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Four Small Universal Turing Machines
, 2009
"... We present universal Turing machines with statesymbol pairs of (5, 5), (6, 4), (9, 3) and (15, 2). These machines simulate our new variant of tag system, the bitag system and are the smallest known singletape universal Turing machines with 5, 4, 3 and 2symbols, respectively. Our 5symbol machin ..."
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Cited by 21 (7 self)
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We present universal Turing machines with statesymbol pairs of (5, 5), (6, 4), (9, 3) and (15, 2). These machines simulate our new variant of tag system, the bitag system and are the smallest known singletape universal Turing machines with 5, 4, 3 and 2symbols, respectively. Our 5symbol machine uses the same number of instructions (22) as the smallest known universal Turing machine by Rogozhin. Also, all of the universal machines we present here simulate Turing machines in polynomial time.
The complexity of small universal Turing machines: a survey, Theor. Comput. Sci
, 2009
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On the computational complexity of spiking neural P systems
 In 7th International Conference on Unconventional Computation (UC 2008), volume 5204 of LNCS
, 2008
"... Abstract. It is shown here that there is no standard spiking neural P system that simulates Turing machines with less than exponential time and space overheads. The spiking neural P systems considered here have a constant number of neurons that is independent of the input length. Following this, we ..."
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Abstract. It is shown here that there is no standard spiking neural P system that simulates Turing machines with less than exponential time and space overheads. The spiking neural P systems considered here have a constant number of neurons that is independent of the input length. Following this, we construct a universal spiking neural P system with exhaustive use of rules that simulates Turing machines in linear time and has only 10 neurons. 1
Information Theory and Computational Thermodynamics: Lessons for Biology from Physics
 INFORMATION
, 2012
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Turing Patterns with Turing Machines: Emergence and StructureFormation from Universal Computation
 In preparation. ACM Communications Ubiquity 10
"... Despite having advanced a reactiondiffusion model of ODE’s in his 1952 paper on morphogenesis, reflecting his interest in mathematical biology, Alan Turing has never been considered to have approached a definition of Cellular Automata. However, his treatment of morphogenesis, and in particular a d ..."
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Cited by 2 (2 self)
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Despite having advanced a reactiondiffusion model of ODE’s in his 1952 paper on morphogenesis, reflecting his interest in mathematical biology, Alan Turing has never been considered to have approached a definition of Cellular Automata. However, his treatment of morphogenesis, and in particular a difficulty he identified relating to the uneven distribution of certain forms as a result of symmetry breaking, are key to connecting his theory of universal computation with his theory of biological pattern formation. Making such a connection would not overcome the particular difficulty that Turing was concerned about, which has in any case been resolved in biology. But instead the approach developed here captures Turing’s initial concern and provides a lowlevel solution to a more general question by way of the concept of algorithmic probability, thus bridging two of his most important contributions to science: Turing pattern formation and universal computation. I will provide experimental results of onedimensional patterns using this approach, with no loss of generality to a ndimensional pattern generalisation.
A Concrete View of Rule 110 Computation
, 906
"... Rule 110 is a cellular automaton that performs repeated simultaneous updates of an infinite row of binary values. The values are updated in the following way: 0s are changed to 1s at all positions where the value to the right is a 1, while 1s are changed to 0s at all positions where the values to th ..."
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Cited by 1 (0 self)
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Rule 110 is a cellular automaton that performs repeated simultaneous updates of an infinite row of binary values. The values are updated in the following way: 0s are changed to 1s at all positions where the value to the right is a 1, while 1s are changed to 0s at all positions where the values to the left and right are both 1. Though trivial to define, the behavior exhibited by Rule 110 is surprisingly intricate, and in [1] we showed that it is capable of emulating the activity of a Turing machine by encoding the Turing machine and its tape into a repeating left pattern, a central pattern, and a repeating right pattern, which Rule 110 then acts on. In this paper we provide an explicit compiler for converting a Turing machine into a Rule 110 initial state, and we present a general approach for proving that such constructions will work as intended. The simulation was originally assumed to require exponential time, but surprising results of Neary and Woods [2] have shown that in fact, only polynomial time is required. We use the methods of Neary and Woods to exhibit a direct simulation of a Turing machine by a tag system in polynomial time. 1 Compiling a Turing machine into a Rule 110 State In this section we give a concrete algorithm for compiling a Turing machine and its tape into an initial state for Rule 110, following the construction given in [1]. We will create an initial state that will eventually
Three small universal spiking neural P systems
"... In this work we give three small spiking neural P systems. We begin by constructing a universal spiking neural P system with extended rules and only 4 neurons. This is the smallest possible number of neurons for a universal system of its kind. We prove this by showing that the set of problems solved ..."
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In this work we give three small spiking neural P systems. We begin by constructing a universal spiking neural P system with extended rules and only 4 neurons. This is the smallest possible number of neurons for a universal system of its kind. We prove this by showing that the set of problems solved by spiking neural P systems with 3 neurons is bounded above by NL, and so there exists no such universal system with 3 neurons. If we generalise the output technique we immediately find a universal spiking neural P system with extended rules that has only 3 neurons. This is also the smallest possible number of neurons for a universal system of its kind. Finally, we give a universal spiking neural P system with standard rules and only 7 neurons. In addition to giving a significant improvement in terms of reducing the number of neurons, our systems also offer an exponential improvement on the time and space overheads of the small universal spiking neural P systems of other authors.
Small Turing universal signal machines
, 906
"... This article aims at providing signal machines as small as possible able to perform any computation (in the classical understanding). After presenting signal machines, it is shown how to get universal ones from Turing machines, cellularautomata and cyclic tag systems. Finally a halting universal si ..."
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This article aims at providing signal machines as small as possible able to perform any computation (in the classical understanding). After presenting signal machines, it is shown how to get universal ones from Turing machines, cellularautomata and cyclic tag systems. Finally a halting universal signal machine with 13 metasignals and 21 collision rules is presented. 1
Creative Commons Attribution License. A Small
"... A universal deterministic inhibitor Petri net with 14 places, 29 transitions and 138 arcs was constructed via simulation of Neary and Woods ’ weakly universal Turing machine with 2 states and 4 symbols; the total time complexity is exponential in the running time of their weak machine. To simulate ..."
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A universal deterministic inhibitor Petri net with 14 places, 29 transitions and 138 arcs was constructed via simulation of Neary and Woods ’ weakly universal Turing machine with 2 states and 4 symbols; the total time complexity is exponential in the running time of their weak machine. To simulate the blank words of the weakly universal Turing machine, a couple of dedicated transitions insert their codes when reaching edges of the working zone. To complete a chain of a given Petri net encoding to be executed by the universal Petri net, a translation of a bitag system into a Turing machine was constructed. The constructed Petri net is universal in the standard sense; a weaker form of universality for Petri nets was not introduced in this work. 1
Small Polynomial Time Universal Petri Nets
"... The time complexity of the presented in 2013 by the author small universal Petri nets with the pairs of places/transitions numbers (14,42) and (14,29) was estimated as exponential. In the present paper, it is shown, that their slight modification and interpretation as timed Petri nets with multichan ..."
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The time complexity of the presented in 2013 by the author small universal Petri nets with the pairs of places/transitions numbers (14,42) and (14,29) was estimated as exponential. In the present paper, it is shown, that their slight modification and interpretation as timed Petri nets with multichannel transitions, introduced by the author in 1991, allows obtaining polynomial time complexity. The modification concerns using only inhibitor arcs to control transitions ’ firing in multiple instances and employing an inverse control flow represented by moving zero. Thus, small universal Petri nets are efficient that justifies their application as models of high performance computations. 1