Abstract
We investigate the decidability of first-order logic extensions. For example, it is established in A. S. Zolotov’s works that a logic with a unary transitive closure operator for the one successor theory is decidable. We show that in a similar case, a logic with a unary partial fixed point operator is undecidable. For this purpose, we reduce the halting problem for the counter machine to the problem of truth of the underlying formula. This reduction uses only one unary non-nested partial fixed operator that is applied to a universal or existential formula.
Reference7 articles.
1. Aho A., Ulman J.D. Universality of data retrieval languages, Proceedings of the 6th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages POPL ’79, 110–119 (1979).
2. Gurevich Y., Shelah S. Fixed-point extensions of first-order logic, Ann. Pure and Appl. Logic 32, 265–280 (1986).
3. Dudakov S.M., Taitslin M. A. Collapse results for query languages in database theory, UMN 61 (2), 3–66 (2006).
4. Zolotov A.S. On decidability of the theory with the transitive closure operator, Lobachevskii J. Math. 36 (4), 434–440 (2015).
5. Sekorin V.S. Ob ekvivalentnosti dvukh semantik PFP-operatora, Vestn. Tversk. gos. un-ta. Ser. Prikl. matem. (3), 41–49 (2020).