This paper introduces a new Ehrenfeucht-Fraı̈ssé type game that is played on two classes of models rather than just two models. This game extends and generalizes the known Ajtai-Fagin game to the case when there are several alternating (coloring) moves played in different models. The game allows Duplicator to delay her choices of the models till (practically) the very end of the game, making it ...

Algebras on the natural numbers and their clones of term operations can be classified according to their descriptive complexity. We give an example of a closed algebra which has only unary operations and whose clone of term operations is not Borel. Moreover, we provide an example of a coatom in the clone lattice whose obvious definition via an ideal of subsets of natural numbers would suggest t...

We investigate ambiguity for symmetric difference nondeterministic finite automata. We show the existence of unambiguous, finitely ambiguous, polynomially ambiguous and exponentially ambiguous symmetric difference nondeterministic finite automata. We show that, for each of these classes, there is a family of n-state nondeterministic finite automata such that the smallest equivalent deterministi...

In this paper, we consider existential monadic second-order logic (monadic Σ1) on finite unary graphs, that is finite graphs with functional edge relation. These can be seen as finite encodings of ultimately periodic words. We show that, on these graphs, the bisimulation invariant fragment of monadic Σ1 equals the bisimulation invariant fragment of monadic second order logic itself, and that MS...

Exploration of the connections between computational complexity, descriptive complexity, and logic remains one of the most active and important areas of theoretical computer science. In this edition of the Computational Complexity Column, we hear from one of the leaders in this area.

We exhibit a close connection between parameterized complexity theory and logic. Our approach is that of descriptive complexity theory. We study the definability of parameterized problems and try to obtain information about the parameterized complexity of the problems through the syntactical structure of the defining sentences. On the one hand, we use this approach to prove that certain problem...

The descriptional complexity of semi-conditional grammars is studied. A proof that every recursively enumerable language is generlji";'I":ff f iru*::?H:"ffi Xl,?li".ffi f iJ,,-,T?,ll,J"'"'n'"

The equivalence of finite automata and regular expressions dates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the present paper we tour a fragment of the literature and summarize results on upper and lower bounds on the conversion of finite automata to regular expressions and vice versa. We also briefly recall the known bounds for the removal of ...

An ω-language is a set of infinite words over a finite alphabet X. We consider the class of recursive ω-languages, i.e. the class of ω-languages accepted by Turing machines with a Büchi acceptance condition, which is also the class Σ1 of (effective) analytic subsets of X for some finite alphabet X. We investigate here the notion of ambiguity for recursive ω-languages with regard to acceptance b...

We consider deterministic and nondeterministic finite automata with acceptance conditions that rely on the whole history of a computation on a given word and not only on the last state of the computation under consideration. Formally, these conditions can be seen as the natural analogies of the Büchi and Muller acceptance for finite automata on infinite words. We study the computational power o...