575 research outputs found
Regularity Preserving but not Reflecting Encodings
Encodings, that is, injective functions from words to words, have been
studied extensively in several settings. In computability theory the notion of
encoding is crucial for defining computability on arbitrary domains, as well as
for comparing the power of models of computation. In language theory much
attention has been devoted to regularity preserving functions.
A natural question arising in these contexts is: Is there a bijective
encoding such that its image function preserves regularity of languages, but
its pre-image function does not? Our main result answers this question in the
affirmative: For every countable class C of languages there exists a bijective
encoding f such that for every language L in C its image f[L] is regular.
Our construction of such encodings has several noteworthy consequences.
Firstly, anomalies arise when models of computation are compared with respect
to a known concept of implementation that is based on encodings which are not
required to be computable: Every countable decision model can be implemented,
in this sense, by finite-state automata, even via bijective encodings. Hence
deterministic finite-state automata would be equally powerful as Turing machine
deciders.
A second consequence concerns the recognizability of sets of natural numbers
via number representations and finite automata. A set of numbers is said to be
recognizable with respect to a representation if an automaton accepts the
language of representations. Our result entails that there is one number
representation with respect to which every recursive set is recognizable
Finite-State Complexity and the Size of Transducers
Finite-state complexity is a variant of algorithmic information theory
obtained by replacing Turing machines with finite transducers. We consider the
state-size of transducers needed for minimal descriptions of arbitrary strings
and, as our main result, we show that the state-size hierarchy with respect to
a standard encoding is infinite. We consider also hierarchies yielded by more
general computable encodings.Comment: In Proceedings DCFS 2010, arXiv:1008.127
Some undecidability results for asynchronous transducers and the Brin-Thompson group 2V
Using a result of Kari and Ollinger, we prove that the torsion problem for
elements of the Brin-Thompson group 2V is undecidable. As a result, we show
that there does not exist an algorithm to determine whether an element of the
rational group R of Grigorchuk, Nekrashevich, and Sushchanskii has finite
order. A modification of the construction gives other undecidability results
about the dynamics of the action of elements of 2V on Cantor Space.
Arzhantseva, Lafont, and Minasyanin prove in 2012 that there exists a finitely
presented group with solvable word problem and unsolvable torsion problem. To
our knowledge, 2V furnishes the first concrete example of such a group, and
gives an example of a direct undecidability result in the extended family of R.
Thompson type groups.Comment: 16 pages, 3 figure
Recommended from our members
Which Classes of Origin Graphs Are Generated by Transducers.
We study various models of transducers equipped with origin information. We consider the semantics of these models as particular graphs, called origin graphs, and we characterise the families of such graphs recognised by streaming string transducers
Self-Similar Law of Energy Release before Materials Fracture
A general law of energy release is derived for stressed heterogeneous
materials, being valid from the starting moment of loading till the moment of
materials fracture. This law is obtained by employing the extrapolation
technique of the self-similar approximation theory. Experiments are
accomplished measuring the energy release for industrial composite samples. The
derived analytical law is confronted with these experimental data as well as
with the known experimental data for other materials.Comment: Latex, 15 pages, no figure
One-way resynchronizability of word transducers
The origin semantics for transducers was proposed in 2014, and it led to various characterizations and decidability results that are in contrast with the classical semantics. In this paper we add a further decidability result for characterizing transducers that are close to one-way transducers in the origin semantics. We show that it is decidable whether a non-deterministic two-way word transducer can be resynchro-nized by a bounded, regular resynchronizer into an origin-equivalent one-way transducer. The result is in contrast with the usual semantics, where it is undecidable to know if a non-deterministic two-way transducer is equivalent to some one-way transducer
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