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Constraint Solving for Proof Planning

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Abstract

Proof planning is an application of AI planning to theorem proving that employs plan operators that encapsulate mathematical proof techniques. Many proofs require the instantiation of variables; that is, mathematical objects with certain properties have to be constructed. This is particularly difficult for automated theorem provers if the instantiations have to satisfy requirements specific for a mathematical theory, for example, for finite sets or for real numbers, because in this case unification is insufficient for finding a proper instantiation. Often, constraint solving can be employed for this task. We describe a framework for the integration of constraint solving into proof planning that combines proof planners and stand-alone constraint solvers. Proof planning has some peculiar requirements that are not met by any off-the-shelf constraint-solving system. Therefore, we extended an existing propagation-based constraint solver in a generic way. This approach generalizes previous work on tackling the problem. It provides a more principled way and employs existing AI technology.

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References

  • Autexier, S. and Hutter, D. (1997) Equational proof-planning by dynamic abstraction, in M. Bonacina and U. Furbach (eds.), Proceedings of the First Workshop on First-Order Theorem Proving (FTP97), RISC-Linz Report Series No. 97-50, Linz, Austria.

  • Baader, F. and Nipkow, T. (1998) Term Rewriting and All That, Cambridge University Press, Cambridge, UK.

    Google Scholar 

  • Bartle, R. and Sherbert, D. (1982) Introduction to Real Analysis, Wiley, New York.

    Google Scholar 

  • Baumgartner, P. and Stolzenburg, F. (1995) Constraint model elimination and a PPT-implementation, in P. Baumgartner, R. Haehnle and J. Posegga (eds.), Proceedings of the 4th Workshop on Theorem Proving with Analytic Tableaux and Related Methods, Springer-Verlag, pp. 201–216.

  • Beeson, M. (1989) Logic and computation in Mathpert: An expert system for learning mathematics, in E. Kaltofen and S. Watt (eds.), Computers and Mathematics ‘89, Springer, pp. 202–214.

  • Beeson, M. (1998) Automatic generation of epsilon-delta proofs of continuity, in J. Calment and J. Plaza (eds.), Artificial Intelligence and Symbolic Computation, Springer-Verlag, pp. 67–83.

  • Bledsoe, W. W. (1990) Challenge problems in elementary calculus, J. Automated Reasoning 6, 341–359.

    Google Scholar 

  • Bledsoe, W. W., Boyer, R. and Henneman, W. (1972) Computer proofs of limit theorems, Artificial Intelligence 3(1), 27–60.

    Google Scholar 

  • Brelaz, D. (1979) New methods to colour the vertices of a graph, Comm. ACM 22(4), 251–256.

    Google Scholar 

  • Bundy, A. (1988) The use of explicit plans to guide inductive proofs, in E. Lusk and R. Overbeek (eds.), Proceedings of the 9th International Conference on Automated Deduction (CADE-9) (Argonne), Lecture Notes in Comput. Sci. 310, Springer, pp. 111–120.

  • Bundy, A., Stevens, A., van Harmelen, F., Ireland, A. and Smaill, A. (1993) Rippling: A heuristic for guiding inductive proofs, Artificial Intelligence 62, 185–253. Also available from Edinburgh as DAI Research Paper No. 567.

    Google Scholar 

  • Bürckert, H.-J. (1994) A resolution principle for constrained logics, Artificial Intelligence 66, 235–271.

    Google Scholar 

  • Collins, G. (1975) Quantifier elimination for real closed fields by cylindrin algebraic decomposition, in Second GI Conference on Automata Theory and Formal Languages, Lecture Notes in Comput. Sci. 33, Springer-Verlag, pp. 134–183.

  • Dechter, R. and Dechter, A. (1988) Belief maintenance in dynamic constraint networks, in Proceedings of the Seventh Annual Conference of the American Association of Artificial Intelligence, pp. 37–42.

  • Dupre, D. (2000) Automated theorem proving by integrating proving, solving and computing, Ph.D. thesis, Forschungsinstitut für symbolisches Rechnen, Technisch-Naturwissenschaftche Fakultät, Johannes Kepler Universität Linz.

  • Fikes, R. and Nilsson, N. (1971) STRIPS: A new approach to the application of theorem proving to problem solving, Logic and Comput. Sci. 2, 189–208.

    Google Scholar 

  • Gentzen, G. (1935) Untersuchungen über das logische Schließen I & II, Math. Zeitschrift 39, 176–210, 572–595.

    Google Scholar 

  • Ghallab, M. and Laruelle, H. (1994) Representation and control in IxTeT, a temporal planner, in Proceedings of the Second International Conference on Artificial Intelligence Planning (AIPS).

  • Giunchiglia, F. and Walsh, T. (1992) A theory of abstraction, Artificial Intelligence 57, 323–390.

    Google Scholar 

  • Haralick, R. and Elliot, G. (1980) Increasing tree search efficiency for constraint satisfaction problems, Artificial Intelligence 14, 263–313.

    Google Scholar 

  • Haslum, P. and Geffner, H. (2001) Heuristic planning with time and resources, in A. Cesta and D. Borrajo (eds.), Proceedings of the 6th European Conference on Planning, pp. 121–132.

  • Inc., I. (1999) ILOG solver user’s manual version 4.4, ILOG Inc., Mountain View, CA.

  • Jaffar, J. and Lassez, J.-L. (1987) Constraint logic programming, in Proceedings 14th ACM Symposium on Principles of Programming Languages.

  • Jonsson, A., Morris, P., Muscettola, N. and Rajan, K. (2000) Planning in interplanetary space: Theory and practice, in S. Chien, S. Kambhampati and C. Knoblock (eds.), Proceedings of the Second International Conference on Artificial Intelligence Planning (AIPS) 2000, AAAI, pp. 177–186.

  • Kambhampati, S. (1996) Refinement planning: Status and prospectus, in Proceedings of the AAAI-96, Morgan Kaufmann, pp. 1331–1336.

  • Kautz, H. and Walser, J. (1999) State-space planning by integer optimization, The Knowledge Engineering Review 15(1), 526–533.

    Google Scholar 

  • Laborie, P. (2001) Algorithms for propagating resource constraints in AI planning and scheduling: Existing approaches and new results, in A. Cesta and D. Borrajo (eds.), Proceedings of the Sixth European Conference on Planning, ECP-01, pp. 205–216.

  • Leron, U. (1985) Heuristic presentations: The role of structuring, For the Learning of Mathematics 5(3), 7–13.

    Google Scholar 

  • Mackworth, A. K. (1977) Consistency in networks of relations, Artificial Intelligence 8, 99–118.

    Google Scholar 

  • McCune, W. (1990) Otter 2.0, in M. E. Stickel (ed.), Proceedings of the 10th International Conference on Automated Deduction (CADE–10) (Kaiserslautern, Deutschland), Lecture Notes in Artif. Intell. 449, Springer-Verlag, Berlin, pp. 663–664.

    Google Scholar 

  • Meier, A., Pollet, M. and Sorge, V. (2002) Comparing approaches to the exploration of the domain of residue classes, J. Symbolic Comput., Special Issue on the Integration of Automated Reasoning and Computer Algebra Systems 34(4), 287–306. S. Linton and R. Sebastiani, eds.

    Google Scholar 

  • Melis, E. (1996) Island planning and refinement, in J. Hendler and J. Koehler (eds.), Control of Search in AI Planning.

  • Melis, E. (1998a) AI-techniques in proof planning, in European Conference on Artificial Intelligence (ECAI-98) (Brighton), Kluwer, pp. 494–498.

  • Melis, E. (1998b) Combining proof planning with constraint solving, in Proceedings of Calculemus and Types’98.

  • Melis, E. (1998c) The “Limit” domain, in R. Simmons, M. Veloso and S. Smith (eds.), Proceedings of the Fourth International Conference on Artificial Intelligence in Planning Systems (AIPS-98).

  • Melis, E. and Meier, A. (2000) Proof planning with multiple strategies, in J. Loyd, V. Dahl, U. Furbach, M. Kerber, K. Lau, C. Palamidessi, L. Pereira and Y. S. P. Stuckey (eds.), First International Conference on Computational Logic, Lecture Notes in Artif. Intell. 1861, Springer-Verlag, pp. 644–659.

  • Melis, E. and Siekmann, J. (1999) Knowledge-based proof-planning, Artificial Intelligence 115(1), 65–105.

    Google Scholar 

  • Monfroy, E. and Ringeissen, C. (1998) SoleX: A domain-independent scheme for constraint solver extension, in J. Calmet and J. Plaza (eds.), Artificial Intelligence and Symbolic Computation AISC’98, Springer, pp. 222–233.

  • Mozart (2001) Mozart Consortium. The Mozart Programming System 1.2.0, Documentation and system available from http://www.mozart-oz.org.

  • Müller, T. (2000) Promoting constraints to first-class status, in Proceedings of the First International Conference on Computational Logic – CL2000 (London), Springer-Verlag.

  • Muscettola, N. (1994) HSTS: Integrating planning and scheduling, in M. Zweben and M. Fox (eds.), Intelligent Scheduling, Morgan Kaufmann, pp. 169–212.

  • Myers, K., Smith, S., Hildrum, D., Jarvis, P. and de Lacaze, R. (2001) Integrating planning and scheduling through adaptation of resource intensity estimates, in Proceedings of the 6th European Conference on Planning, pp. 133–144.

  • Penberthy, J. and Weld, D. (1994) Temporal planning with continuous change, in Proceedings of the 12th National Conference on AI, pp. 1010–1015.

  • Schulte, C. (2000) Programming constraint services, Doctoral dissertation, Universität des Saarlandes, Naturwissenschaftlich-Technische Fakultät I, Fachrichtung Informatik, Saarbrücken, Germany.

  • Siekmann, J., Benzmüller, C., Brezhnev, V., Cheikhrouhou, L., Fiedler, A., Franke, A., Horacek, H., Kohlhase, M., Meier, A., Melis, E., Moschner, M., Normann, I., Pollet, M., Sorge, V., Ullrich, C., Wirth, C. and Zimmer, J. (2002) Proof development with ΩMEGA, in A. Voronkov (ed.), Proceedings of the 18th Conference on Automated Deduction (CADE-18) (Copenhagen, Denmark), Lecture Notes in Artif. Intell. 2392, Springer-Verlag, Berlin, pp. 144–149.

    Google Scholar 

  • Smith, D., Frank, J. and Jonsson, A. (2000) Bridging the gap between planning and scheduling, Knowledge Engineering Review 15(1), 61–94.

    Google Scholar 

  • Smolka, G. (1995) The Oz programming model, in J. van Leeuwen (ed.), Current Trends in Computer Science, Lecture Notes in Comput. Sci. 1000, Springer, Berlin, pp. 324–343.

    Google Scholar 

  • Sorge, V. (2000) Non-trivial computations in proof planning, in H. Kirchner and C. Ringeissen (eds.), Frontiers of Combining Systems: Third International Workshop, FroCoS 2000 (Nancy, France), Lecture Notes in Comput. Sci. 1794, Springer-Verlag, Berlin, pp. 121–135.

    Google Scholar 

  • Stickel, M. (1985) Automated deduction by theory resolution, in Proceedings of the 9th International Joint Conference on Artificial Intelligence, pp. 1181–1186.

  • Sutcliffe, G., Suttner, C. and Yemenis, T. (1994) The TPTP problem library, in A. Bundy (ed.), Proceedings of the 12th Conference on Automated Deduction (Nancy, France), Springer-Verlag.

  • Tate, A. (1996) Representing plans as a set of constraints – the I-N-OVA model, in Proceedings of the Third International Conference on Artificial Intelligence Planning Systems (AIPS-96).

  • Wolfman, S. and Weld, D. (1999) Combining linear programming and satisfiability solving for resource planning, The Knowledge Engineering Review 15(1).

  • Wolfram, S. (1996) The Mathematica Book: Version 3.0, 3rd edn, Wolfram Media, Inc., Champaign, IL.

    Google Scholar 

  • Yoshida, T., Bundy, A., Green, I., Walsh, T. and Basin, D. (1994) Coloured rippling: An extension of a theorem proving heuristic, in Proceedings of the 11th ECAI (Amsterdam, The Netherlands), pp. 85–89.

  • Zimmer, J. (2000) Constraintlösen für Beweisplanung, Master’s thesis, Fachbereich Informatik, Universität des Saarlandes, Saarbrücken, Germany. In German.

  • Zimmer, J. and Kohlhase, M. (2002) System description: The MathWeb software bus for distributed mathematical reasoning, in A. Voronkov (ed.), Proceedings of the 18th International Conference on Automated Deduction, Springer-Verlag, pp. 139–143.

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Authors and Affiliations

  1. Arbeitsgruppe Siekmann (AGS), Fachbereich Informatik (FB 14), Universität des Saarlandes, D-66041, Saarbrücken, Germany

    Jürgen Zimmer

  2. Competence Centre for eLearning, German Research Center for Artificial Intelligence (DFKI), D-66123, Saarbrücken, Germany

    Erica Melis

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  1. Jürgen Zimmer
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  2. Erica Melis
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Correspondence to Jürgen Zimmer.

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Zimmer, J., Melis, E. Constraint Solving for Proof Planning. J Autom Reasoning 33, 51–88 (2004). https://doi.org/10.1007/s10817-004-2725-6

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