Abstract
We investigate properties of the coprimality relation within the family of finite models being initial segments of the standard model for coprimality, denoted by \(\mathrm{FM}((\omega,\bot))\).
Within \(\mathrm{FM}((\omega,\bot))\) we construct an interpretation of addition and multiplication on indices of prime numbers. Consequently, the first order theory of \(\mathrm{FM}((\omega,\bot))\) is Π\(^{\rm 0}_{\rm 1}\)–complete (in contrast to the decidability of the theory of multiplication in the standard model). This result strengthens an analogous theorem of Marcin Mostowski and Anna Wasilewska, 2004, for the divisibility relation.
As a byproduct we obtain definitions of addition and multiplication on indices of primes in the model \((\omega,\bot,\leq_{P_2})\), where P 2 is the set of primes and products of two different primes and ≤ X is the ordering relation restricted to the set X. This can be compared to the decidability of the first order theory of \((\omega,\bot,\leq_P)\), for P being the set of primes (Maurin, 1997) and to the interpretation of addition and multiplication in \((\omega,\bot,\leq_{P^2})\), for P 2 being the set of primes and squares of primes, given by Bès and Richard, 1998.
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References
Bès, A., And Richard, D.: Undecidable extensions of Skolem arithmetic. Journal of Symbolic Logic 63, 379–401 (1998)
Gurevich, Y.: Logic and the Challenge of Computer Science. In: Börger, E. (ed.) Current Trends in Theoretical Computer Science, pp. 1–7. Computer Science Press, Rockville (1988)
Hoare, C.A.R.: An axiomatic basis for computer programming. Communications of the ACM 12, 576–583 (1969)
Jameson, G.J.O.: The prime number theorem. Cambridge University Press, Cambridge (2003)
Kołodziejczyk, L.A.: Truth definitions in finite models. Journal of Symbolic Logic 69, 183–200 (2004)
Kołodziejczyk, L.A.: A finite model-theoretical proof of a property of bounded query classes within PH. Journal of Symbolic Logic 69, 1105–1116 (2004)
Krynicki, M., And Zdanowski, K.: Theories of arithmetics in finite models. Journal of Symbolic Logic 70, 1–28 (2005)
Lee, T.: Arithmetical definability over finite structures. Mathematical Logic Quarterly 49, 385–393 (2003)
Maurin, F.: The theory of integer multiplication with order restricted to primes is decidable. Journal of Symbolic Logic 62, 123–130 (1997)
Mostowski, M.: On representing concepts in finite models. Mathematical Logic Quarterly 47, 513–523 (2001)
Mostowski, M.: On representing semantics in finite models. In: Rojszczak, A., Cachro, J., Kurczewski, G. (eds.) Philosophical Dimensions of Logic and Science, pp. 15–28. Kluwer Academic Publishers, Dordrecht (2003)
Mostowski, M., Andwasilewska, A.: Arithmetic of divisibility in finite models. Mathematical Logic Quarterly 50, 169–174 (2004)
Mostowski, M., And Zdanowski, K.: FM–representability and beyond. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 358–367. Springer, Heidelberg (2005)
Schweikardt, N.: Arithmetic, First-Order Logic, and Counting Quantifiers. ACMTransactions on Computational Logic 5, 1–35 (2004)
Sierpiński, W.: Elementary Theory of Numbers. PWN (Polish Scientific Publishers), North Holland (1964)
Szczerba, L.W.: Interpretability of elementary theories. In: Butts, Hintikka (eds.) Proceedings 15th ICALP 88 Logic, foundations of mathematics and computability theory, pp. 129–145. Reidel Publishing, Dordrecht (1977)
Trachtenbrot, B.: The impossibility of an algorithm for the decision problem for finite domains. In: Doklady Akademii Nauk SSSR, vol. 70, pp. 569–572 (1950) (in Russian)
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Mostowski, M., Zdanowski, K. (2005). Coprimality in Finite Models. In: Ong, L. (eds) Computer Science Logic. CSL 2005. Lecture Notes in Computer Science, vol 3634. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538363_19
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DOI: https://doi.org/10.1007/11538363_19
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