Klaus W. Wagner

On ω-regular sets

The investigation of the acceptional power of finite automata with respect to several notions of acceptance for ω-sets, done in the literature, has exhibited six subclasses of the class of ω-regular sets. These classes are well characterized by means of topology and set representation. First we give an overview of these results. The main aim of this paper is the investigation of further natural subclasses of the class of ω-regular sets. We define such subclasses by three methods: by the structural complexity of accepting automata, by the m-reducibility with finite automata, and by topological difficulty. It turns out that these classifications coincide or are at least comparable to each other.

On the power of polynomial time bit-reductions

For a nondeterministic polynomial-time Turing machine M and an input string x, the leaf string of M on x is the 0-1-sequence of leaf-values (0 approximately reject, 1 approximately accept) of the computation tree of M with input x. The set A is said to be bit-reducible to B if there exists and M as above such that every input x is in A if and only if the leaf string of M on x is in B. A class C is definable via leaf language B, if C is the class of all languages that are bit-reducible to B. The question of how complex a leaf language must be in order to characterize some given class C is investigated. This question leads to the examination of the closure of different language classes under bit-reducibility. The question is settled for subclasses of regular languages, context free languages, and a number of time and space bounded classes, resulting in a number of surprising characterizations for PSPACE.<<ETX>>

Bounded query computations

A survey is given of directions, results, and methods in the study of complexity-bounded computations with a restricted number of queries to an oracle. In particular, polynomial-time-bounded computations with an NP oracle are considered. The main topics are: the relationship between the number of adaptive and parallel queries, connections to the closure of NP under polynomial-time truth-table reducibility, the Boolean hierarchy, the power of one more query, sparse oracles versus few queries, and natural complete problems for the most important bounded query classes.<<ETX>>

On the power of number-theoretic operations with respect to counting

We investigate function classes /sub f/ which are defined as the closure of P under the operation f and a set of known closure properties of P, e.g. summation over an exponential range. First, we examine operations f under which P is closed (i.e., /sub f/=P) in every relativization. We obtain the following complete characterization of these operations: P is closed under f in every relativization if and only if f is a finite sum of binomial coefficients over constants. Second, we characterize operations f with respect to their power in the counting context in the unrelativized case. For closure properties f of P, we have /sub f/= P. The other end of the range is marked by operations f for which /sub f/ corresponds to the counting hierarchy. We call these operations counting hard and give general criteria for hardness. For many operations f we show that /sub f/ corresponds to some subclass C of the counting hierarchy. This will then imply that P is closed under f if and only if UP=C; and on the other hand f is counting hard if and only if C contains the counting hierarchy.

THE COMPLEXITY OF FINDING MIDDLE ELEMENTS

Seinosuke Toda introduced the class Mid P of functions that yield the middle element in the set of output values over all paths of nondeterministic polynomial time Turing machines. We define two related classes: Med P consists of those functions that yield the middle element in the ordered sequence of output values of nondeterministic polynomial time Turing machines (i.e. we take into account that elements may occur with multiplicities greater than one). P consists of those functions that yield the middle element of all accepting paths (in some resonable encoding) of nondeterministic polynomial time Turing machines. We exhibit similarities and differences between these classes and completely determine the inclusion structure between these classes and some other well-known classes of functions like Valiant’s # P and Kobler, Schoning, and Toran’s span-P, that hold under general accepted complexity theoretic assumptions such as the counting hierarchy does not collapse. Our results help in clarifying the status of Toda’s very important class Mid P in showing that it is closely related to the class PPNP.

ON THE POWER OF ONE-WAY SYNCHRONIZED ALTERNATING MACHINES WITH SMALL SPACE

This paper continues the investigation of the concept of synchronized alternation. The open problems from Ref. 4 are solved by showing that one-way synchronized alternating (multihead) automata are as powerful as two-way ones. More precisely it is shown that: (i) one-way synchronized alternating finite automata recognize exactly context-sensitive languages, and (ii) NSPACE(nk) is exactly the family of languages recognized by one-way (two-way) synchronized alternating k-head finite automata, for k≥1. Finaly, the synchronization complexity of one-way synchronized Turing machines (1satms) is investigated and an infinite hierarchy among classes of sets accepted by 1satms with space and synchronization bounds between log log n and log n is established. Some closure properties of the classes in this hierarchy are also proved.

A hierarchy of regular sequence sets

In the paper STAIGER/WAGNER [I] several topological classes of regular sequence sets are characterized both by suitable notions of acceptance for finite automata and also without the notion of automaton, namely the classes of regular open, closed, G~ F~ , G~ and F~ -setss respectively. As an information about further investigations in this field, in the present paper a survey about a hierarchy of reducibility degress of regular sequence sets generated by m-reducing with finite automata is given. These degrees can be characterized as complexity classes with respect to suitable complexity measures of finite automata.

Reversal-Bounded and Visit-Bounded Realtime Computations

First it is dealt with the class RBQ (also sometimes called BNP) of all languages acceptable in linear time by reversal-bounded nondeterministic multitape Turing machines.