Domino Games and Complexity

It would be of interest to separate nondeterminism from determinism i.e., to show that for all nice functions t(n), NTIME (t(n)), (the class of languages accepted by multitape nondeterministic Turing machines in time O(t(n))) strictly contains DTIME (t(n)) (the class of languages accepted by multitape deterministic Turing machines in time O(t(n)). We establish a weaker form of the statement. We show that there is a universal constant k such that for all nice functions t(n), the class of languages that can be accepted simultaneously in deterministic time O(t(n)) and space o((t(n))1/k) is strictly contained in NTIME (t(n)). (We will use the notation SPACE, TIME (s(n),t(n)) to denote the class of languages accepted by a deterministic TM in time O(t(n)) and simultaneously space O(s(n)).) This result is proved using a time-alternation trade-off and several other applications of this trade-off are presented. For example, we show that for each language L in SPACE, TIME (nl-e, ni) (where o≪e≪l, e, i constants) there exists a j such that L is accepted by a O(n) time bounded alternating Turing machine with j alternations. The trade-off also leads to the separation ∪SeSt SPACE, TIME (s,t)⊂+Σ2 TIME(t) where t(n) is any nice function and St is a class of nice functions in o(t). Here St includes most natural functions for natural t. For example, nj/log*n is in Snj.

Wang’s B machines are efficiently universal, as is Hasenjaeger’s small universal electromechanical toy

Structures of polynomial-time computable distributions and polynomial-time many-one reductions on randomized decision problems are investigated. The scope is widened from the most studied class DNP (distributional-NP) to the class ANP (average-NP), which consists of randomized decision problems accepted by nondeterministic Turing machines in average polynomial time. Results indicate that the structures of average-case complexity classes are very different from their counterparts in worst-case complexity due to the complex structures of probability distributions and distribution functions. >

CHAPTER 2 - A Catalog of Complexity Classes

Let Σ be an arbitrary alphabet. denotes ɛ∪Σ∪...∪Σn. We say that a function t, t : is f-sparse iff card for every natural n. The main theorem of this paper establishes that if CLIQUE has some f-sparse translation into another set, which is calculable by a deterministic Turing machine in time bounded by f, then all the sets belonging to NP are calculable in time bounded by a function polynomially related to f. The proof is constructive and shows the way of constructing a proper algorithm. The simplest and most significant corollary says that if there is an NP-complete language over a single letter alphabet, then P=NP.

The paper gives a logical characterisation of the class NTIME(n) of problems that can be solved on a nondeterministic Turing machine in linear time. It is shown that a set L of strings is in this class if and only if there is a formula of the form ∃f 1··∃f k ∃R 1··∃R m ∀xϕ that is true exactly for all strings in L. In this formula the f i are unary function symbols, the R i are unary relation symbols and ϕ is a quantifier-free formula. Furthermore, the quantification of functions is restricted to non-crossing, decreasing functions and in ϕ no equations in which different functions occur are allowed. There are a number of variations of this statement, e.g., it holds also for k = 3. From these results we derive an Ehrenfeucht game characterisation of NTIME(n).KeywordsLinear TimeTuring MachineDerivation TreeRelational AtomLogical CharacterisationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We construct a first-order formula φ such that all finite models of φ are non-narrow rectangular grids without using any binary relations other than the grid neighborship relations. As a corollary, we prove that a set A ⊆ ℕ is a spectrum of a formula which has only planar models if numbers n ∈ A can be recognized by a non-deterministic Turing machine (or a one-dimensional cellular automaton) in time t(n) and space s(n), where t(n)s(n) ≤ n and t(n); s(n) = Ω(log(n)).

There are essentially two results described in this paper. First, it is shown that for any random access machine (RAM) time-constructable function $T(n) \geqq n$, there are languages L such that L can be recognized in time $O(T(n))$ by a RAM (using the unit cost measure), but L cannot be recognized by any deterministic multitape Turing machine in time $O(T(n))$. Secondly, a family of random access stored program machines (RASP’S) are considered. For these RASP’S it is shown that there is an arbitrarily complex (infinitely often) partial recursive function $f(n)$. which has only 0–1 values (whenever defined) such that $f(n)$ can be computed in time $F(n)$ by some RASP, but cannot be computed in time $(1 - \varepsilon )F(n)$, for any $\varepsilon > 0$, by any RASP in this family.

On 1-inkdot alternating Turing machines with small space

Realtime subshifts

This paper introduces a two-dimensional alternating Turing machine (2-ATM) which is an extension of an alternating Turing machine to two-dimensions. This paper also introduces a three-way two-dimensional alternating Turing machine (TR2-ATM) which is an alternating version of a three-way two-dimensional Turing machine. We first investigate a relationship between the accepting powers of space-bounded 2-ATM's (or TR2-ATM's) and ordinary space-bounded two-dimensional Turing machines (or three-way two-dimensional Turing machines). We then introduce a simple, natural complexity measure for 2-ATM's (or TR2-ATM's), called -&-ldquo;leaf-size-&-rdquo;, and provides a spectrum of complexity classes based on leaf-size bounded computations. We finally investigate the recognizability of connected patterns by 2-ATM's (or TR2-ATM's).

Two-dimensional alternating turing machines

A neural network composed of a finite number of continuous value output neurons is known to be able to simulate any deterministic Turing machine. Several simulation models have been proposed, but there remains the problem that a massive number of neurons is needed to simulate the tape part in real time. In this paper the authors demonstrate that a recurrent higher‐order neural network can simulate any Turing machine, then specify the lower limit for the number of neurons required and the time needed for the simulation. First, the authors show that a recurrent higher‐order neural network consisting of a limited number of neurons using a threshold input/output function and a linear input/output function can simulate any deterministic Turing machine in real time, and that the number of neurons required to simulate the tape can be reduced to six. Then the authors show that if a saw‐type function is used, the number of neurons required for the simulation can be reduced to four. A simplified simulation model is then offered as a viable framework for learning formal language using a neural network. © 2001 Scripta Technica, Syst Comp Jpn, 32(10): 42–50, 2001

Turing machines, transition systems, and interaction

It is known that 1-tape-1-counter alternating Turing machines (ATMs) making a constant number of reversals recognize all recursively enumerable languages. Yamamoto and Noguchi raised the question whether 1-tape off-line ATMs with a constant reversal bound have the same computational power. Recently, Jiang has given a negative answer to this question by defining a recursive functionh(s,r,n) such that, for everys-state 1-tape off-line ATMM running inr reversals, the language accepted byM is inASPACE(h(s, r, n)). In this paper we continue the investigation of the reversal complexity of 1-tape off-line ATMs. We improve the result of Jiang showing that the functionh can be reduced to a functionf(s, r, n)≈ log log log logh(s,r,n). We also prove that 1-tape off-line ATMs recognize only recursive languages even if an arbitrary recursive bound (instead of only a constant) on the number of reversals is set. We conclude the paper by giving an astonishing reversal hierarchy theorem for the machines considered.