Optimal state amalgamation is NP-hard

2017 ◽  
Vol 39 (7) ◽  
pp. 1857-1869 ◽  
Author(s):  
RAFAEL M. FRONGILLO
Keyword(s):  
Finite Type ◽  
Directed Graph ◽  
Symbolic Dynamics ◽  
Hitting Set ◽  
State Splitting ◽  
Continuous Maps ◽  
Np Complete ◽  
Subshifts Of Finite Type ◽  
Hitting Set Problem ◽  
Inverse Operation

A state amalgamation of a directed graph is a node contraction which is only permitted under certain configurations of incident edges. In symbolic dynamics, state amalgamation and its inverse operation, state splitting, play a fundamental role in the theory of subshifts of finite type (SFT): any conjugacy between SFTs, given as vertex shifts, can be expressed as a sequence of symbol splittings followed by a sequence of symbol amalgamations. It is not known whether determining conjugacy between SFTs is decidable. We focus on conjugacy via amalgamations alone and consider the simpler problem of deciding whether one can perform $k$ consecutive amalgamations from a given graph. This problem also arises when using symbolic dynamics to study continuous maps, where one seeks to coarsen a Markov partition in order to simplify it. We show that this state amalgamation problem is NP-complete by reduction from the hitting set problem, thus giving further evidence that classifying SFTs up to conjugacy may be undecidable.

scholarly journals Lower entropy factors of sofic systems

1983 ◽  
Vol 3 (4) ◽  
pp. 541-557 ◽  
Author(s):  
Mike Boyle
Keyword(s):  
Periodic Point ◽  
Finite Type ◽  
Embedding Theorem ◽  
Subshift Of Finite Type ◽  
Sofic Shift ◽  
Sofic Shifts ◽  
Continuous Maps ◽  
Lower Entropy ◽  
Subshifts Of Finite Type ◽  
General Method

AbstractA mixing subshift of finite type T is a factor of a sofic shift S of greater entropy if and only if the period of any periodic point of S is divisible by the period of some periodic point of T. Mixing sofic shifts T satisfying this theorem are characterized, as are those mixing sofic shifts for which Krieger's Embedding Theorem holds. These and other results rest on a general method for extending shift-commuting continuous maps into mixing subshifts of finite type.

scholarly journals Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Silvère Gangloff
Keyword(s):  
Finite Type ◽  
Symbolic Dynamics ◽  
Recent Trend ◽  
Higher Dimensions ◽  
Topological Invariant ◽  
Entropy Dimension ◽  
Zero Entropy ◽  
Subshifts Of Finite Type ◽  
Present Text ◽  
Made In

<p style='text-indent:20px;'>In this text I study the asymptotics of the complexity function of <i>minimal</i> multidimensional subshifts of finite type through their entropy dimension, a topological invariant that has been introduced in order to study zero entropy dynamical systems. Following a recent trend in symbolic dynamics I approach this using concepts from computability theory. In particular it is known [<xref ref-type="bibr" rid="b12">12</xref>] that the possible values of entropy dimension for d-dimensional subshifts of finite type are the <inline-formula><tex-math id="M1">\begin{document}$ \Delta_2 $\end{document}</tex-math></inline-formula>-computable numbers in <inline-formula><tex-math id="M2">\begin{document}$ [0, d] $\end{document}</tex-math></inline-formula>. The kind of constructions that underlies this result is however quite complex and minimality has been considered thus far as hard to achieve with it. In this text I prove that this is possible and use the construction principles that I developped in order to prove (in principle) that for all <inline-formula><tex-math id="M3">\begin{document}$ d \ge 2 $\end{document}</tex-math></inline-formula> the possible values for entropy dimensions of <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional SFT are the <inline-formula><tex-math id="M5">\begin{document}$ \Delta_2 $\end{document}</tex-math></inline-formula>-computable numbers in <inline-formula><tex-math id="M6">\begin{document}$ [0, d-1] $\end{document}</tex-math></inline-formula>. In the present text I prove formally this result for <inline-formula><tex-math id="M7">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>. Although the result for other dimensions does not follow directly, it is enough to understand this construction to see that it is possible to reproduce it in higher dimensions (I chose dimension three for optimality in terms of exposition). The case <inline-formula><tex-math id="M8">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula> requires some substantial changes to be made in order to adapt the construction that are not discussed here.</p>

Medvedev degrees of two-dimensional subshifts of finite type

2012 ◽  
Vol 34 (2) ◽  
pp. 679-688 ◽  
Author(s):  
STEPHEN G. SIMPSON
Keyword(s):  
Finite Type ◽  
Equivalence Class ◽  
Symbolic Dynamics ◽  
Recursion Theory ◽  
Two Dimensional ◽  
Infinite Collection ◽  
Subshifts Of Finite Type ◽  
Medvedev Degrees

AbstractIn this paper, we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new example in symbolic dynamics. Two sets X and Y are said to be Medvedev equivalent if there exist partial computable functionals from X into Y and vice versa. The Medvedev degree of X is the equivalence class of X under Medvedev equivalence. There is an extensive recursion-theoretic literature on the lattices ℰs and ℰw of Medvedev degrees and Muchnik degrees of non-empty effectively closed subsets of {0,1}ℕ. We now prove that ℰs and ℰwconsist precisely of the Medvedev degrees and Muchnik degrees of two-dimensional subshifts of finite type. We apply this result to obtain an infinite collection of two-dimensional subshifts of finite type which are, in a certain sense, mutually incompatible.

THE COMPUTATIONAL COMPLEXITY OF AVOIDING FORBIDDEN SUBMATRICES BY ROW DELETIONS

2006 ◽  
Vol 17 (06) ◽  
pp. 1467-1484 ◽  
Author(s):  
SEBASTIAN WERNICKE ◽  
JOCHEN ALBER ◽  
JENS GRAMM ◽  
JIONG GUO ◽  
ROLF NIEDERMEIER
Keyword(s):  
Close Relation ◽  
Constant Factor ◽  
Input Matrix ◽  
Hitting Set ◽  
Positive Side ◽  
Column Permutation ◽  
Fixed Parameter ◽  
Minimum Number ◽  
Np Complete ◽  
Hitting Set Problem

We initiate a systematic study of the ROW DELETION(B) problem on matrices: Given an input matrix A and a fixed "forbidden submatrix" B, the task is to remove a minimum number of rows from A such that no row or column permutation of B occurs as a submatrix in the resulting matrix. An application of this problem can be found, for instance, in the construction of perfect phylogenies. Establishing a strong connection to variants of the NP-complete HITTING SET problem, we describe and analyze structural properties of B that make ROW DELETION(B)NP-complete. On the positive side, the close relation with HITTING SET problems yields constant-factor polynomial-time approximation algorithms and fixed-parameter tractability results.

Perturbations of multidimensional shifts of finite type

2010 ◽  
Vol 31 (2) ◽  
pp. 483-526 ◽  
Author(s):  
RONNIE PAVLOV
Keyword(s):  
Finite Type ◽  
Lower Bounds ◽  
Symbolic Dynamics ◽  
Upper And Lower Bounds ◽  
Shift Of Finite Type ◽  
Strongly Irreducible ◽  
Dimensional Cube

AbstractIn this paper, we study perturbations of multidimensional shifts of finite type. Specifically, for any ℤd shift of finite type X with d>1 and any finite pattern w in the language of X, we denote by Xw the set of elements of X not containing w. For strongly irreducible X and patterns w with shape a d-dimensional cube, we obtain upper and lower bounds on htop (X)−htop (Xw) dependent on the size of w. This extends a result of Lind for d=1 . We also apply our methods to an undecidability question in ℤd symbolic dynamics.

Subshifts of finite type

1976 ◽  
pp. 117-130
Author(s):  
Manfred Denker ◽  
Christian Grillenberger ◽  
Karl Sigmund
Keyword(s):  
Finite Type ◽  
Subshifts Of Finite Type
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