A formalisation of nominal α-equivalence with A, C, and AC function symbols

Ayala-Rincón, Maurício; Carvalho-Segundo, Washington de; Fernández, Maribel; Nantes-Sobrinho, Daniele; Rocha-Oliveira, Ana Cristina

doi:10.1016/j.tcs.2019.02.020

Cited by 8 publications

(4 citation statements)

References 31 publications

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“…the support) of π is the set dom(π) := {a ∈ A | π • a = a}. We will assume, as in Ayala-Rincón et al (2019a), countable sets of function symbols with different equational properties such as associativity, commutativity, idempotence. Function symbols have superscripts that indicate their equational properties; thus, f C k will denote the kth function symbol that is commutative and f ∅ j the jth function symbol without any equational property.…”

Section: Consider Countable Disjoint Sets Of Variablesmentioning

confidence: 99%

“…Key properties of the nominal freshness and α-equivalence relations have been extensively explored in previous works (Ayala-Rincón et al 2019a, 2016bUrban 2010;Urban et al 2004). Amongst them we have freshness preservation: if ∇ a # s and ∇ s ≈ α t, then ∇ a # t; equivariance: for all permutations π, if ∇ s ≈ α t then ∇ π • s ≈ α π • t; and equivalence: ∇ _ ≈ α _ is an equivalence relation.…”

Section: Consider Countable Disjoint Sets Of Variablesmentioning

confidence: 99%

“…We investigated the literature but could not find a database for unification problems. In Ayala-Rincón et al (2019a), experiments are made for nominal equality-check in the presence of A, C and AC function symbols, while in Calvès and Fernández (2010), experiments are made for syntactic nominal matching and nominal α-equivalence. In Ayala-Rincón et al (2019a), the terms generated are ground and arbitrary choices were made with respect to the size of unification problems, the number of different atoms and the different function symbols.…”

Section: Testing the Python Algorithmmentioning

confidence: 99%

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Formalising nominal C-unification generalised with protected variables

Ayala-Rincón

Carvalho-Segundo

Fernández

et al. 2021

Math. Struct. Comp. Sci.

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This work extends a rule-based specification of nominal C-unification formalised in Coq to include ‘protected variables’ that cannot be instantiated during the unification process. By introducing protected variables, we are able to reuse the C-unification simplification rules to solve nominal C-matching (as well as equality check) problems. From the algorithmic point of view, this extension is sufficient to obtain a generalised C-unification procedure; however, it cannot be formally checked by simple reuse of the original formalisation. This paper describes the additional effort necessary in order to adapt the specification of the inference rules and reuse previous formalisations. We also generalise a functional recursive nominal C-unification algorithm specified in PVS with protected variables, effectively adapting this algorithm to the tasks of nominal C-matching and nominal equality check. The PVS formalisation is applied to test the correctness of a Python manual implementation of the algorithm.

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Section: Consider Countable Disjoint Sets Of Variablesmentioning

confidence: 99%

Section: Consider Countable Disjoint Sets Of Variablesmentioning

confidence: 99%

Section: Testing the Python Algorithmmentioning

confidence: 99%

See 1 more Smart Citation

Formalising nominal C-unification generalised with protected variables

Ayala-Rincón

Carvalho-Segundo

Fernández

et al. 2021

Math. Struct. Comp. Sci.

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“…In the case in which E = A, C, AC, or a combination of these theories, an algorithm to check E-α-equality via freshness constraints was proposed in [AdCSFN17,AdCSMFR19], where correctness results were formally verified using the Coq proof assistant, and an implementation in Ocaml was given. Unification was considered only for C theories, for which it was shown that in general there is no finitary representation of the set of solutions if solutions are represented by freshness contexts and substitutions.…”

Section: Nominal Alpha-equivalence Modulo Equational Theoriesmentioning

confidence: 99%

On Nominal Syntax and Permutation Fixed Points

Ayala-Rincón,

Fernández,

Nantes-Sobrinho

2019

Preprint

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We propose a new axiomatisation of the alpha-equivalence relation for nominal terms, based on a primitive notion of fixed-point constraint. We show that the standard freshness relation between atoms and terms can be derived from the more primitive notion of permutation fixed-point, and use this result to prove the correctness of the new αequivalence axiomatisation. This gives rise to a new notion of nominal unification, where solutions for unification problems are pairs of a fixed-point context and a substitution. Although it may seem less natural than the standard notion of nominal unifier based on freshness constraints, the notion of unifier based on fixed-point constraints behaves better when equational theories are considered: for example, nominal unification remains finitary in the presence of commutativity, whereas this is not the case when unifiers are expressed using freshness contexts. We provide a definition of α-equivalence modulo equational theories that takes into account A, C and AC theories. Based on this notion of equivalence, we show that C-unification is finitary and we provide a sound and complete C-unification algorithm, as a first step towards the development of nominal unification modulo AC and other equational theories with permutative properties.

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