Automated Theorem Proving in a First-Order Logic with First class Boolean Sort

Licentiate thesis, 2016

Automated theorem proving is one of the central areas of computer mathematics. It studies methods and techniques for establishing validity of mathematical problems using a computer. The problems are expressed in a variety of formal logics, including first-order logic. Algorithms of automated theorem proving are implemented in computer programs called theorem provers. They find significant application in formal methods of system development and as a mean of automation in proof assistants. This thesis contributes to automated theorem proving with an ex- tension of many-sorted first-order logic called FOOL. In FOOL boolean sort has a fixed interpretation and boolean terms are treated as formulas. In addition, FOOL contains if-then-else and let-in constructs. We argue that these extensions are useful for expressing problems coming from program analysis and interactive theorem proving. We give a formalisation of FOOL and a translation of FOOL formulas to ordinary first-order logic. This translation can be used for proving theorems of FOOL using a first-order theorem prover. We describe our implementation of this translation in the Vampire theorem prover. We extend TPTP, the standard input language of first-order provers, to sup- port formulas of FOOL. We simplify TPTP by providing more powerful and uniform representations of if-then-else and let-in expressions. We discuss a modification of superposition calculus that can reason efficiently about formulas with interpreted boolean sort. We present a superposition-friendly translation of FOOL formulas to clausal normal form. We demonstrate usability and high performance of these modifications in Vampire on a series of benchmarks coming from various libraries of problems for automated provers. Finally, we present an extension of FOOL, aimed to be used for auto- mated program analysis. With this extension, the next state relation of a program can be expressed as a boolean formula which is linear in the size of the program.