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

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

Non-disjoint Combined Unification and Closure by Equational Paramodulation (Extended Version)

Short version published in the Proceedings of FroCoS 2021Closure properties such as forward closure and closure via paramodulation have proven to be very useful in equational logic, especially for the formal analysis of security protocols. In this paper, we consider the non-disjoint unification problem in conjunction with these closure properties. Given a base theory E, we consider classes of theory extensions of E admitting a unification algorithm built in a hierarchical way. In this context, a hierarchical unification procedure is obtained by extending an E-unification algorithm with some additional inference rules to take into account the rest of the theory. We look at hierarchical unification procedures by investigating an appropriate notion of E-constructed theory, defined in terms of E-paramodulation. We show that any E-constructed theory with a finite closure by E-paramodulation admits a terminating hierarchical unification procedure. We present modularity results for the unification problem modulo the union of E-constructed theories sharing only symbols in E. Finally, we also give sufficient conditions for obtaining terminating (combined) hierarchical unification procedures in the case of regular and collapse-free E-constructed theories

Non-disjoint Combined Unification and Closure by Equational Paramodulation

Extended version available at https://hal.inria.fr/hal-03329075International audienceClosure properties such as forward closure and closure via paramodulation have proven to be very useful in equational logic, especially for the formal analysis of security protocols. In this paper, we consider the non-disjoint unification problem in conjunction with these closure properties. Given a base theory E, we consider classes of theory extensions of E admitting a unification algorithm built in a hierarchical way. In this context, a hierarchical unification procedure is obtained by extending an E-unification algorithm with some additional inference rules to take into account the rest of the theory. We look at hierarchical unification procedures by investigating an appropriate notion of E-constructed theory, defined in terms of E-paramodulation. We show that any E-constructed theory with a finite closure by E-paramodulation admits a terminating hierarchical unification procedure. We present modularity results for the unification problem modulo the union of E-constructed theories sharing only symbols in E. Finally, we also give sufficient conditions for obtaining terminating (combined) hierarchical unification procedures in the case of regular and collapse-free E-constructed theories

Termination of rewriting strategies: a generic approach

We propose a generic termination proof method for rewriting under strategies, based on an explicit induction on the termination property. Rewriting trees on ground terms are modeled by proof trees, generated by alternatively applying narrowing and abstracting steps. The induction principle is applied through the abstraction mechanism, where terms are replaced by variables representing any of their normal forms. The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof. Abstraction constraints can be used to control the narrowing mechanism, well known to easily diverge. The generic method is then instantiated for the innermost, outermost and local strategies.Comment: 49 page

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

Variant-Based Satisfiability

Although different satisfiability decision procedures can be combined by algorithms such as those of Nelson-Oppen or Shostak, current tools typically can only support a finite number of theories to use in such combinations. To make SMT solving more widely applicable, generic satisfiability algorithms that can allow a potentially infinite number of decidable theories to be user-definable, instead of needing to be built in by the implementers, are highly desirable. This work studies how folding variant narrowing, a generic unification algorithm that offers good extensibility in unification theory, can be extended to a generic variant-based satisfiability algorithm for the initial algebras of its user-specified input theories when such theories satisfy Comon-Delaune's finite variant property (FVP) and some extra conditions. Several, increasingly larger infinite classes of theories whose initial algebras enjoy decidable variant-based satisfiability are identified, and a method based on descent maps to bring other theories into these classes and to improve the generic algorithm's efficiency is proposed and illustrated with examples.Partially supported by NSF Grant CNS 13-19109.Ope

Pseudo-contractions as Gentle Repairs

Updating a knowledge base to remove an unwanted consequence is a challenging task. Some of the original sentences must be either deleted or weakened in such a way that the sentence to be removed is no longer entailed by the resulting set. On the other hand, it is desirable that the existing knowledge be preserved as much as possible, minimising the loss of information. Several approaches to this problem can be found in the literature. In particular, when the knowledge is represented by an ontology, two different families of frameworks have been developed in the literature in the past decades with numerous ideas in common but with little interaction between the communities: applications of AGM-like Belief Change and justification-based Ontology Repair. In this paper, we investigate the relationship between pseudo-contraction operations and gentle repairs. Both aim to avoid the complete deletion of sentences when replacing them with weaker versions is enough to prevent the entailment of the unwanted formula. We show the correspondence between concepts on both sides and investigate under which conditions they are equivalent. Furthermore, we propose a unified notation for the two approaches, which might contribute to the integration of the two areas