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  • Print publication year: 2005
  • Online publication date: March 2017

Resolution theorem proving: a logical point of view



§1. Introduction. Logical calculi were invented to model mathematical thinking and to formalize mathematical arguments. The calculi of Boole [8] and of Frege [15] can be considered as the first mathematical models of logical inference. Their work paved the way for the discipline of metamathematics, where mathematical reasoning itself is the object of mathematical investigation. The early calculi, the so-called Hilbert type- and Gentzen-type calculi [25], [17] developed in the 20th century served the main purpose to analyze and to reconstruct mathematical proofs and to investigate provability. A practical use of these calculi, i.e., using them for solving actual problems (e.g., for proving theorems in “real” mathematics), was not intended and even did not make sense.

But the idea of a logical calculus as a problem solver is in fact much older than the origin of propositional and predicate logic in the 19th century. Indeed this idea can be traced back toG.W. Leibniz with his brave vision of a calculusratiocinator [29], a calculus which would allow solution of arbitrary problems by purely mechanical computation, once they have been represented in a special formalism. Today we know that, even for restricted languages, this dream of a complete mechanization is not realizable—not even in principle (we just refer to the famous results of Gödel [20] and Turing [39]). That does not imply that we have to reject the idea altogether. Still it makes sense to search for a lean version of the calculus ratiocinator. Concerning the logical language, the ideal candidate is first-order logic; it is axiomatizable (and thus semidecidable), well-understood and sufficiently expressive to represent relevant mathematical structures. By Church's result [10] we know that there is no decision procedure for the validity problem of first-order logic; thus there is no procedure which is 1. capable of verifying the validity of all valid formulas and 2. terminating on all formulas. So, even in first-order logic, we have to be content with the verification of problems. The only thing we can hope for is a calculus which offers a basis for efficient proof search. It is not surprising that the invention of the computer lead to a revival of Leibniz's dream.

[1] W., Ackermann, ü ber die Erfüllbarkeit gewisser Zählausdrücke, Mathematische Annalen, vol. 100 (1928), pp. 638–649.
[2] F., Baader and W., Snyder, Unification theory, Handbook of automated reasoning (A., Robinson and A., Voronkov, editors), vol. I, Elsevier Science, 2001, pp. 445–532.
[3] M., Baaz, C., Ferüller, and A., Leitsch, A non-elementary speed-up in proof length by structural clause formtransformation, Proceedings of the symposium on Logic in Computer Science, 1994, pp. 213–219.
[4] M., Baaz and A., Leitsch, Cut normal forms and proof complexity, Annals of Pure and Applied Logic, vol. 97 (1999), pp. 127–177.
[5] M., Baaz, Cut-elimination and redundancy-elimination by resolution, Journal of Symbolic Computation, vol. 29 (2000), pp. 149–176.
[6] M., Baaz, A., Leitsch, and G., Moser, System description: CutRes 01, cut elimination by resolution, Conference on automated deduction, CADE-16, LectureNotes in Artificial Intelligence, Springer, 1999, pp. 212–216.
[7] P., Bernays and M., Scönfinkel, Zum Entscheidungsproblem der mathematischen Logik, Mathematische Annalen, vol. 99 (1928), pp. 342–372.
[8] G., Boole, An investigation on the laws of thought, Dover Publications, 1958.
[9] E., Örger, E., Grädel, and Y., Gurevich, The Classical Decision Problem, Perspectives in Mathematical Logic, Springer, 1997.
[10] A., Church, A note on the Entscheidungsproblem, The Journal of Symbolic Logic, vol. 1 (1936), pp. 40–44.
[11] Martin, Davis and Hilary, Putnam, A computing procedure for quantification theory, Journal of the ACM, vol. 7 (1960), no. 3, pp. 201–215.
[12] E., Eder, Relative complexities of first-order calculi, Vieweg, 1992.
[13] U., Egly and T., Rath, On the practical value of different definitional translations to normal form, Proceedings of CADE-13, Lecture Notes in Artificial Intelligence, vol. 1104, Springer, 1996, pp. 403–417.
[14] C., Ferüller, A., Leitsch, T., Tammet, and N., Zamov, Resolution methods for the decision problem, Lecture Notes in Artificial Intelligence, vol. 679, Springer, 1993.
[15] G., Frege, Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, From Frege to Gödel: A sourcebook in mathematical logic 1879–1931 (J., van Heijenoort, editor), Harvard University Press, 1967.
[16] H., Gelernter, Realization of a geometry theorem proving machine, Proceedings of the international conference on information processing, June 1959, pp. 273–282.
[17] G., Gentzen, Untersuchungen über das logische Schließen, Mathematische Zeitschrift, vol. 39 (1934), pp. 405–431.
[18] P. C., Gilmore, A proof method for quantification theory: its justification and realization, IBM Journal of Research and Development, (1960), pp. 28–35.
[19] Y., Girard, Proof theory and logical complexity, Studies in Proof Theory, Bibliopolis, 1987.
[20] K., Gödel, ü ber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 175–198.
[21] K., Gödel, Ein Spezialfall des Entscheidungsproblems der theoretischen Logik, Ergebnisse Mathematischer Kolloquien, vol. 2 (1932), pp. 27–28.
[22] R., ähnle, Tableaux and related methods, Handbook of automated reasoning (A., Robinson and A., Voronkov, editors), vol. I, Elsevier Science, 2001, pp. 101–178.
[23] J., Herbrand, Sur le proble`eme fondamental de la logique mathèmatique, Sprawozdania z posiedzen Towarzysta Naukowego Warszawskiego,Wydzial III, vol. 24 (1931), pp. 12–56.
[24] D., Hilbert and W., Ackermann, Grundzüge der theoretischen Logik, Springer, Berlin, 1928.
[25] D., Hilbert and P., Bernays, Grundlagen der Mathematik, Springer, 1970.
[26] W., Joyner, Automated theorem proving and the decision problem, Ph.D. thesis, Harvard University, 1973.
[27] W. H., Joyner, Resolution strategies as decision procedures, Journal of the ACM, vol. 23 (1976), pp. 398–417.
[28] R., Kowalski and P., Hayes, Semantic trees in automatic theorem proving, Machine intelligence (B., Meltzer and D., Michie, editors), vol. 7, American Elsevier, 1969, pp. 87–101.
[29] G. W., Leibniz, Calculus ratiocinator, Sämtliche Schriften und Briefe (Preussische Akademie der, Wissenschaften, editor), Reichel, Darmstadt, 1923.
[30] A., Leitsch, The resolution calculus, Texts in Theoretical Computer Science, Springer, 1997.
[31] John W., Lloyd, Foundations of logic programming, second ed., Springer, 1987.
[32] L., Löwenheim, ü ber Möglichkeiten im Relativkalkül, Mathematische Annalen, vol. 68 (1915), pp. 169–207.
[33] A., Martelli and U., Montanari, Unification in linear time and space: a structured presentation, Technical Report B76-16, Instituto di elaborazione della informazione, 1976.
[34] S. Y., Maslov, The inverse method for establishing deducibility for logical calculi, Proceedings of the Steklov Institute ofMathematics, vol. 98 (1968), pp. 25–96.
[35] W., McCune, Solution of the Robbins problem, Journal of Automated Reasoning, vol. 19 (1997), no. 3, pp. 263–276.
[36] J. A., Robinson, Automatic deduction with hyperresolution, International Journal of Computer Mathematics, vol. 1 (1965), pp. 227–234.
[37] J. A., Robinson, A machine-oriented logic based on the resolution principle, Journal of the ACM, vol. 12 (1965), pp. 23–41.
[38] G., Takeuti, Proof theory, second ed., North-Holland, 1987.
[39] A., Turing, On computable numbers with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, Series 2, vol. 42 (1936/37), pp. 230–265.