Relationship between density and deterministic complexity of MP-complete languages

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

Domino games which describe computations of alternating Turing machines in the same way as dominoes (tiling systems) encode computations of deterministic and nondeterministic Turing machines are considered. The domino games are two-person games in the course of which the players build up domino tilings of a square of prescribed size. Acceptance of an alternating Turing machine corresponds to a winning strategy for one player—the number of moves in the game is the number of alternations of the Turing machine. Let $\operatorname{ATIME}(T(n), m)$ denote the class of all sets that are accepted by some alternating Turing machine in time $T(n)$ with at most m alternations. It is shown that any problem in such a complexity class can be reduced to the strategy problem for some domino game. In particular domino games which are complete in the classes $\Sigma_{m}^{p}$, and $\Pi_{m}^{p}$, of the polynomial time hierarchy are found. This corresponds to the approach of Savelsbergh and van Emde Boas and of Lewis and Papadimitriou, who have shown that the theory of NP-completeness may also be founded on a finite domino problem instead of the satisfiability problem for propositional formulae. Similar generalizations are possible for domino connectability problems: Domino thread games where the players build up a thread of dominoes are introduced; the first player wins if he can ultimately connect two given points. It is shown that a certain class of such games captures deterministic time complexity. In particular, games are found whose strategy problems are P-complete. In the last section applications to the complexity of simple prefix classes in logical theories are briefly discussed.

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.

Realtime subshifts

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

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. >

Regular languages are languages recognised by finite automata; automatic structures are a generalisation of regular languages where one also uses automatic relations (which are relations recognised by synchronous finite automata) and automatic functions (which are functions whose graph is an automatic relation). Functions and relations first-order definable from other automatic functions and relations are again automatic. Automatic functions coincide with the functions computed by position-faithful one-tape Turing machines in linear time. This survey addresses recent results and open questions on topics related to automatic structures: How difficult is the isomorphism problem for various types of automatic structures? Which groups are automatic? When are automatic groups Abelian or orderable? How can one overcome some of the limitations to represent rings and fields by weakening the automaticity requirements of a structure?

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.

Totalistic cellular automata, introduced by S. Wolfram, are cellular automata in which the state transition function depends only on the sum of the states in a cell's neighborhood. Each state is considered as a nonnegative integer and the sum includes the cell's own state. It is shown that one-dimensional totalistic cellular automata can simulate an arbitrary Turing machine in linear time, even when the neighborhood is restricted to one cell on each side. This result settles Wolfram's conjecture that totalistic cellular automata are computation-universal.

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)).

We consider depth 2 unbounded fan-in circuits with symmetric and linear threshold gates. We present a deterministic algorithm that, given such a circuit with n variables and m gates, counts the number of satisfying assignments in time $$2^{n-\Omega \left( \left( \frac{n}{\sqrt{m} \cdot \textrm{poly}(\log n)}\right) ^a\right) }$$ 2 n - Ω n m · poly ( log n ) a for some constant $$a>0$$ a > 0 . Our algorithm runs in time super-polynomially faster than $$2^n$$ 2 n if $$m=O(n^2/\log ^b n)$$ m = O ( n 2 / log b n ) for some constant $$b>0$$ b > 0 . Previously, such algorithms were only known for bounded depth circuits with linear threshold gates and a slightly super-linear number of wires [Impagliazzo-Paturi-Schneider, FOCS 2013 and Chen-Santhanam-Srinivasan, CCC 2016]. We also show that depth 2 circuits with $$O(n^2/\log ^b n)$$ O ( n 2 / log b n ) symmetric and linear threshold gates in total cannot compute an explicit function computable by a deterministic $$2^{O(n)}$$ 2 O ( n ) -time Turing machine with an NP oracle. Previously, even slightly super-linear lower bounds on the number of gates were not known until recently Kane and Williams [STOC 2016] showed that depth 2 linear threshold circuits with $$o(n^{3/2}/\log ^3 n)$$ o ( n 3 / 2 / log 3 n ) gates cannot compute an explicit function computable in linear time.

Constructible functions in cellular automata and their applications to hierarchy results

Functions computable in polynomial space

Bounded rationality aims to understand the effects of how limited rationality affects decision-making. The traditional models in game theory and multiagent system research, such as finite automata or unrestricted Turing machine, fall short of capturing how intelligent agents make decision in realistic applications. To address this problem, we model bounded rational agents as restricted Turing machines: restrictions on running time and on storage space. We then study our model in two-person repeated games. In the case where the running time of Turing machines is restricted, we show that computing the best response of a given strategy is much harder than the strategy itself. In the case where the storage space of the Turing machines is restricted, we show the best response of a space restricted strategy can not be implemented by machines within the same size (up to a constant factor). Finally, we study how these restrictions affect the set of Nash equilibria in infinitely repeated games. We show restricting the agent's computational resources will give rise to new Nash equilibria.