Integer Induction in Saturation

Abstract

Integers are ubiquitous in programming and therefore also in applications of program analysis and verification. Such applications often require some sort of inductive reasoning. In this paper we analyze the challenge of automating inductive reasoning with integers. We introduce inference rules for integer induction within the saturation framework of first-order theorem proving. We implemented these rules in the theorem prover Vampire and evaluated our work against other state-of-the-art theorem provers. Our results demonstrate the strength of our approach by solving new problems coming from program analysis and mathematical properties of integers.

Rewriting and Inductive Reasoning

“…While auxiliary lemmas might be provided by users in interactive theorem proving [14,31], saturation-based automated theorem provers, by design, do not support user guidance during proof search. Automation of induction in saturation therefore implements inductive generalizations [1,12,24,33], uses failed proof attempts [8], specialized sound inferences with common patterns [38], or integrates induction directly into saturation [16,27]. Our work extends these techniques by guiding proof search with auxiliary lemmas generated during proof search.…”

“…Applying induction only when triggered means that only premises of schema instances directly related to a clause selected during proof search are added to the search space. This is a strong heuristic method for automating induction, particularly in the presence of full first-order logic and theories [21,23,24].…”

Integrating Answer Literals with AVATAR for Program Synthesis

“…We give a simplified overview of our method from [2]. The method synthesizes programs for specifications given as formulas in first-order logic having the form…”

“…We add an answer literal ans(y) to the negated and clausified specification, obtaining 1) is found, we construct a program for (1) by composing the programs with conditions recorded during the proof search using a (nested) if−then−else construction. For more details on the synthesis framework, see [2].…”

“…In [2], we present a new framework for program synthesis based on a functional specification using a saturation-based theorem prover. The main idea is to piggyback on theorem proving: search for a proof of the specification, and in the process construct a program satisfying the specification.…”