Positive and negative results for higher-order disunification

This paper is devoted to higher-order disunification which is the process of solving quantified formulae built on simply-typed lambda-terms, the equality induced by the $\eta$ and the $\beta$ reductions, boolean connectives and the negation. This problem is motivated by tests of completeness of definitions in algebraic higher-order specification languages which combine the advantages of algebraic specification languages and higher-order programming languages. We show that higher-order disunification is not semi-decidable and we prove the undecidability of second-order complement problems which are the formulae expressing the completeness of some scheme, by encoding Minsky machines. On the other hand, we show that second-order complement problems are decidable if second-order variables and bound variables satisfy some (reasonable) conditions and that the validity of any quantified equational formula can be checked when all the terms occurring in this formula are patterns, i.e. s.t. the arguments of free variables are distinct bound variables. Both cases are decided using quantifier elimination techniques

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

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

Understanding Addiction

The addiction literature is fraught with conceptual confusions, stalled debates, and an unfortunate lack of clear and careful attempts to delineate the phenomenon of addiction in a way that might lead to consensus. My dissertation has two overarching aims, one metaphysical and one practical. The first aim is to defend an account of addiction as the systematic disposition to fail to control one’s desires to engage in certain types of behaviors. I defend the inclusion of desires and impaired control in the definition, and I flesh out the notion of systematicity central to my dispositionalist framework. I engage the so-called ‘disease vs. choice’ debate, criticizing its presupposition that we are dealing here with a dichotomy and arguing that the movement towards a middle ground is the right track to take. I explain how the dispositionalist account can capture this middle ground and how it serves to expand upon existing views, in particular by filling in the metaphysical details. The second aim is to show how the account I defend can help to unify the extant views and disciplinary perspectives in the literature. Both the dispositionalist aspect of my framework and the methodology adopted (applied ontology and systematic metaphysics) can move the literature towards both substantive and methodological unification. This will help to clear up conceptual confusions, resolve (or sometimes dissolve) apparently intractable disputes, situate different research perspectives with respect to each other, facilitate interdisciplinary dialogue, and help to frame important questions about addiction. Finally, I offer the beginnings of an ontology of addiction, which will provide a terminologically well-structured guide to the addiction literature in a way that will facilitate more effective and efficient communication and data management across disciplines

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

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

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

Vigilance and control

We sometimes fail unwittingly to do things that we ought to do. And we are, from time to time, culpable for these unwitting omissions. We provide an outline of a theory of responsibility for unwitting omissions. We emphasize two distinctive ideas: (i) many unwitting omissions can be understood as failures of appropriate vigilance, and; (ii) the sort of self-control implicated in these failures of appropriate vigilance is valuable. We argue that the norms that govern vigilance and the value of self-control explain culpability for unwitting omissions