Automated proof synthesis for propositional logic with deep neural networks

Needless to say, mathematics has become the reliable foundation of modern natural science, including several branches of theoretical computer science, by justifying theorems with proofs. The importance of correct proofs leads to the study of software called proof assistants [Nipkow et al. 2002; Norell 2009; The Coq Development Team 2017], which allow users to state theorems and their proofs formally in the form of certain programming languages and automatically check that the proofs correctly prove the theorems. The realm of the areas that rely on theorem proving is expanding beyond mathematics; for example, it is being applied for system verification [Klein et al. 2009; Leroy 2009], where one states the correctness of a system as a theorem and justifies it in the form of proofs. Automated theorem proving (ATP) [Bibel 2013; Fitting 2012; Pfenning 2004] is a set of techniques that prove logical formulas automatically. We are concerned with the following form of ATP called automated proof synthesis (APS): Given a logical formula P, if P holds, return a proof M of P. In the light of the importance of theorem proving, APS serves as a useful tool for activities based on formal reasoning. For example, from the perspective of the aforementioned system verification, APS serves for automating system verification; indeed, various methods for (semi)automated static program verification [Barnett et al. 2005; Chalin et al. 2007; Filliâtre and Paskevich 2013] can be seen as APS procedures. We also remark another important application of APS: automated program synthesis. An APS algorithm can be seen as an automated program synthesis procedure via the Curry-Howard isomorphism [Sørensen and Urzyczyn 2006], in which M can be seen as a program and P can be seen as a specification. Not only is APS interesting from the practical viewpoint, it is also interesting from the theoretical perspective in that it investigates the algorithmic aspect of theorem proving.