# Provability and Decidability of Arithmetical Sentences

Conference paper
Part of the Logic in Asia: Studia Logica Library book series (LIAA)

## Abstract

Gödel’s incompleteness theorem tells that the elementary theories of $${\mathbf N}$$ and $${\mathbf Z}$$ are not axiomatizable, respectively. J. Robinson proved that the elementary theory of $${\mathbf Q}$$ is undecidable, hence not axiomatizable. We may ask what subsets of these elementary theories are decidable or axiomatizable. We call an arithmetical sentence $$\phi$$ an $$\exists ^n\forall \exists$$ sentence if it is logically equivalent to a sentence of the form $$\exists x_1\cdots \exists x_n\forall y\exists z \,\psi (x_1, \ldots ,x_n,y,z),$$ where $$\psi (x_1, x_2,\ldots ,x_n,y,z)$$ is a quantifier-free formula. In this talk we show that the set of all true $$\exists ^n\forall \exists$$ sentences in $$\mathbf N,$$ $$\mathbf Z$$, and $$\mathbf Q$$ are axiomatizable, respectively. Also, let $$f(x,y)\in {\mathbf Z}[x,y]$$ and $$a\in {\mathbf Z}$$. The sets of all sentences of the form $$\forall z\exists x \forall y\, f(x,y)-az \ne 0$$ true in $$\mathbf N,$$ $$\mathbf Z$$, and $$\mathbf Q$$ are also axiomatizable, respectively. Therefore, the sets of sentences of the form $$\exists z\forall x \exists y\, f(x,y)-az =0$$ true in $$\mathbf N,$$ $$\mathbf Z$$, and $$\mathbf Q$$ are decidable, respectively. Over $$\mathbf N$$, we have shown that this decision problem is NP-hard, co-NP-hard, and in PSPACE. Therefore, if NP $$\ne$$ co-NP, then this decision problem is in PSPACE $$\setminus$$ (NP $$\cup$$ co-NP).

## References

1. Baker, A. (1990). Transcendental number theory (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
2. Davis, M. (1973). Hilbert’s tenth problem is unsolvable. The American Mathematical Monthly, 80, 233–269.
3. Enderton, H. B. (2001). A mathematical introduction to logic (2nd ed.). San Diego: Harcourt/Academic Press.Google Scholar
4. Fried, M. D., & Jarden, M. (2005). Field arithmetic (2nd ed.). Berlin: Springer.Google Scholar
5. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A Guide to theory of NP-completeness. San Francisco: W. H. Freeman.Google Scholar
6. Hungerford, T. W. (1974). Algebra. New York: Springer.Google Scholar
7. Jones, J. P. (1981). Classification of quantifier prefixes over diophantine equations. Z. Math. Logik Grundlag. Math., 27, 403–410.
8. Jones, J. P. (1982). Universal diophantine equation. Journal of Symbolic Logic, 47, 549–571.
9. Lang, S. (1997). Survey of diophantine geometry. Berlin: Springer.Google Scholar
10. Manders, K., & Adleman, L. (1978). NP-complete decision problems for binary quadratics. Journal of Computer and System Sciences, 16, 168–184.
11. Matijasevich, Y. V., & Robinson, J. (1974). Two universal 3-quantifier representations of recursively enumerable sets. Teoriya Algorifmov i Matematicheskaya Logika, (volume dedicated to A. A. Markov), 112–123, Vychislitel’nyĭ Tsentr, Akademiya Nauk SSSR, Moscow, (Russian).Google Scholar
12. Robinson, J. (1949). Definability and decision problems in arithmetic. Journal of Symbol Logic, 14, 98–114.
13. Schinzel, A. (1982). Diophantine equations with parameters. In J. V. Arimitage (Ed.), Journées arithmetiques 1980 (pp. 211–217)., London mathematical society lecture note series Cambridge: Cambridge University Press.
14. Schinzel, A. (2000). Polynomials with special regard to reducibility. Cambridge: Cambridge University Press.
15. Tung, S. P. (1985). On weak number theories. Japanese Journal of Mathematics, 11, 203–232.
16. Tung, S. P. (1986). Provability and decidability of arithmetical universal-existential sentences. Bulletin of the London Mathematical Society, 18, 241–247.
17. Tung, S. P. (1987). Computational complexities of diophantine equations with parameters. Journal of Algorithms, 8, 324–336.
18. Tung, S. P. (1995). Computational complexity of arithmetical sentences. Information and Computation, 120, 315–325.
19. Tung, S. P. (2017). Diophantine equations with three quantifiers, manuscript.Google Scholar
20. Wüstholz, G. (Ed.). (2002). A Panorama of number theory or the view from Baker’s garden. Cambridge: Cambridge University Press.Google Scholar