scholarly journals The Word Problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable

2011 ◽  
Vol 345 (1) ◽  
pp. 324-342 ◽  
Author(s):  
Alexei Myasnikov ◽  
Alexander Ushakov ◽  
Dong Wook Won
Keyword(s):  
Word Problem ◽  
Polynomial Time ◽  
Dehn Function

scholarly journals Taming the hydra: The word problem and extreme integer compression

2018 ◽  
Vol 28 (07) ◽  
pp. 1299-1381
Author(s):  
W. Dison ◽  
E. Einstein ◽  
T. R. Riley
Keyword(s):  
Word Problem ◽  
Polynomial Time ◽  
Complexity Measure ◽  
Dehn Function ◽  
Dehn Functions ◽  
Fast Growing ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
Direct Attack ◽  
Finitely Presented

For a finitely presented group, the word problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is a complexity measure of a direct attack on the word problem by applying the defining relations. Dison and Riley showed that a “hydra phenomenon” gives rise to novel groups with extremely fast growing (Ackermannian) Dehn functions. Here, we show that nevertheless, there are efficient (polynomial time) solutions to the word problems of these groups. Our main innovation is a means of computing efficiently with enormous integers which are represented in compressed forms by strings of Ackermann functions.

scholarly journals Groups with undecidable word problem and almost quadratic Dehn function

2012 ◽  
Vol 5 (4) ◽  
pp. 785-886
Author(s):  
A.Yu. Ol'shanskii ◽  
Mark V. Sapir
Keyword(s):  
Word Problem ◽  
Dehn Function

scholarly journals COMPRESSED WORDS AND AUTOMORPHISMS IN FULLY RESIDUALLY FREE GROUPS

2010 ◽  
Vol 20 (03) ◽  
pp. 343-355 ◽  
Author(s):  
JEREMY MACDONALD
Keyword(s):  
Automorphism Group ◽  
Word Problem ◽  
Free Group ◽  
Polynomial Time ◽  
Free Groups ◽  
Finitely Generated

We show that the compressed word problem in a finitely generated fully residually free group ([Formula: see text]-group) is decidable in polynomial time, and use this result to show that the word problem in the automorphism group of an [Formula: see text]-group is decidable in polynomial time.

scholarly journals Context-freeness of the languages of Schützenberger automata of HNN-extensions of finite inverse semigroups

2016 ◽  
Vol 99 (113) ◽  
pp. 177-191
Author(s):  
Mohammed Ayyash ◽  
Emanuele Rodaro
Keyword(s):  
Inverse Semigroup ◽  
Word Problem ◽  
Polynomial Time ◽  
Inverse Semigroups ◽  
Free Graph ◽  
Hnn Extensions ◽  
Language L ◽  
Hnn Extension ◽  
Context Free ◽  
Standard Presentation

We prove that the Sch?tzenberger graph of any element of the HNN-extension of a finite inverse semigroup S with respect to its standard presentation is a context-free graph in the sense of [11], showing that the language L recognized by this automaton is context-free. Finally we explicitly construct the grammar generating L, and from this fact we show that the word problem for an HNN-extension of a finite inverse semigroup S is decidable and lies in the complexity class of polynomial time problems.

scholarly journals The word problem for free adequate semigroups

2014 ◽  
Vol 24 (06) ◽  
pp. 893-907 ◽  
Author(s):  
Mark Kambites ◽  
Alexandr Kazda
Keyword(s):  
Word Problem ◽  
Polynomial Time ◽  
Normal Forms ◽  
Finitely Generated ◽  
Model Of Computation ◽  
Free Left ◽  
Computation Algorithms

We study the complexity of computation in finitely generated free left, right and two-sided adequate semigroups and monoids. We present polynomial time (quadratic in the RAM model of computation) algorithms to solve the word problem and compute normal forms in each of these, and hence to test whether any given identity holds in the classes of left, right and/or two-sided adequate semigroups.

scholarly journals The conjugacy problem for Higman’s group

2020 ◽  
Vol 30 (06) ◽  
pp. 1211-1235
Author(s):  
Owen Baker
Keyword(s):  
Fixed Points ◽  
Polynomial Time ◽  
Conjugacy Problem ◽  
Simple Groups ◽  
Finite State Automata ◽  
Dehn Function ◽  
Power Circuit ◽  
Finite State Transducers ◽  
Finite State ◽  
Circuit Technology

Higman’s group [Formula: see text] is a remarkable group with large (non-elementary) Dehn function. Higman constructed the group in 1951 to produce the first examples of infinite simple groups. Using finite state automata, and studying fixed points of certain finite state transducers, we show the conjugacy problem in [Formula: see text] is decidable for all inputs. Diekert, Laun and Ushakov have recently shown the word problem in [Formula: see text] is solvable in polynomial time, using the power circuit technology of Myasnikov, Ushakov and Won. Building on this work, we also show in a strongly generic setting that the conjugacy problem for [Formula: see text] has a polynomial time solution.

COMPRESSED DECISION PROBLEMS FOR GRAPH PRODUCTS AND APPLICATIONS TO (OUTER) AUTOMORPHISM GROUPS

2012 ◽  
Vol 22 (08) ◽  
pp. 1240007 ◽  
Author(s):  
NIKO HAUBOLD ◽  
MARKUS LOHREY ◽  
CHRISTIAN MATHISSEN
Keyword(s):  
Automorphism Group ◽  
Word Problem ◽  
Polynomial Time ◽  
Coxeter Group ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Conjugacy Problem ◽  
Artin Group ◽  
Graph Products ◽  
Direct Corollary

It is shown that the compressed word problem of a graph product of finitely generated groups is polynomial time Turing-reducible to the compressed word problems of the vertex groups. A direct corollary of this result is that the word problem for the automorphism group of a right-angled Artin group or a right-angled Coxeter group can be solved in polynomial time. Moreover, it is shown that a restricted variant of the simultaneous compressed conjugacy problem is polynomial time Turing-reducible to the same problem for the vertex groups. A direct corollary of this result is that the word problem for the outer automorphism group of a right-angled Artin group or a right-angled Coxeter group can be solved in polynomial time. Finally, it is shown that the compressed variant of the ordinary conjugacy problem can be solved in polynomial time for right-angled Artin groups.

Dehn Functions

2017 ◽  
Author(s):  
Timothy Riley
Keyword(s):  
Word Problem ◽  
Riemannian Manifolds ◽  
Isoperimetric Problem ◽  
Complexity Measure ◽  
Research Projects ◽  
Dehn Function ◽  
Continuous Version ◽  
Dehn Functions ◽  
Jigsaw Puzzles ◽  
Combinatorial Space

This chapter is concerned with Dehn functions. It begins by presenting jigsaw puzzles that are somewhat different from the conventional kind and explains how to solve them. It then considers a complexity measure for the word problem and shows that, for a word w, the problem of finding a sequence of free reductions, free expansions, and applications of defining relators that carries it to the empty word is equivalent to solving the puzzle where, starting from some vertex υ‎, one reads w around the initial circle of rods. The chapter also explains how the Dehn function corresponds to an isoperimetric problem in a combinatorial space, the Cayley 2-complex, and describes a continuous version of this, via group actions, along with the isoperimetry in Riemannian manifolds. Finally, it defines the Dehn function as a quasi-isometry invariant. The discussion includes exercises and research projects.

THE PROPERTY FDT IS UNDECIDABLE FOR FINITELY PRESENTED MONOIDS THAT HAVE POLYNOMIAL-TIME DECIDABLE WORD PROBLEMS

2000 ◽  
Vol 10 (03) ◽  
pp. 285-307 ◽  
Author(s):  
F. OTTO ◽  
A. SATTLER-KLEIN
Keyword(s):  
Word Problem ◽  
Polynomial Time ◽  
Word Problems ◽  
Doctoral Dissertation ◽  
New Technique ◽  
A New Technique ◽  
Finite Derivation Type ◽  
Finitely Presented ◽  
Undecidability Result

By exploiting a new technique for proving undecidability results developed by A. Sattler-Klein in her doctoral dissertation (1996) it is shown that it is undecidable in general whether or not a finitely presented monoid with a polynomial-time decidable word problem has finite derivation type (FDT). This improves upon the undecidability result of R. Cremanns and F. Otto (1996), which was based on the undecidability of the word problem for the finitely presented monoids considered.

The 2-closure of a 32-transitive group in polynomial time

2018 ◽  
Vol 60 (2) ◽  
pp. 360-375
Author(s):  
A. V. Vasil'ev ◽  
D. V. Churikov
Keyword(s):  
Polynomial Time ◽  
Transitive Group
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