Licentiate thesis 2016-012

New Techniques for Handling Quantifiers in Boolean and First-Order Logic

Peter Backeman

12 December 2016

The automation of reasoning has been an aim of research for a long time. Already in 17th century, the famous mathematician Leibniz invented a mechanical calculator capable of performing all four basic arithmetic operators. Although automatic reasoning can be done in different fields, many of the procedures for automated reasoning handles formulas of first-order logic. Examples of use cases includes hardware verification, program analysis and knowledge representation.

One of the fundamental challenges in first-order logic is handling quantifiers and the equality predicate. On the one hand, SMT-solvers (Satisfiability Modulo Theories) are quite efficient at dealing with theory reasoning, on the other hand they have limited support for complete and efficient reasoning with quantifiers. Sequent, tableau and resolution calculi are methods which are used to construct proofs for first-order formulas and can use more efficient techniques to handle quantifiers. Unfortunately, in contrast to SMT, handling theories is more difficult.

In this thesis we investigate methods to handle quantifiers by restricting search spaces to finite domains which can be explored in a systematic manner. We present this approach in two different contexts.

First we introduce a function synthesis based on template-based quantifier elimination, which is applied to gene interaction computation. The function synthesis is shown to be capable of generating smaller representations of solutions than previous solvers, and by restricting the constructed functions to certain forms we can produce formulas which can more easily be interpreted by a biologist.

Secondly we introduce the concept of Bounded Rigid E-Unification (BREU), a finite form of unification that can be used to define a complete and sound sequent calculus for first-order logic with equality. We show how to solve this bounded form of unification in an efficient manner, yielding a first-order theorem prover utilizing BREU that is competitive with other state-of-the-art tableau theorem provers.

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