7,293 research outputs found
New Size Hierarchies for Two Way Automata
© 2018, Pleiades Publishing, Ltd. We introduce a new type of nonuniform two-way automaton that can use a different transition function for each tape square. We also enhance this model by allowing to shuffle the given input at the beginning of the computation. Then we present some hierarchy and incomparability results on the number of states for the types of deterministic, nondeterministic, and bounded-error probabilistic models. For this purpose, we provide some lower bounds for all three models based on the numbers of subfunctions and we define two witness functions
Multi-Head Finite Automata: Characterizations, Concepts and Open Problems
Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg,
1966). Since that time, a vast literature on computational and descriptional
complexity issues on multi-head finite automata documenting the importance of
these devices has been developed. Although multi-head finite automata are a
simple concept, their computational behavior can be already very complex and
leads to undecidable or even non-semi-decidable problems on these devices such
as, for example, emptiness, finiteness, universality, equivalence, etc. These
strong negative results trigger the study of subclasses and alternative
characterizations of multi-head finite automata for a better understanding of
the nature of non-recursive trade-offs and, thus, the borderline between
decidable and undecidable problems. In the present paper, we tour a fragment of
this literature
Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages
The purpose of this paper is to provide efficient algorithms that decide
membership for classes of several Boolean hierarchies for which efficiency (or
even decidability) were previously not known. We develop new forbidden-chain
characterizations for the single levels of these hierarchies and obtain the
following results: - The classes of the Boolean hierarchy over level
of the dot-depth hierarchy are decidable in (previously only the
decidability was known). The same remains true if predicates mod for fixed
are allowed. - If modular predicates for arbitrary are allowed, then
the classes of the Boolean hierarchy over level are decidable. - For
the restricted case of a two-letter alphabet, the classes of the Boolean
hierarchy over level of the Straubing-Th\'erien hierarchy are
decidable in . This is the first decidability result for this hierarchy. -
The membership problems for all mentioned Boolean-hierarchy classes are
logspace many-one hard for . - The membership problems for quasi-aperiodic
languages and for -quasi-aperiodic languages are logspace many-one complete
for
Universality of two-dimensional critical cellular automata
We study the class of monotone, two-state, deterministic cellular automata,
in which sites are activated (or 'infected') by certain configurations of
nearby infected sites. These models have close connections to statistical
physics, and several specific examples have been extensively studied in recent
years by both mathematicians and physicists. This general setting was first
studied only recently, however, by Bollob\'as, Smith and Uzzell, who showed
that the family of all such 'bootstrap percolation' models on
can be naturally partitioned into three classes, which they termed subcritical,
critical and supercritical.
In this paper we determine the order of the threshold for percolation
(complete occupation) for every critical bootstrap percolation model in two
dimensions. This 'universality' theorem includes as special cases results of
Aizenman and Lebowitz, Gravner and Griffeath, Mountford, and van Enter and
Hulshof, significantly strengthens bounds of Bollob\'as, Smith and Uzzell, and
complements recent work of Balister, Bollob\'as, Przykucki and Smith on
subcritical models.Comment: 83 pages, 9 figures. This version contains significant changes to
Section 8, correcting an error in the proof, and numerous additional minor
change
Complexity Hierarchies Beyond Elementary
We introduce a hierarchy of fast-growing complexity classes and show its
suitability for completeness statements of many non elementary problems. This
hierarchy allows the classification of many decision problems with a
non-elementary complexity, which occur naturally in logic, combinatorics,
formal languages, verification, etc., with complexities ranging from simple
towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep
updating the catalogue of problems from Section 6 in future revision
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