On Solving Nominal Fixpoint Equations

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

doi:10.1007/978-3-319-66167-4_12

Cited by 10 publications

(14 citation statements)

References 15 publications

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“…General solutions for fixpoint problems. To compute the set of solutions for a fixpoint equation, we use a method described in [4], which is based on the computation of unitary extended pseudo-cycles (epc). We refer to [4] for the definition of extended pseudo-cycles and an algorithm to enumerate all the solutions of a successful leaf in the derivation tree.…”

Section: Generation Of Solutions For Successful Leaves Of T ∇Pmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The formalisation and a full version of the present paper with all details of the proofs are available at http://ayala.mat.unb.br/publications.html. Related work: To generate the set of combinatorial solutions for fixpoint equations we can use an enumeration procedure given in [4], which is based on the combinatorics of permutations. By combining the simplification and enumeration methods, we obtain a nominal C-unification procedure in two phases: a simplification phase, described in this paper, which outputs a finite set of most general solutions that may include fixpoint constraints, and a generation phase, which eliminates the fixpoint constraints according to [4].Several extensions of the nominal unification algorithm have been defined, in addition to the equational extensions already mentioned.An algorithm for nominal unification of higher-order expressions with recursive let was proposed in [23]; as for nominal C-unification, fixpoint equations are obtained in the process.…”

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confidence: 99%

“…Related work: To generate the set of combinatorial solutions for fixpoint equations we can use an enumeration procedure given in [4], which is based on the combinatorics of permutations. By combining the simplification and enumeration methods, we obtain a nominal C-unification procedure in two phases: a simplification phase, described in this paper, which outputs a finite set of most general solutions that may include fixpoint constraints, and a generation phase, which eliminates the fixpoint constraints according to [4].Several extensions of the nominal unification algorithm have been defined, in addition to the equational extensions already mentioned.An algorithm for nominal unification of higher-order expressions with recursive let was proposed in [23]; as for nominal C-unification, fixpoint equations are obtained in the process. Using the techniques in [4], it is possible to proceed further and generate the combinatorial solutions of fixpoint equations.Recently, Aoto and Kikuchi [1] proposed a rule-based procedure for nominal equivariant unication [13], an extension of nominal unification that is useful in confluence analysis of nominal rewriting systems [2,16].Furthermore, several formalisations and implementations of the nominal unification algorithm are available.…”

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confidence: 99%

“…Organisation: Section 2 presents basic concepts and notations. Section 3 introduces the formalised equational and freshness inference rules for nominal C-unification, and also discusses NP-completeness; Section 4 shows how single fixpoint equations are solved giving rise to infinite independent solutions and also we briefly explain how fixpoint solutions are combined in [4] in order to produce solutions. Section 5 concludes and describes future work.…”

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confidence: 99%

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Nominal C-Unification

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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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Section: Generation Of Solutions For Successful Leaves Of T ∇Pmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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A Certified Functional Nominal C-Unification Algorithm

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Interest in anti-unification, the dual problem of unification, is on the rise due to applications within the field of software analysis and related areas. For example, anti-unification-based techniques have found uses within clone detection and automatic program repair methods. While syntactic forms of anti-unification are enough for many applications, some aspects of software analysis methods are more appropriately modeled by reasoning modulo an equational theory. Thus, extending existing anti-unification methods to deal with important equational theories is the natural step forward. This paper considers anti-unification modulo pure absorption theories, i.e., some operators are associated with a special constant satisfying the axiom f (x, ε f ) ≈ f (ε f , x) ≈ ε f . We provide a sound and complete rule-based algorithm for such theories. Furthermore, we show that anti-unification modulo absorption is infinitary. Despite this, our algorithm terminates and produces a finitary algorithmic representation of the minimal complete set of solutions. We also show that the linear variant is finitary.

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On Solving Nominal Fixpoint Equations

Cited by 10 publications

References 15 publications

Nominal C-Unification

Nominal C-Unification

A Certified Functional Nominal C-Unification Algorithm

Nominal AC-Matching

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