Describing Groups

2007 ◽  
Vol 13 (3) ◽  
pp. 305-339 ◽  
Author(s):  
André Nies
Keyword(s):  
Finite Automaton ◽  
Finite Automata ◽  
Finite Alphabet ◽  
Prime Model ◽  
Finitely Generated ◽  
Ring Of Integers ◽  
Group Operation ◽  
First Order

AbstractTwo ways of describing a group are considered. 1. A group is finite-automaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g. group is quasi-finitely axiomatizable if there is a description consisting of a single first-order sentence, together with the information that the group is finitely generated. In the first part of the paper we survey examples of FA-presentable groups, but also discuss theorems restricting this class. In the second part, we give examples of quasi-finitely axiomatizable groups, consider the algebraic content of the notion, and compare it to the notion of a group which is a prime model. We also show that if a structure is bi-interpretable in parameters with the ring of integers, then it is prime and quasi-finitely axiomatizable.

A note on finitely generated models

1983 ◽  
Vol 48 (1) ◽  
pp. 163-166
Author(s):  
Anand Pillay
Keyword(s):  
Countable Model ◽  
Stable Theory ◽  
Prime Model ◽  
Finitely Generated ◽  
First Order ◽  
Elementary Chain ◽  
Skolem Functions ◽  
Order Language ◽  
Finite Set

A model M (of a countable first order language) is said to be finitely generated if it is prime over a finite set, namely if there is a finite tuple ā in M such that (M, ā) is a prime model of its own theory. Similarly, if A ⊂ M, then M is said to be finitely generated overA if there is finite ā in M such that M is prime over A ⋃ ā. (Note that if Th(M) has Skolem functions, then M being prime over A is equivalent to M being generated by A in the usual sense, that is, M is the closure of A under functions of the language.) We show here that if N is ā model of an ω-stable theory, M ≺ N, M is finitely generated, and N is finitely generated over M, then N is finitely generated. A corollary is that any countable model of an ω-stable theory is the union of an elementary chain of finitely generated models. Note again that all this is trivial if the theory has Skolem functions.The result here strengthens the results in [3], where we show the same thing but assuming in addition that the theory is either nonmultidimensional or with finite αT. However the proof in [3] for the case αT finite actually shows the following which does not assume ω-stability): Let A be atomic over a finite set, tp(ā / A) have finite Cantor-Bendixson rank, and B be atomic over A ⋃ ā. Then B is atomic over a finite set.

scholarly journals STALLINGS FOLDINGS AND SUBGROUPS OF AMALGAMS OF FINITE GROUPS

2007 ◽  
Vol 17 (08) ◽  
pp. 1493-1535 ◽  
Author(s):  
L. MARKUS-EPSTEIN
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Finite Automaton ◽  
Finite Automata ◽  
Free Groups ◽  
Minimal Immersion ◽  
Finitely Generated ◽  
Inverse Monoids ◽  
Folding Algorithm ◽  
Algorithmic Problems

Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In this paper, we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for "sewing" on relations of non-free groups. We look at the class of groups that are amalgams of finite groups. It is known that these groups are locally quasiconvex and thus, all finitely generated subgroups are represented by finite automata. We present an algorithm to compute such a finite automaton and use it to solve various algorithmic problems.

Membership Problem for Two-Dimensional General Row Jumping Finite Automata

2020 ◽  
Vol 31 (04) ◽  
pp. 527-538
Author(s):  
Grzegorz Madejski ◽  
Andrzej Szepietowski
Keyword(s):  
Computational Model ◽  
Turing Machine ◽  
Finite Automaton ◽  
Finite Automata ◽  
The Other ◽  
Membership Problem ◽  
Two Dimensional ◽  
Np Complete ◽  
Nondeterministic Turing Machine ◽  
Membership Problems

Two-dimensional general row jumping finite automata were recently introduced as an interesting computational model for accepting two-dimensional languages. These automata are nondeterministic. They guess an order in which rows of the input array are read and they jump to the next row only after reading all symbols in the previous row. In each row, they choose, also nondeterministically, an order in which segments of the row are read. In this paper, we study the membership problem for these automata. We show that each general row jumping finite automaton can be simulated by a nondeterministic Turing machine with space bounded by the logarithm. This means that the fixed membership problems for such automata are in NL, and so in P. On the other hand, we show that the uniform membership problem is NP-complete.

MAGIC NUMBERS FOR SYMMETRIC DIFFERENCE NFAS

2005 ◽  
Vol 16 (05) ◽  
pp. 1027-1038 ◽  
Author(s):  
LYNETTE VAN ZIJL
Keyword(s):  
Finite Automaton ◽  
Finite Automata ◽  
Symmetric Difference ◽  
Magic Numbers ◽  
Deterministic Finite Automaton ◽  
Nondeterministic Finite Automata ◽  
Binary Case ◽  
Nondeterministic Finite Automaton

Iwama et al. showed that there exists an n-state binary nondeterministic finite automaton such that its equivalent minimal deterministic finite automaton has exactly 2n - α states, for all n ≥ 7 and 5 ≤ α ≤ 2n-2, subject to certain coprimality conditions. We investigate the same question for both unary and binary symmetric difference nondeterministic finite automata. In the binary case, we show that for any n ≥ 4, there is an n-state symmetric difference nondeterministic finite automaton for which the equivalent minimal deterministic finite automaton has 2n - 1 + 2k - 1 - 1 states, for 2 < k ≤ n - 1. In the unary case, we consider a large practical subclass of unary symmetric difference nondeterministic finite automata: for all n ≥ 2, we argue that there are many values of α such that there is no n-state unary symmetric difference nondeterministic finite automaton with an equivalent minimal deterministic finite automaton with 2n - α states, where 0 < α < 2n - 1. For each n ≥ 2, we quantify such values of α precisely.

LOGICAL CHARACTERIZATION OF RECOGNIZABLE SETS OF POLYNOMIALS OVER A FINITE FIELD

2011 ◽  
Vol 22 (07) ◽  
pp. 1549-1563 ◽  
Author(s):  
MICHEL RIGO ◽  
LAURENT WAXWEILER
Keyword(s):  
Finite Field ◽  
Order Theory ◽  
Ring Of Integers ◽  
First Order ◽  
Polynomial Coefficients ◽  
Bounded Number ◽  
First Order Theory ◽  
Ring Of Polynomials

The ring of integers and the ring of polynomials over a finite field share a lot of properties. Using a bounded number of polynomial coefficients, any polynomial can be decomposed as a linear combination of powers of a non-constant polynomial P playing the role of the base of the numeration. Having in mind the theorem of Cobham from 1969 about recognizable sets of integers, it is natural to study P-recognizable sets of polynomials. Based on the results obtained in the Ph.D. thesis of the second author, we study the logical characterization of such sets and related properties like decidability of the corresponding first-order theory.

DETERMINATIONS OF WEIGHTED FINITE AUTOMATA OVER COMMUTATIVE IDEMPOTENT MF-SEMIRINGS

2012 ◽  
Vol 22 (03) ◽  
pp. 1250020
Author(s):  
YONG HE ◽  
GONGCAI XIN ◽  
ZHIXI WANG
Keyword(s):  
Finite Automaton ◽  
Finite Dimension ◽  
Finite Automata ◽  
Commutative Semiring ◽  
Special Cases ◽  
Weighted Finite Automata

The semirings admitting maximal factorizations of any finite dimension are called MF-semirings. We first show that a commutative semiring K is an MF-semiring if and only if K admits a maximal factorization of dimension n ≥ 2, and if and only if K is a multiplicatively cancellative semiring satisfying the g.c.d. condition. And then, by using above result, we prove that a weighted finite automaton [Formula: see text] over a commutative idempotent MF-semiring has a determination if [Formula: see text] has the victory property and twins property. Also, some special cases are considered.

Error bounds on multivariate Normal approximations for word count statistics

2002 ◽  
Vol 34 (03) ◽  
pp. 559-586 ◽  
Author(s):  
Haiyan Huang
Keyword(s):  
Covariance Matrix ◽  
Error Bounds ◽  
Multivariate Distribution ◽  
Finite Alphabet ◽  
Multivariate Normal ◽  
Word Count ◽  
First Order ◽  
Normal Approximations ◽  
Necessary And Sufficient ◽  
The Given

Given a sequence S and a collection Ω of d words, it is of interest in many applications to characterize the multivariate distribution of the vector of counts U = (N(S,w 1), …, N(S,w d )), where N(S,w) is the number of times a word w ∈ Ω appears in the sequence S. We obtain an explicit bound on the error made when approximating the multivariate distribution of U by the normal distribution, when the underlying sequence is i.i.d. or first-order stationary Markov over a finite alphabet. When the limiting covariance matrix of U is nonsingular, the error bounds decay at rate O ((log n) / √n) in the i.i.d. case and O ((log n)3 / √n) in the Markov case. In order for U to have a nondegenerate covariance matrix, it is necessary and sufficient that the counted word set Ω is not full, that is, that Ω is not the collection of all possible words of some length k over the given finite alphabet. To supply the bounds on the error, we use a version of Stein's method.

STATE COMPLEXITY OF ADDITIVE WEIGHTED FINITE AUTOMATA

2007 ◽  
Vol 18 (06) ◽  
pp. 1407-1416 ◽  
Author(s):  
KAI SALOMAA ◽  
PAUL SCHOFIELD
Keyword(s):  
Upper Bound ◽  
Finite Automaton ◽  
Regular Language ◽  
Finite Automata ◽  
The State ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Weighted Finite Automata

It is known that the neighborhood of a regular language with respect to an additive distance is regular. We introduce an additive weighted finite automaton model that provides a conceptually simple way to reprove this result. We consider the state complexity of converting additive weighted finite automata to deterministic finite automata. As our main result we establish a tight upper bound for the state complexity of the conversion.

On the power of two-way multihead quantum finite automata

2019 ◽  
Vol 53 (1-2) ◽  
pp. 19-35 ◽  
Author(s):  
Amandeep Singh Bhatia ◽  
Ajay Kumar
Keyword(s):  
Prime Number ◽  
Finite Automaton ◽  
Finite Automata ◽  
Language Recognition ◽  
Quantum Finite Automata ◽  
Recognition Capability

This paper introduces a variant of two-way quantum finite automata named two-way multihead quantum finite automata. A two-way quantum finite automaton is more powerful than classical two-way finite automata. However, the generalizations of two-way quantum finite automata have not been defined so far as compared to one-way quantum finite automata model. We have investigated the newly introduced automata from two aspects: the language recognition capability and its comparison with classical and quantum counterparts. It has been proved that a language which cannot be recognized by any one-way and multi-letter quantum finite automata can be recognized by two-way quantum finite automata. Further, it has been shown that a language which cannot be recognized by two-way quantum finite automata can be recognized by two-way multihead quantum finite automata with two heads. Furthermore, it has been investigated that quantum variant of two-way deterministic multihead finite automata takes less number of heads to recognize a language containing of all words whose length is a prime number.

scholarly journals THE AUTOMORPHISM GROUP OF A SHIFT OF LINEAR GROWTH: BEYOND TRANSITIVITY

2015 ◽  
Vol 3 ◽  
Author(s):  
VAN CYR ◽  
BRYNA KRA
Keyword(s):  
Automorphism Group ◽  
Finite Groups ◽  
Linear Growth ◽  
Finite Alphabet ◽  
Finitely Generated ◽  
Complexity Function ◽  
Algebraic Properties

For a finite alphabet ${\mathcal{A}}$ and shift $X\subseteq {\mathcal{A}}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group $\text{Aut}(X)$. For such systems, we show that every finitely generated subgroup of $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if $X$ is transitive, then $\text{Aut}(X)$ is virtually $\mathbb{Z}$; if $X$ has dense aperiodic points, then $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$. We also classify all finite groups that arise as the automorphism group of a shift.

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