Abstract
We carry out a study of definability issues in the standard models of Presburger and Skolem arithmetics (henceforth referred to simply as Presburger and Skolem arithmetics, for short, because we only deal with these models, not the theories, thus there is no risk of confusion) supplied with free unary predicates—which are strongly related to definability in the monadic SOA (second-order arithmetic) without × or + , respectively. As a consequence, we obtain a very direct proof for \({\Pi^1_1}\) -completeness of Presburger, and also Skolem, arithmetic with a free unary predicate, generalize it to all \({\Pi^1_n}\) -levels, and give an alternative description of the analytical hierarchy without × or + . Here ‘direct’ means that one explicitly m-reduces the truth of \({\Pi^1_1}\) -formulae in SOA to the truth in the extended structures. Notice that for the case of Presburger arithmetic, the \({\Pi^1_1}\) -completeness was already known, but the proof was indirect and exploited some special \({\Pi^1_1}\) -completeness results on so-called recurrent nondeterministic Turing machines—for these reasons, it was hardly able to shed any light on definability issues or possible generalizations.
Similar content being viewed by others
References
Abadi M., Halpern J.Y.: Decidability and expressiveness for first-order logics of probability. Inf. Comput. 112(1), 1–36 (1994)
Bès, A.: A survey of arithmetical definability, A tribute to Maurice Boffa, Soc. Math. Belgique, 1–54 (2002)
Bès A., Richard D.: Undecidable extensions of Skolem Arithmetic. J. Symb. Log. 63(2), 379–401 (1998)
Downey, P.: Undecidability of Presburger Arithmetic with a Single Monadic Predicate Letter, Technical Report 18-72. Center for Research in Computing Technology, Harvard University, Cambridge (1972)
Ershov Yu.L., Lavrov I.A., Taimanov A.D., Taitslin M.A.: Elementary theories. Russ. Math. Surv. 20(4), 35–105 (1965)
Gaifman, H.: On local and nonlocal properties. In: Proceedings of the Herbrand Symposium ’81, Studies in Logic and the Foundations of Mathematics 107, pp. 105–135. North-Holland, Amsterdam (1982)
Garfunkel S., Schmerl J.H.: The undecidability of theories of groupoids with an extra predicate. Proc. AMS 42(1), 286–289 (1974)
Gurevich Yu.: Monadic second-order theories. In: Barwise, J., Feferman, S. (eds.) Model-Theoretic Logics, pp. 479–506. Springer, Berlin (1985)
Halpern J.Y.: Presburger arithmetic with unary predicates is \({\Pi_1^1}\) complete. J. Symb. Log. 56(2), 637–642 (1991)
Harel D., Pnueli A., Stavi J.: Propositional dynamic logic of nonregular programs. J. Comput. Syst. Sci. 26(2), 222–243 (1983)
Korec I.: A list of arithmetical structures complete with respect to the first-order definability. Theor. Comput. Sci. 257, 115–151 (2001)
Putnam H.: Decidability and essential undecidability. J. Symb. Log. 22, 39–54 (1957)
Robinson J.: Definability and decision problems in arithmetic. J. Symb. Log. 14, 98–114 (1949)
Rogers H.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)
Simpson S.G.: Subsystems of Second Order Arithmetic. Cambridge University Press, Cambridge (2009)
Speranski, S.O.: Complexity for probability logic with quantifiers over propositions. J. Log. Comput. (2012). doi:10.1093/logcom/exs041
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Speranski, S.O. A note on definability in fragments of arithmetic with free unary predicates. Arch. Math. Logic 52, 507–516 (2013). https://doi.org/10.1007/s00153-013-0328-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-013-0328-9
Keywords
- Definability
- Expressiveness
- Decidability
- Computational complexity
- Presburger arithmetic
- Skolem arithmetic