Abstract
Let \({\cal A}\) be a partial algebra on a finite signature. We say that \({\cal A}\) has decidable query evaluation problem if there exists an algorithm that given a first order formula \(\phi(\bar{x})\) and a tuple \(\bar{a}\) from the domain of \({\cal A}\) decides whether or not \(\phi(\bar{a})\) holds in \({\cal A}\). Denote by \(E({\cal A})\) the total algebra freely generated by \({\cal A}\). We prove that if \({\cal A}\) has a decidable query evaluation problem then so does \(E({\cal A})\). In particular, the first order theory of \(E({\cal A})\) is decidable. In addition, if \({\cal A}\) has elimination of quantifiers then so does \(E({\cal A})\) extended by finitely many definable selector functions and tester predicates. Our proof is a refinement of the quantifier elimination procedure for free term algebras. As an application we show that any finitely presented term algebra has a decidable query evaluation problem. This extends the known result that the word problem for finitely presented term algebras is decidable.
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References
Astesiano, E., Bidoit, M., Kirchner, H., Krieg-Brückner, B., Mosses, P.D., Sannella, D., Tarlecki, A.: Casl: The Common Algebraic Specification Language. Theoretical Computer Science 286(2), 153–196 (2002)
Blumensath, A., Gradel, E.: Automatic structures. In: Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science (LICS), pp. 51–62 (2000)
Comon, H.: Complete axiomatization of some quotient term algebras. Theoretical Computer Science 122(1-2), 165–200 (1993)
Compton, K., Henson, C.: A uniform method for proving lower bounds on computational complexity of logical theories. Annals of Pure and Applied Logic 48, 1–79 (1990)
Hodges, W.: Model theory. Cambridge University Press, Cambridge (1993)
Khoussainov, B., Rubin, S., Stephan, F.: Automatic Linear Orders and Trees. In: Kolaitis, P.G. (ed.) Transactions on Computational Logic (TOCL) special issue (selected papers from the LICS 2003 conference) (2003) (accepted)
Khoussainov, B., Nerode, A.: Automatic Presentations of Structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–393. Springer, Heidelberg (1995)
Korovin, K., Voronkov, A.: A decision procedure for the existential theory of term algebra with the Knuth-Bendix ordering. In: Proceedings IEEE Conference on Logic in Computer Science, pp. 291–302 (2000)
Grätzer, G.: Universal Algebra. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London (1968)
Kozen, D.: Complexity of finitely presented algebras. In: Proceedings of the 9th ACM symposium on theory of computing, pp. 164–177 (1977)
Kozen, D.: Partial automata and finitely generated congruences: an extension of Nerode’s theorem. In: Crossley, J., Remmel, J., Shore, R., Sweedler, M. (eds.) Logical methods: in honour of Anil Nerode’s Siztieth Birthday, Birkhauser, pp. 490–511 (1993)
Kuncak, V., Rinard, M.C.: Structural sybtyping of non-recursive types is decidable. In: Proceedings IEEE Conference on Logic in Computer Science, pp. 96–107 (2003)
Lohrey, M.: Automatic structures of bounded degree. In: Y. Vardi, M., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, pp. 346–360. Springer, Heidelberg (2003)
Maher, M.: A CLP view of logic programming. In: Algebraic and Logic Programming, pp. 364–383 (1992)
A. Mal’cev: Axiomatizable classes of locally free algebras of various types. In The Metamathimatics of Algebraic Systems. AnatoliÄ Ivanovic̆ Mal’cev. Collected papers: 1936-1967, B. Wells III, Ed. Vol. 66. North Holland. Chapter 23, 262–281 (1971)
Zhang, T., Sipma, H., Manna, Z.: Term algebras with length function and bounded quantifier alternation. In: Slind, K., Bunker, A., Gopalakrishnan, G.C. (eds.) TPHOLs 2004. LNCS, vol. 3223, pp. 321–336. Springer, Heidelberg (2004)
Rybina, T., Voronkov, A.: A Decision procedure for term algebras with queues. ACM Transaction on Computational Logic 2(2), 155–181 (2001)
Vorobyov, S.: An improved lower bound for the elementary theories of trees. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS, vol. 1104, pp. 275–287. Springer, Heidelberg (1996)
Walukiewicz, I.: Monadic second order logic on tree-like structures. Theoretical computer science 275(1-2), 311–346 (2002)
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Khoussainov, B., Rubin, S. (2005). Decidability of Term Algebras Extending Partial Algebras. In: Ong, L. (eds) Computer Science Logic. CSL 2005. Lecture Notes in Computer Science, vol 3634. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538363_21
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DOI: https://doi.org/10.1007/11538363_21
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