A Certified Functional Nominal C-Unification Algorithm

Ayala-Rincón, Maurício; Fernández, Maribel; Silva, Gabriel Ferreira; Nantes-Sobrinho, Daniele

doi:10.1007/978-3-030-45260-5_8

Cited by 5 publications

(8 citation statements)

References 15 publications

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“…Content may change prior to final publication. work to construct a counterpart of various extensions of nominal algorithms [28], [29] in our graph-based approach.…”

Section: Discussionmentioning

confidence: 99%

Revisiting Graph Types in HyperLMNtal: A Modeling Language for Hypergraph Rewriting

Yasen

Ueda

2021

IEEE Access

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Hypergraphs are a highly expressive data structure for modeling and programming, for which high-level language constructs are yet to be established. HyperLMNtal is a modeling language based on hypergraph rewriting. Rewrite rules can copy and remove subgraphs identified by graph types which serve as a wildcard construct that enables powerful handling of subgraphs. HyperLMNtal has featured several graph types over the years, enabling it to encode various computational models. Important applications of graph types include the modeling of formal systems involving name binding including the λ-calculus. However, the concept of graph types for hypergraphs has not been studied in sufficient detail, and our recent work revealed that the encoding of a unification algorithm modulo α-equivalence requires further evolution of graph types. This paper describes the motivation and redesign of a graph type for handling subgraphs appearing in the above applications and conduct experiments to show performance improvements. We believe that the idea of graph types could be useful in programming and modeling languages in general and is worth further investigation and deployment.

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“…Content may change prior to final publication. work to construct a counterpart of various extensions of nominal algorithms [28], [29] in our graph-based approach.…”

Section: Discussionmentioning

confidence: 99%

Revisiting Graph Types in HyperLMNtal: A Modeling Language for Hypergraph Rewriting

Yasen

Ueda

2021

IEEE Access

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“…The set FormpΣq of equational Σ-formulas is then inductively built from atoms by: conjunction (^), disjunction (_), negation ( ), and universal (@x 1 : s 1 , ... , x n : s n ) and existential (Dx 1 : s 1 , ... , x n : s n ) quantification with distinct sorted variables x 1 : s 1 ,..., x n : s n , with s 1 ,..., s n P S (by convention, for H the empty set of variables and ϕ a formula, we define p@Hq ϕ " pDHq ϕ " ϕ). A literal pt " t 1 q is denoted t t 1 . Given a Σ-algebra A, a formula ϕ P FormpΣq, and an assignment α P rYÑAs, where Y Ě fvarspϕq, with fvarspϕq the free variables of ϕ, the satisfaction relation A,α |ù ϕ is defined inductively as usual: for atoms, A,α |ù t " t 1 iff tα " t 1 α; for Boolean connectives it is the corresponding Boolean combination of the satisfaction relations for subformulas; and for quantifiers: A,α |ù p@x 1 : s 1 ,...,x n : s n q ϕ (resp.…”

Section: Background On Order-sorted First-order Logicmentioning

confidence: 99%

“…In inductive theorem proving one should try to use induction hypotheses as much as possible to simplify conjectures. But this raises two obvious questions: (1) In what sense should a conjecture be made simpler? (2) How can an induction hypothesis be used for this purpose?…”

Section: Ordered Rewritingmentioning

confidence: 99%

“…where x,y have sort Elt, V has sort NeMSet, and the constant U has also sort NeMSet. So in this example our Γ is xYU " yYV, X Γ " tUu, X Γ " tUu, which we assume is fresh, i.e., it appears nowhere else, and Γ ˝" x Y U " yY V. One of the variant constructor unifiers of Γ ˝is the unifier α 1 ,Wu, and Y α " ty 1 ,Wu. Therefore, α is the composed substitution:…”

Section: Constructor Variant Unification Left (Cvul)mentioning

confidence: 99%

“…A key challenge is the large body of symbolic algorithms involved that need to be certified. For example, only very recently has certification for unsorted associativecommutative unification become possible after a very large effort formalizing and verifying Stickel's algorithm in the PVS prover [1]. Several inference rules use either ordersorted B-unification for any combination of associative and/or commutative and/or unit axioms, or variant E Y B-unification, for which no machine-assisted formalizations such as that in [1] currently exist to the best of my knowledge.…”

Section: Related Work and Conclusionmentioning

confidence: 99%

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Inductive Reasoning with Equality Predicates, Contextual Rewriting and Variant-Based Simplification

Meseguer

Skeirik

2020

Lecture Notes in Computer Science

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An inductive inference system for proving validity of formulas in the initial algebra T E of an order-sorted equational theory E is presented. It has 20 inference rules, but only 9 of them require user interaction; the remaining 11 can be automated as simplification rules. In this way, a substantial fraction of the proof effort can be automated. The inference rules are based on advanced equational reasoning techniques, including: equationally defined equality predicates, narrowing, constructor variant unification, variant satisfiability, order-sorted congruence closure, contextual rewriting, ordered rewriting, and recursive path orderings. All these techniques work modulo axioms B, for B any combination of associativity and/or commutativity and/or identity axioms. Most of these inference rules have already been implemented in Maude's NuITP inductive theorem prover.

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