Linear time and the power of one first-order universal quantifier

Authors:
Arnaud Durand
Affiliations:
LACL, Départment Informatique, Faculté des Sciences et technologie, Univesité Paris XII-Val de Marne, 61 avenue du général de Gaulle, Créteil Cedex, France
Venue:
Information and Computation
Year:
2002

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Abstract

The aim of this paper is to give two new logical characterizations of NLIN (nondeterministic linear time) improving significantly a previous characterization of E. Grandjean (1994, SIAM J. Comput. 23, 573-597; and F. Olive, 1998, Comput. Complexity 7, 54-97). It is known that NLIN coincides with the class of problems definable by formulas of the prenex form f1... fk ψ where the fi are unary function symbols and ψ is first-order, prenex, with only one universal quantifier. We show that the characterization remains true in the two following cases: (a) Unary functions are replaced by a single binary relation whose outdegree is bounded by some fixed constant h. (b) When only ordered structures are considered, unary functions are restricted to be bijective.