2,252 research outputs found
Computing the Longest Common Prefix of a Context-free Language in Polynomial Time
We present two structural results concerning the longest common prefixes of non-empty languages.
First, we show that the longest common prefix of the language generated by a context-free grammar of size N
equals the longest common prefix of the same grammar where the heights of the derivation trees are bounded by
4N.
Second, we show that each non-empty language L has a representative subset of at most three elements which behaves
like L w.r.t. the longest common prefix as well as w.r.t. longest common prefixes of L after unions or
concatenations with arbitrary other languages.
From that, we conclude
that the longest common prefix, and thus the longest common suffix, of a context-free language can be computed in polynomial time
Deciding Equivalence of Linear Tree-to-Word Transducers in Polynomial Time
We show that the equivalence of deterministic linear top-down tree-to-word
transducers is decidable in polynomial time. Linear tree-to-word transducers
are non-copying but not necessarily order-preserving and can be used to express
XML and other document transformations. The result is based on a partial normal
form that provides a basic characterization of the languages produced by linear
tree-to-word transducers.Comment: short version of this paper will be published in the proceedings of
the 20th Conference on Developments in Language Theory (DLT 2016), Montreal,
Canad
It Is NL-complete to Decide Whether a Hairpin Completion of Regular Languages Is Regular
The hairpin completion is an operation on formal languages which is inspired
by the hairpin formation in biochemistry. Hairpin formations occur naturally
within DNA-computing. It has been known that the hairpin completion of a
regular language is linear context-free, but not regular, in general. However,
for some time it is was open whether the regularity of the hairpin completion
of a regular language is is decidable. In 2009 this decidability problem has
been solved positively by providing a polynomial time algorithm. In this paper
we improve the complexity bound by showing that the decision problem is
actually NL-complete. This complexity bound holds for both, the one-sided and
the two-sided hairpin completions
Streamability of nested word transductions
We consider the problem of evaluating in streaming (i.e., in a single
left-to-right pass) a nested word transduction with a limited amount of memory.
A transduction T is said to be height bounded memory (HBM) if it can be
evaluated with a memory that depends only on the size of T and on the height of
the input word. We show that it is decidable in coNPTime for a nested word
transduction defined by a visibly pushdown transducer (VPT), if it is HBM. In
this case, the required amount of memory may depend exponentially on the height
of the word. We exhibit a sufficient, decidable condition for a VPT to be
evaluated with a memory that depends quadratically on the height of the word.
This condition defines a class of transductions that strictly contains all
determinizable VPTs
Transition Property For Cube-Free Words
We study cube-free words over arbitrary non-unary finite alphabets and prove
the following structural property: for every pair of -ary cube-free
words, if can be infinitely extended to the right and can be infinitely
extended to the left respecting the cube-freeness property, then there exists a
"transition" word over the same alphabet such that is cube free. The
crucial case is the case of the binary alphabet, analyzed in the central part
of the paper.
The obtained "transition property", together with the developed technique,
allowed us to solve cube-free versions of three old open problems by Restivo
and Salemi. Besides, it has some further implications for combinatorics on
words; e.g., it implies the existence of infinite cube-free words of very big
subword (factor) complexity.Comment: 14 pages, 5 figure
Deciding Regularity of Hairpin Completions of Regular Languages in Polynomial Time
The hairpin completion is an operation on formal languages that has been
inspired by the hairpin formation in DNA biochemistry and by DNA computing. In
this paper we investigate the hairpin completion of regular languages.
It is well known that hairpin completions of regular languages are linear
context-free and not necessarily regular. As regularity of a (linear)
context-free language is not decidable, the question arose whether regularity
of a hairpin completion of regular languages is decidable. We prove that this
problem is decidable and we provide a polynomial time algorithm.
Furthermore, we prove that the hairpin completion of regular languages is an
unambiguous linear context-free language and, as such, it has an effectively
computable growth function. Moreover, we show that the growth of the hairpin
completion is exponential if and only if the growth of the underlying languages
is exponential and, in case the hairpin completion is regular, then the hairpin
completion and the underlying languages have the same growth indicator
Coding-theorem Like Behaviour and Emergence of the Universal Distribution from Resource-bounded Algorithmic Probability
Previously referred to as `miraculous' in the scientific literature because
of its powerful properties and its wide application as optimal solution to the
problem of induction/inference, (approximations to) Algorithmic Probability
(AP) and the associated Universal Distribution are (or should be) of the
greatest importance in science. Here we investigate the emergence, the rates of
emergence and convergence, and the Coding-theorem like behaviour of AP in
Turing-subuniversal models of computation. We investigate empirical
distributions of computing models in the Chomsky hierarchy. We introduce
measures of algorithmic probability and algorithmic complexity based upon
resource-bounded computation, in contrast to previously thoroughly investigated
distributions produced from the output distribution of Turing machines. This
approach allows for numerical approximations to algorithmic
(Kolmogorov-Chaitin) complexity-based estimations at each of the levels of a
computational hierarchy. We demonstrate that all these estimations are
correlated in rank and that they converge both in rank and values as a function
of computational power, despite fundamental differences between computational
models. In the context of natural processes that operate below the Turing
universal level because of finite resources and physical degradation, the
investigation of natural biases stemming from algorithmic rules may shed light
on the distribution of outcomes. We show that up to 60\% of the
simplicity/complexity bias in distributions produced even by the weakest of the
computational models can be accounted for by Algorithmic Probability in its
approximation to the Universal Distribution.Comment: 27 pages main text, 39 pages including supplement. Online complexity
calculator: http://complexitycalculator.com
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