# Modeling and solving constraint satisfaction problems through Petri nets

## Abstract

Constraint satisfaction problems (CSP) represent one of the most studied areas in Artificial Intelligence and related disciplines. A lot of theoretical problems and applications, including computer vision, job-shop scheduling, planning, design and others, can be formulated as problems related to the satisfaction of constraints. Classical approaches to the solution of CSP are usually based on some form of backtracking search; such approaches suffer, in the general case, of some drawbacks essentially represented by the so-called *thrashing behavior*, a problem arising when the search algorithm repeatedly explores parts of the search space not leading to any solution. In this paper an alternative approach to backtracking search is proposed by modeling a system of constraints through an ordinary Petri net model. The Petri net model is able to capture every finite domain CSP, representing a wide and significant class of CSP used in applications. Algebraic analysis is then exploited for solving the CSP corresponding to the net model. In particular, even if a direct application of the *fundamental equation* of Petri nets would be theoretically possible, such a possibility could be unsatisfactory in practice. Because of that, a cyclic net model for CSP is introduced, such that the whole set of solutions can be characterized by means of a subset of the set of minimal support T-invariants of the net model. An algorithm able to directly compute only such a subset of invariants is then proposed as the core process for solving a CSP.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.H. Anton and C. Rorres.
*Elementary Linear Algebra*. John Wiley, 1991.Google Scholar - 2.L. Bernardinello and F. De Cindio. A survey of basic net models and modular net classes. In G. Rozenberg, editor,
*Advances in Petri Nets 1992, LNCS 609*, pages 304–351. Springer Verlag, 1992.Google Scholar - 3.J.M. Colom and M. Silva. Convex geometry and semiflows in P-T nets: a comparative study of algorithms for computation of minimal semiflows. In
*Advances in Petri Nets 1990, LNCS 483*, pages 79–112. Springer Verlag, 1990.Google Scholar - 4.R. Dechter and J. Pearl. Network-based heuristics for constraint satisfaction problems.
*Artificial Intelligence*, 34(1):1–38, 1988.Google Scholar - 5.J. deKleer and G.J. Sussman. Propagation of constraints applied to circuit synthesis.
*Circuit Theory and Applications*, 8:127–144, 1980.Google Scholar - 6.M.S. Fox, B. Allen, and G. Strohm. Job-shop scheduling: an investigation in constraint-directed reasoning. In
*Proc. 2nd AAAI 82*, pages 155–158, Pittsburgh, PA, 1982.Google Scholar - 7.H. Geffner and J. Pearl. An improved constraint propagation algorithm for diagnosis. In
*Proc 10th IJCAI*, pages 1105–1111, Milano, 1987.Google Scholar - 8.H.J. Genrich and K. Lautenbach. System modeling with high level petri nets.
*Theoretical Computer Science*, 13:109–136, 1981.CrossRefGoogle Scholar - 9.K. Jensen. Coloured Petri Nets and the invariant method.
*Theoretical Computer Science*, 14:317–336, 1981.CrossRefGoogle Scholar - 10.V. Kumar. Algorithms for constraint satisfaction problems: a survey.
*AI Magazine*, Spring 1992:32–44, 1992.Google Scholar - 11.A.K. Mackworth. Consistency in networks of relations.
*Artificial Intelligence*, 8(1):99–118, 1977.CrossRefGoogle Scholar - 12.A.K. Mackworth. On seeing things, again. In
*Proc. 8th IJCAI 83*, pages 1187–1191, Karslruhe, GE, 1983.Google Scholar - 13.A.K. Mackworth. Constraint satisfaction. In S.C. Shapiro, editor,
*Encyclopedia of Artificial Intelligence*, pages 205–211. John Wiley, 1990.Google Scholar - 14.A.K. Mackworth and E.C. Freuder. The complexity of some polynomial network consistency algorithms for constraint satisfactions problems.
*Artificial Intelligence*, 25(1):65–74, 1984.Google Scholar - 15.J. Martinez and M. Silva. A simple and fast algorithm to obtain all invariants of a generalized Petri net. In W. Reisig C. Girault, editor,
*Informatik-Fachberichte, Applications and Theory of Petri Nets*, pages 301–310. Springer Verlag, 1982.Google Scholar - 16.G. Memmi and G. Roucairol. Linear algebra in net theory. In
*Lecture Notes in Computer Science*, volume 84, pages 213–223. Springer Verlag, 1980.Google Scholar - 17.U. Montanari. Networks of constraints: fundamental properties and applications to picture processing.
*Information Science*, 7:95–132, 1974.CrossRefGoogle Scholar - 18.T. Murata. Petri nets: Properties, analysis and applications.
*Proceedings of the IEEE*, 77(4):541–580, 1989.Google Scholar - 19.B. Nadel. Tree search and arc consistency in constraint satisfaction algorithms. In L Kanal and V. Kumar, editors,
*Search in Artificial Intelligence*, pages 287–342. Springer Verlag, 1988.Google Scholar - 20.B. Nadel and J. Lin. Automobile transmission design as a constraint satisfaction problem. In
*Artificial Intelligence for Engineering Design, Analysis and Manufactirung*. 1991.Google Scholar - 21.M. Stefik. Planning with constraints.
*Artificial Intelligence*, 16:111–140, 1981.Google Scholar - 22.N. Treves. A comparative study of different techniques for semi-flows computation in Place/Transition nets. In
*Advances in Petri Nets 1989*, pages 433–452. LNCS 424, Springer Verlag, 1988.Google Scholar