Theory of linear and integer programming
Theory of linear and integer programming
The Omega test: a fast and practical integer programming algorithm for dependence analysis
Proceedings of the 1991 ACM/IEEE conference on Supercomputing
Mixed real-integer linear quantifier elimination
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Minimal Solutions of Linear Diophantine Systems: Bounds and Algorithms
RTA '91 Proceedings of the 4th International Conference on Rewriting Techniques and Applications
Combining Multisets with Integers
CADE-18 Proceedings of the 18th International Conference on Automated Deduction
Deciding Boolean Algebra with Presburger Arithmetic
Journal of Automated Reasoning
Theoretical Computer Science - Foundations of software science and computation structures
Towards Efficient Satisfiability Checking for Boolean Algebra with Presburger Arithmetic
CADE-21 Proceedings of the 21st international conference on Automated Deduction: Automated Deduction
CAV '08 Proceedings of the 20th international conference on Computer Aided Verification
An online proof-producing decision procedure for mixed-integer linear arithmetic
TACAS'03 Proceedings of the 9th international conference on Tools and algorithms for the construction and analysis of systems
Decision procedures for multisets with cardinality constraints
VMCAI'08 Proceedings of the 9th international conference on Verification, model checking, and abstract interpretation
A fast linear-arithmetic solver for DPLL(T)
CAV'06 Proceedings of the 18th international conference on Computer Aided Verification
Carathéodory bounds for integer cones
Operations Research Letters
Ordered sets in the calculus of data structures
CSL'10/EACSL'10 Proceedings of the 24th international conference/19th annual conference on Computer science logic
Building a calculus of data structures
VMCAI'10 Proceedings of the 11th international conference on Verification, Model Checking, and Abstract Interpretation
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We present decision procedures for logical constraints involving collections such as sets, multisets, and fuzzy sets. Element membership in our collections is given by characteristic functions from a finite universe (of unknown size) to a user-defined subset of rational numbers. Our logic supports standard operators such as union, intersection, difference, or any operation defined pointwise using mixed linear integer-rational arithmetic. Moreover, it supports the notion of cardinality of the collection, defined as the sum of occurrences of all elements. Deciding formulas in such logic has applications in software verification.Our decision procedure reduces satisfiability of formulas with collections to satisfiability of formulas in an extension of mixed linear integer-rational arithmetic with a "star" operator. The star operator computes the integer cone (closure under vector addition) of the solution set of a given formula. We give an algorithm for eliminating the star operator, which reduces the problem to mixed linear integer-rational arithmetic. Star elimination combines naturally with quantifier elimination for mixed integer-rational arithmetic. Our decidability result subsumes previous special cases for sets and multisets. The extension with star is interesting in its own right because it can encode reachability problems for a simple class of transition systems.