Equicardinality on linear orders

Linear orders are of inherent interest infinite model theory, especially in descriptive complexity theory. Here, the class of ordered structures is approached from a novel point of view, using generalized quantifiers as a means of analysis. The main technical result is a characterization of the cardinality quantifiers which can express equicardinality on ordered structures. This result can be viewed as a dichotomy: the cardinality quantifier either shows a lot of periodicity, or is quite non-periodic, the equicardinality quantifier being definable only in the latter case. The main result shows, once more, that there is a drastic difference between definability among ordered structures and definability on unordered structures. Connections of the result to the descriptive complexity of low-level complexity classes are discussed.

[1]  Kerkko Luosto Hierarchies of Monadic Generalized Quantifiers , 2000, J. Symb. Log..

[2]  Serge Abiteboul,et al.  Procedural and declarative database update languages , 1988, PODS '88.

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

[4]  Anuj Dawar Generalized Quantifiers and Logical Reducibilities , 1995, J. Log. Comput..

[5]  Ronald Fagin Generalized first-order spectra, and polynomial. time recognizable sets , 1974 .

[6]  Phokion G. Kolaitis,et al.  Generalized quantifiers and pebble games on finite structures , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.

[7]  X. Caicedo Back-and-Forth Systems for Arbitrary Quantifiers , 1980 .

[8]  Thomas Schwentick,et al.  Locality of order-invariant first-order formulas , 1998, TOCL.

[9]  Neil Immerman,et al.  Descriptive Complexity , 1999, Graduate Texts in Computer Science.

[10]  E. Lander,et al.  Describing Graphs: A First-Order Approach to Graph Canonization , 1990 .

[11]  Matthias Ruhl,et al.  Counting and addition cannot express deterministic transitive closure , 1999, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158).

[12]  Perlindström First Order Predicate Logic with Generalized Quantifiers , 1966 .

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

[14]  Nicole Schweikardt,et al.  On the expressive power of first-order logic with built-in predicates , 2001 .

[15]  Lauri Hella,et al.  The expressive Power of Finitely Many Generalized Quantifiers , 1995, Inf. Comput..

[16]  Kousha Etessami,et al.  Counting quantifiers, successor relations, and logarithmic space , 1995, Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference.

[17]  Lauri Hella,et al.  The hierarchy theorem for generalized quantifiers , 1996, Journal of Symbolic Logic.

[18]  Lauri Hella,et al.  Enhancing Fixed Point Logic with Cardinality Quantifiers , 1998, J. Log. Comput..

[19]  Lauri Hella Logical Hierarchies in PTIME , 1996, Inf. Comput..

[20]  Juha Nurmonen,et al.  Counting Modulo Quantifiers on Finite Structures , 2000, Inf. Comput..

[21]  Martin Weese Generalized Ehrenfeucht games , 1980 .

[22]  Kerkko Luosto Ramsey Theory Is Needed for Solving Definability Problems of Generalized Quantifiers , 1997, ESSLLI.

[23]  Jouko Väänänen Unary Quantifiers on Finite Models , 1997, J. Log. Lang. Inf..