Defining the meaning of TPTP formatted proofs

International audienceThe TPTP library is one of the leading problem libraries in the automated theorem proving community. Over time, support was added for problems beyond those in first-order clausal form. TPTP has also been augmented with support for various proof formats output by theorem provers. Such proofs can also be maintained in the TSTP proof library. In this paper we propose an extension of this framework to support the semantic specification of the inference rules used in proofs

A Proof-Theoretic Approach to Certifying Skolemization

International audienceWhen presented with a formula to prove, most theorem provers for classical first-order logic process that formula following several steps, one of which is commonly called skolemization. That process eliminates quantifier alternation within formulas by extending the language of the underlying logic with new Skolem functions and by instantiating certain quantifiers with terms built using Skolem functions. In this paper, we address the problem of checking (i.e., certifying) proof evidence that involves Skolem terms. Our goal is to do such certification without using the mathematical concepts of model-theoretic semantics (i.e., preservation of satisfiability) and choice principles (i.e., epsilon terms). Instead , our proof checking kernel is an implementation of Gentzen's sequent calculus, which directly supports quanti-fier alternation by using eigenvariables. We shall describe deskolemization as a mapping from client-side terms, used in proofs generated by theorem provers, into kernel-side terms, used within our proof checking kernel. This mapping which associates skolemized terms to eigenvariables relies on using outer skolemization. We also point out that the removal of Skolem terms from a proof is influenced by the polarities given to propositional connectives

A Semantic Framework for Proof Evidence

International audienceTheorem provers produce evidence of proof in many different formats, such as proof scripts, natural deductions, resolution refutations, Herbrand expansions, and equational rewritings. In implemented provers, numerous variants of such formats are actually used: consider, for example, such variants of or restrictions to resolution refu-tations as binary resolution, hyper-resolution, ordered-resolution, paramodulation, etc. We propose the foundational proof certificates (FPC) framework for defining the semantics of a broad range of proof evidence. This framework allows both producers of proof certificates and the checkers of those certificates to have a clear formal definition of the semantics of a wide variety of proof evidence. Employing the FPC framework will allow one to separate a proof from its provenance and to allow anyone to construct their own proof checker for a given style of proof evidence. The foundation on which FPC relies is that of proof theory, particularly recent work into focused proof systems: such proof systems provide protocols by which a checker extracts information from the certificate (mediated by the so called clerks and experts) as well as performs various deterministic and non-deterministic computations. While we shall limit ourselves to first-order logic in this paper, we shall not limit ourselves in many other ways. The FPC framework is described for both classical and intuitionistic logics and for proof structures as diverse as resolution refutations, natural deduction, Frege proofs, and equality proofs

Nominal commutative narrowing

Dissertação (mestrado) — Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2022.Modelagem e raciocínio equacional são onipresentes na Matemática e na Ciência da Computação. Técnicas de reescrita têm sido aplicadas com sucesso para formalizar e implementar inferência automatizada em estruturas matemáticas dedutivas. Apresentar teorias equacionais por meio da reescrita dá origem a um mecanismo para decidir a redução equacional da teoria sempre que o sistema de reescrita for terminante e confluente, ou seja, sempre que for convergente. Resolver problemas equacionais é um passo adiante que requer mais esforço do que apenas usar reescrita. De fato, “estreitar” problemas equacionais é uma técnica bem conhecida que adiciona à reescrita o poder necessário para buscar soluções; em outras palavras, adiciona o poder de buscar instâncias das variáveis que ocorrem em um problema equacional que “unifica” as equações. Por sua vez, a lógica nominal foi desenvolvida para contornar as inconveniências apresentadas quando as variáveis são instanciadas. A abordagem nominal usa átomos nominais em vez de variáveis para evitar a necessidade de renomeação de variáveis ao lidar com equações na abordagem notacional padrão. A sintaxe nominal também inclui permutações de átomos para distinguir algebricamente os átomos evitando colisões e capturas destes. Neste trabalho, estudamos a reescrita nominal módulo comutatividade. Desenvolvemos o método estreitamento nominal comutativo (nominal commutative narrowing) para lidar com o problema de unificação nominal módulo teorias equacionais que incluem comutatividade, o qual não é finitário dependendo da representação das soluções.Equational modelling and reasoning are ubiquitous in Mathematics and Computer Science. Rewriting techniques have been applied successfully to formalize and implement automated inference in mathematical deductive frameworks. Presenting equational theories by rewriting gives rise to a mechanism to decide the equational reduct of the theory whenever the rewriting system is terminating and confluent, i.e., whenever it is convergent. Solving equational problems is a step further that requires more effort than just rewriting. Indeed, “narrowing” equational problems is a well-known technique that adds to rewriting the required power to search for solutions; in other words, it adds the power to search for instantiations of the variables occurring in an equational problem that “unify” the equations. On its side, the nominal logic has been developed to contour inconveniences presented when variables are instantiated. The nominal approach uses nominal atoms instead of variables to avoid the requirement of variable renaming when dealing with equations in the standard notational approach. The nominal syntax also includes atom permutations to algebraically distinguish atoms avoiding atom collisions and captures. In this work, we study nominal rewriting modulo commutativity. We develop nominal commutative narrowing to deal with the problem of nominal unification modulo equational theories that include commutativity, which is not finitary depending on the representation of solutions

Healthy snacks consumption and the Theory of Planned Behaviour. The role of anticipated regret

Two empirical studies explored the role of anticipated regret (AR) within the Theory of Planned Behavior (TPB) framework (Ajzen, 1991), applied to the case of healthy snacks consumption. AR captures affective reactions and it can be defined as an unpleasant emotion experienced when people realize or imagine that the present situation would be better if they had made a different decision. In this research AR refers to the expected negative feelings for not having consumed healthy snacks (i.e., inaction regret). The aims were: a) to test whether AR improves the TPB predictive power; b) to analyze whether it acts as moderator within the TPB model relationships. Two longitudinal studies were conducted. Target behaviors were: consumption of fruit and vegetables as snacks (Study 1); consumption of fruit as snacks (Study 2). At time 1, the questionnaire included measures of intention and its antecedents, according to the TPB. Both the affective and evaluative components of attitude were assessed. At time 2, self-reported consumption behaviors were surveyed. Two convenience samples of Italian adults were recruited. In hierarchical regressions, the TPB variables were added at the first step; AR was added at the second step, and the interactions at the last step. Results showed that AR significantly improved the TPB ability to predict both intentions and behaviours, also after controlling for intention. In both studies AR moderated the effect of affective attitude on intention: affective attitude was significant only for people low in AR