On Polynomial Time Computation over Unordered Structures

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 plausible candidate, involving determinants, for the task of separating polynomial time from choiceless polynomial time plus counting. Partially supported by NSF grant DMS–0070723 and a grant from Microsoft Research. Address: Mathematics Department, University of Michigan, Ann Arbor, MI 48109–1109, U.S.A. email: [email protected] Microsoft Research, One Microsoft Way, Redmond, WA 98052, U.S.A. e-mail: [email protected] Partially supported by U.S.-Israel Binational Science Foundation. Address: Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel, and Mathematics Department, Rutgers University, New Brunswick, NJ 08903, U.S.A. e-mail: [email protected]