In this work a complete problem for an unambiguous logspace class is presented. This is surprising since unambiguity is a `promise' or `semantic' concept. These usually lead to classes apparently without complete problems. 1 Introduction One of the most central questions of complexity theory is to compare determinism with nondeterminism. Our inability to exhibit the precise relationship between these two features motivates the investigation of intermediate features such as symmetry or unambiguity. In this paper we will concentrate on the notion of unambiguity. Unfortunately, unambiguity of a device or of a language is in general an undecidable property. Unambiguous classes are not defined by a `syntactical' machine property but rather by a `semantical' restriction. A nasty consequence is the apparent lack of complete sets. In the case of time bounded computations there are relativizations of unambiguity which provably have no complete problem ([10]). For space bounded computations t...

. In this paper we prove that if there exists an optimal quantified propositional proof system then there exists a complete language for NP " co-NP. 1 Introduction There is a lot of activity around the complexity of propositional proof systems (see [7, 12]). Research into propositional proof systems is motivated by the open problem NP=co-NP? Research into quantified propositional proof systems is not so popular. The study of quantified propositional proof systems may be related to the problem NP=PSPACE? Some deep results about connections between quantified propositional proof systems and bounded arithmetic are contained in [8]. We propose to study the problem of the existence of an optimal quantified propositional proof system. The similar problem for propositional proof systems has been studied in [9]. It is not known whether complete languages exist for NP " co-NP and Sipser has shown in [10] that there are relativizations so that NP " co-NP has no complete languages (see also [4...

Smullyan's notion of effectively inseparable pairs of sets is not the best effective /constructive analog of Kleene's notion of pairs of sets inseparable by a recursive set. We present a corrected notion of effectively inseparable pairs of sets, prove a characterization of our notion, and show that the pairs of index sets effectively inseparable in Smullyan's sense are the same as those effectively inseparable in ours. In fact we characterize the pairs of index sets effectively inseparable in either sense thereby generalizing Rice's Theorem. For subrecursive index sets we have sufficient conditions for various inseparabilities to hold. For inseparability by sets in the same subrecursive class we have a characterization. The latter essentially generalizes Kozen's (and Royer's later) Subrecursive Rice Theorem, and the proof of each result about subrecursive index sets is presented "Rogers style" with care to observe subrecursive restrictions. There are pairs of sets effectively inseparab...

this paper we develope a connection between optimal propositional proof systems and structural complexity theory - specifically there exists an optimal propositional proof system if and only if there is a suitable recursive presentation of the class of all easy (polynomial time recognizable) subsets of TAUT. As a corollary we have obtained the result that if there does not exist an optimal propositional proof system, then for every theory T there exists an easy subset of TAUT which is not T-provably easy.

In this paper we show that the problem of the existence of a p-optimal proof system for SAT can be characterized in a similar manner as J. Hartmanis and L. Hemachandra characterized the problem of the existence of complete languages for UP. Namely, there exists a p-optimal proof system for SAT if and only if there is a suitable recursive presentation of the class of all easy (polynomial time recognizable) subsets of SAT . Using this characterization we prove that if there does not exist a p-optimal proof system for SAT , then for every theory T there exists an easy subset of SAT which is not T-provably easy.