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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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Cited by 8 (2 self)
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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
Complexity of Fractran and productivity
 In Proceedings of the 22th Conference on Automated Deduction (CADE’09
, 2009
"... Abstract. In functional programming languages the use of infinite structures is common practice. For total correctness of programs dealing with infinite structures one must guarantee that every finite part of the result can be evaluated in finitely many steps. This is known as productivity. For prog ..."
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Cited by 5 (1 self)
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Abstract. In functional programming languages the use of infinite structures is common practice. For total correctness of programs dealing with infinite structures one must guarantee that every finite part of the result can be evaluated in finitely many steps. This is known as productivity. For programming with infinite structures, productivity is what termination in welldefined results is for programming with finite structures. Fractran is a simple Turingcomplete programming language invented by Conway. We prove that the question whether a Fractran program halts on all positive integers is Π 0 2complete. In functional programming, productivity typically is a property of individual terms with respect to the inbuilt evaluation strategy. By encoding Fractran programs as specifications of infinite lists, we establish that this notion of productivity is Π 0 2complete even for some of the most simple specifications. Therefore it is harder than termination of individual terms. In addition, we explore generalisations of the notion of productivity, and prove that their computational complexity is in the analytical hierarchy, thus exceeding the expressive power of firstorder logic. 1
UNDECIDABLE PROBLEMS: A SAMPLER
"... Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence ..."
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Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number strictly between ℵ0 and 2 ℵ0, is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [Göd40, Coh63, Coh64].) The first examples of statements independent of a “natural ” axiom system were constructed by K. Gödel [Göd31]. 2. Decision problem: A family of problems with YES/NO answers is called undecidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert’s tenth problem, to decide whether a multivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 1.1. In modern literature, the word “undecidability ” is used more commonly in sense 2, given that “independence ” adequately describes sense 1. To make 2 precise, one needs a formal notion of algorithm. Such notions were introduced by A. Church [Chu36a] and A. Turing [Tur36] independently in the 1930s. From now on, we interpret algorithm to mean Turing machine, which, loosely speaking, means that it is a computer program that takes as input a finite string of 0s and 1s. The role of the finite string is to specify which problem in the family is to be solved. Remark 1.2. Often in describing a family of problems, it is more convenient to use higherlevel mathematical objects such as polynomials or finite simplicial complexes as input. This is acceptable if these objects can be encoded as finite binary strings. It is not necessary to specify the encoding as long as it is clear that a Turing machine could convert between reasonable encodings imagined by two different readers.
Can't decide? Undecide!
"... In my mathematical youth, when I first learned of Gödel’s Theorem, and computational undecidability, I was at once fascinated and strangely reassured of our limited place in the grand universe: incredibly mathematics itself establishes limits on mathematical knowledge. At the same time, as one digs ..."
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In my mathematical youth, when I first learned of Gödel’s Theorem, and computational undecidability, I was at once fascinated and strangely reassured of our limited place in the grand universe: incredibly mathematics itself establishes limits on mathematical knowledge. At the same time, as one digs into the formalisms, this area can seem remote from most areas of mathematics and irrelevant to the efforts of most workaday mathematicians. But that’s just not so! Undecidable problems surround us, everywhere, even in recreational mathematics!
Mortality of Iterated Piecewise Affine Functions over the Integers: Decidability and Complexity
, 2013
"... In the theory of discretetime dynamical systems one studies the limiting behaviour of processes defined by iterating a fixed function f over a given space. A muchstudied case involves piecewise affine functions on Rn. Blondel et al. (2001) studied the decidability of questions such as global conve ..."
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In the theory of discretetime dynamical systems one studies the limiting behaviour of processes defined by iterating a fixed function f over a given space. A muchstudied case involves piecewise affine functions on Rn. Blondel et al. (2001) studied the decidability of questions such as global convergence and mortality for such functions with rational coefficients. Mortality means that every trajectory includes a 0; if the iteration is implemented as a loop while (x = 0) x: = f(x), mortality means that the loop is guaranteed to terminate. Checking the termination of simple loops (under various restrictions of the guard and the update function) is a muchstudied topic in automated program analysis. Blondel et al. proved that the problems are undecidable when the state space is R n (or Q n), and the dimension n is at least two. From a program analysis (and discrete Computability) viewpoint, it is more natural to consider functions over the integers. This paper establishes (un)decidability results for the integer setting. We show that also over integers, undecidability (moreover, Π 0 2 completeness) begins at two dimensions. We further investigate the effect of several restrictions on the iterated functions. Specifically, we consider bounding the size of the partition defining f, and restricting the coefficients of the linear components. In the decidable cases, we give complexity results. The complexity is PTIME for affine functions, but for piecewiseaffine ones it is PSPACEcomplete. The undecidability proofs use some variants of the Collatz problem, which may be of independent interest. 1
3x + 1 Minus the +
, 2001
"... We use Conway’s Fractran language to derive a function R: Z + → Z + of the form R(n) = rin if n ≡ i mod d where d is a positive integer, 0 ≤ i < d and r0,r1,...rd−1 are rational numbers, such that the famous 3x+1 conjecture holds if and only if the Rorbit of 2 n contains 2 for all positive integer ..."
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We use Conway’s Fractran language to derive a function R: Z + → Z + of the form R(n) = rin if n ≡ i mod d where d is a positive integer, 0 ≤ i < d and r0,r1,...rd−1 are rational numbers, such that the famous 3x+1 conjecture holds if and only if the Rorbit of 2 n contains 2 for all positive integers n. We then show that the Rorbit of an arbitrary positive integer is a constant multiple of an orbit that contains a power of 2. Finally we apply our main result to show that any cycle {x0,...,xm−1} of positive integers for the 3x + 1 function must satisfy i∈E xi
TuringCompleteness of Polymorphic Stream Equation Systems
"... Polymorphic stream functions operate on the structure of streams, infinite sequences of elements, without inspection of the contained data, having to work on all streams over all signatures uniformly. A natural, yet restrictive class of polymorphic stream functions comprises those definable by a sys ..."
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Polymorphic stream functions operate on the structure of streams, infinite sequences of elements, without inspection of the contained data, having to work on all streams over all signatures uniformly. A natural, yet restrictive class of polymorphic stream functions comprises those definable by a system of equations using only stream constructors and destructors and recursive calls. Using methods reminiscent of prior results in the field, we first show this class consists of exactly the computable polymorphic stream functions. Using much more intricate techniques, our main result states this holds true even for unary equations free of mutual recursion, yielding an elegant model of Turingcompleteness in a severely restricted environment and allowing us to recover previous complexity results in a much more restricted setting.