Modal Logic S5 Satisfiability in Answer Set Programming

Modal logic S5 has attracted significant attention and has led to several practical applications, owing to its simplified approach to dealing with nesting modal operators. Efficient implementations for evaluating satisfiability of S5 formulas commonly rely on Skolemisation to convert them into propositional logic formulas, essentially by introducing copies of propositional atoms for each set of interpretations (possible worlds). This approach is simple, but often results into large formulas that are too difficult to process, and therefore more parsimonious constructions are required. In this work, we propose to use Answer Set Programming for implementing such constructions, and in particular for identifying the propositional atoms that are relevant in every world by means of a reachability relation. The proposed encodings are designed to take advantage of other properties such as entailment relations of subformulas rooted by modal operators. An empirical assessment of the proposed encodings shows that the reachability relation is very effective and leads to comparable performance to a state-of-the-art S5 solver based on SAT, while entailment relations are possibly too expensive to reason about and may result in overhead.</p

Provas de teoremas em lógica três-valorada

Monografia (graduação)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Ciência da Computação, 2017.Existem poucas ferramentas disponíveis atualmente capazes de avaliar fórmulas em Lógica Três-Valorada. Este trabalho teve como objetivo a codificação de um provador de teoremas para Lógica Três-Valorada usando a Tradução Bivalente e Resolução Clausal formulados em [1]. O provador foi submetido a uma série de testes de desempenho e comparado com outro provador construído em linguagem Prolog, o MUltseq [2]. Os desempenhos de ambos foram comparados tanto quanto ao tempo consumido quando ao uso de memória RAM. São apresentados os fundamentos da Lógica infinitamente-valorada de Łukasiewicz, as definições para a formulação da Lógica Três-Valorada, a definição de Tradução Bivalente e o método de Resolução Clausal. Em seguida, explica-se como foi feita a implementação do programa tradutor proposto, as ferramentas e programas utilizados, a saída esperada, a realização dos testes de comparação e, por fim, a verificação de um melhor desempenho e de um melhor uso de memória do provador construído em relação ao MUltseq.To the best of our knowledge, there are very few tools currently available that can evaluate formulas in Three-Valued Logic. This work provides an implementation of a prover for Three-Valued Logic using the Bivalent Translation and the Clausal Resolution formulated in [1]. Our prover has undergone a series of performance tests, in comparison with another prover built in Prolog language, the MUltseq [2]. The performance of both provers took in consideration their efficiency in terms of runtime and RAM memory usage. We present the foundations of the infinitely-valued Logic of Łukasiewicz, the definitions for the formulation of Three-Valued Logic, the definition of Bivalent Translation and the method of Clausal Resolution for this logic. We then explain how the implementation of the proposed prover was done, the tools and programs used, the expected output, the realization of comparison tests and, finally, the verification of a better performance and better memory usage of the our prover in comparison to MUltseq

Modal Resolution: Proofs, Layers and Refinements

Resolution-based provers for multimodal normal logics require pruning of the search space for a proof in order to ameliorate the inherent intractability of the satisfiability problem for such logics. We present a clausal modal-layered hyper-resolution calculus for the basic multimodal logic, which divides the clause set according to the modal level at which clauses occur in order to reduce the number of possible inferences. We show that the calculus is complete for the logics being considered. We also show that the calculus can be combined with other strategies. In particular, we discuss the completeness of combining modal layering with negative and ordered resolution and provide experimental results comparing the different refinements

Métodos polinomiais para simplificação de fórmulas modais

Trabalho de Conclusão de Curso (graduação)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Ciência da Computação, 2016.A simplificação de fórmulas lógicas busca melhorar a eficiência do tratamento destas. Este projeto desenvolve um algoritmo baseado na eliminação de literal puro e propagação de constante para realizar simplificação. O trabalho é realizado em cima do provador KSP que já implementa os métodos de simplificação citados, porém acreditamos que esse novo algoritmo desenvolvido trará maior desempenho na execução. Como melhorias foram propostas uma lista de localização para as proposições das fórmulas e a realização de eliminação de literal puro enquanto isso for possível. A implementação é explicada detalhadamente neste trabalho. As comparações realizadas entre as versões dos provadores mostra que a nova versão é melhor para algumas fórmulas e o KSP para outras. Apesar disso, a nova implementação pode produzir fórmulas mais simplificadas e sua manipulação das estruturas de dados é mais eficiente, porém os tempos de execução não foram afetados de maneira substancial.Simplification of logical formulae is a preprocessing procedure which seeks to reduce the size of the input formula in order to improve the overall efficiency of theorem provers. This project implements two procedures for simplification: one for pure literal elimination and other for constant propagation. The work is based on an existing prover, KSP, which already implements the mentioned simplification methods. However, our hypothesis was that our implementation would improve performance by changing the underlying data structures. One of the implementation improvements proposed was a localization list for the propositions of the formulae. Also, pure literal elimination is executed until a fixed-point is reached. We explain the implementation in detail. Comparison between the provers versions show that there is no time difference pattern, but the new implementation usually produces results in less time. The new implementation produces better formulae in some cases and makes good use of data structures, but execution times have not been substantially affected

Tableaux clausal para lógica modal

Trabalho de Conclusão de Curso (graduação)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Ciência da Computação, 2016.Este trabalho apresenta a definição de um cálculo tableaux clausal proposto para lógicas modais, além de apresentar uma implementação do mesmo. O objetivo geral é aplicar o método refutacional tableaux utilizando a forma normal construída para resolução clausal na tentativa de gerar o máximo de informações possíveis sobre conjuntos de fórmulas na linguagem modal, sem que seja necessário a especificação de um número elevado de regras de inferência no cálculo, o que tornaria a tarefa de implementar razoavelmente mais difícil. Estas informações extraídas contêm singularidades dos conjuntos de fórmulas que podem auxiliar em decisões relacionadas aos métodos de provas aplicados aos mesmos. As provas de correção para o cálculo são apresentadas. A implementação realizada, apesar de bem sucedida, revelou imperfeições com relação ao desempenho de execução. Esforços direcionados à melhoria de tais imperfeições, bem como uma análise das informações extraídas, são deixados como trabalhos futuros.This work presents a clausal tableaux calculus for modal logics and na implementation for it. The main goal is to combine aspects of two different proof methods, the inference rules of a tableaux calculus and the normal form employed in a clausal resolution method. The goal is to generate as much information as possible about sets of formulae in modal logic, without the need to define a large number of inference rules in the calculus, which would make the job of implementing the calculus reasonably harder. The pieces of extracted information may help to characterise singularities of the set of formulae, which could then help to make decisions about what proof method to apply to such a set. The implementation is correct, but does not perform well. Efforts in the direction of a more robust implementation and the analysis of the extracted information are left as future work

19th Brazilian Logic Conference: Book of Abstracts

This is the book of abstracts of the 19th Brazilian Logic Conferences. The Brazilian Logic Conferences (EBL) is one of the most traditional logic conferences in South America. Organized by the Brazilian Logic Society (SBL), its main goal is to promote the dissemination of research in logic in a broad sense. It has been occurring since 1979, congregating logicians of different fields — mostly philosophy, mathematics and computer science — and with different backgrounds — from undergraduate students to senior researchers. The meeting is an important moment for the Brazilian and South American logical community to join together and discuss recent developments of the field. The areas of logic covered in the conference spread over foundations and philosophy of science, analytic philosophy, philosophy and history of logic, mathematics, computer science, informatics, linguistics and artificial intelligence. Previous editions of the EBL have been a great success, attracting researchers from all over Latin America and elsewhere. The 19th edition of EBL takes place from May 6-10, 2019, in the beautiful city of João Pessoa, at the northeast coast of Brazil. It is conjointly organized by Federal University of Paraíba (UFPB), whose main campus is located in João Pessoa, Federal University of Campina Grande (UFCG), whose main campus is located in the nearby city of Campina Grande (the second-largest city in Paraíba state) and SBL. It is sponsored by UFPB, UFCG, the Brazilian Council for Scientific and Technological Development (CNPq) and the State Ministry of Education, Science and Technology of Paraíba. It takes place at Hotel Luxxor Nord Tambaú, privileged located right in front Tambaú beach, one of João Pessoa’s most famous beaches

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions