Decision procedures for term algebras with integer constraints
Information and Computation - Special issue: Combining logical systems
Combinable Extensions of Abelian Groups
CADE-22 Proceedings of the 22nd International Conference on Automated Deduction
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Decision procedures are algorithms that can determine validity or satisfiability of first-order formulas in a given decidable theory. They can efficiently discharge a large number of formulas in specific theories without any user guidance. Therefore they are at the heart of virtually every verification system, reasoning about the system behaviors, proving their safety or finding situations that could lead to bugs. This thesis offers novel solutions to an important class of decision problems, the mixed constraints on data structures and their quantitative properties. It includes three major contributions. (1) Decision Procedures for Term Algebras with Integer Arithmetic. Term Algebras can model a variety of tree-like data types such as records, lists, stacks, etc., which are essential constructs in programming languages. We developed the basic reduction technique, namely, extraction of accurate integer constraints from data constraints. From the construction of accurate integer constraints that precisely characterize data constraints, we can derive decision procedures for combined constraints by utilizing decision procedures for data structures and decision procedures for integer arithmetic. (2) Decision Procedures for Queues with Integer Arithmetic. Queue is a typical data type of linear structure and is widely used in programming languages and forms the basis for many concurrent algorithms and communication protocols. As a queue can grow at both ends, it does not fall in the category of recursive data structures, which can be modeled as term algebras. For this reason, we have further improved the reduction technique and developed new normalization procedures to handle the distinguished properties of queues. (3) Decision Procedures for Knuth-Bendix Order. Using quantifier elimination and the reduction technique developed for solving the combined constraints of term algebras and integer arithmetic, we proved the decidability of the first-order theory of Knuth-Bendix Order, thereby solving a long-standing open problem in term rewriting (officially listed as RTA open problem 99 since 2000). Knuth-Bendix order is widely used in term rewriting and theorem proving, along with lexicographic path order. Unfortunately, the first-order theory of lexicographic path order is undecidable. Therefore, our result on the decidability of Knuth-Bendix order may greatly benefit future algorithm design in term rewriting and theorem proving.