Logical Theory of the Additive Monoid of Subsets of Natural Integers
We consider the logical theory of the monoid of subsets of ℕ endowed solely with addition lifted to sets: no other set theoretical predicate or function, no constant (contrarily to previous work by J̇ez and Okhotin cited below). We prove that the class of true Σ5 formulas is undecidable and that the whole theory is recursively isomorphic to second-order arithmetic. Also, each ultimately periodic set A (viewed as a predicate X = A) is Π4 definable and their collection is Σ6. Though these undecidability results are not surprising, they involve technical difficulties witnessed by the following facts: 1) no elementary predicate or operation on sets (inclusion, union, intersection, complementation, adjunction of 0) is definable, 2) The class of subsemigroups is not definable though that of submonoids is easily definable. To get our results, we code integers by a Π3 definable class of submonoids and arithmetic operations on ℕ by Δ5 operations on this class.
KeywordsMinimum Generate Order Theory Logical Theory Great Common Divisor Numerical Semigroup
Unable to display preview. Download preview PDF.
- 1.Choffrut, C., Grigorieff, S.: Logical theory of the monoid of languages over a non tally alphabet (in preparation)Google Scholar
- 2.Cohn, L.: On the submonoids of the additive group of integers, http://www.macs.citadel.edu/cohnl/submonoids2002.pdf
- 3.Fischer, M.J., Rabin, M.O.: Super-exponential complexity of presburger arithmetic. In: SIAM-AMS Symposium in Applied Mathematics, vol. 7, pp. 27–41 (1974)Google Scholar
- 4.García-Sánchez, P.A.: Numerical semigroups minicourse, http://www.ugr.es/~pedro/minicurso-porto.pdf
- 6.J̇ez, A., Okhotin, A.: Equations over sets of natural numbers with addition only. In: STACS, pp. 577–588 (2009)Google Scholar
- 7.Odifreddi, P.: Classical recursion theory. The theory of functions and sets of natural integers, vol. 1. North Holland (1989)Google Scholar
- 8.Ramaré, O.: On Shnirelman’s constant 22(4), 645–706 (1995)Google Scholar
- 9.Ramírez-Alfonsín, J.L.: The Diophantine Frobenius Problem. Oxford University Press (2005)Google Scholar
- 10.Rogers, H.: Theory of recursive functions and effective computability. McGraw Hill (1967)Google Scholar
- 11.Rosales, J.C., García-Sánchez, P.A.: Numerical semigroups. Developments in Mathematics, vol. 20. Springer (2009)Google Scholar