# THE SOLUTION OF DISCRETE CONSTRAINT PROBLEMS USING BOOLEAN MODELS - The Use of Ternary Vectors for Parallel SAT-Solving

### Christian Posthoff, Bernd Steinbach

#### Abstract

The use of Boolean models for discrete constraint problems has been tried at several occasions, it was, however, not recognized as efficient (Rossi et al., 2006). The solution methods were dominated by using decision trees together with depth-first or breadth-first search and/or resolution algorithms. In this paper we will show the use of ternary vectors for the solution of SAT-problems and all the problems that can be modeled by means of SAT-equations. They are an appropriate data structure representing sets of Boolean vectors. They also allow to include problem-relevant knowledge into the problem-solving process at an early point of time. The respective set operations (mainly the intersection) can be executed in a bit-parallel way (64 bits at present). For larger problems the processing can be transferred to processors working fully in parallel. There is no need for any search algorithms. The approach always finds all solutions of the problem without consideration of special cases (i.e. no solution, one solution, all solutions). Some examples are used to illustrate the approach or have been published before (Sudoku, Queen's problems on the chessboard, node bases in graphs, graph-coloring problems).

#### References

- Lynce, I. and Ouaknine, J. (2006). Sudoku as a sat problem. In Proceedings of the 9 th International Symposium on Artificial Intelligence and Mathematics, AIMATH 2006, Fort Lauderdale. Springer.
- Posthoff, C. and Steinbach, B. (2004). Logic Functions and Equations - Binary Models for Computer Science. Springer, Dordrecht, The Netherlands.
- Posthoff, C. and Steinbach, B. (2006). A multi-processor approach to sat-problems. In Proceedings of the 7th International Workshop on Boolean Problems, pages 49-62, Freiberg University of Mining and Technology, Freiberg, Germany.
- Rossi, F., Van Beek, P., and Walsh, T. (2006). Handbook of Constraint Programming. Elsevier.
- Steinbach, B. and Posthoff, C. (2009). Logic Functions and Equations - Examples and Exercises. Springer Science + Business Media B.V.
- Weber, T. (2005). A SAT-based Sudoku solver. In Sutcliffe, G. and Voronkov, A., editors, LPAR-12, The 12th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, Short Paper Proceedings, pages 11-15.

#### Paper Citation

#### in Harvard Style

Posthoff C. and Steinbach B. (2010). **THE SOLUTION OF DISCRETE CONSTRAINT PROBLEMS USING BOOLEAN MODELS - The Use of Ternary Vectors for Parallel SAT-Solving** . In *Proceedings of the 2nd International Conference on Agents and Artificial Intelligence - Volume 1: ICAART,* ISBN 978-989-674-021-4, pages 487-493. DOI: 10.5220/0002703804870493

#### in Bibtex Style

@conference{icaart10,

author={Christian Posthoff and Bernd Steinbach},

title={THE SOLUTION OF DISCRETE CONSTRAINT PROBLEMS USING BOOLEAN MODELS - The Use of Ternary Vectors for Parallel SAT-Solving},

booktitle={Proceedings of the 2nd International Conference on Agents and Artificial Intelligence - Volume 1: ICAART,},

year={2010},

pages={487-493},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0002703804870493},

isbn={978-989-674-021-4},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 2nd International Conference on Agents and Artificial Intelligence - Volume 1: ICAART,

TI - THE SOLUTION OF DISCRETE CONSTRAINT PROBLEMS USING BOOLEAN MODELS - The Use of Ternary Vectors for Parallel SAT-Solving

SN - 978-989-674-021-4

AU - Posthoff C.

AU - Steinbach B.

PY - 2010

SP - 487

EP - 493

DO - 10.5220/0002703804870493