Induction in Saturation-Based Proof Search

Abstract

Many applications of theorem proving, for example program verification and analysis, require first-order reasoning with both quantifiers and theories such as arithmetic and datatypes. There is no complete procedure for reasoning in such theories but the state-of-the-art in automated theorem proving is still able to reason effectively with real-world problems from this rich domain. In this paper we introduce a missing part of the puzzle: automated induction inside a saturation-based theorem prover. Our goal is to incorporate lightweight automated induction in a way that complements the saturation-based approach, allowing us to solve problems requiring a combination of first-order reasoning, theory reasoning, and inductive reasoning. We implement a number of techniques and heuristics and evaluate them within the Vampire theorem prover. Our results show that these new techniques enjoy practical success on real-world problems.

Integer Induction in Saturation

“…In what follows, Vampire refers to the (default) version of Vampire, as in [10,16]. By Vampire-I we denote our new version of Vampire, using integer induction rules (--induction int).…”

“…The first two ideas were already used in the first papers underlying our approach to induction in saturation theorem proving [10,16]. For example, they can be implemented by using only induction formulas that are obtained from ground literals L[t] in the search space, where t is a ground term.…”

“…Induction in saturation-based theorem proving is a new exciting direction in the automation of induction, recently introduced in [5,10,16]. This work focused on induction on inductively defined data types, also called algebraic data types [12], such as natural numbers or lists.…”

The Complex Approach of the C-lightVer System to the Automated Error Localization in C-programs

“…Отметим, что Регеру и др. для проекта по расширению возможностей системы Vampire [28] пришлось разрабатывать новые стратегии автоматизации доказательства по индукции.…”

Aiming for the Goal with SInE

“…, A n , ¬C}. While the subsequent theorem proving technology does not, in principle, need to keep track of the "distinction in origin" between the axioms A i and the negated conjecture ¬C, doing the opposite and treating the descendants of ¬C-typically referred to as the goal here-in a special way has been the core idea behind several successful proving heuristics, most notably the set of support strategy [19,8], applied in various contexts [11,1], and others [15,14,10,12]. The main reason why such goal-directed search often succeeds is that since the axioms A 1 , .…”