Our starting point is the observation that if graphs in a class C have low descriptive complexity, then the isomorphism problem for C is solvable by a fast parallel algorithm. More precisely, we prove that if every graph in C is definable in a finite-variable first order logic with counting quantifiers within logarithmic quantifier depth, then Graph Isomorphism for C is in TC1 ⊆ NC2. If no coun...

We address the problems of separation and description in some fragments of modal logics. The former consists in finding a formula that is true in some given subset of the domain and false in another. The latter is a special case when one separates a singleton from the rest. We are interested in the shortest size of both separations and descriptions. This is motivated by applications in computat...

We construct non-isomorphic models M,N , e.g. of cardinality א1 such that in the Ehrenfeucht-Fraissé game of any length ζ < ω1 the isomorphism player wins. Research supported by German-Israeli Foundation for Scientific Research and Development (Grant No.I-706-54.6/2001) I would like to thank Alice Leonhardt for the beautiful typing. First Typed 06/Oct/28 Latest Revision 08/Sept/25 Typeset by AM...

We introduce the inverted prefix tries (a variation of suffix tries) as a convenient formalism for stating and proving properties of the EhrenfeuchtMycielski sequence ([3]). We also prove an upper bound on the position in the sequence by which all strings of a given length will have appeared; our bound is given by the Ackermann function, which, in light of experimental data, may be a gross over...

The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and...

The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and...

Motivated by recent applications of finite automata to theoretical physics, we study the minimization problem for nondeterministic automata (with outputs, but no initial states). We use Ehrenfeucht–Fräısse-like games to model automata responses and simulations. The minimal automaton is constructed and, in contrast with the classical case, proved to be unique up to an isomorphism. Finally, we in...

Recently, Boutonnet, Chifan, and Ioana proved that McDuff’s family of continuum many pairwise nonisomorphic separable II1 factors are in fact pairwise non-elementarily equivalent by proving that any ultrapowers of two distinct members of the family are nonsiomorphic. We use Ehrenfeucht-Fraisse games to provide an upper bound on the quantifier-depth of sentences which distinguish these theories.