On the computational power of totalistic cellular automata

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.

Constructible functions in cellular automata and their applications to hierarchy results

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.

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.

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.

Artificial genetic circuits have been recently proposed to realize unprecedented biological systems which do not exist in nature by combining several genes. For example, recombinant Escherichia coli provides novel artificial functions such as blinking periodically or invading cancer cells. In protein engineering, new functional proteins have been created by site-directed mutagenesis methods. The main purpose of our research is to implement the Cellular Automata (CA) in vivo employing these methods. Computational capacity of CA is equivalent to the universal Turing Machine. CA are multi-cellular systems in which uniform cells are allocated on lattice grid. Each cell has a finite number of states. The state transition is determined by the current state and the states of neighbor cells. Three mechanisms are required to implement CA in bacteria: (1) Sending and receiving signals between cells, (2) Sustaining the state, (3) Sensing input signals and changing the state following the state transition rules. To encode signals for the mechanism (1), we use small molecules inducing transcription. For the mechanism (2), the toggle switch circuit was employed to represent a finite number of states using gene expression. If one gene does express and the other does not, the state is 1, and vice versa, 0. For the mechanism (3), the state transition functions were implemented as logic gates using transcriptional regulatory proteins which bind to specific signal molecules (chemicals). In this paper, we applied this method to in vivo implementation of one-dimensional CA and corroborated whether it behaves correctly. In principle, this method can be applied to any dimensional CA by employing more signal molecules Permission to make digital or hard copies of all or part of this work for to construct more complex logic gate. Furthermore, we succeeded to execute one step calculation of one-dimensional CA in vivo in a laboratory experiment.

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.

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.

Characterization of 1-d Periodic Boundary Reversible CA

Authors propose a hardware based approach for reliable encryption in wireless network using Cellular Automata (CA). Distinct layers of encryption have been utilized in proposed model for enhancing data security. Transmitter and receiver modules are designed for performing state transition based encrypted activities. Transmitter module is required for capturing environmental turbulences (noise) from a specific geographical location and send encrypted signal to receiver module through wireless network. Receiver module analyses received signal after decryption for desired activities based on real time situations. Different hardware components are selected and compared based on market study targeting efficient construction of proposed modules. Time complexity and code break complexity are measured using different key lengths, keysets, and data sizes for proposed CA based encryption scheme achieving unique results in hardware based cryptography system. Encryption key length, index, and keyset are required for computing code break complexity in design model. Large data size is effectively managed to calculate protection ratio and reliability factor for realizing efficient protectiveness and accuracy. Comparative study between existing encryption technique and proposed CA based encryption technique is established based on time complexity and code break complexity.

In this paper a fast and space efficient method for simulating a d′-dimensional cellular automaton (CA) on a d-dimensional CA (d<d′) is introduced. For d′=2 and d=1 this method is optimal (under certain assumptions) with respect to time as well as space complexity. Let in this case t(n) be the time complexity and r(n) the side length of the smallest square enclosing all used cells. Then the simulation does not need more than O(r 2) cells and has a running time of d′ = d+1 a version with the time and space complexity of O(t 2) will be presented. Finally it will be shown, how it is possible to simulate a 2d-dimensional CA on a d-dimensional CA in a similarly efficient way.KeywordsTime ComplexityCellular AutomatonSpace ComplexityQuiescent StateComputation StepThese 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.

Fast language acceptance by shrinking cellular automata

Realtime subshifts