Automatically Proving Mathematical Theorems with Evolutionary Algorithms and Proof Assistants

Mathematical theorems are human knowledge able to be accumulated in the form of symbolic representation, and proving theorems has been considered intelligent behavior. Based on the BHK interpretation and the Curry-Howard isomorphism, proof assistants, software capable of interacting with human for constructing formal proofs, have been developed in the past several decades. Since proofs can be considered and expressed as programs, proof assistants simplify and verify a proof by computationally evaluating the program corresponding to the proof. Thanks to the transformation from logic to computation, it is now possible to generate or search for formal proofs directly in the realm of computation. Evolutionary algorithms, known to be flexible and versatile, have been successfully applied to handle a variety of scientific and engineering problems in numerous disciplines for also several decades. Examining the feasibility of establishing the link between evolutionary algorithms, as the program generator, and proof assistants, as the proof verifier, in order to automatically find formal proofs to a given logic sentence is the primary goal of this study. In the article, we describe in detail our first, ad-hoc attempt to fully automatically prove theorems as well as the preliminary results. Ten simple theorems from various branches of mathematics were proven, and most of these theorems cannot be proven by using the tactic auto alone in Coq, the adopted proof assistant. The implication and potential influence of this study are discussed, and the developed source code with the obtained experimental results are released as open source.Comment: Accepted by 2016 IEEE Congress on Evolutionary Computation (CEC 2016), part of 2016 IEEE World Congress on Computational Intelligence (IEEE WCCI 2016

A Survey on Theorem Provers in Formal Methods

Mechanical reasoning is a key area of research that lies at the crossroads of mathematical logic and artificial intelligence. The main aim to develop mechanical reasoning systems (also known as theorem provers) was to enable mathematicians to prove theorems by computer programs. However, these tools evolved with time and now play vital role in the modeling and reasoning about complex and large-scale systems, especially safety-critical systems. Technically, mathematical formalisms and automated reasoning based-approaches are employed to perform inferences and to generate proofs in theorem provers. In literature, there is a shortage of comprehensive documents that can provide proper guidance about the preferences of theorem provers with respect to their designs, performances, logical frameworks, strengths, differences and their application areas. In this work, more than 40 theorem provers are studied in detail and compared to present a comprehensive analysis and evaluation of these tools. Theorem provers are investigated based on various parameters, which includes: implementation architecture, logic and calculus used, library support, level of automation, programming paradigm, programming language, differences and application areas