A Formalisation of Nominal α -equivalence with A and AC Function Symbols

Ayala-Rincón, Maurício; Carvalho-Segundo, Washington de; Fernández, Maribel; Nantes-Sobrinho, Daniele

doi:10.1016/j.entcs.2017.04.003

Cited by 9 publications

(18 citation statements)

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“…, E = C or both s and t are not pairs (≈ {α,C} app) Key properties of the nominal freshness and α-equivalence relations have been extensively explored in previous works [3,6,14,15]. In [4] we also have formalised analogous properties for ≈ {α,C} .…”

Section: 2mentioning

confidence: 97%

“…In [3], the relation ≈ α was extended to deal with associative and commutative theories. Here we will consider α-equivalence modulo commutativity, denoted ≈ {α,C} .…”

Section: 2mentioning

confidence: 99%

“…Alternatively, Definition 5 could be restricted to ground terms (by removing the first case in the construction of epc's), and then instead of computing C-matchers via C-unification, one could use an α-C-equivalence checker (for example, the one specified in [3]). This would also simplify case iv in Definition 5, since it would be sufficient to consider just one atom e not in dom(π).…”

Section: Theorem 1 (Soundness Of Solutions Of Singleton Fixpoint Probmentioning

confidence: 99%

“…We will assume as in [3] countable sets of function symbols with different equational properties such as associativity, commutativity, idempotence, etc. Function symbols have superscripts that indicate their equational properties; thus, f C k will denote the k th function symbol that is commutative and f ∅ j the j th function symbol without any equational property.…”

Section: Nominal Syntaxmentioning

confidence: 99%

“…In previous work [3], we studied α-AC-equivalence of nominal terms, and nominal C-unification [4], that is, nominal unification in languages with commutative operators. It is well-known that C-unification is an NP-complete problem (see Chapter 10 in [7]).…”

Section: Introductionmentioning

confidence: 99%

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On Solving Nominal Fixpoint Equations

Ayala-Rincón

Carvalho-Segundo

Fernández

et al. 2017

Frontiers of Combining Systems

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Abstract. In nominal syntax, variable binding is specified using atomabstraction constructors, and alpha-equivalence is formalised using freshness constraints and atom swappings, which implement variable renamings. Composition of swappings gives rise to atom permutations. Algorithms to check equivalence, match and unify nominal terms have been extended to deal with terms where some operators are associative and/or commutative. In the case of nominal C-unification, problems are transformed into finite and complete families of fixpoint equations of the form π.X ≈ ? X, where π is a permutation. To generate nominal C-unifiers, a technique to obtain a sound and complete set of solutions for these equations is needed. In this work we show how complete sets of solutions for nominal fixpoint problems are built and discuss efficient techniques to generate solutions based on algebraic properties of permutations.

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Section: 2mentioning

confidence: 97%

“…In [3], the relation ≈ α was extended to deal with associative and commutative theories. Here we will consider α-equivalence modulo commutativity, denoted ≈ {α,C} .…”

Section: 2mentioning

confidence: 99%

Section: Theorem 1 (Soundness Of Solutions Of Singleton Fixpoint Probmentioning

confidence: 99%

Section: Nominal Syntaxmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Solving Nominal Fixpoint Equations

Ayala-Rincón

Carvalho-Segundo

Fernández

et al. 2017

Frontiers of Combining Systems

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Nominal Unification of Higher Order Expressions with Recursive Let

Schmidt-Schauß¹,

Kutsia

Levy

et al. 2017

Logic-Based Program Synthesis and Transformation

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Nominal C-Unification

Ayala-Rincón

Carvalho-Segundo

Fernández

et al. 2018

Logic-Based Program Synthesis and Transformation

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Nominal unification is an extension of first-order unification that takes into account the α-equivalence relation generated by binding operators, following the nominal approach. We propose a sound and complete procedure for nominal unification with commutative operators, or nominal C-unification for short, which has been formalised in Coq. The procedure transforms nominal C-unification problems into simpler (finite families) of fixpoint problems, whose solutions can be generated by algebraic techniques on combinatorics of permutations. IntroductionUnification, where the goal is to solve equations between first-order terms, is a key notion in logic programming systems, type inference algorithms, protocol analysis tools, theorem provers, etc. Solutions to unification problems are represented by substitutions that map variables (X, Y, . . . ) to terms.When terms include binding operators, a more general notion of unification is needed: unification modulo α-equivalence. In this paper, we follow the nominal approach to the specification of binding operators [20,30,26], where the syntax of terms includes, in addition to variables, also atoms (a, b, . . . ), which can be abstracted, and α-equivalence is axiomatised by means of a freshness relation a#t and name-swappings (a b). For example, the first-order logic formula ∀a.a ≥ 0 can be written as a nominal term ∀([a]geq(a, 0)), using function symbols ∀ and geq and an abstracted atom a. Nominal unification [30] is the problem of solving equations between nominal terms modulo α-equivalence; it is a decidable problem and efficient nominal unification algorithms are available [11,9,24], that compute solutions consisting of freshness contexts (containing freshness constraints of the form a#X) and substitutions.In many applications, operators obey equational axioms. Nominal reasoning and unification have been extended to deal with equational theories presented by rewrite rules (see, e.g., [18,17,5]) or defined by equational axioms (see, e.g., [14,19]). The case of associative and commutative nominal theories was considered in [3], where a parametric {α, AC}-equivalence relation was formalised in ⋆ Work supported by the Brazilian agencies FAPDF (DE 193.001.369/2016), CAPES (Proc. 88881.132034/2016-01, 2nd author) and CNPq (PQ 307009/2013, 1st author). M. Ayala-Rincón, W. de Carvalho, M. Fernández, D. Nantes-SobrinhoCoq. However, only equational deduction was considered (not unification). In this paper, we study nominal C-unification.Contributions: We present a nominal C-unification algorithm, based on a set of simplification rules, which transforms a given nominal C-unification problem ∆, Q , where ∆ is a freshness context and Q a set of freshness constraints and equations, respectively of the form a# ? s and s ≈ ? t, into a finite set of triples of the form ∇, σ, P , consisting of a freshness context ∇, a substitution σ and a set of fixpoint equations P , of the form π.X ≈ ? X. The simplifications are based on the deduction rules for freshness and α-C-equivalence (denoted as ≈...

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A Formalisation of Nominal α -equivalence with A and AC Function Symbols

Cited by 9 publications

References 17 publications

On Solving Nominal Fixpoint Equations

On Solving Nominal Fixpoint Equations

Nominal Unification of Higher Order Expressions with Recursive Let

Nominal C-Unification

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