IT Licentiate theses 2016-012 New Techniques for Handling Quantifiers in Boolean and First-Order Logic PETER BACKEMAN UPPSALA UNIVERSITY Department of Information Technology New Techniques for Handling Quantifiers in Boolean and First-Order Logic Peter Backeman
[email protected] December 2016 Division of Computer Systems Department of Information Technology Uppsala University Box 337 SE-751 05 Uppsala Sweden http://www.it.uu.se/ Dissertation for the degree of Licentiate of Philosophy in Computer Science c Peter Backeman 2016 ISSN 1404-5117 Printed by the Department of Information Technology, Uppsala University, Sweden u Abstract 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 reas- oning can be done in di↵erent 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 hand- ling 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 quanti- fiers. 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 re- stricting search spaces to finite domains, explorable in a system- atic manner.