# Interpretations of Presburger Arithmetic in Itself

• Alexander Zapryagaev
• Fedor Pakhomov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10703)

## Abstract

Presburger arithmetic $$\mathop {\mathbf {PrA}}\nolimits$$ is the true theory of natural numbers with addition. We study interpretations of $$\mathop {\mathbf {PrA}}\nolimits$$ in itself. We prove that all one-dimensional self-interpretations are definably isomorphic to the identity self-interpretation. In order to prove the results we show that all linear orders that are interpretable in $$(\mathbb {N},+)$$ are scattered orders with the finite Hausdorff rank and that the ranks are bounded in terms of the dimension of the respective interpretations. From our result about self-interpretations of $$\mathop {\mathbf {PrA}}\nolimits$$ it follows that $$\mathop {\mathbf {PrA}}\nolimits$$ isn’t one-dimensionally interpretable in any of its finite subtheories. We note that the latter was conjectured by A. Visser.

## Keywords

Presburger Arithmetic Interpretations Scattered linear orders

## Notes

### Acknowledgments

The authors wish to thank Lev Beklemishev for suggesting to study Visser’s conjecture, number of discussions of the subject, and his useful comments on the paper.

## References

1. 1.
Apelt, H.: Axiomatische Untersuchungen über einige mit der Presburgerschen Arithmetik verwandte Systeme. MLQ Math. Log. Q. 12(1), 131–168 (1966)Google Scholar
2. 2.
Barrington, D., Immerman, N., Straubing, H.: On uniformity within NC1. J. Comput. System Sci. 41(3), 274–306 (1990)
3. 3.
Blakley, G.R.: Combinatorial remarks on partitions of a multipartite number. Duke Math. J. 31(2), 335–340 (1964)
4. 4.
Ginsburg, S., Spanier, E.: Semigroups, Presburger formulas, and languages. Pacific J. Math. 16(2), 285–296 (1966)
5. 5.
Hájek, P., Pudlák, P.: Metamathematics of First-Order Arithmatic. Springer, New York (1993)
6. 6.
Fueter, R., Pólya, G.: Rationale Abzhlung der Gitterpunkte, Vierteljschr. Naturforsch. Ges. Zrich 58, 280–386 (1923)Google Scholar
7. 7.
Hausdorff, F.: Grundzüge einer Theorie der geordneten Mengen. Math. Ann. 65(4), 435–505 (1908)
8. 8.
Hodges, W.: Model Theory, vol. 42. Cambridge University Press, Cambridge (1993)
9. 9.
Ito, R.: Every semilinear set is a finite union of disjoint linear sets. J. Comput. Syst. Sci. 3(2), 221–231 (1969)
10. 10.
Khoussainov, B., Rubin, S., Stephan, F.: Automatic linear orders and trees. ACM Trans. Comput. Log. 6(4), 675–700 (2005)
11. 11.
Nathanson, M.B.: Cantor polynomials and the Fueter-Pólya theorem. Am. Math. Monthly 123(10), 1001–1012 (2016)
12. 12.
Tarski, A., Mostowski, A., Robinson, R.M.: Undecidable Theories. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1953)
13. 13.
Presburger, M.: Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. Comptes Rendus du I congrès de Mathématiciens des Pays Slaves 92101 (1929). English translation in [16]Google Scholar
14. 14.
Rosenstein, J.: Linear Orderings, vol. 98. Academic Press, New York (1982)
15. 15.
Schweikardt, N.: Arithmetic, first-order logic, and counting quantifiers. ACM Trans. Comput. Log. 6(3), 634–671 (2005)
16. 16.
Stansifer, R.: Presburger’s Article on Integer Arithmetic: Remarks and Translation (Technical report). Cornell University (1984)Google Scholar
17. 17.
Sturmfels, B.: On vector partition functions. J. Combin. Theory Ser. A 72(2), 302–309 (1995)
18. 18.
Visser, A.: An overview of interpretability logic. In: Kracht, M., de Rijke, M., Wansing, H., Zakharyaschev, M. (eds.) Advances in Modal Logic. CSLI Lecture Notes, vol. 87, pp. 307–359 (1998)Google Scholar
19. 19.
Zoethout, J.: Interpretations in Presburger Arithmetic. BS thesis. Utrecht University (2015)Google Scholar