313 research outputs found
Propagating chain-free normal forms for EOL systems
We establish two types of normal forms for EOL systems. We first show that each ε-free EOL language can be generated by a propagating EOL system in which each derivation tree is chain-free. By this we mean that it contains at least one path from the root to the grandfather of a leaf in which each node has more than one son. We use this result to prove that each ε-free EOL language can be generated by a propagating EOL system in which each production has a right side of length at most two and which does not contain nonterminal chainproductions, i.e., productions A → B for nonterminals A and B. As applications of our results we give a simple proof for the decidability of the finiteness problem for EOL systems and solve an open problem concerning completeness of EOL forms
On the decidability of homomorphism equivalence for languages
AbstractWe consider decision problems of the following type. Given a language L and two homomorphisms h1 and h2, one has to determine to what extent h1 and h2 agree on L. For instance, we say that h1 and h2 are equivalent on L if h1(ω) = h2(ω) holds for each ω ε L. In our main theorem we present an algorithm for deciding whether two given homomorphisms are equivalent on a given context-free language. This result also gives an algorithm for deciding whether the translations defined by two deterministic gsm mappings agree on a given context-free language
The decidability of the equivalence problem for DOL-systems
The language and sequence equivalence problem for DOL-systems is shown to be decidable. In an algebraic formulation the sequence equivalence problem for DOL-systems can be stated as follows: Given homomorphisms h1 and h2 on a free monoid Σ* and a word σ from Σ*, is h1n(σ) = h2n(σ) for all n ⩾ 0
Aperiodic tilings and entropy
In this paper we present a construction of Kari-Culik aperiodic tile set -
the smallest known until now. With the help of this construction, we prove that
this tileset has positive entropy. We also explain why this result was not
expected
Finite automata for caching in matrix product algorithms
A diagram is introduced for visualizing matrix product states which makes
transparent a connection between matrix product factorizations of states and
operators, and complex weighted finite state automata. It is then shown how one
can proceed in the opposite direction: writing an automaton that ``generates''
an operator gives one an immediate matrix product factorization of it. Matrix
product factorizations have the advantage of reducing the cost of computing
expectation values by facilitating caching of intermediate calculations. Thus
our connection to complex weighted finite state automata yields insight into
what allows for efficient caching in matrix product algorithms. Finally, these
techniques are generalized to the case of multiple dimensions.Comment: 18 pages, 19 figures, LaTeX; numerous improvements have been made to
the manuscript in response to referee feedbac
Compressing Random Microstructures via Stochastic Wang Tilings
This paper presents a stochastic Wang tiling based technique to compress or
reconstruct disordered microstructures on the basis of given spatial
statistics. Unlike the existing approaches based on a single unit cell, it
utilizes a finite set of tiles assembled by a stochastic tiling algorithm,
thereby allowing to accurately reproduce long-range orientation orders in a
computationally efficient manner. Although the basic features of the method are
demonstrated for a two-dimensional particulate suspension, the present
framework is fully extensible to generic multi-dimensional media.Comment: 4 pages, 6 figures, v2: minor changes as suggested by reviewers, v3:
corrected two typos in the revised versio
Sofic-Dyck shifts
We define the class of sofic-Dyck shifts which extends the class of
Markov-Dyck shifts introduced by Inoue, Krieger and Matsumoto. Sofic-Dyck
shifts are shifts of sequences whose finite factors form unambiguous
context-free languages. We show that they correspond exactly to the class of
shifts of sequences whose sets of factors are visibly pushdown languages. We
give an expression of the zeta function of a sofic-Dyck shift
Fixed Point and Aperiodic Tilings
An aperiodic tile set was first constructed by R.Berger while proving the
undecidability of the domino problem. It turned out that aperiodic tile sets
appear in many topics ranging from logic (the Entscheidungsproblem) to physics
(quasicrystals) We present a new construction of an aperiodic tile set that is
based on Kleene's fixed-point construction instead of geometric arguments. This
construction is similar to J. von Neumann self-reproducing automata; similar
ideas were also used by P. Gacs in the context of error-correcting
computations. The flexibility of this construction allows us to construct a
"robust" aperiodic tile set that does not have periodic (or close to periodic)
tilings even if we allow some (sparse enough) tiling errors. This property was
not known for any of the existing aperiodic tile sets.Comment: v5: technical revision (positions of figures are shifted
Conjugacy of one-dimensional one-sided cellular automata is undecidable
Two cellular automata are strongly conjugate if there exists a
shift-commuting conjugacy between them. We prove that the following two sets of
pairs of one-dimensional one-sided cellular automata over a full shift
are recursively inseparable: (i) pairs where has strictly larger
topological entropy than , and (ii) pairs that are strongly conjugate and
have zero topological entropy.
Because there is no factor map from a lower entropy system to a higher
entropy one, and there is no embedding of a higher entropy system into a lower
entropy system, we also get as corollaries that the following decision problems
are undecidable: Given two one-dimensional one-sided cellular automata and
over a full shift: Are and conjugate? Is a factor of ? Is
a subsystem of ? All of these are undecidable in both strong and weak
variants (whether the homomorphism is required to commute with the shift or
not, respectively). It also immediately follows that these results hold for
one-dimensional two-sided cellular automata.Comment: 12 pages, 2 figures, accepted for SOFSEM 201
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