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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 Int ..."
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Cited by 37 (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...
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
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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Cited by 11 (0 self)
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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...
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.
Networks of Relations
, 2005
"... 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 ..."
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Cited by 6 (2 self)
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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
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
Semanticsbased obfuscationresilient binary code similarity comparison with applications to software plagiarism detection
 in Proceedings of the 22nd ACM SIGSOFT International Symposium on the Foundations of Software Engineering (FSE
, 2014
"... Existing code similarity comparison methods, whether source or binary code based, are mostly not resilient to obfuscations. In the case of software plagiarism, emerging obfuscation techniques have made automated detection increasingly difficult. In this paper, we propose a binaryoriented, obfuscat ..."
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Existing code similarity comparison methods, whether source or binary code based, are mostly not resilient to obfuscations. In the case of software plagiarism, emerging obfuscation techniques have made automated detection increasingly difficult. In this paper, we propose a binaryoriented, obfuscationresilient method based on a new concept, longest common subsequence of semantically equivalent basic blocks, which combines rigorous program semantics with longest common subsequence based fuzzy matching. We model the semantics of a basic block by a set of symbolic formulas representing the inputoutput relations of the block. This way, the semantics equivalence (and similarity) of two blocks can be checked by a theorem prover. We then model the semantics similarity of two paths using the longest common subsequence with basic blocks as elements. This novel combination has resulted in strong resiliency to code obfuscation. We have developed a prototype and our experimental results show that our method is effective and practical when applied to realworld software.
Cyclic Structure of Dynamical Systems Associated with 3x+d Extensions of Collatz Problem.
, 2000
"... . We study here, from both theoretical and experimental points of view, the cyclic structures, both general and primitive, of dynamical systems D d generated by iterations of the functions T d acting, for all d # 1 relatively prime to 6, on positive integers : T d : N # N; T d (n) = n 2 ..."
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. We study here, from both theoretical and experimental points of view, the cyclic structures, both general and primitive, of dynamical systems D d generated by iterations of the functions T d acting, for all d # 1 relatively prime to 6, on positive integers : T d : N # N; T d (n) = n 2 , if n is even; 3n+d 2 , if n is odd. In the case d = 1, the properties of the system D = D 1 are the subject of the wellknown 3x + 1 conjecture. For every one of 6667 systems D d , 1 # d # 19999, we calculate its (complete, as we argue) list of primitive cycles. We unite in a single conceptual framework of primitive memberships, and we experimentally confirm three primitive cycles conjectures of Je# Lagarias. An indeep analysis of the diophantine formulae for primitive cycles, together with new rich experimental data, suggest several new conjectures, theoretically studied and experimentally confirmed in the present paper. As a part of this program, we prove a new upper bound t...
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,