Abstract: We consider uniform circuit complexity, introduced by Borodin as a model of parallel complexity. Three main results are presented. First, we show that simultaneous size/depth of uniform circuits is the same as space/time of alternating Turing machines, with depth and time within a constant factor and likewise log(size) and space. Second, we apply this to characterize the class of polynomial size and polynomial-in-log depth circuits in terms of tree-size bounded alternating TM's, in particular showing that context-free recognition is in this class of circuits. Third, we investigate various definitions of uniform circuit complexity, showing that it is fairly insensitive to the choice of definition.

Abstract: We consider two-person games of incomplete information in which certain portions of positions are private to each player and cannot be viewed by the opponent. We present various games of incomplete information which are universal for all reasonable games. The problem of determining the outcome of these universal games from a given initial position is shown to be complete in doubly-exponential time. We also define “private alternating Turing machines” which are alternating Turing machines with certain tapes and portions of states private to universal states. The time and space complexity of these machines is characterized in terms of the time complexity of deterministic Turing machines, with single and double exponential jumps. We also consider blindfold games, which are restricted games in which the second player is not allowed to modify the common position. We show various blindfold games to have exponential space complete outcome problems and to be universal for reasonable blindfold games. We define “blind alternating Turing machines” which are private alternating Turing machines with the restriction that the universal states cannot modify the public tapes and public portion of states. A single exponential jump characterizes the relation between the space complexity of deterministic Turing machines.

Abstract: This paper introduces a new, finer space complexity measure of computations on Turing machines. The complexity of a computation on a Turing machine now takes into account also the capacity of the finite control. It is proved that a slight enlarging (by an additive constant) of the complexity bound increases the computing power. The proofs are based on a new principle of diagonalization. The results are similar for deterministic and nondeterministic off-line Turing machines, auxiliary pushdown automata, auxiliary counter automata and also for their versions with an oracle.

Abstract: A computation universal Turing machine, U, with 2 states, 4 letters, 1 head and 1 two-dimensional tape is constructed by a translation of a universal register-machine language into networks over some simple abstract automata and, finally, of such networks into U. As there exists no universal Turing machine with 2 states, 2 letters, 1 head and 1 two-dimensional tape only the 2-state, 3-letter case for such machines remains an open problem. An immediate consequence of the construction of U is the existence of a universal 2-state, 2-letter, 2-head, 1 two-dimensional tape Turing machine, giving a first sharp boundary of the necessary complexity of universal Turing machines.

Abstract: Logarithmically t(n)-time bounded RAMs can be simulated by t(n)/log t(n)-tape bounded Turing machines, t(n)-time bounded multidimensional multitape Turing machines can be simulated by t(n) loglog t(n)/log t(n)-tape bounded Turing machines.

Abstract: It is shown that concatenable double-ended queues can be simulated in real-time by double-ended queues without concatenation. Consequently, every multihead Turing machine with head-to-head jumps can be simulated in real-time by multitape Turing machines.

Abstract: We consider relativizing the constructions of Cook in [4] characterizing space-bounded auxiliary pushdown automata in terms of timebounded computers. LetS(n) ≥ logn be a measurable space bound. LetDTA[NTA] be the class of setsS such that there exists a machineM such thatM with oracleA recognizes the setS andM is a deterministic [nondeterministic] oracle Turing machine acceptor that runs in time 2cS(n) for some constantc. LetDBiA[NBiA] be the class of setsS such that there exists a machineM such thatM with oracleA recognizes the setS andM is a deterministic [non-deterministic] oracle Turing machine acceptor with auxiliary pushdown that runs in spaceS(n) and never queries the oracle about strings longer than:S(n) ifi = 1, 2cS(n) for some constantc, ifi = 2, + ∞ ifi = 3.

Abstract: We study the power of RAM acceptors with several instruction sets. We exhibit several instances where the availability of the division operator increases the power of the acceptors. We also show that in certain situations parallelism and stochastic features ('distributed random choices') are provably more powerful than either parallelism or randomness alone. We relate the class of probabilistic Turing machine computations to random access machines with multiplication (but without boolean vector operations). Again, the availability of integer division seems to play a crucial role in these results.

Abstract: We study the time relationships between several models of computation (variants of counter machines, Turing machines, and random access machines). It is shown that counter machines augmented by a “copy” instruction can be simulated in linear time by counter machines without such an instruction, and that these counter machines can be simulated by RAM's with speedup by a fixed polynomial. Since the difference between augmented counter machines and RAM's lies partly in the latter's indirect addressing capabilities, we obtain bounds on the extent to which these capabilities speed up computations. We also show that unit-cost RAM's can simulate multi-dimensional Turing machines with speedup using their addressing capabilities to efficiently implement multidimensional arrays. Evidence is presented to show that on a restricted class of RAM's, “successor” RAM's, efficient implementation of multi-dimensional arrays is not possible.

Abstract: We investigate the relative computing power of Turing machines with differences in the num/3er of work tapes, heads pro work tape, instruction repertoire etc. We concentrate on the k-tape, k-head and k-head jump models as well as the 2-way multihead finite automata with and without jumps. Differences in computing power between machines of unlike specifications emerge under the real-time restriction. In particular it is shown that k+l heads are more powerful than k heads for real-time Turing machines.

Abstract: We define a complexity measure for 2-tape Taring machines (2TM) that generalizes the usual crossing measure for l-tape Turing machines (ITM). We prove that the constant resource bound functions yield an infinite hierarchy of complexity classes of the new measur~ This shows a completely different behaviour of the new measure compared to the crossing measure for ITM. In a similar way as in the l-tape case the new measure allows to prove lower time bounds on 2TM computations.