Temporal Answer Set Programming

[Abstract] Commonsense temporal reasoning is full of situations that require drawing default conclusions, since we rarely have all the information available. Unfortunately, most modal temporal logics cannot accommodate default reasoning, since they typically deal with a monotonic inference relation. On the other hand, non-monotonic approaches are very expensive and their treatment of time is not so well delimited and studied as in modal logic. Temporal Equilibrium Logic (TEL) is the first non-monotonic temporal logic which fully covers the syntax of some standard modal temporal approach without requiring further constructions. TEL shares the syntax of Linear-time Temporal Logic (LTL) (first proposed by Arthur Prior and later extended by Hans Kamp) which has become one of the simplest, most used and best known temporal logics in Theoretical Computer Science. Although TEL had been already defined, few results were known about its fundamental properties and nothing at all on potential computational methods that could be applied for practical purposes. This situation unfavourably contrasted with the huge body of knowledge available for LTL, both in well-known formal properties and in computing methods with practical implementations. In this thesis we have mostly filled this gap, following a research program that has systematically analysed different essential properties of TEL and, simultaneously, built computational tools for its practical application. As an overall, this thesis collects a corpus of results that constitutes a significant breakthrough in the knowledge about TEL.[Resumen] El razonamiento temporal del sentido común está lleno de situaciones que requieren suponer conclusiones por defecto, puesto que raramente contamos con toda la información disponible. Lamentablemente, la mayoría de lógicas modales temporales no permiten modelar este tipo de razonamiento por defecto debido a que, típicamente, se definen por medio de relaciones de inferencia monótonas. Por el contrario, las aproximaciones no monótonas existentes son típicamente muy costosas pero su manejo del tiempo no está tan bien delimitado como en lógica modal. Temporal Equilibrium Logic (TEL) es la primera lógica temporal no monótona que cubre totalmente la sintaxis de alguna de las lógicas modales tradicionales sin requerir el uso de más construcciones. TEL comparte la sintaxis de Linear-time Temporal Logic (LTL) (formalismo propuesto por Arthur Prior y posteriormente extendido por Hans Kamp), que es una de las lógicas más simples, utilizadas y mejor conocidas en Teoría de la Computación. Aunque TEL había sido definido, muy pocas propiedades eran conocidas, lo que contrastaba con el vasto conocimiento de LTL que está presente en el estado del arte. En esta tesis hemos estudiado diferentes aspectos de TEL, una novedosa combinación de lógica modal temporal y un formalismo no monótono. A grandes rasgos, esta tesis recoge un conjunto de resultados, tanto desde el punto de vista teórico como práctico, que constituye un gran avance en lo relativo al conocimiento sobre TEL.[Resumo] O razoamento do sentido común aplicado ao caso temporal está cheo de situacións que requiren supoñer conclusións por defecto, posto que raramente contamos con toda a información dispoñible. Lamentablemente a maioría de lóxicas modais temporáis non permiten modelar este tipo de razoamento por defecto debido a que, típicamente, están definidas por medio de relacións de inferencia monótonas. Pola contra, as aproximacións non monótonas existentes son moi costosos e o seu tratamento do tempo non está ben tan delimitado nin estudiado como nas lóxicas modais. Temporal Equilibrium Logic (TEL) é a primeira aproximación non monótona que cubre totalmente a sintaxe dalgunha das lóxicas modais traidicionáis sen requerir o uso de máis construccións. TEL comparte a sintaxe de Lineartime Temporal Logic (LTL) (formalismo proposto por Arthur Prior e extendido posteriormente por Hans Kamp), que é considerada unha das lóxicas modais máis simples, utilizadas e coñecidas dentro da Teoría da Computación. Aínda que TEL xa fora definido previamente, moi poucas das súas propiedades eran coñecidas, dato que contrasta co vasto coñecemento de LTL existente no estado da arte. Nesta tese, estudiamos diferentes aspectos de TEL, unha novidosa combinación de lóxica modal temporal e un formalimo non monótono. A grandes rasgos, esta tese recolle un conxunto de resultados, tanto dende o punto de vista teórico como práctico, que constitúe un gran avance no relativo ó coñecemento sobre o formalismo TEL

Probability Logic for Harsanyi Type Spaces

Probability logic has contributed to significant developments in belief types for game-theoretical economics. We present a new probability logic for Harsanyi Type spaces, show its completeness, and prove both a de-nesting property and a unique extension theorem. We then prove that multi-agent interactive epistemology has greater complexity than its single-agent counterpart by showing that if the probability indices of the belief language are restricted to a finite set of rationals and there are finitely many propositional letters, then the canonical space for probabilistic beliefs with one agent is finite while the canonical one with at least two agents has the cardinality of the continuum. Finally, we generalize the three notions of definability in multimodal logics to logics of probabilistic belief and knowledge, namely implicit definability, reducibility, and explicit definability. We find that S5-knowledge can be implicitly defined by probabilistic belief but not reduced to it and hence is not explicitly definable by probabilistic belief

Paving the Way for Temporal Grounding

In this paper we consider the problem of introducing variables in temporal logic programs under the formalism of Temporal Equilibrium Logic (TEL), an extension of Answer Set Programming (ASP) for dealing with linear-time modal operators. We provide several fundamental contributions that pave the way for the implementation of a grounding process, that is, a method that allows replacing variables by ground instances in all the possible (or better, relevant) ways

Fragments and frame classes:Towards a uniform proof theory for modal fixed point logics

This thesis studies the proof theory of modal fixed point logics. In particular, we construct proof systems for various fragments of the modal mu-calculus, interpreted over various classes of frames. With an emphasis on uniform constructions and general results, we aim to bring the relatively underdeveloped proof theory of modal fixed point logics closer to the well-established proof theory of basic modal logic. We employ two main approaches. First, we seek to generalise existing methods for basic modal logic to accommodate fragments of the modal mu-calculus. We use this approach for obtaining Hilbert-style proof systems. Secondly, we adapt existing proof systems for the modal mu-calculus to various classes of frames. This approach yields proof systems which are non-well-founded, or cyclic.The thesis starts with an introduction and some mathematical preliminaries. In Chapter 3 we give hypersequent calculi for modal logic with the master modality, building on work by Ori Lahav. This is followed by an Intermezzo, where we present an abstract framework for cyclic proofs, in which we give sufficient conditions for establishing the bounded proof property. In Chapter 4 we generalise existing work on Hilbert-style proof systems for PDL to the level of the continuous modal mu-calculus. Chapter 5 contains a novel cyclic proof system for the alternation-free two-way modal mu-calculus. Finally, in Chapter 6, we present a cyclic proof system for Guarded Kleene Algebra with Tests and take a first step towards using it to establish the completeness of an algebraic counterpart

Forgetting Auxiliary Atoms in Forks

©2019 Elsevier B.V. All rights reserved. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/bync-nd/4.0/. This version of the article has been accepted for publication in Artificial Intelligence. The Version of Record is available online at https://doi.org/10.1016/j.artint.2019.07.005Versión final aceptada de: F. Aguado, P. Cabalar, J. Fandinno, D. Pearce, G. Pérez, and, C. Vidal, "Forgetting Auxiliary Atoms in Forks", Artificial Intelligence, Vol. 275, pp. 575-601, Oct. 2019, doi: 10.1016/j.artint.2019.07.005[Abstract]: In this work we tackle the problem of checking strong equivalence of logic programs that may contain local auxiliary atoms, to be removed from their stable models and to be forbidden in any external context. We call this property projective strong equivalence (PSE). It has been recently proved that not any logic program containing auxiliary atoms can be reformulated, under PSE, as another logic program or formula without them – this is known as strongly persistent forgetting. In this paper, we introduce a conservative extension of Equilibrium Logic and its monotonic basis, the logic of Here-and-There, in which we deal with a new connective ‘|’ we call fork. We provide a semantic characterisation of PSE for forks and use it to show that, in this extension, it is always possible to forget auxiliary atoms under strong persistence. We further define when the obtained fork is representable as a regular formula.We are grateful to the anonymous reviewers of the Artificial Intelligence Journal, and previously, to the reviewers of the workshop ASPOCP'17, for their comments and suggestions that have helped improve the paper substantially. This work has been partially supported by MINECO (grant TIN2017-84453-P) and Xunta de Galicia (grants GPC ED431B 2019/03 and 2016-2019 ED431G/01 for CITIC center), Spain; by the Salvador de Madariaga programme, Spain; by the European Regional Development Fund (ERDF); and by the Centre International de Mathématiques et d'Informatique de Toulouse (CIMI) through contract ANR-11-LABEX-0040-CIMI within the programme ANR-11-IDEX-0002-02.Xunta de Galicia; ED431B 2019/03Xunta de Galicia; 2016-2019 ED431G/0