Abstract
We present a uniform algorithm for proving automatically a fairly wide class of elementary facts connected with integer divisibility. The assertions that can be handled are those with a limited quantifier structure involving addition, multiplication and certain number-theoretic predicates such as ‘divisible by’, ‘congruent’ and ‘coprime’; one notable example in this class is the Chinese Remainder Theorem (for a specific number of moduli). The method is based on a reduction to ideal membership assertions that are then solved using Gröbner bases. As well as illustrating the usefulness of the procedure on examples, and considering some extensions, we prove a limited form of completeness for properties that hold in all rings.
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References
Aschenbrenner, M.: Ideal membership in polynomial rings over the integers. Journal of the American Mathematical Society 17, 407–441 (2004)
Baker, A.: A Concise Introduction to the Theory of Numbers. Cambridge University Press, Cambridge (1985)
Beltyokov, A.P.: Decidability of the universal theory of natural numbers with addition and divisibility (Russian). Sem. Leningrad Otd. Mat. Inst. Akad. Nauk SSSR 40, 127–130 (1974), English translation in Journal Of Mathematical Sciences 14, 1436–1444 (1980)
Boyer, R.S., Moore, J.S.: A Computational Logic. In: ACM Monograph Series, Academic Press, San Diego (1979)
Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Mathematisches Institut der Universität Innsbruck (1965), English translation to appear in Journal of Symbolic Computation (2006)
Bundy, A.: A science of reasoning. In: Lassez, J.-L., Plotkin, G. (eds.) Computational Logic: Essays in Honor of Alan Robinson, pp. 178–198. MIT Press, Cambridge (1991)
Cooper, D.C.: Theorem proving in arithmetic without multiplication. In: Melzer, B., Michie, D. (eds.) Machine Intelligence 7, pp. 91–99. Elsevier, Amsterdam (1972)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Clarendon Press, Oxford (1979)
Harrison, J.: HOL Light: A tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996)
Hodges, W.: Logical features of Horn clauses. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming (logical foundations), vol. 1, pp. 449–503. Oxford University Press, Oxford (1993)
Kandri-Rody, A., Kapur, D.: Algorithms for computing Gröbner bases of polynomial ideals over various Euclidean rings. In: Fitch, J. (ed.) EUROSAM 1984. LNCS, vol. 174, pp. 195–206. Springer, Heidelberg (1984)
Kandri-Rody, A., Kapur, D., Narendran, P.: An ideal-theoretic approach to word problems and unification problems over finitely presented commutative algebras. In: Jouannaud, J.-P. (ed.) Rewriting Techniques and Applications. LNCS, vol. 202, pp. 345–364. Springer, Heidelberg (1985)
Kreisel, G., Krivine, J.-L.: Elements of mathematical logic: model theory. Studies in Logic and the Foundations of Mathematics. North-Holland, revised second edition, 1971. First edition 1967. Translation of the French ‘Eléments de logique mathématique, théorie des modeles’ published by Dunod, Paris (1964)
Lifschitz, V.: Semantical completeness theorems in logic and algebra. Proceedings of the American Mathematical Society 79, 89–96 (1980)
Lipshitz, L.: The Diophantine problem for addition and divisibility. Transactions of the American Mathematical Society 235, 271–283 (1978)
Matiyasevich, Y.V.: Enumerable sets are Diophantine. Soviet Mathematics Doklady 11, 354–358 (1970)
Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen In welchem die Addition als einzige Operation hervortritt. In: Sprawozdanie z I Kongresu metematyków slowiańskich, Warszawa 1929, pp. 92–101, 395. Warsaw, 1930. Annotated English version by [20]
Robinson, J.: Definability and decision problems in arithmetic. Journal of Symbolic Logic. Author’s PhD thesis 14, 98–114 (1949)
Simmons, H.: The solution of a decision problem for several classes of rings. Pacific Journal of Mathematics 34, 547–557 (1970)
Stansifer, R.: Presburger’s article on integer arithmetic: Remarks and translation. Technical Report CORNELLCS:TR84-639, Cornell University Computer Science Department (1984)
Weispfenning, V., Becker, T.: Groebner bases: a computational approach to commutative algebra. In: Graduate Texts in Mathematics, Springer, Heidelberg (1993)
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Harrison, J. (2007). Automating Elementary Number-Theoretic Proofs Using Gröbner Bases. In: Pfenning, F. (eds) Automated Deduction – CADE-21. CADE 2007. Lecture Notes in Computer Science(), vol 4603. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73595-3_5
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DOI: https://doi.org/10.1007/978-3-540-73595-3_5
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