Combination techniques and decision problems for disunification

Previous work on combination techniques considered the question of how to combine unification algorithms for disjoint equational theories E_{1} ,...,E_{n} in order to obtain a unification algorithm for the union E1 unified ... unified En of the theories. Here we want to show that variants of this method may be used to decide solvability and ground solvability of disunification problems in E_{1}cup...cup E_{n}. Our first result says that solvability of disunification problems in the free algebra of the combined theory E_{1}cup...cup E_{n} is decidable if solvability of disunification problems with linear constant restrictions in the free algebras of the theories E_{i}(i = 1,...,n) is decidable. In order to decide ground solvability (i.e., solvability in the initial algebra) of disunification problems in E_{1}cup...cup E_{n} we have to consider a new kind of subproblem for the particular theories Ei, namely solvability (in the free algebra) of disunification problems with linear constant restriction under the additional constraint that values of variables are not Ei-equivalent to variables. The correspondence between ground solvability and this new kind of solvability holds, (1) if one theory Ei is the free theory with at least one function symbol and one constant, or (2) if the initial algebras of all theories Ei are infinite. Our results can be used to show that the existential fragment of the theory of the (ground) term algebra modulo associativity of a finite number of function symbols is decidable; the same result follows for function symbols which are associative and commutative, or associative, commutative and idempotent

Nominal disunification

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2019.Propõe-se uma extensão para problemas de disunificação de primeira-ordem adicionando suporte a operadores de ligação de acordo com a abordagem nominal. Nesta abordagem, abstração é implementada usando átomos nominais ao invés de variáveis de ligação como na representação clássica de termos e renomeamento de átomos é implementado por permutações. Em lógica nominal problemas de unificação consistem de perguntas equacionais da forma s ≈α ? t (lê-se: s é α-equivalente a t?) consideradas sobre problemas de freshness da forma a# ? t (lê-se: a é fresco em t?) que restringem soluções proibindo ocorrências livres de átomos na instanciação de variáveis. Além dessas questões equacionais e freshness, problemas de disunificação nominal incluem restrições na forma de disequações s ̸≈α ? t (lê-se: s é αdiferente de t?) com soluções dadas por pares consistindo de uma substituição σ e um conjunto de restrições de freshness na forma a#X tal que sobre estas restrições a σ-instanciação de equações, disequações, e problemas de freshness são válidas. Mostra-se, reutilizando noções de unificação nominal, como decidir se dois termos nominais podem ser feitos diferentes módulo α-equivalência. Isso é feito extendendo resultados anteriores sobre disunificação de primeira ordem e definindo a noção de soluções com exceção na linguagem nominal. Uma discussão sobre a semântica de restrições em forma de disequações também é apresentada.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).An extension of first-order disunification problems is proposed by taking into account binding operators according to the nominal approach. In this approach, bindings are implemented through nominal atoms used instead of binding variables and renaming of atoms are implemented by atom permutations. In the nominal setting, unification problems consist of equational questions of the form s ≈α ? t (read: is s α-equivalent to t?) considered under freshness problems a# ? t (read: is a fresh for t?) that restrict solutions by forbidding free occurrences of atoms in the instantiations of variables. In addition to equational and freshness problems, nominal disunification problems also include nominal disunification constraints in the form of disequations s ̸≈α ? t (read: is s α-different to t?) and their solutions consist of pairs of a substitution σ and a finite set of freshness constraints in the form of a#X such that under these restrictions the σ-instantiation of the equations, disequations, and freshness problems holds. By re-using nominal unification techniques, it is shown how to decide whether two nominal terms can be made different modulo α-equivalence. This is done by extending previous results on first-order disunification and by defining the notion of solutions with exceptions in the nominal syntax. A discussion on the semantics of disunification constraints is also given

Dismatching and Local Disunification in EL

Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic EL to disunification since negative constraints on unifiers can be used to avoid unwanted unifiers. While decidability of the solvability of general EL-disunification problems remains an open problem, we obtain NP-completeness results for two interesting special cases: dismatching problems, where one side of each negative constraint must be ground, and local solvability of disunification problems, where we restrict the attention to solutions that are built from so-called atoms occurring in the input problem. More precisely, we first show that dismatching can be reduced to local disunification, and then provide two complementary NP-algorithms for finding local solutions of (general) disunification problems

Dismatching and Local Disunification in EL

Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic EL to disunification since negative constraints on unifiers can be used to avoid unwanted unifiers. While decidability of the solvability of general EL-disunification problems remains an open problem, we obtain NP-completeness results for two interesting special cases: dismatching problems, where one side of each negative constraint must be ground, and local solvability of disunification problems, where we restrict the attention to solutions that are built from so-called atoms occurring in the input problem. More precisely, we first show that dismatching can be reduced to local disunification, and then provide two complementary NP-algorithms for finding local solutions of (general) disunification problems

Combination techniques and decision problems for disunification

Previous work on combination techniques considered the question of how to combine unification algorithms for disjoint equational theories E_{1} ,...,E_{n} in order to obtain a unification algorithm for the union E1 unified ... unified En of the theories. Here we want to show that variants of this method may be used to decide solvability and ground solvability of disunification problems in E_{1}cup...cup E_{n}. Our first result says that solvability of disunification problems in the free algebra of the combined theory E_{1}cup...cup E_{n} is decidable if solvability of disunification problems with linear constant restrictions in the free algebras of the theories E_{i}(i = 1,...,n) is decidable. In order to decide ground solvability (i.e., solvability in the initial algebra) of disunification problems in E_{1}cup...cup E_{n} we have to consider a new kind of subproblem for the particular theories Ei, namely solvability (in the free algebra) of disunification problems with linear constant restriction under the additional constraint that values of variables are not Ei-equivalent to variables. The correspondence between ground solvability and this new kind of solvability holds, (1) if one theory Ei is the free theory with at least one function symbol and one constant, or (2) if the initial algebras of all theories Ei are infinite. Our results can be used to show that the existential fragment of the theory of the (ground) term algebra modulo associativity of a finite number of function symbols is decidable; the same result follows for function symbols which are associative and commutative, or associative, commutative and idempotent

A study on unification and disunification modulo

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Ciência da Computação, 2020.Estuda-se a comparação entre unificação assimétrica e desunificação módulo teorias equa- cionais em relação às suas complexidades, como desenvolvida por Ravishankar, Narendran e Gero. A unificação assimétrica é um tipo de unificação equacional em que as soluções devem fornecer o lado direito dos problemas apresentados na forma normal. E a desunifi- cação é resolver problemas com equações e “disequações” em relação à uma teoria equaci- onal dada. As soluções para os problemas de desunificação são substituições que tornam os dois termos de cada equação iguais, mas os dois termos de cada “disequação” diferen- tes. Unificação e desunificação equacional foram comparadas por os autores mencionados com relação as suas complexidades de tempo para duas teorias equacionais: a primeira associativa (A), comutativa (C), com unidade (U) e nilpotente (N), como (ACUN) e a segunda com tais propriedades, mas adicionando um homomorfismo (h), como (ACUNh), mostrando que desunificação pode ser resolvida em tempo polinomial enquanto unificação assimétrica é NP-difícil para ambas as teorias equacionais. Além disso, foi estudada a abordagem introduzidas por Zhiqiang Liu, em sua dissertação de doutorado, para converter osunificadores módulo ACUN em assimétricos, com símbolos de função não interpretados, usando as regras de inferência. Para a teoria associativa comutativa com homomorfismo (ACh), estudou-se a prova de que unificação módulo ACh é indecidível, assim como o algoritmo de semi-decisão, recentemente introduzido por Ajay Kumar Eeralla e Christopher Lynch, que apresenta um conjunto de regras de inferência para resolver o problema com limitações.Comparisons between asymmetric unification and disunification modulo AC concerning their complexities, as developed by Ravishankar, Narendran and Gero are studied. Asym- metric unification is a type of equational unification problem in which the solutions must give as right-hand sides of the input problem, normal forms regarding some rewriting sys- tem. And disunification problems require solving equations and "disequations" for a given equational theory. Solutions to the disunification problems are substitutions that make the two terms of each equation equal, but the two terms of each “disequation” different. These authors compared the complexity of the unification and disunification problems for two equational theories. The properties of the first equational theory are associativity (A), commutativity (C), the existence of unity (U), and nilpotence (N), abbreviated as ACUN. And, the second equational theory has the same properties but adds a homomorphism (h), for short, ACUNh. For such equational theories, details of the proof that disunification can be solved in polynomial time while the asymmetric unification is NP-hard have been studied. Besides, the approach for converting ACUN unifiers to asymmetric ones, with uninterpreted function symbols using the inference rules introduced by Zhiqiang Liu, in his Ph.D. dissertation, was studied. Narendran’s proof of the undecidability of the unifi- cation problem modulo the associative commutative theory with homomorphism ACh is studied. Also, the semi-decision algorithm, recently introduced by Ajay Kumar Eeralla and Christopher Lynch, is studied, which presents a set of inference rules for solving a bounded version of ACh unification

A resolution principle for clauses with constraints

We introduce a general scheme for handling clauses whose variables are constrained by an underlying constraint theory. In general, constraints can be seen as quantifier restrictions as they filter out the values that can be assigned to the variables of a clause (or an arbitrary formulae with restricted universal or existential quantifier) in any of the models of the constraint theory. We present a resolution principle for clauses with constraints, where unification is replaced by testing constraints for satisfiability over the constraint theory. We show that this constrained resolution is sound and complete in that a set of clauses with constraints is unsatisfiable over the constraint theory if we can deduce a constrained empty clause for each model of the constraint theory, such that the empty clauses constraint is satisfiable in that model. We show also that we cannot require a better result in general, but we discuss certain tractable cases, where we need at most finitely many such empty clauses or even better only one of them as it is known in classical resolution, sorted resolution or resolution with theory unification

Proceedings of Sixth International Workshop on Unification

Swiss National Science Foundation; Austrian Federal Ministry of Science and Research; Deutsche Forschungsgemeinschaft (SFB 314); Christ Church, Oxford; Oxford University Computing Laborator