2 research outputs found
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