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

[1]  Françoise Gire,et al.  An Extension of Fixpoint Logic with a Symmetry-Based Choice Construct , 1998, Inf. Comput..

[2]  Neil Immerman,et al.  An optimal lower bound on the number of variables for graph identification , 1992, Comb..

[3]  Neil Immerman,et al.  Relational Queries Computable in Polynomial Time , 1986, Inf. Control..

[4]  Serge Abiteboul,et al.  Generic Computation and its complexity , 1991, STOC '91.

[5]  Saharon Shelah,et al.  On finite rigid structures , 1994, Journal of Symbolic Logic.

[6]  Martin Otto,et al.  Bounded Variable Logics and Counting , 1997 .

[7]  David Harel,et al.  Structure and complexity of relational queries , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[8]  Yuri Gurevichz.  Metaanite Model Theory , 1997 .

[9]  Saharon Shelah,et al.  Choiceless Polynomial Time , 1997, Ann. Pure Appl. Log..

[10]  Jan Van den Bussche,et al.  Abstract State Machines and Computationally Complete Query Languages , 2000, Abstract State Machines.

[11]  Yuri Gurevich,et al.  From Invariants to Canonization , 2001, Bull. EATCS.

[12]  Saharon Shelah.  Choiceless Polynominal Time Logic: Inability to Express , 2000, CSL.

[13]  Andreas Blass,et al.  Strong extension axioms and Shelah's zero-one law for choiceless polynomial time , 2003, Journal of Symbolic Logic.

[14]  Moshe Y. Vardi.  The complexity of relational query languages (Extended Abstract) , 1982, STOC '82.

[15]  Yuri Gurevich,et al.  Metafinite Model Theory , 1994, LCC.

[16]  Andreas Blass,et al.  Equivalence relations, invariants, and normal forms , 1983, Logic and Machines.

[17]  Egon Börger,et al.  Logic and Machines: Decision Problems and Complexity , 1984, Lecture Notes in Computer Science.

[18]  Andreas Blass,et al.  Equivalence Relations, Invariants, and Normal Forms , 1983, SIAM J. Comput..

[19]  Yuri Gurevich,et al.  Evolving algebras 1993: Lipari guide , 1995, Specification and validation methods.