Abstract: The computing machine Z3, built by Konrad Zuse between 1938 and 1941, could execute only fixed sequences of floating point arithmetical operations (addition, subtraction, multiplication, division, and square root) coded in a punched tape. An interesting question to ask, from the viewpoint of the history of computing, is whether or not these operations are sufficient for universal computation. The paper shows that, in fact, a single program loop containing these arithmetical instructions can simulate any Turing machine whose tape is of a given finite size. This is done by simulating conditional branching and indirect addressing by purely arithmetical means. Zuse's Z3 is therefore, at least in principle, as universal as today's computers that have a bounded addressing space. A side effect of this result is that the size of the program stored on punched tape increases enormously.

Abstract: In this dissertation, we investigate the computational power of quantum Turing machines operating in bounded space. First, we consider space-bounds that are space-constructible and at least logarithmic in the input size. For such space-bounds, it is shown that quantum Turing machines and probabilistic Turing machines are equivalent in power in the unbounded-error setting, in the sense that each model may simulate the other with at most a constant factor increase in space. From this, it follows that any quantum Turing machine computation can be simulated deterministically with at most a quadratic increase in space, and can be simulated deterministically in time at most exponential in the space-bound. Several other facts regarding quantum complexity classes defined in terms of such space-bounds are also proved. Second, we consider the power of quantum Turing machines restricted to constant space. In this case, we first prove that quantum Turing machines having one-sided error and running in linear time are strictly more powerful than probabilistic Turing machines having either one-sided error or having two-sided bounded error and running in polynomial time. Second, we prove that exact (i.e., accepting with probability 0 or 1) constant-space quantum Turing machines upon which no restrictions on running time are placed can recognize languages that cannot be recognized by any bounded-error constant-space probabilistic Turing machine.

Abstract: We show that restricting a number of characterizations of the complexity class P to be positive (in natural ways) results in the same class of (monotone) problems, which we denote by posP . By a well-known result of Razborov, posP is a proper subclass of the class of monotone problems in P . We exhibit complete problems for posP via weak logical reductions, as we do for other logically defined classes of problems. Our work is a continuation of research undertaken by Grigni and Sipser, and subsequently Stewart; indeed, we introduce the notion of a positive deterministic Turing machine and consequently solve a problem posed by Grigni and Sipser.

Abstract: The following simulations by machines equipped with a one-way input tape and additional queue storage are shown: Every single-tape Turing machine (no separate input-tape) with time bound t(n) can be simulated by one queue in O(t(n)) time. Every pushdown automaton can be simulated by one queue in time O(n√n). Every deterministic machine with a one-turn pushdown store can be simulated deterministically by one queue in O(n√n) time. Every Turing machine with several multi-dimensional tapes accepting with time bound t(n) can be simulated by two queues in time O(t(n) log2 t(n)). Every deterministic Turing machine with several linear tapes accepting with time bound t(n) can be simulated deterministically by a queue and a pushdown store in O(t(n) log t(n)) time.

Abstract: Using ideas introduced by Buhrman et al. ([2], [3]) to separate various completeness notions for NEXP = NTIME(2poly), positive Turing complete sets for NEXP are studied. In contrast to many-one completeness and bounded truth-table completeness with norm 1 which are known to coincide on NEXP ([3]), whence any such set for NEXP is positive Turing complete, we give sets A and B such that (1) A is ≤ bT(2) P -complete but not ≤ posT P -complete for NEXP (2) B is ≤ posT P -complete but not ≤ tt P -complete for NEXP. These results come close to optimality since a further strengthening of (1), as was done by Buhrman in [1] for EXP = DTIME(2poly), seems to require the assumption NEXP = co-NEXP.

Abstract: In 1950, Alan Turing wrote about a test that would ``refute anyone who doubts that a computer can really think: if an observer cannot distinguish the responses of a programmed machine from those of a human being, the machine is said to have passed the Turing test'' [2]. In 1991, the first Turing test competition was conducted and the Loebner Prize was awarded to the winner. (For information, see http://acm.org/~loebner/loebner-prize.htmlx) Charles Platt was a judge at a Turing test competition a few years later, and you can read about his experience at www.wired.com/wired/3.04/features/turing. html. To make these Turing tests practical, the programs had to respond to only a small subset of topics, including women's clothing, Burgundy wine, and romantic relationships [1]. The restriction of discussion topics takes the teeth out the Turing test, for it is much easier to converse about a restricted range of topics than to demonstrate general conversational ability.