Reachability and Mortality Problems for Restricted Hierarchical Piecewise Constant Derivatives

Bell, Paul C.; Chen, Shang; Jackson, Lisa M.

doi:10.1007/978-3-319-11439-2_3

Cited by 2 publications

(4 citation statements)

References 10 publications

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“…Note that in the β-expansion PAMs all linear coefficients are the same, so the density of the orbit correspond to the density of the following sequence x(n) = f n (x 0 ), where f (x) = {βx}. For example when β = 5 2 and x 0 = 1 we get the sequence:…”

Section: Proof Let Us Define the Piecewise Affine Mappingmentioning

confidence: 99%

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Reachability Problems for PAMs

Kurganskyy

Potapov

2016

Lecture Notes in Computer Science

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Abstract. Piecewise affine maps (PAMs) are frequently used as a reference model to show the openness of the reachability questions in other systems. The reachability problem for one-dimentional PAM is still open even if we define it with only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on p-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM's orbits, reachability problems and representation of numbers in a rational base system. Finally we show a particular instance where the uniform distribution of the original orbit may not remain uniform or even dense after making regular shifts and taking a fractional part in that sequence.

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Section: Proof Let Us Define the Piecewise Affine Mappingmentioning

confidence: 99%

“…It also has a very natural geometrical interpretation as pseudo-billiard system [17] and Hierarchical Piecewise Constant Derivative (HPCD) system [3]. The reachability problem for one-dimensional PAM is still open even if we define it with only two intervals [2,3,5,12].…”

Section: Introductionmentioning

confidence: 99%

Reachability Problems for PAMs

Kurganskyy

Potapov

2016

Lecture Notes in Computer Science

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“…The computations in PAMs have a natural geometrical interpretation as pseudo-billiard system [21] and Hierarchical Piecewise Constant Derivative (HPCD) system [3]. The reachability problem for one-dimensional PAMs is still open even if restricted to a map with only two intervals [2,3,6,13].…”

Section: Introductionmentioning

confidence: 99%

“…Later the result was improved by constructing undecidability proof for a map with a finite number of bounded intervals and only one unbounded interval (l, + 8 ) in [5]. On the other hand some simplifications or additional constrains on affine maps can make the system to be more predictable with decidable reachability problems, see [2,3,6,13].…”

Section: Introductionmentioning

confidence: 99%

Reachability Problems for One-Dimensional Piecewise Affine Maps

Bournez

Kurganskyy

Potapov

2018

Int. J. Found. Comput. Sci.

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Piecewise affine maps (PAMs) are frequently used as a reference model to discuss the frontier between known and open questions about the decidability for reachability questions. In particular, the reachability problem for one-dimensional PAM is still an open problem, even if restricted to only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on [Formula: see text]-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM’s orbits, reachability problems and representation of numbers in a rational base system. Finally we construct an example where the distribution properties of well studied sequences can be significantly disrupted by taking fractional parts after regular shifts. The study of such sequences could help with understanding similar sequences generated in PAMs or in well known Mahler’s [Formula: see text] problem.

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Reachability and Mortality Problems for Restricted Hierarchical Piecewise Constant Derivatives

Cited by 2 publications

References 10 publications

Reachability Problems for PAMs

Reachability Problems for PAMs

Reachability Problems for One-Dimensional Piecewise Affine Maps

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