Journal of Symbolic Logic

On polynomial time computation over unordered structures

Andreas Blass, Yuri Gurevich, and Saharon Shelah

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This paper is motivated by the question whether there exists a logic capturing polynomial time computation over unordered structures. We consider several algorithmic problems near the border of the known, logically defined complexity classes contained in polynomial time. We show that fixpoint logic plus counting is stronger than might be expected, in that it can express the existence of a complete matching in a bipartite graph. We revisit the known examples that separate polynomial time from fixpoint plus counting. We show that the examples in a paper of Cai, Fürer, and Immerman, when suitably padded, are in choiceless polynomial time yet not in fixpoint plus counting. Without padding, they remain in polynomial time but appear not to be in choiceless polynomial time plus counting. Similar results hold for the multipede examples of Gurevich and Shelah, except that their final version of multipedes is, in a sense, already suitably padded. Finally, we describe another possible candidate, involving determinants, for the task of separating polynomial time from choiceless polynomial time plus counting.

Article information

J. Symbolic Logic, Volume 67, Issue 3 (2002), 1093-1125.

First available in Project Euclid: 18 September 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 68Q19: Descriptive complexity and finite models [See also 03C13]
Secondary: 03B70: Logic in computer science [See also 68-XX] 03C13: Finite structures [See also 68Q15, 68Q19] 03D15: Complexity of computation (including implicit computational complexity) [See also 68Q15, 68Q17]


Blass, Andreas; Gurevich, Yuri; Shelah, Saharon. On polynomial time computation over unordered structures. J. Symbolic Logic 67 (2002), no. 3, 1093--1125. doi:10.2178/jsl/1190150152.

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