scholarly journals Finite automata for Schreier graphs of virtually free groups

2016 ◽  
Vol 19 (1) ◽  
Author(s):  
Pedro V. Silva ◽  
Xaro Soler-Escrivà ◽  
Enric Ventura
Keyword(s):  
Finite Automata ◽  
Free Groups ◽  
Efficient Computation ◽  
Constructive Approach ◽  
The Core ◽  
Schreier Graphs

AbstractThe Stallings construction for f.g. subgroups of free groups is generalized by introducing the concept of Stallings section, which allows efficient computation of the core of a Schreier graph based on edge folding. It is proved that the groups that admit Stallings sections are precisely the f.g. virtually free groups, this is proved through a constructive approach based on Bass–Serre theory. Complexity issues and applications are also discussed.

scholarly journals PSPACE-complete problems for subgroups of free groups and inverse finite automata

2000 ◽  
Vol 242 (1-2) ◽  
pp. 247-281 ◽  
Author(s):  
J.-C. Birget ◽  
S. Margolis ◽  
J. Meakin ◽  
P. Weil
Keyword(s):  
Finite Automata ◽  
Free Groups ◽  
Complete Problems

scholarly journals AUTOMATA OVER A BINARY ALPHABET GENERATING FREE GROUPS OF EVEN RANK

2011 ◽  
Vol 21 (01n02) ◽  
pp. 329-354 ◽  
Author(s):  
BENJAMIN STEINBERG ◽  
MARIYA VOROBETS ◽  
YAROSLAV VOROBETS
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Finite Automata ◽  
Free Groups ◽  
Free Products ◽  
Cyclic Groups ◽  
Binary Alphabet

We construct automata over a binary alphabet with 2n states, n ≥ 2, whose states freely generate a free group of rank 2n. Combined with previous work, this shows that a free group of every finite rank can be generated by finite automata over a binary alphabet. We also construct free products of cyclic groups of order two via such automata.

scholarly journals Faithful group actions and Schreier graphs

2018 ◽  
Vol 9 (2) ◽  
pp. 202-207
Author(s):  
M. Fedorova
Keyword(s):  
Group Action ◽  
Finite Groups ◽  
Finite Automaton ◽  
Finite Automata ◽  
Group Actions ◽  
Sufficient Condition ◽  
Free Products ◽  
Schreier Graphs ◽  
Products Of Groups ◽  
Dynamical Properties

Each action of a finitely generated group on a set uniquely defines a labelled directed graph called the Schreier graph of the action. Schreier graphs are used mainly as a tool to establish geometrical and dynamical properties of corresponding group actions. In particilar, they are widely used in order to check amenability of different classed of groups. In the present paper Schreier graphs are utilized to construct new examples of faithful actions of free products of groups. Using Schreier graphs of group actions a sufficient condition for a group action to be faithful is presented. This result is applied to finite automaton actions on spaces of words i.e. actions defined by finite automata over finite alphabets. It is shown how to construct new faithful automaton presentations of groups upon given such a presentation. As an example a new countable series of faithful finite automaton presentations of free products of finite groups is constructed. The obtained results can be regarded as another way to construct new faithful actions of  groups  as soon as at least one such an action is provided.

scholarly journals Realizable ranks of joins and intersections of subgroups in free groups

2019 ◽  
Vol 30 (03) ◽  
pp. 625-666
Author(s):  
Ignat Soroko
Keyword(s):  
Fundamental Group ◽  
Free Group ◽  
Upper Bound ◽  
Free Groups ◽  
Combinatorial Structure ◽  
The Core ◽  
Main Conjecture ◽  
Open Case

The famous Hanna Neumann Conjecture (now the Friedman–Mineyev theorem) gives an upper bound for the ranks of the intersection of arbitrary subgroups [Formula: see text] and [Formula: see text] of a non-abelian free group. It is an interesting question to “quantify” this bound with respect to the rank of [Formula: see text], the subgroup generated by [Formula: see text] and [Formula: see text]. We describe a set of realizable values [Formula: see text] for arbitrary [Formula: see text], [Formula: see text], and conjecture that this locus is complete. We study the combinatorial structure of the topological pushout of the core graphs for [Formula: see text] and [Formula: see text] with the help of graphs introduced by Dicks in the context of his Amalgamated Graph Conjecture. This allows us to show that certain conditions on ranks of [Formula: see text], [Formula: see text] are not realizable, thus resolving the remaining open case [Formula: see text] of Guzman’s “Group-Theoretic Conjecture” in the affirmative. This in turn implies the validity of the corresponding “Geometric Conjecture” on hyperbolic 3-manifolds with a 6-free fundamental group. Finally, we prove the main conjecture describing the locus of realizable values for the case when [Formula: see text].

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.

The core-free groups of Sylow-2-type M24

1979 ◽  
Vol 167 (1) ◽  
pp. 15-23 ◽  
Author(s):  
Hedwig Sandl�bes ◽  
Ulrich Schoenwaelder
Keyword(s):  
Free Groups ◽  
The Core

Efficient Computation of the Core Functions of Exponential Itegrators

2011 ◽  
Author(s):  
Paolo Novati ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras ◽  
Zacharias Anastassi
Keyword(s):  
Efficient Computation ◽  
The Core

FINITE AUTOMATA OVER FREE GROUPS

2000 ◽  
Vol 10 (06) ◽  
pp. 725-737 ◽  
Author(s):  
JÜRGEN DASSOW ◽  
VICTOR MITRANA
Keyword(s):  
Free Group ◽  
Finite Automata ◽  
Free Groups ◽  
Input String ◽  
Neutral Element ◽  
Deterministic Case ◽  
Final Configuration ◽  
Context Free

Finite automata are extended by adding an element of a given group to each of their configurations. An input string is accepted if and only if the neutral element of the group is associated to a final configuration reached by the automaton. We get a new characterization of the context-free languages as soon as the considered group is the binary free group. The result cannot be carried out in the deterministic case. Some remarks about finite automata over other groups are also presented.

scholarly journals The Strong Minimalist Thesis

Philosophies ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 97
Author(s):  
Robert Freidin
Keyword(s):  
Minimalist Program ◽  
Cognitive Systems ◽  
Interface Conditions ◽  
Efficient Computation ◽  
Additional Structure ◽  
Form And Function ◽  
Structure Building ◽  
The Core ◽  
Computational Operation ◽  
And Function

This article reviews and attempts to evaluate the various proposals for a strong minimalist thesis that have been at the core of the minimalist program for linguistic theory since its inception almost three decades ago. These proposals have involved legibility conditions for the interface between language and the cognitive systems that access it, the simplest computational operation Merge (its form and function), and principles of computational efficiency (including inclusiveness, no-tampering, cyclic computation, and the deletion of copies). This evaluation attempts to demonstrate that reliance on interface conditions encounters serious long-standing problems for the analysis of language. It also suggests that the precise formulation of Merge may, in fact, subsume the effects of those principles of efficient computation currently under investigation and might possibly render unnecessary proposals for additional structure building operations (e.g., Pair-Merge and FormSequence).

What face familiarity feelings say about the lateralization of specific entities within the core system

2019 ◽  
Vol 42 ◽  
Author(s):  
Guido Gainotti
Keyword(s):  
Temporal Lobe ◽  
Memory System ◽  
Core System ◽  
The Core ◽  
Memory Traces ◽  
Specific Memory ◽  
Target Article ◽  
Temporal Structures ◽  
The Right

Abstract The target article carefully describes the memory system, centered on the temporal lobe that builds specific memory traces. It does not, however, mention the laterality effects that exist within this system. This commentary briefly surveys evidence showing that clear asymmetries exist within the temporal lobe structures subserving the core system and that the right temporal structures mainly underpin face familiarity feelings.

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