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Complete problems for fixed-point logics

  • Martin Grohe (a1)

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The notion of logical reducibilities is derived from the idea of interpretations between theories. It was used by Lovász and Gács [LG77] and Immerman [Imm87] to give complete problems for certain complexity classes and hence establish new connections between logical definability and computational complexity.

However, the notion is also interesting in a purely logical context. For example, it is helpful to establish nonexpressibility results.

We say that a class of τ-structures is a >complete problem for a logic under L-reductions if it is definable in [τ] and if every class definable in can be ”translated” into by L-formulae (cf. §4).

We prove the following theorem:

1.1. Theorem. There are complete problemsfor partial fixed-point logic andfor inductive fixed-point logic under quantifier-free reductions.

The main step of the proof is to establish a new normal form for fixed-point formulae (which might be of some interest itself). To obtain this normal form we use theorems of Abiteboul and Vianu [AV91a] that show the equivalence between the fixed-point logics we consider and certain extensions of the database query language Datalog.

In [Dah87] Dahlhaus gave a complete problem for least fixed-point logic. Since least fixed-point logic equals inductive fixed-point logic by a well-known result of Gurevich and Shelah [GS86], this already proves one part of our theorem.

However, our class gives a natural description of the fixed-point process of an inductive fixed-point formula and hence sheds some light on completely different aspects of the logic than Dahlhaus's construction, which is strongly based on the features of least fixed-point formulae.

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References

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[AV91a] Abiteboul, S. and Vianu, V., Datalog extensions for database queries and updates, Journal of Computer and System Sciences, vol. 43 (1991), pp. 62124.
[AV91b] Abiteboul, S. and Vianu, V., Generic computation and its complexity, Proceedings of 23rd ACM symposium on theory of computing (1991), pp. 209219.
[Dah87] Dahlhaus, E., Skolem normal forms concerning the least fixed point, Computation theory and logic (Börger, E., editor), Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1987, pp. 2036.
[Daw93] Dawar, A., Feasible computation through model theory, Ph.D. thesis, University of Pennsylvania, University Park, Pennsylvania, 1993.
[EF] Ebbinghaus, H. D. and Flum, J., Finite model theory (to appear).
[EFT94] Ebbinghaus, H. D., Flum, J., and Thomas, W., Mathematical logic, 2nd ed, Springer-Verlag, Berlin, 1994.
[GS86] Gurevich, Y. and Shelah, S., Fixed point extensions of first-order logic, Annals of Pure and Applied Logic, vol. 32 (1986), pp. 265280.
[Hel94] Hella, L., Fixpoint logic vs generalized quantifiers, manuscript (1994).
[Imm87] Immerman, N., Languages that capture complexity classes, SIAM Journal on Computing, vol. 16 (1987), pp. 760778.
[LG77] Lovász, L. and Gacs, P., Some remarks on generalized spectra, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 23 (1977), pp. 27144.
[Mos74] Moschovakis, Y. N., Elementary induction on abstract structures, North-Holland, Amsterdam, 1974.
[MP93] Makowsky, J. A. and Pnueli, Y. B., Oracles and quantifiers, Proceedings of the 7th workshop on computer science logic, Lecture Notes in Computer Science, vol. 832, Springer-Verlag, Berlin, 1993, pp. 150164.

Complete problems for fixed-point logics

  • Martin Grohe (a1)

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