82,281 research outputs found
Syndeticity and independent substitutions
We associate in a canonical way a substitution to any abstract numeration
system built on a regular language. In relationship with the growth order of
the letters, we define the notion of two independent substitutions. Our main
result is the following. If a sequence is generated by two independent
substitutions, at least one being of exponential growth, then the factors of
appearing infinitely often in appear with bounded gaps. As an
application, we derive an analogue of Cobham's theorem for two independent
substitutions (or abstract numeration systems) one with polynomial growth, the
other being exponential
Asymptotic properties of free monoid morphisms
Motivated by applications in the theory of numeration systems and
recognizable sets of integers, this paper deals with morphic words when erasing
morphisms are taken into account. Cobham showed that if an infinite word is the image of a fixed point of a morphism under another
morphism , then there exist a non-erasing morphism and a coding
such that .
Based on the Perron theorem about asymptotic properties of powers of
non-negative matrices, our main contribution is an in-depth study of the growth
type of iterated morphisms when one replaces erasing morphisms with non-erasing
ones. We also explicitly provide an algorithm computing and
from and .Comment: 25 page
Existence versus Exploitation: The Opacity of Backbones and Backdoors Under a Weak Assumption
Backdoors and backbones of Boolean formulas are hidden structural properties.
A natural goal, already in part realized, is that solver algorithms seek to
obtain substantially better performance by exploiting these structures.
However, the present paper is not intended to improve the performance of SAT
solvers, but rather is a cautionary paper. In particular, the theme of this
paper is that there is a potential chasm between the existence of such
structures in the Boolean formula and being able to effectively exploit them.
This does not mean that these structures are not useful to solvers. It does
mean that one must be very careful not to assume that it is computationally
easy to go from the existence of a structure to being able to get one's hands
on it and/or being able to exploit the structure.
For example, in this paper we show that, under the assumption that P
NP, there are easily recognizable families of Boolean formulas with strong
backdoors that are easy to find, yet for which it is hard (in fact,
NP-complete) to determine whether the formulas are satisfiable. We also show
that, also under the assumption P NP, there are easily recognizable sets
of Boolean formulas for which it is hard (in fact, NP-complete) to determine
whether they have a large backbone
Complexity of Two-Dimensional Patterns
In dynamical systems such as cellular automata and iterated maps, it is often
useful to look at a language or set of symbol sequences produced by the system.
There are well-established classification schemes, such as the Chomsky
hierarchy, with which we can measure the complexity of these sets of sequences,
and thus the complexity of the systems which produce them.
In this paper, we look at the first few levels of a hierarchy of complexity
for two-or-more-dimensional patterns. We show that several definitions of
``regular language'' or ``local rule'' that are equivalent in d=1 lead to
distinct classes in d >= 2. We explore the closure properties and computational
complexity of these classes, including undecidability and L-, NL- and
NP-completeness results.
We apply these classes to cellular automata, in particular to their sets of
fixed and periodic points, finite-time images, and limit sets. We show that it
is undecidable whether a CA in d >= 2 has a periodic point of a given period,
and that certain ``local lattice languages'' are not finite-time images or
limit sets of any CA. We also show that the entropy of a d-dimensional CA's
finite-time image cannot decrease faster than t^{-d} unless it maps every
initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy
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