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Closedform Analytic Maps in One and Two Dimensions Can Simulate Turing Machines
, 1996
"... We show closedform analytic functions consisting of a finite number of trigonometric terms can simulate Turing machines, with exponential slowdown in one dimension or in real time in two or more. 1 A part of this author's work was done when he was visiting DIMACS at Rutgers University. 1 Introduc ..."
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Cited by 31 (4 self)
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We show closedform analytic functions consisting of a finite number of trigonometric terms can simulate Turing machines, with exponential slowdown in one dimension or in real time in two or more. 1 A part of this author's work was done when he was visiting DIMACS at Rutgers University. 1 Introduction Various authors have independently shown [9, 12, 4, 14, 1] that finitedimensional piecewiselinear maps and flows can simulate Turing machines. The construction is simple: associate the digits of the x and y coordinates of a point with the left and right halves of a Turing machine's tape. Then we can shift the tape head by halving or doubling x and y, and write on the tape by adding constants to them. Thus two dimensions suffice for a map, or three for a continuoustime flow. These systems can be thought of as billiards or optical ray tracing in three dimensions, recurrent neural networks, or hybrid systems. However, piecewiselinear functions are not very realistic from a physical p...
Halting Problem of One Binary Horn Clause is Undecidable.
, 1993
"... . This paper proposes a codification of the halting problem of any Turing machine in the form of only one rightlinear binary Horn clause as follows : p(t) / p(tt) : where t (resp. tt) is any (resp. linear) term. Recursivity is wellknown to be a crucial and fundamental concept in programming th ..."
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. This paper proposes a codification of the halting problem of any Turing machine in the form of only one rightlinear binary Horn clause as follows : p(t) / p(tt) : where t (resp. tt) is any (resp. linear) term. Recursivity is wellknown to be a crucial and fundamental concept in programming theory. This result proves that in Horn clause languages there is no hope to control it without additional hypotheses even for the simplest recursive schemes. Some direct consequences are presented here. For instance, there exists an explicitly constructible rightlinear binary Horn clause for which no decision algorithm, given a goal, always decides in a finite number of steps whether or not the resolution using this clause is finite. The halting problem of derivations w.r.t. one binary Horn clause had been shown decidable if the goal is ground [SS88] or if the goal is linear [Dev88, Dev90, DLD90]. The undecidability in the nonlinear case is an unexpected extension. The proof of the main r...
Programmability of Chemical Reaction Networks
"... Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard c ..."
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Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard chemical kinetics equations. SCRNs have been widely used for describing naturally occurring (bio)chemical systems, and with the advent of synthetic biology they become a promising language for the design of artificial biochemical circuits. Our interest here is the computational power of SCRNs and how they relate to more conventional models of computation. We survey known connections and give new connections between SCRNs and
Networks of Relations
, 2005
"... Project, and my advisor Shuki Bruck for supporting me during my studies. I would also like to thank Shuki for being a good advisor and collaborator. I am grateful not only to Shuki but to all the people I have worked with, including Erik Winfree and David Soloveichik, in collaboration with whom the ..."
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Cited by 5 (2 self)
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Project, and my advisor Shuki Bruck for supporting me during my studies. I would also like to thank Shuki for being a good advisor and collaborator. I am grateful not only to Shuki but to all the people I have worked with, including Erik Winfree and David Soloveichik, in collaboration with whom the material in section 3.4.2 was produced. My family has supported my adventure of being a student, especially my wife Éva, my children András, Adam, and Emma, my mother Sarah, and my grandfather Howard, and to them I am very grateful. iv Relations are everywhere. In particular, we think and reason in terms of mathematical and English sentences that state relations. However, we teach our students much more about how to manipulate functions than about how to manipulate relations. Consider functions. We know how to combine functions to make new functions, how to evaluate functions efficiently, and how to think about compositions of functions. Especially in the area of boolean functions, we have become experts in the theory and art of designing combinations of functions to yield what we want, and this expertise has led to techniques that enable
Decidability and universality in symbolic dynamical systems
 Fund. Inform
"... Abstract. Many different definitions of computational universality for various types of dynamical systems have flourished since Turing’s work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. Universality of a system is defined as un ..."
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Abstract. Many different definitions of computational universality for various types of dynamical systems have flourished since Turing’s work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. Universality of a system is defined as undecidability of a modelchecking problem. For Turing machines, counter machines and tag systems, our definition coincides with the classical one. It yields, however, a new definition for cellular automata and subshifts. Our definition is robust with respect to initial condition, which is a desirable feature for physical realizability. We derive necessary conditions for undecidability and universality. For instance, a universal system must have a sensitive point and a proper subsystem. We conjecture that universal systems have infinite number of subsystems. We also discuss the thesis according to which computation should occur at the ‘edge of chaos ’ and we exhibit a universal chaotic system. 1.
Computational universality in symbolic dynamical systems
 Fundamenta Informaticae
"... Abstract. Many different definitions of computational universality for various types of systems have flourished since Turing’s work. In this paper, we propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. For Turing machines and tag systems, ..."
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Abstract. Many different definitions of computational universality for various types of systems have flourished since Turing’s work. In this paper, we propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. For Turing machines and tag systems, our definition coincides with the usual notion of universality. It however yields a new definition for cellular automata and subshifts. Our definition is robust with respect to noise on the initial condition, which is a desirable feature for physical realizability. We derive necessary conditions for universality. For instance, a universal system must have a sensitive point and a proper subsystem. We conjecture that universal systems have an infinite number of subsystems. We also discuss the thesis that computation should occur at the ‘edge of chaos ’ and we exhibit a universal chaotic system. 1
One Binary Horn Clause is Enough
, 1994
"... . This paper proposes an equivalent form of the famous BohmJacopini theorem for declarative languages. C. Bohm and G. Jacopini [1] proved that all programming can be done with at most one single whiledo. That result is cited as a mathematical justification for structured programming. A similar r ..."
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. This paper proposes an equivalent form of the famous BohmJacopini theorem for declarative languages. C. Bohm and G. Jacopini [1] proved that all programming can be done with at most one single whiledo. That result is cited as a mathematical justification for structured programming. A similar result can be shown for declarative programming. Indeed the simplest class of recursive programs in Horn clause languages can be defined by the following scheme : ( A1 / : A2 / A3 : that is 8x1 \Delta \Delta \Delta 8xm [A1 (A2 :A3) :A4 ] / A4 : where A i are positive firstorder literals. This class is shown here to be as expressive as Turing machines and all simpler classes would be trivial. The proof is based on a remarkable and not enough known codification of any computable function by unpredictable iterations proposed by [5]. Then, we prove effectively by logical transformations that all conjunctive formulas of Horn clauses can be translated into an equivalent conjuctive ...
Two Cellular Automata for the 3x+1 Map
, 2005
"... Two simple Cellular Automata, which mimic the CollatzUlam iterated map (3x+ 1 map), are introduced. These Cellular Automata allow to test efficiently the Collatz conjecture for very large numbers. Mathematics Subject Classification: 37B15; 68Q80; 68Q10; 11B85;11B99 1 1 ..."
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Two simple Cellular Automata, which mimic the CollatzUlam iterated map (3x+ 1 map), are introduced. These Cellular Automata allow to test efficiently the Collatz conjecture for very large numbers. Mathematics Subject Classification: 37B15; 68Q80; 68Q10; 11B85;11B99 1 1
Tag systems and Collatzlike functions
"... Tag systems were invented by Emil Leon Post and proven recursively unsolvable by Marvin Minsky. These production systems have shown very useful in constructing small universal (Turing complete) systems for several different classes of computational systems, including Turing machines, and are thus im ..."
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Tag systems were invented by Emil Leon Post and proven recursively unsolvable by Marvin Minsky. These production systems have shown very useful in constructing small universal (Turing complete) systems for several different classes of computational systems, including Turing machines, and are thus important instruments for studying limits or boundaries of solvability and unsolvability. Although there are some results on tag systems and their limits of solvability and unsolvability, there are hardly any that consider both the shift number v, as well as the number of symbols µ. This paper aims to contribute to research on limits of solvability and unsolvability for tag systems, taking into account these two parameters. The main result is the reduction of the 3n + 1problem to a surprisingly small tag system. It indicates that the present unsolvability line – defined in terms of µ and v – for tag systems might be significantly decreased. Key words: Tag Systems, limits of solvability and unsolvability, universality,