We survey recent results on the topological complexity of context-free ω-languages which form the second level of the Chomsky hierarchy of languages of infinite words. In particular, we consider the Borel hierarchy and the Wadge hierarchy of non-deterministic or deterministic context-free ω-languages. We study also decision problems, the links with the notions of ambiguity and of degrees of amb...

The operators min·, max·, and #· translate classes of the polynomial-time hierarchy to function classes. Although the inclusion relationships between these function classes have been studied in depth, some questions concerning separations remained open. We provide oracle constructions that answer most of these open questions in the relativized case. As a typical instance for the type of results...

We give machine characterizations of the complexity classes of the W-hierarchy. Moreover, for every class of this hierarchy, we present a parameterized halting problem complete for this class.

We survey recent results on the topological complexity of context-free ω-languages which form the second level of the Chomsky hierarchy of languages of infinite words. In particular, we consider the Borel hierarchy and the Wadge hierarchy of non-deterministic or deterministic context-free ω-languages. We study also decision problems, the links with the notions of ambiguity and of degrees of amb...

It is shown for any constant c < 1, that if NPMV have re nements in the class of multivalued functions that are polynomial time computable with c(log n) access to NPSV, then the polynomial time hierarchy collapses to its second level.

(1) Many typos have been corrected (2) Some redoundancy has been eliminated Notes on polynomially bounded arithmetic Abstract We characterize the collapse of Buss' bounded arithmetic in terms of the prov-able collapse of the polynomial time hierarchy. We include also some general model-theoretical investigations on fragments of bounded arithmetic.

We show that the bounded arithmetic theory V does not prove that the polynomial time hierarchy collapses to the linear time hierarchy (without parameters). The result follows from a lower bound for bounded depth circuits computing prefix parity, where the circuits are allowed some auxiliary input; we derive this from a theorem of Ajtai.

The star height hierarchy (resp. the variable hierarchy) results in classifying μ-terms into classes according to the nested depth of fixed point operators (resp. to the number of bound variables). We prove, under some assumptions, that the variable hierarchy is a proper refinement of the star height hierarchy. We mean that the non collapse of the variable hierarchy implies the non collapse of ...