value, higher-order, typed meta-programming language.…”

Section: Lam (Lam (Var X))mentioning

confidence: 99%

“…Both this constraint problem and the equivariant unification problem are known to be NP-complete (see [3] and [19,Section 3] for proofs). Thus they are both polynomial-time reducible to each other and hence our constraint problem is equivalent to equivariant unification.…”

Section: Lam (Lam (Var X))mentioning

confidence: 99%

“…Thus they are both polynomial-time reducible to each other and hence our constraint problem is equivalent to equivariant unification. We showed in [19] that proof-search in αML is complete for a simple yet powerful model of inductive definitions involving binders. Hence, the motivation for adopting NPNAS comes from the (constraint) logic programming side of αML.…”

Section: Lam (Lam (Var X))mentioning

confidence: 99%

“…In this section we give a brief overview of the syntax, type system and operational semantics of the αML meta-language, which was introduced in [19].…”

Section: The αMl Languagementioning

confidence: 99%

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Encoding Abstract Syntax Without Fresh Names

Lakin

Pitts

2011

J Autom Reasoning

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This paper introduces a variant of nominal abstract syntax in which bindable names are represented by normal meta-variables as opposed to a separate class of globally fresh names. Distinct meta-variables can be instantiated with the same concrete name, which we call aliasing. The possible aliasing patterns are controlled by explicit constraints on the distinctness (freshness) of names. This approach has already been used in the nominal meta-programming language αML. We recap that language and develop a theory of contextual equivalence for it. The central result of the paper is that abstract syntax trees (ASTs) involving binders can be encoded into αML in such a way that α-equivalence of ASTs corresponds with contextual equivalence of their encodings. This is novel because the encoding does not rely on the existence of globally fresh names and fresh name generation, which are fundamental to the correctness of the pre-existing encoding of abstract syntax into FreshML.

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Section: Lam (Lam (Var X))mentioning

confidence: 97%

Section: Lam (Lam (Var X))mentioning

confidence: 99%

Section: Lam (Lam (Var X))mentioning

confidence: 99%

Section: Lam (Lam (Var X))mentioning

confidence: 99%

“…In this section we give a brief overview of the syntax, type system and operational semantics of the αML meta-language, which was introduced in [19].…”

Section: The αMl Languagementioning

confidence: 99%

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Encoding Abstract Syntax Without Fresh Names

Lakin

Pitts

2011

J Autom Reasoning

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Equivariant Unification

Cheney

2009

J Autom Reasoning

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Nominal logic is a variant of first-order logic with special facilities for reasoning about names and binding based on the underlying concepts of swapping and freshness. It serves as the basis of logic programming, term rewriting, and automated theorem proving techniques that support reasoning about languages with name-binding. These applications often require nominal unification, or equational reasoning and constraint solving in nominal logic. Urban, Pitts and Gabbay developed an algorithm for a broadly applicable class of nominal unification problems. However, because of nominal logic's equivariance property, these applications also require a different form of unification, which we call equivariant unification. In this article, we first study the complexity of the decision problem for equivariant unification and equivariant matching. We show that these problems are NP-hard in general, as is nominal unification without the ground-name restrictions employed in previous work on nominal unification. Moreover, we present an exponentialtime algorithm for equivariant unification that can be used to decide satisfiability, or produce a complete finite set of solutions. We also study special cases that can be solved efficiently. In particular, we present a polynomial time algorithm for swapping-free equivariant matching, that is, for matching problems in which the swapping operation does not appear.

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Contextual equivalence for inductive definitions with binders in higher order typed functional programming

Lakin¹,

Pitts²

2013

J. Funct. Prog.

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Correct handling of names and binders is an important issue in meta-programming. This paper presents an embedding of constraint logic programming into the αML functional programming language, which provides a provably correct means of implementing proof search computations over inductive definitions involving names and binders modulo α-equivalence. We show that the execution of proof search in the αML operational semantics is sound and complete with regard to the model-theoretic semantics of formulae, and develop a theory of contextual equivalence for the subclass of αML expressions which correspond to inductive definitions and formulae. In particular, we prove that αML expressions, which denote inductive definitions, are contextually equivalent precisely when those inductive definitions have the same model-theoretic semantics. This paper is a revised and extended version of the conference paper (Lakin, M. R. & Pitts, A. M. (2009) Resolving inductive definitions with binders in higher-order typed functional programming. InProceedings of the 18th European Symposium on Programming (ESOP 2009), Castagna, G. (ed), Lecture Notes in Computer Science, vol. 5502. Berlin, Germany: Springer-Verlag, pp. 47–61) and draws on material from the first author's PhD thesis (Lakin, M. R. (2010)An Executable Meta-Language for Inductive Definitions with Binders. University of Cambridge, UK).

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Resolving Inductive Definitions with Binders in Higher-Order Typed Functional Programming

Cited by 6 publications

References 24 publications

Encoding Abstract Syntax Without Fresh Names

Encoding Abstract Syntax Without Fresh Names

Equivariant Unification

Contextual equivalence for inductive definitions with binders in higher order typed functional programming

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