Asymmetric Unification and Disunification

Ravishankar, Veena; Cornell, Kimberly A.; Narendran, Paliath

doi:10.1007/978-3-030-22102-7_23

Cited by 1 publication

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“…Comon and Lescanne [10] introduced first-order equational problems and studied their solutions in the algebra of rational trees, the initial term algebra, and the ground term algebra. A restricted version of equational problems, called disunification problems, which do not contain quantified variables, has been extensively studied in the first-order framework [8,3,11,2,22]. More recently, a nominal approach to disunification problems was proposed by Ayala et.al [1], including only conjunctions of equations and disequations and freshness constraints, without quantified variables.…”

Section: Introductionmentioning

confidence: 99%

Nominal Equational Problems

Ayala-Rincón

Fernández

Nantes-Sobrinho

et al. 2021

Lecture Notes in Computer Science

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We define nominal equational problems of the form $$\exists \overline{W} \forall \overline{Y} : P$$ ∃ W ¯ ∀ Y ¯ : P , where $$P$$ P consists of conjunctions and disjunctions of equations $$s\approx _\alpha t$$ s ≈ α t , freshness constraints $$a\#t$$ a # t and their negations: $$s \not \approx _\alpha t$$ s ≉ α t and "Equation missing", where $$a$$ a is an atom and $$s, t$$ s , t nominal terms. We give a general definition of solution and a set of simplification rules to compute solutions in the nominal ground term algebra. For the latter, we define notions of solved form from which solutions can be easily extracted and show that the simplification rules are sound, preserving, and complete. With a particular strategy for rule application, the simplification process terminates and thus specifies an algorithm to solve nominal equational problems. These results generalise previous results obtained by Comon and Lescanne for first-order languages to languages with binding operators. In particular, we show that the problem of deciding the validity of a first-order equational formula in a language with binding operators (i.e., validity modulo $$\alpha $$ α -equality) is decidable.

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