scholarly journals FUNCTIONS ON GROUPS AND COMPUTATIONAL COMPLEXITY

2004 ◽  
Vol 14 (04) ◽  
pp. 409-429 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Computational Complexity ◽  
Symmetric Space ◽  
Word Problem ◽  
Exponential Inequality ◽  
Isoperimetric Function ◽  
Length Functions ◽  
Finitely Presented Groups ◽  
Double Exponential ◽  
Context Free ◽  
Finitely Presented

We give some connections between various functions defined on finitely presented groups (isoperimetric, isodiametric, Todd–Coxeter radius, filling length functions, etc.), and we study the relation between those functions and the computational complexity of the word problem (deterministic time, nondeterministic time, symmetric space). We show that the isoperimetric function can always be linearly decreased (unless it is the identity map). We present a new proof of the Double Exponential Inequality, based on context-free languages.

A. A. Fridman. Stépéni nérazréšimosti problémy toždéstva v konéčno oprédélénnyh gruppah. Doklady Akadémii Nauk SSSR, vol. 147 (1962), pp. 805–808. - A. A. Fridman. Degrees of insolvability of the word problem in finitely defined groups. English translation of the preceding by Sue Ann Walker. Soviet mathematics, vol. 3 no. 6 (1962), pp. 1733–1737. - C. R. J. Clapham. Finitely presented groups with word problems of arbitrary degrees of insolubility. Proceedings of the London Mathematical Society, ser. 3 vol. 14 (1964), pp. 633–676. - William W. Boone. Finitely presented group whose word problem has the same degree as that of an arbitrarily given Thue system (an application of methods of Britton). Proceedings of the National Academy of Sciences, vol. 53 (1965), pp. 265–269. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A first paper on Thue systems. Annals of mathematics, ser. 2 vol. 83 (1966), pp. 520–571. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A sequel on finitely presented groups. Annals of mathematics, ser. 2 vol. 84 (1966), pp. 49–84.

1968 ◽  
Vol 33 (2) ◽  
pp. 296-297
Author(s):  
J. C. Shepherdson
Keyword(s):  
Word Problem ◽  
Word Problems ◽  
Nauk Sssr ◽  
London Mathematical Society ◽  
National Academy Of Sciences ◽  
Doklady Akademii Nauk Sssr ◽  
Finitely Presented Groups ◽  
Recursively Enumerable ◽  
Finitely Presented ◽  
Degrees Of Unsolvability

Separating Classes of Groups by First-Order Sentences

2003 ◽  
Vol 13 (03) ◽  
pp. 287-302 ◽  
Author(s):  
André Nies
Keyword(s):  
Nilpotent Group ◽  
Word Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
First Order ◽  
Solvable Word Problem ◽  
Rank 2 ◽  
Finitely Presented ◽  
Solvable Word

For various proper inclusions of classes of groups [Formula: see text], we obtain a group [Formula: see text] and a first-order sentence φ such that H⊨φ but no G∈ C satisfies φ. The classes we consider include the finite, finitely presented, finitely generated with and without solvable word problem, and all countable groups. For one separation, we give an example of a f.g. group, namely ℤp ≀ ℤ for some prime p, which is the only f.g. group satisfying an appropriate first-order sentence. A further example of such a group, the free step-2 nilpotent group of rank 2, is used to show that true arithmetic Th(ℕ,+,×) can be interpreted in the theory of the class of finitely presented groups and other classes of f.g. groups.

scholarly journals Embeddings into groups with only a few defining relations

1974 ◽  
Vol 18 (1) ◽  
pp. 1-7 ◽  
Author(s):  
W. W. Boone ◽  
D. J. Collins
Keyword(s):  
Word Problem ◽  
Finitely Presented Groups ◽  
Fixed Group ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
One Relator Groups ◽  
Finitely Presented ◽  
Unsolvable Word Problem ◽  
Trivial Consequence

It is a trivial consequence of Magnus' solution to the word problem for one-relator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there is a fixed group with twenty-six defining relations in which every recursively presented group is embedded.

Relations between Control Mechanisms for Sequential Grammars1

2021 ◽  
Vol 181 (2-3) ◽  
pp. 239-271
Author(s):  
Artiom Alhazov ◽  
Rudolf Freund ◽  
Sergiu Ivanov ◽  
Marion Oswald
Keyword(s):  
Partial Order ◽  
General Framework ◽  
Control Mechanisms ◽  
Computational Power ◽  
Finitely Presented Groups ◽  
New Variant ◽  
Context Free ◽  
Finitely Presented

We extend and refine previous results within the general framework for regulated rewriting based on the applicability of rules in sequential grammars [3]. Besides the well-known control mechanisms as control graphs, matrices, permitting and forbidden rules, partial order on rules, and priority relations on rules we also consider the new variant of activation and blocking of rules as investigated in [1, 2, 4]. Moreover, we exhibit special results for strings and multisets as well as for arrays in the general variant defined on Cayley grids of finitely presented groups. Especially we prove that array grammars defined on Cayley grids of finitely presented groups using #-context-free array productions together with control mechanisms as control graphs, matrices, permitting and forbidden rules, partial order on rules, priority relations on rules, or activation and blocking of rules have the same computational power as such array grammars using arbitrary array productions.

scholarly journals THE GROUPS OF RICHARD THOMPSON AND COMPLEXITY

2004 ◽  
Vol 14 (05n06) ◽  
pp. 569-626 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Word Problem ◽  
Simple Groups ◽  
Direct Products ◽  
Turing Machines ◽  
Group V ◽  
Dehn Functions ◽  
Finitely Presented Groups ◽  
Distortion Functions ◽  
The 1960S ◽  
Finitely Presented

We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We give a faithful representation in the Cuntz C⋆-algebra. For the finitely presented simple group V we show that the word-length and the table size satisfy an n log n relation. We show that the word problem of V belongs to the parallel complexity class AC1 (a subclass of P), whereas the generalized word problem of V is undecidable. We study the distortion functions of V and show that V contains all finite direct products of finitely generated free groups as subgroups with linear distortion. As a consequence, up to polynomial equivalence of functions, the following three sets are the same: the set of distortions of V, the set of Dehn functions of finitely presented groups, and the set of time complexity functions of nondeterministic Turing machines.

Time-Complexity of the Word Problem for Semigroups and the Higman Embedding Theorem

1998 ◽  
Vol 08 (02) ◽  
pp. 235-294 ◽  
Author(s):  
Jean-Camille Birget
Keyword(s):  
Word Problem ◽  
Time Complexity ◽  
Linear Time ◽  
Algorithm Design ◽  
Embedding Theorem ◽  
Algebraic Characterization ◽  
Algorithmic Problem ◽  
Finitely Generated ◽  
Isoperimetric Function ◽  
Finitely Presented

The following algebraic characterization of the computational complexity of the word problem for finitely generated semigroups is proved, in the form of a refinement of the Higman Embedding Theorem: Let S be a finitely generated semigroup whose word problem has nondeterministic time complexity T (where T is a function on the positive integers which is superadditive, i.e. T(n+m) ≥T(n)+T(m)). Then S can be embedded in a finitely presented semigroup H in which the derivation distance between any two equivalent words x and y (and hence the isoperimetric function) is O (T(∣x∣+∣y∣)2). Moreover, there is a conjunctive linear-time reduction from the word problem of H to the word problem of S, so the word problems of S and H have the same nondeterministic time complexity (and also the same deterministic time complexity). Thus, a finitely generated semigroup S has a word problem in NTIME(T) (or in DTIME(To)) iff S is embeddable into a finitely presented semigroup H whose word problem is in NTIME(T) (respectively in DTIME(To)). In the other direction, if a finitely generated semigroup S is embeddable in a finitely presented semigroup H with isoperimetric function ≤ D (where D(n) ≥ n), then the word problem of S has nondeterministic time complexity O(D). The word problem of a finitely generated semigroup S is in NP (or more generally, in NTIME((T) O(1) )) iff S can be embedded in a finitely presented semigroup H with polynomial (respectively (T) O(1) ) isoperimetric function. An algorithmic problem L is in NP (or more generally, in NTIME((T) O(1) )) iff L is reducible (via a linear-time one-to-one reduction) to the word problem of a finitely presented semigroup with polynomial (respectively (T) O(1) ) isoperimetric function. In essence, this shows: (1) Finding embeddings into finitely presented semigroups or groups is an algebraic analogue of nondeterministic algorithm design; (2) the isoperimetric function is an algebraic analogue of nondeterministic time complexity.

Friends and relatives of BS(1,2)

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
Charles F. Miller III
Keyword(s):  
Group Theory ◽  
Word Problem ◽  
Teaching Experience ◽  
Geometric Group Theory ◽  
Point Of View ◽  
Fundamental Importance ◽  
Finitely Presented Groups ◽  
Historical Essay ◽  
Central Interest ◽  
Finitely Presented

AbstractAlgorithms, constructions and examples are of central interest in combinatorial and geometric group theory. Teaching experience and, more recently, preparing a historical essay have led me to think the familiar group BS(1,2) is an example of fundamental importance. The purpose of this note is to make a case for this point of view. We recall several interesting constructions and important examples of groups related to BS(1,2), and indicate why certain of these groups played a key role in showing the word problem for finitely presented groups is unsolvable.

scholarly journals The computability of group constructions II

1973 ◽  
Vol 8 (1) ◽  
pp. 27-60 ◽  
Author(s):  
R.W. Gatterdam
Keyword(s):  
Characteristic Function ◽  
Word Problem ◽  
Elementary Functions ◽  
Recursively Enumerable Set ◽  
Finitely Presented Groups ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Recursively Enumerable ◽  
Finitely Presented ◽  
Computational Level

Finitely presented groups having word, problem solvable by functions in the relativized Grzegorczyk hierarchy, {En(A)| n ε N, A ⊂ N (N the natural numbers)} are studied. Basically the class E3 consists of the elementary functions of Kalmar and En+1 is obtained from En by unbounded recursion. The relativization En(A) is obtained by adjoining the characteristic function of A to the class En.It is shown that the Higman construction embedding, a finitely generated group with a recursively enumerable set of relations into a finitely presented group, preserves the computational level of the word problem with respect to the relativized Grzegorczyk hierarchy. As a corollary it is shown that for every n ≥ 4 and A ⊂ N recursively enumerable there exists a finitely presented group with word problem solvable at level En(A) but not En-1(A). In particular, there exist finitely presented groups with word problem solvable at level En but not En-1 for n ≥ 4, answering a question of Cannonito.

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