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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 ...
Observability of hybrid systems and Turing machines
 PROCEEDINGS OF THE 43RD IEEE CONFERENCE ON DECISION AND CONTROL
, 2004
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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.
A General Theory of SemiUnification
, 1993
"... Various restrictions on the terms allowed for substitution give rise to different cases of semiunification. Semiunification on finite and regular terms has already been considered in the literature. We introduce a general case of semiunification where substitutions are allowed on nonregular term ..."
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Various restrictions on the terms allowed for substitution give rise to different cases of semiunification. Semiunification on finite and regular terms has already been considered in the literature. We introduce a general case of semiunification where substitutions are allowed on nonregular terms, and we prove the equivalence of this general case to a wellknown undecidable data base dependency problem , thus establishing the undecidability of general semiunification. We present a unified way of looking at the various problems of semiunification. We give some properties that are common to all the cases of semiunification. We also the principality property and the solution set for those problems. We prove that semiunification on general terms has the principality property. Finally, we present a recursive inseparability result between semiunification on regular terms and semiunification on general terms. Partly supported by NSF grant CCR9113196. Address: Department of Compu...
Undecidable Boundedness Problems for Datalog Programs
 Journal of Logic Programming
, 1995
"... A given Datalog program is bounded if its depth of recursion is independent of the input database. Deciding boundedness is a basic task for the analysis of database logic programs. The undecidability of Datalog boundedness was first demonstrated by Gaifman et al. We introduce new techniques for p ..."
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A given Datalog program is bounded if its depth of recursion is independent of the input database. Deciding boundedness is a basic task for the analysis of database logic programs. The undecidability of Datalog boundedness was first demonstrated by Gaifman et al. We introduce new techniques for proving the undecidability of (various kinds of) boundedness, which allow us to considerably strengthen the results of Gaifman et al. In particular: (1) We use a new generic reduction technique to show that program boundedness is undecidable for arity 2 predicates, even with linear rules. (2) We use the mortality problem of Turing machines to show that uniform boundedness is undecidable for arity 3 predicates and for arity 1 predicates when is also allowed. (3) By encoding all possible transitions of a twocounter machine in a single rule, we show that program (resp., predicate) boundedness is undecidable for two linear rules (resp., one rule and a projection) and one initialization rule, where all predicates have small arities (6 or 7). 1
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
Printed in U.S.A. ON DECIDING STABILITYOF CONSTRAINED HOMOGENEOUS RANDOM WALKS AND QUEUEING SYSTEMS
"... We investigate stability of scheduling policies in queueing systems. To this day no algorithmic characterization exists for checking stability of a given policy in a given queueing system. In this paper we introduce a certain generalized priority policy and prove that the stability of this policy is ..."
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We investigate stability of scheduling policies in queueing systems. To this day no algorithmic characterization exists for checking stability of a given policy in a given queueing system. In this paper we introduce a certain generalized priority policy and prove that the stability of this policy is algorithmically undecidable. We also prove that stability of a homogeneous random walk in � d is undecidable. Finally, we show that the problem of computing a fluid limit of a queueing system or of a constrained homogeneous random walk is undecidable. To the best of our knowledge these are the first undecidability results in the area of stability of queueing systems and random walks in �d +. We conjecture that stability of common policies like FirstInFirstOut and priority policy is also an undecidable problem. 1. Introduction. We
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.