1,568 research outputs found
The subpower membership problem for semigroups
Fix a finite semigroup and let be tuples in a direct
power . The subpower membership problem (SMP) asks whether can be
generated by . If is a finite group, then there is a
folklore algorithm that decides this problem in time polynomial in . For
semigroups this problem always lies in PSPACE. We show that the SMP for a full
transformation semigroup on 3 letters or more is actually PSPACE-complete,
while on 2 letters it is in P. For commutative semigroups, we provide a
dichotomy result: if a commutative semigroup embeds into a direct product
of a Clifford semigroup and a nilpotent semigroup, then SMP(S) is in P;
otherwise it is NP-complete
Von Neumann Regular Cellular Automata
For any group and any set , a cellular automaton (CA) is a
transformation of the configuration space defined via a finite memory set
and a local function. Let be the monoid of all CA over .
In this paper, we investigate a generalisation of the inverse of a CA from the
semigroup-theoretic perspective. An element is von
Neumann regular (or simply regular) if there exists
such that and , where is the composition of functions. Such an
element is called a generalised inverse of . The monoid
itself is regular if all its elements are regular. We
establish that is regular if and only if
or , and we characterise all regular elements in
when and are both finite. Furthermore, we study
regular linear CA when is a vector space over a field ; in
particular, we show that every regular linear CA is invertible when is
torsion-free elementary amenable (e.g. when ) and , and that every linear CA is regular when
is finite-dimensional and is locally finite with for all .Comment: 10 pages. Theorem 5 corrected from previous versions, in A.
Dennunzio, E. Formenti, L. Manzoni, A.E. Porreca (Eds.): Cellular Automata
and Discrete Complex Systems, AUTOMATA 2017, LNCS 10248, pp. 44-55, Springer,
201
Finite Computational Structures and Implementations
What is computable with limited resources? How can we verify the correctness
of computations? How to measure computational power with precision? Despite the
immense scientific and engineering progress in computing, we still have only
partial answers to these questions. In order to make these problems more
precise, we describe an abstract algebraic definition of classical computation,
generalizing traditional models to semigroups. The mathematical abstraction
also allows the investigation of different computing paradigms (e.g. cellular
automata, reversible computing) in the same framework. Here we summarize the
main questions and recent results of the research of finite computation.Comment: 12 pages, 3 figures, will be presented at CANDAR'16 and final version
published by IEEE Computer Societ
Automaton semigroup constructions
The aim of this paper is to investigate whether the class of automaton
semigroups is closed under certain semigroup constructions. We prove that the
free product of two automaton semigroups that contain left identities is again
an automaton semigroup. We also show that the class of automaton semigroups is
closed under the combined operation of 'free product followed by adjoining an
identity'. We present an example of a free product of finite semigroups that we
conjecture is not an automaton semigroup. Turning to wreath products, we
consider two slight generalizations of the concept of an automaton semigroup,
and show that a wreath product of an automaton monoid and a finite monoid
arises as a generalized automaton semigroup in both senses. We also suggest a
potential counterexample that would show that a wreath product of an automaton
monoid and a finite monoid is not a necessarily an automaton monoid in the
usual sense.Comment: 13 pages; 2 figure
Automaton semigroups: new construction results and examples of non-automaton semigroups
This paper studies the class of automaton semigroups from two perspectives:
closure under constructions, and examples of semigroups that are not automaton
semigroups. We prove that (semigroup) free products of finite semigroups always
arise as automaton semigroups, and that the class of automaton monoids is
closed under forming wreath products with finite monoids. We also consider
closure under certain kinds of Rees matrix constructions, strong semilattices,
and small extensions. Finally, we prove that no subsemigroup of arises as an automaton semigroup. (Previously, itself was
the unique example of a finitely generated residually finite semigroup that was
known not to arise as an automaton semigroup.)Comment: 27 pages, 6 figures; substantially revise
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