### Abstract

We consider the problem of deciding the theory of finite sets with cardinality constraints using a satisfiability modulo theories solver. Sets are a common high-level data structure used in programming; thus, such a theory is useful for modeling program constructs directly. More importantly, sets are a basic construct of mathematics and thus natural to use when formalizing the properties of computational systems. We develop a calculus describing a modular combination of a procedure for reasoning about membership constraints with a procedure for reasoning about cardinality constraints. Cardinality reasoning involves tracking how different sets overlap. For efficiency, we avoid considering Venn regions directly, as done previous work. Instead, we develop a novel technique wherein potentially overlapping regions are considered incrementally as needed. We use a graph to track the interaction among the different regions. Initial experimental results demonstrate that the new technique is competitive with previous techniques and scales much better on certain classes of problems.

Original language | English (US) |
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Title of host publication | Automated Reasoning - 8th International Joint Conference, IJCAR 2016, Proceedings |

Publisher | Springer Verlag |

Pages | 82-98 |

Number of pages | 17 |

Volume | 9706 |

ISBN (Print) | 9783319402284 |

DOIs | |

State | Published - 2016 |

Event | 8th International Joint Conference on Automated Reasoning, IJCAR 2016 - Coimbra, Portugal Duration: Jun 27 2016 → Jul 2 2016 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9706 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 8th International Joint Conference on Automated Reasoning, IJCAR 2016 |
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Country | Portugal |

City | Coimbra |

Period | 6/27/16 → 7/2/16 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Automated Reasoning - 8th International Joint Conference, IJCAR 2016, Proceedings*(Vol. 9706, pp. 82-98). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9706). Springer Verlag. https://doi.org/10.1007/978-3-319-40229-1_7

**A new decision procedure for finite sets and cardinality constraints in SMT.** / Bansal, Kshitij; Reynolds, Andrew; Barrett, Clark; Tinelli, Cesare.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Automated Reasoning - 8th International Joint Conference, IJCAR 2016, Proceedings.*vol. 9706, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9706, Springer Verlag, pp. 82-98, 8th International Joint Conference on Automated Reasoning, IJCAR 2016, Coimbra, Portugal, 6/27/16. https://doi.org/10.1007/978-3-319-40229-1_7

}

TY - GEN

T1 - A new decision procedure for finite sets and cardinality constraints in SMT

AU - Bansal, Kshitij

AU - Reynolds, Andrew

AU - Barrett, Clark

AU - Tinelli, Cesare

PY - 2016

Y1 - 2016

N2 - We consider the problem of deciding the theory of finite sets with cardinality constraints using a satisfiability modulo theories solver. Sets are a common high-level data structure used in programming; thus, such a theory is useful for modeling program constructs directly. More importantly, sets are a basic construct of mathematics and thus natural to use when formalizing the properties of computational systems. We develop a calculus describing a modular combination of a procedure for reasoning about membership constraints with a procedure for reasoning about cardinality constraints. Cardinality reasoning involves tracking how different sets overlap. For efficiency, we avoid considering Venn regions directly, as done previous work. Instead, we develop a novel technique wherein potentially overlapping regions are considered incrementally as needed. We use a graph to track the interaction among the different regions. Initial experimental results demonstrate that the new technique is competitive with previous techniques and scales much better on certain classes of problems.

AB - We consider the problem of deciding the theory of finite sets with cardinality constraints using a satisfiability modulo theories solver. Sets are a common high-level data structure used in programming; thus, such a theory is useful for modeling program constructs directly. More importantly, sets are a basic construct of mathematics and thus natural to use when formalizing the properties of computational systems. We develop a calculus describing a modular combination of a procedure for reasoning about membership constraints with a procedure for reasoning about cardinality constraints. Cardinality reasoning involves tracking how different sets overlap. For efficiency, we avoid considering Venn regions directly, as done previous work. Instead, we develop a novel technique wherein potentially overlapping regions are considered incrementally as needed. We use a graph to track the interaction among the different regions. Initial experimental results demonstrate that the new technique is competitive with previous techniques and scales much better on certain classes of problems.

UR - http://www.scopus.com/inward/record.url?scp=84976613923&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976613923&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-40229-1_7

DO - 10.1007/978-3-319-40229-1_7

M3 - Conference contribution

SN - 9783319402284

VL - 9706

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 82

EP - 98

BT - Automated Reasoning - 8th International Joint Conference, IJCAR 2016, Proceedings

PB - Springer Verlag

ER -