Semantics modulo satisfiability with applications: function representation, probabilities and game theory

• Ano de defesa: 2021
• Autor(a) principal: Preto, Sandro Márcio da Silva
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Coerência de restrições; Coherence of constraints; Equilíbrio de Nash; Formal methods; Funções lineares por partes; Funções racionais de McNaughton; Function representation; Jogos com incerteza; Lógica infinito-valorada de Lukasiewicz; Lógicas proposicionais; Lukasiewicz infinitely-valued logic; Métodos formais; Nash equilibrium; Neural networks; Non-classical probabilities; Piecewise linear functions; Probabilidades não clássicas; Probabilistic constraints; Probabilistic satisfiability; Propositional logics; Rational McNaughton functions; Redes neurais; Representação de funções; Restrições probabilísticas; Satisfatibilidade probabilística; Semânticas de valoração; Uncertain games; Valuation semantics
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/45/45134/tde-17062021-163257/

In the context of propositional logics, we apply semantics modulo satisfiability - a restricted semantics which comprehends only valuations that satisfy some specific set of formulas - with the aim to efficiently solve some computational tasks. Three possible such applications are developed. We begin by studying the possibility of implicitly representing rational McNaughton functions in Lukasiewicz Infinitely-valued Logic through semantics modulo satisfiability. We theoretically investigate some approaches to such representation concept, called representation modulo satisfiability, and describe a polynomial algorithm that builds representations in the newly introduced system. An implementation of the algorithm, test results and ways to randomly generate rational McNaughton functions for testing are presented. Moreover, we propose an application of such representations to the formal verification of properties of neural networks by means of the reasoning framework of Lukasiewicz Infinitely-valued Logic. Then, we move to the investigation of the satisfiability of joint probabilistic assignments to formulas of Lukasiewicz Infinitely-valued Logic, which is known to be an NP-complete problem. We provide an exact decision algorithm derived from the combination of linear algebraic methods with semantics modulo satisfiability. Also, we provide an implementation for such algorithm for which the phenomenon of phase transition is empirically detected. Lastly, we study the game theory situation of observable games, which are games that are known to reach a Nash equilibrium, however, an external observer does not know what is the exact profile of actions that occur in a specific instance; thus, such observer assigns subjective probabilities to players actions. We study the decision problem of determining if a set of these probabilistic constraints is coherent by reducing it to the problems of satisfiability of probabilistic assignments to logical formulas both in Classical Propositional Logic and Lukasiewicz Infinitely-valued Logic depending on whether only pure equilibria or also mixed equilibria are allowed. Such reductions rely upon the properties of semantics modulo satisfiability. We provide complexity and algorithmic discussion for the coherence problem and, also, for the problem of computing maximal and minimal probabilistic constraints on actions that preserves coherence.