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E-mail: davorenahybrid . edu. An abstract transition system is a structure I = (X, { $ ) i , t ~ c t ) . where X # 0 is the state space and for each a E Act. 5 is a binary relation on X . Such structures are used in many areas of computer science to model dynamic changes of state. They provide a Kripke semantics for propositional modal languages with box modalities [a] for a E Act. Given an assignment of a subset of X to each atomic proposition, extend the valuation to all formulas by interpreting the propositional connectives by their set-theoretic counterparts.

BELTIUKOV, Alternating time complexity bounds for protothetics. , Izhevsk, 426034, Russia. ru. ,propositional formulas with quantifiers on propositional variables) are considered. The sets under consideration are composed o f formulas with bounded densities o f quantifiers. Machines o f both these classes are considered t o have random access t o input tape using special address tape or register. THEOREM. Let f be a nondecreasing function, f ( n ) 5 n. , for some Turing machine p that computes f with binary output.

Ru. ,propositional formulas with quantifiers on propositional variables) are considered. The sets under consideration are composed o f formulas with bounded densities o f quantifiers. Machines o f both these classes are considered t o have random access t o input tape using special address tape or register. THEOREM. Let f be a nondecreasing function, f ( n ) 5 n. , for some Turing machine p that computes f with binary output. Let Prot( f ) be the set of all trueprototheticsformulas with number ofquantijiers f ( n )where n is the length of theformula.

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