Nominal Unification

Urban, Christian; Pitts, Andrew M.; Gabbay, Murdoch J.

doi:10.1007/978-3-540-45220-1_41

Cited by 69 publications

(182 citation statements)

References 14 publications

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“…For simplicity's sake αML has a monomorphic type system with a single, top-level data type declaration of some name sorts (N ), of some data sorts (S, including a distinguished one bool with constructors T () and F ()) whose recursive definitions may only involve equality types (E), and of some general data types (D) whose recursive definitions may involve function types and prop. Note that in accord with point (a) above, such a declaration subsumes the notion of nominal signature [28] that we used in Sect. 2; in particular, the signature's nominal arities (1) coincide with equality types.…”

Section: 4] To Achieve Point (B) We Note That the Meaning D Of D Ismentioning

confidence: 78%

“…Alpha-equivalence arises from the presence of binding constructs in the expressions and we will deal with this in a generic way by using nominal signatures [28], Σ. These generalize the usual notion of many-sorted algebraic signature to encompass constructors that bind names of various sorts.…”

Section: α-Inductive Definitionsmentioning

confidence: 99%

“…(These trees are the ground nominal terms from [28], that is, the ones not involving variables.) The third operation, name-abstraction, is the generic binding form provided by nominal signatures: renaming n (−)-bound names in trees gives an equivalence relation = α [24, Fig.…”

Section: α-Inductive Definitionsmentioning

confidence: 99%

“…Remark 2.2(d)) complicates unification-see for example the use of terms involving explicit namepermutations in nominal [28] and equivariant [6] unification algorithms. αML's operational semantics is specified in Fig.…”

Section: Fig 3 αMl Operational Semanticsmentioning

confidence: 99%

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

Lakin

Pitts

2009

Programming Languages and Systems

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Abstract. This paper studies inductive definitions involving binders, in which aliasing between free and bound names is permitted. Such aliasing occurs in informal specifications of operational semantics, but is excluded by the common representation of binding as meta-level λ-abstraction. Drawing upon ideas from functional logic programming, we represent such definitions with aliasing as recursively defined functions in a higher-order typed functional programming language that extends core ML with types for name-binding, a type of "semi-decidable propositions" and existential quantification for types with decidable equality. We show that the representation is sound and complete with respect to the language's operational semantics, which combines the use of evaluation contexts with constraint programming. We also give a new and simple proof that the associated constraint problem is NP-complete.

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Section: 4] To Achieve Point (B) We Note That the Meaning D Of D Ismentioning

confidence: 78%

Section: α-Inductive Definitionsmentioning

confidence: 99%

Section: α-Inductive Definitionsmentioning

confidence: 99%

Section: Fig 3 αMl Operational Semanticsmentioning

confidence: 99%

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

Lakin

Pitts

2009

Programming Languages and Systems

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“…Our approach is to formulate the π-calculus using ideas from nominal logic developed by Pitts et al [37,21,42]. This is a first order logic designed to work with calculi using binders.…”

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confidence: 99%

Formalising the pi-calculus using nominal logic

Bengtson¹,

Parrow²

2009

Logical Methods in Computer Science

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Abstract. We formalise the pi-calculus using the nominal datatype package, based on ideas from the nominal logic by Pitts et al., and demonstrate an implementation in Isabelle/HOL. The purpose is to derive powerful induction rules for the semantics in order to conduct machine checkable proofs, closely following the intuitive arguments found in manual proofs. In this way we have covered many of the standard theorems of bisimulation equivalence and congruence, both late and early, and both strong and weak in a uniform manner. We thus provide one of the most extensive formalisations of a process calculus ever done inside a theorem prover.A significant gain in our formulation is that agents are identified up to alpha-equivalence, thereby greatly reducing the arguments about bound names. This is a normal strategy for manual proofs about the pi-calculus, but that kind of hand waving has previously been difficult to incorporate smoothly in an interactive theorem prover. We show how the nominal logic formalism and its support in Isabelle accomplishes this and thus significantly reduces the tedium of conducting completely formal proofs. This improves on previous work using weak higher order abstract syntax since we do not need extra assumptions to filter out exotic terms and can keep all arguments within a familiar first-order logic.

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Avoiding Equivariance in Alpha-Prolog

Urban

Cheney

2005

Lecture Notes in Computer Science

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Abstract. αProlog is a logic programming language which is well-suited for rapid prototyping of type systems and operational semantics of typed λ-calculi and many other languages involving bound names. In αProlog, the nominal unification algorithm of Urban, Pitts and Gabbay is used instead of first-order unification. However, although αProlog can be viewed as Horn-clause logic programming in Pitts' nominal logic, proof search using nominal unification is incomplete in nominal logic. Because of nominal logic's equivariance principle, complete proof search would require solving NP-hard equivariant unification problems. Nevertheless, the αProlog programs we studied run correctly without equivariant unification. In this paper, we give several examples of αProlog programs that do not require equivariant unification, develop a test for identifying such programs, and prove the correctness of this test via a proof-theoretic argument.

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Nominal Unification

Cited by 69 publications

References 14 publications

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

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

Formalising the pi-calculus using nominal logic

Avoiding Equivariance in Alpha-Prolog

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