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

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

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

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

Saturation-based decision procedures for fixed domain and minimal model validity

Superposition is an established decision procedure for a variety of first-order logic theories represented by sets of clauses. A satisfiable theory, saturated by superposition, implicitly defines a minimal Herbrand model for the theory. This raises the question in how far superposition calculi can be employed for reasoning about such minimal models. This is indeed often possible when existential properties are considered. However, proving universal properties directly leads to a modification of the minimal model's termgenerated domain, as new Skolem functions are introduced. For many applications, this is not desired because it changes the problem. In this thesis, I propose the first superposition calculus that can explicitly represent existentially quantified variables and can thus compute with respect to a given fixed domain. It does not eliminate existential variables by Skolemization, but handles them using additional constraints with which each clause is annotated. This calculus is sound and refutationally complete in the limit for a fixed domain semantics. For saturated Horn theories and classes of positive formulas, the calculus is even complete for proving properties of the minimal model itself, going beyond the scope of known superpositionbased approaches. The calculus is applicable to every set of clauses with equality and does not rely on any syntactic restrictions of the input. Extensions of the calculus lead to various new decision procedures for minimal model validity. A main feature of these decision procedures is that even the validity of queries containing one quantifier alternation can be decided. In particular, I prove that the validity of any formula with at most one quantifier alternation is decidable in models represented by a finite set of atoms and that the validity of several classes of such formulas is decidable in models represented by so-called disjunctions of implicit generalizations. Moreover, I show that the decision of minimal model validity can be reduced to the superposition-based decision of first-order validity for models of a class of predicative Horn clauses where all function symbols are at most unary.Superposition ist eine bewährte Entscheidungsprozedur für eine Vielzahl von Theorien in Prädikatenlogik erster Stufe, die durch Klauseln repräsentiert sind. Eine erfüllbare und bezüglich Superposition saturierte Theorie definiert ein minimales Herbrand-Modell dieser Theorie. Dies wirft die Frage auf, inwiefern Superpositionskalküle zur Argumentation in solchen minimalen Modellen verwendet werden können. Das ist bei der Betrachtung existenziell quantifizierter Eigenschaften tatsächlich oft möglich. Die Analyseuniversell quantifizierter Eigenschaften führt jedoch unmittelbar zu einer Modifizierung der termgenerierten Domäne des minimalen Modells, da neue Skolemfunktionen eingeführt werden. Für viele Anwendungen ist dies unerwünscht, da es die Problemstellung verändert. In dieser Arbeit stelle ich den ersten Superpositionskalkül vor, der existenziell quantifizierte Variablen explizit darstellen und daher Berechnungen über einer gegebenen festen Domäne anstellen kann. In ihm werden existenziell quantifizierte Variablen nicht durch Skolemisierung eliminiert sondern mithilfe zusätzlicher Constraints gehandhabt, mit denen jede Klausel versehen wird. Dieser Kalkül ist korrekt und im Grenzwert widerspruchsvollständig für eine domänenspezifische Semantik. Für saturierte Horntheorien und Klassen positiver Formeln ist der Kalkül sogar korrekt für den Beweis von Eigenschaften des minimalen Modells selbst. Dies übersteigt die Möglichkeiten bisheriger superpositionsbasierter Ansätze. Der Kalkül ist auf beliebige Klauselmengen mit Gleichheit anwendbar und erlegt der Eingabe keine syntaktischen Beschränkungen auf. Erweiterungen des Kalküls führen zu verschiedenen neuen Entscheidungsverfahren für die Gültigkeit in minimalen Modellen. Ein Hauptmerkmal dieser Verfahren ist es, dass selbst die Gültigkeit von Anfragen entscheidbar ist, die einen Quantorenwechsel enthalten. Insbesondere beweise ich, dass die Gültigkeit jeder Formel mit höchstens einem Quantorenwechsel in durch endlich viele Atome repräsentierten Modellen entscheidbar ist, und gleiches gilt für die Gültigkeit mehrerer Klassen solcher Formeln in durch so genannte Disjunktionen impliziter Verallgemeinerungen repräsentieren Modellen. Außerdem zeige ich, dass für eine Klasse prädikativer Hornklauseln, bei denen alle vorkommenden Funktionssymbole maximal einstellig sind, die Entscheidbarkeit der Gültigkeit in minimalen Modellen auf superpositionsbasierte Entscheidbarkeit in Prädikatenlogik erster Stufe reduziert werden kann

Terminological reasoning with constraint handling rules

Constraint handling rules (CHRs) are a flexible means to implement \u27user-defined\u27 constraints on top of existing host languages (like Prolog and Lisp). Recently, M. Schmidt-Schauß and G. Smolka proposed a new methodology for constructing sound and complete inference algorithms for terminological knowledge representation formalisms in the tradition of KLONE. We propose CHRs as a flexible implementation language for the consistency test of assertions, which is the basis for all terminological reasoning services. The implementation results in a natural combination of three layers: (i) a constraint layer that reasons in well- understood domains such as rationals or finite domains, (ii) a terminological layer providing a tailored, validated vocabulary on which (iii) the application layer can rely. The flexibility of the approach will be illustrated by extending the formalism, its implementation and an application example (solving configuration problems) with attributes, a new quantifier and concrete domains