scholarly journals Native Chemical Computation. A Generic Application of Oscillating Chemistry Illustrated With the Belousov-Zhabotinsky Reaction. A Review

2021 ◽  
Vol 9 ◽  
Author(s):  
Marta Dueñas-Díez ◽  
Juan Pérez-Mercader
Keyword(s):  
Entropy Production ◽  
Turing Machine ◽  
Finite Automata ◽  
Logic Gates ◽  
Context Sensitive ◽  
Chemical Computation ◽  
Oscillatory Chemical ◽  
New Interpretation ◽  
Synthesized Materials ◽  
Belousov Zhabotinsky Reaction

Computing with molecules is at the center of complex natural phenomena, where the information contained in ordered sequences of molecules is used to implement functionalities of synthesized materials or to interpret the environment, as in Biology. This uses large macromolecules and the hindsight of billions of years of natural evolution. But, can one implement computation with small molecules? If so, at what levels in the hierarchy of computing complexity? We review here recent work in this area establishing that all physically realizable computing automata, from Finite Automata (FA) (such as logic gates) to the Linearly Bound Automaton (LBA, a Turing Machine with a finite tape) can be represented/assembled/built in the laboratory using oscillatory chemical reactions. We examine and discuss in depth the fundamental issues involved in this form of computation exclusively done by molecules. We illustrate their implementation with the example of a programmable finite tape Turing machine which using the Belousov-Zhabotinsky oscillatory chemistry is capable of recognizing words in a Context Sensitive Language and rejecting words outside the language. We offer a new interpretation of the recognition of a sequence of chemicals representing words in the machine's language as an illustration of the “Maximum Entropy Production Principle” and concluding that word recognition by the Belousov-Zhabotinsky Turing machine is equivalent to extremal entropy production by the automaton. We end by offering some suggestions to apply the above to problems in computing, polymerization chemistry, and other fields of science.

Space-bounded OTMs and REG ∞

Computability ◽  
2021 ◽  
pp. 1-16
Author(s):  
Merlin Carl
Keyword(s):  
Complexity Theory ◽  
Turing Machine ◽  
Finite Automata ◽  
Regular Languages ◽  
The Other ◽  
Deterministic Finite Automata ◽  
Logarithmic Space ◽  
Working Space ◽  
Important Theorem

An important theorem in classical complexity theory is that REG = LOGLOGSPACE, i.e., that languages decidable with double-logarithmic space bound are regular. We consider a transfinite analogue of this theorem. To this end, we introduce deterministic ordinal automata (DOAs) and show that they satisfy many of the basic statements of the theory of deterministic finite automata and regular languages. We then consider languages decidable by an ordinal Turing machine (OTM), introduced by P. Koepke in 2005 and show that if the working space of an OTM is of strictly smaller cardinality than the input length for all sufficiently long inputs, the language so decided is also decidable by a DOA, which is a transfinite analogue of LOGLOGSPACE ⊆ REG; the other direction, however, is easily seen to fail.

The First Computational Theory of Cognition

2020 ◽  
pp. 107-127
Author(s):  
Gualtiero Piccinini
Keyword(s):  
Neural Networks ◽  
Body Problem ◽  
Digital Circuits ◽  
Finite Automata ◽  
Cognitive Activity ◽  
Logic Gates ◽  
Computer Design ◽  
Computational Theory ◽  
Modern Computer ◽  
The Mind

McCulloch and Pitts were the first to use and Alan Turing’s notion of computation to understand neural, and thus cognitive, activity. McCulloch and Pitts’s contributions included (i) a formalism whose refinement and generalization led to the notion of finite automata, which is an important formalism in computability theory, (ii) a technique that inspired the notion of logic design, which is a fundamental part of modern computer design, (iii) the first use of computation to address the mind–body problem, and (iv) the first modern computational theory of cognition, which posits that neurons are equivalent to logic gates and neural networks are digital circuits.

Membership Problem for Two-Dimensional General Row Jumping Finite Automata

2020 ◽  
Vol 31 (04) ◽  
pp. 527-538
Author(s):  
Grzegorz Madejski ◽  
Andrzej Szepietowski
Keyword(s):  
Computational Model ◽  
Turing Machine ◽  
Finite Automaton ◽  
Finite Automata ◽  
The Other ◽  
Membership Problem ◽  
Two Dimensional ◽  
Np Complete ◽  
Nondeterministic Turing Machine ◽  
Membership Problems

Two-dimensional general row jumping finite automata were recently introduced as an interesting computational model for accepting two-dimensional languages. These automata are nondeterministic. They guess an order in which rows of the input array are read and they jump to the next row only after reading all symbols in the previous row. In each row, they choose, also nondeterministically, an order in which segments of the row are read. In this paper, we study the membership problem for these automata. We show that each general row jumping finite automaton can be simulated by a nondeterministic Turing machine with space bounded by the logarithm. This means that the fixed membership problems for such automata are in NL, and so in P. On the other hand, we show that the uniform membership problem is NP-complete.

scholarly journals Growth in Baumslag–Solitar groups I: subgroups and rationality

2011 ◽  
Vol 14 ◽  
pp. 34-71 ◽  
Author(s):  
Eric M. Freden ◽  
Teresa Knudson ◽  
Jennifer Schofield
Keyword(s):  
Turing Machine ◽  
Time Complexity ◽  
Machine Construction ◽  
Convolution Product ◽  
Growth Series ◽  
Context Sensitive ◽  
Context Free Language ◽  
Time And Space Complexity ◽  
Context Free ◽  
Free Language

AbstractThe computation of growth series for the higher Baumslag–Solitar groups is an open problem first posed by de la Harpe and Grigorchuk. We study the growth of the horocyclic subgroup as the key to the overall growth of these Baumslag–Solitar groups BS(p,q), where 1<p<q. In fact, the overall growth series can be represented as a modified convolution product with one of the factors being based on the series for the horocyclic subgroup. We exhibit two distinct algorithms that compute the growth of the horocyclic subgroup and discuss the time and space complexity of these algorithms. We show that when p divides q, the horocyclic subgroup has a geodesic combing whose words form a context-free (in fact, one-counter) language. A theorem of Chomsky–Schützenberger allows us to compute the growth series for this subgroup, which is rational. When p does not divide q, we show that no geodesic combing for the horocyclic subgroup forms a context-free language, although there is a context-sensitive geodesic combing. We exhibit a specific linearly bounded Turing machine that accepts this language (with quadratic time complexity) in the case of BS(2,3) and outline the Turing machine construction in the general case.

TWO TOPICS CONCERNING TWO-DIMENSIONAL AUTOMATA OPERATING IN PARALLEL

1992 ◽  
Vol 06 (02n03) ◽  
pp. 211-225 ◽  
Author(s):  
KATSUSHI INOUE ◽  
ITSUO SAKURAMOTO ◽  
MAKOTO SAKAMOTO ◽  
ITSUO TAKANAMI
Keyword(s):  
Turing Machine ◽  
Finite Automata ◽  
Space Complexity ◽  
Two Dimensional ◽  
Turing Machines ◽  
Small Space ◽  
Bottom Up ◽  
Constructible Function ◽  
Deterministic Turing Machine ◽  
Alternating Turing Machines

This paper deals with two topics concerning two-dimensional automata operating in parallel. We first investigate a relationship between the accepting powers of two-dimensional alternating finite automata (2-AFAs) and nondeterministic bottom-up pyramid cellular acceptors (NUPCAs), and show that Ω ( diameter × log diameter ) time is necessary for NUPCAs to simulate 2-AFAs. We then investigate space complexity of two-dimensional alternating Turing machines (2-ATMs) operating in small space, and show that if L (n) is a two-dimensionally space-constructible function such that lim n → ∞ L (n)/ loglog n > 1 and L (n) ≤ log n, and L′ (n) is a function satisfying L′ (n) =o (L(n)), then there exists a set accepted by some strongly L (n) space-bounded two-dimensional deterministic Turing machine, but not accepted by any weakly L′ (n) space-bounded 2-ATM, and thus there exists a rich space hierarchy for weakly S (n) space-bounded 2-ATMs with loglog n ≤ S (n) ≤ log n.

scholarly journals ENTROPY PRODUCTION OF ENTIRELY DIFFUSIONAL LAPLACIAN TRANSFER AND THE POSSIBLE ROLE OF FRAGMENTATION OF THE BOUNDARIES

Fractals ◽  
2015 ◽  
Vol 23 (03) ◽  
pp. 1550026 ◽  
Author(s):  
K. KARAMANOS ◽  
S. I. MISTAKIDIS ◽  
T. J. MASSART ◽  
I. S. MISTAKIDIS
Keyword(s):  
Entropy Production ◽  
Active Zone ◽  
Numerical Study ◽  
Near Field ◽  
Irreversible Thermodynamics ◽  
Koch Curve ◽  
Variational Functional ◽  
Numerical Resolution ◽  
New Interpretation

The entropy production and the variational functional of a Laplacian diffusional field around the first four fractal iterations of a linear self-similar tree (von Koch curve) is studied analytically and detailed predictions are stated. In a next stage, these predictions are confronted with results from numerical resolution of the Laplace equation by means of Finite Elements computations. After a brief review of the existing results, the range of distances near the geometric irregularity, the so-called "Near Field", a situation never studied in the past, is treated exhaustively. We notice here that in the Near Field, the usual notion of the active zone approximation introduced by Sapoval et al. [M. Filoche and B. Sapoval, Transfer across random versus deterministic fractal interfaces, Phys. Rev. Lett. 84(25) (2000) 5776;1 B. Sapoval, M. Filoche, K. Karamanos and R. Brizzi, Can one hear the shape of an electrode? I. Numerical study of the active zone in Laplacian transfer, Eur. Phys. J. B. Condens. Matter Complex Syst. 9(4) (1999) 739-753.]2 is strictly inapplicable. The basic new result is that the validity of the active-zone approximation based on irreversible thermodynamics is confirmed in this limit, and this implies a new interpretation of this notion for Laplacian diffusional fields.

Weak cardinality theorems

2005 ◽  
Vol 70 (3) ◽  
pp. 861-878
Author(s):  
Till Tantau
Keyword(s):  
Turing Machine ◽  
Polynomial Time ◽  
Computational Models ◽  
Finite Automata ◽  
Good Reason ◽  
Recursion Theory ◽  
Middle Ground ◽  
Logical Structures

AbstractKummer's Cardinality Theorem states that a language A must be recursive if a Turing machine can exclude for any n words , …, one of the n + 1 possibilities for the cardinality of {, …, }⋂ A. There was good reason to believe that this theorem is a peculiarity of recursion theory: neither the Cardinality Theorem nor weak forms of it hold for resource-bounded computational models like polynomial time. This belief may be flawed. In this paper it is shown that weak cardinality theorems hold for finite automata and also for other models. An explanation is proposed as to why recursion-theoretic and automata-theoretic weak cardinality theorems hold, but not corresponding 'middle-ground theorems': The recursion- and automata-theoretic weak cardinality theorems are instantiations of purely logical weak cardinality theorems. The logical theorems can be instantiated for logical structures characterizing recursive computations and finite automata computations. A corresponding structure characterizing polynomial time computations does not exist.

ON THE POWER OF ONE-WAY SYNCHRONIZED ALTERNATING MACHINES WITH SMALL SPACE

1992 ◽  
Vol 03 (01) ◽  
pp. 65-79 ◽  
Author(s):  
JURAJ HROMKOVIČ ◽  
KATSUSHI INOUE ◽  
BRANISLAV ROVAN ◽  
ANNA SLOBODOVÁ ◽  
ITSUO TAKANAMI ◽  
...  
Keyword(s):  
Finite Automata ◽  
Turing Machines ◽  
Small Space ◽  
Closure Properties ◽  
Open Problems ◽  
Infinite Hierarchy ◽  
Context Sensitive ◽  
The Family

This paper continues the investigation of the concept of synchronized alternation. The open problems from Ref. 4 are solved by showing that one-way synchronized alternating (multihead) automata are as powerful as two-way ones. More precisely it is shown that: (i) one-way synchronized alternating finite automata recognize exactly context-sensitive languages, and (ii) NSPACE(nk) is exactly the family of languages recognized by one-way (two-way) synchronized alternating k-head finite automata, for k≥1. Finaly, the synchronization complexity of one-way synchronized Turing machines (1satm's) is investigated and an infinite hierarchy among classes of sets accepted by 1satm's with space and synchronization bounds between log log n and log n is established. Some closure properties of the classes in this hierarchy are also proved.

PARALLEL FINITE AUTOMATA SYSTEMS COMMUNICATING BY STATES

2002 ◽  
Vol 13 (05) ◽  
pp. 733-749 ◽  
Author(s):  
CARLOS MARTÍN-VIDE ◽  
ALEXANDRU MATEESCU ◽  
VICTOR MITRANA
Keyword(s):  
Finite Automata ◽  
Point Of View ◽  
Computational Power ◽  
Open Problems ◽  
Context Sensitive ◽  
Recursively Enumerable ◽  
Power Point ◽  
Grammar Systems ◽  
Recursively Enumerable Languages ◽  
Parallel Communicating Grammar Systems

An accepting device based on the communication between finite automata working in parallel is introduced. It consists of several finite automata working independently but communicating states to each other by request. Several variants of parallel communicating finite automata systems are investigated from their computational power point of view. We prove that all of them are at most as powerful as multi-head finite automata. Homomorphical characterizations of recursively enumerable languages are obtained starting from languages recognized by all variants of parallel communicating finite automata systems having at most three components. We present a brief comparison with the parallel communicating grammar systems. Some remarks suggesting that these devices might be mildly context-sensitive ones as well as a few open problems and directions for further research are also discussed.

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