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36
Simple Termination is Difficult
 Applicable Algebra in Engineering, Communication and Computing
, 1993
"... A terminating term rewriting system is called simply terminating if its termination can be shown by means of a simplification ordering, an ordering with the property that a term is always bigger than its proper subterms. Almost all methods for proving termination yield, when applicable, simple termi ..."
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A terminating term rewriting system is called simply terminating if its termination can be shown by means of a simplification ordering, an ordering with the property that a term is always bigger than its proper subterms. Almost all methods for proving termination yield, when applicable, simple termination. We show that simple termination is an undecidable property, even for onerule systems. This contradicts a result by Jouannaud and Kirchner. The proof is based on the ingenious construction of Dauchet who showed the undecidability of termination for onerule systems. Our results may be summarized as follows: being simply terminating, (non)selfembedding, and (non)looping are undecidable properties of orthogonal, variable preserving, onerule constructor systems. 1. Introduction It is wellknown that termination is an undecidable property of term rewriting systems. This result was obtained by Huet and Lankford [9] in 1978. They showed that every Turing machine can be coded as a strin...
Observability of hybrid systems and Turing machines
 PROCEEDINGS OF THE 43RD IEEE CONFERENCE ON DECISION AND CONTROL
, 2004
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On the presence of periodic configurations in Turing machines and in counter machines
"... . A configuration of a Turing machine is given by a tape content together with a particular state of the machine. Petr Kurka has conjectured that every Turing machine  when seen as a dynamical system on the space of its configurations  has at least one periodic orbit. In this paper, we provide a ..."
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. A configuration of a Turing machine is given by a tape content together with a particular state of the machine. Petr Kurka has conjectured that every Turing machine  when seen as a dynamical system on the space of its configurations  has at least one periodic orbit. In this paper, we provide an explicit counterexample to this conjecture. We also consider counter machines and prove that, in this case, the problem of determining if a given machine has a periodic orbit in configuration space is undecidable. 1 Introduction A Turing machine is an abstract deterministic computer with a finite set Q of internal states. The machine operates on a doublyinfinite tape of cells ind exed by an integer i # Z. Symbols taken from a finite alphabet # are written on every cell; a tape content can thus be seen as an element of # Z . fait ceci. At every discrete time step, the Turing machine scans the cell indexed by 0 and, depending upon its internal state and the scanned symbol, the machine ...
Computing Stationary Probability Distributions and Large Deviation Rates for Constrained Random Walks. The Undecidability Results
, 2002
"... Our model is a constrained homogeneous random walk in + . The convergence to stationarity for such a random walk can often be checked by constructing a Lyapunov function. The same Lyapunov function can also be used for computing approximately the stationary distribution of this random walk, us ..."
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Our model is a constrained homogeneous random walk in + . The convergence to stationarity for such a random walk can often be checked by constructing a Lyapunov function. The same Lyapunov function can also be used for computing approximately the stationary distribution of this random walk, using methods developed by Meyn and Tweedie in [34]. In this paper we show that, for stationary homogeneous random walks, computing the stationary probability exactly is an undecidable problem, even if a Lyapunov function is available. That is no algorithm can exist to achieve this task. We then prove that computing large deviation rates for this model is also an undecidable problem. We extend these results to a certain type of queueing systems. The implication of these results is that no useful formulas for computing stationary probabilities and large deviations rates can exist in these systems.
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
On Immortal Configurations in Turing Machines
, 2012
"... Abstract We investigate the immortality problem for Turing machines and prove that there exists a Turing Machine that is immortal but halts on every recursive configuration. The result is obtained by combining a new proof of Hooper’s theorem [11] with recent results on effective symbolic dynamics. ..."
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Abstract We investigate the immortality problem for Turing machines and prove that there exists a Turing Machine that is immortal but halts on every recursive configuration. The result is obtained by combining a new proof of Hooper’s theorem [11] with recent results on effective symbolic dynamics.
THE PERIODIC DOMINO PROBLEM REVISITED
"... NOTICE: This is the author’s version of a work accepted for publication by Elsevier. Changes resulting from the publishing process, including peer review, editing, corrections, structural formatting and other quality control mechanisms, may not be reflected in this document. Changes may have been m ..."
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NOTICE: This is the author’s version of a work accepted for publication by Elsevier. Changes resulting from the publishing process, including peer review, editing, corrections, structural formatting and other quality control mechanisms, may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. The definitive version has been published in Theoretical Computer Science, 411:40104016, 2010. doi:10.1016/j.tcs.2010.08.017 Abstract. In this article we give a new proof of the undecidability of the periodic domino problem. The main difference with the previous proofs is that this one does not start from a proof of the undecidability of the (general) domino problem but only from the existence of an aperiodic tileset. The formalism of Wang tiles was introduced in [Wan61] to study decision procedures for the ∀∃ ∀ fragment of the firstorder logic. The earliest and most fundamental question is the domino problem: decide, given a finite set of Wang tiles if it tiles the plane. It turns out that this is not possible, this socalled domino problem was proven undecidable [Ber64]. There are until now to the author’s knowledge 6