Modular Machines and The Higman-Clapham-Valiev Embedding Theorem

Aanderaa, Stål; Cohen, Daniel E.

doi:10.1016/s0049-237x(08)71328-4

Cited by 11 publications

(25 citation statements)

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“…Therefore, after a nite number of steps, one of these processes will end and we will decide if u = v in A. 2 Thus if, in a variety V, all nitely presented algebras are residually nite then the word problem is solvable in V. Moreover the McKinsey algorithm is \uniform": it does not depend on the presentation of an algebra and so it solves the uniform word problem also. By virtue of a result by Mekler, Nelson, Shelah, and Wells 265], this implies that the decidability of the word problem is weaker than residual niteness: the varieties constructed in 265] have solvable word problem and unsolvable uniform word problem.…”

Section: Connection 24 Let a Be An Algebra Nitely Presented In A Nitmentioning

confidence: 99%

Algorithmic Problems in Varieties

Kharlampovich

Sapir

1995

Int. J. Algebra Comput.

151

120

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Section: Connection 24 Let a Be An Algebra Nitely Presented In A Nitmentioning

confidence: 99%

Algorithmic Problems in Varieties

Kharlampovich

Sapir

1995

Int. J. Algebra Comput.

151

120

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“…For the convenience of the reader we recall the definition of modular machines (introduced in [1], [2]). Let T be a directed state Turing machine (we do not need to have directed states but this simplifies matters, e.g., see [9]).…”

Section: Modular Machinesmentioning

confidence: 99%

“…Simplifications and generalizations (e.g., to degrees of unsolvability) were subsequently obtained by various workers, e.g., Bokut' [4], Collins [10] and Miller [13] (for finitely generated recursively presented groups-see also Miller [14]). Aaanderaa and Cohen [1], [2] introduced modular machines and applied them to the area of decision problems in group theory (see also Kalorkoti [12]). These are easy to "embed" into groups and result in further simplifications.…”

Section: Introductionmentioning

confidence: 99%

A Finitely Presented Group With Almost Solvable Conjugacy Problem

Kalorkoti

2009

Asian-European J. Math.

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The algorithmic unsolvability of the conjugacy problem for finitely presented groups was demonstrated by Novikov in the early 1950s. Various simplifications and alternative proofs were found by later researchers and further questions raised. Recent work by Borovik, Myasnikov and Remeslennikov has considered the question of what proportion of the number of elements of a group (obtained by standard constructions) falls into the realm of unsolvability. In this paper we provide a straightforward construction, as a Britton tower, of a finitely presented group with solvable word problem but unsolvable conjugacy problem of any r.e. (recursively enumerable) Turing degree a. The question of whether two elements are conjugate is bounded truth-table reducible to the question of whether the elements are both conjugate to a single generator of the group. We also define computable normal forms, based on the method of Bokut', that are suitable for the conjugacy problem. We consider (ordered) pairs of normal words U , V for the conjugacy problem whose lengths add to l and show that the proportion of such pairs for which conjugacy is undecidable (in the case a = 0) is strictly less than l 2 /(2λ−1) l where λ > 4.The construction is based on modular machines, introduced by Aanderaa and Cohen. For the purposes of this construction it was helpful to extend the notion of configuration to include pairs of m-adic integers. The notion of computation step was also extended and is referred to as s-fold computation where s ∈ Z (the usual notion coresponds to s = 1). If gcd(m, s) = 1 then determinism is preserved, i.e., if the modular machine is deterministic then it remains so under the extended notion. Furthermore there is a simple correspondence between s-fold and standard computation in this case. Otherwise computation is non-deterministic and there does not seem to be any straightforward correspondence between s-fold and standard computation.

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“…, wn such that wo is u , wn is v and, for i < n, w i t 1 is obtained from W j by one of the following: (1) X-insertions, X-deletions, or R-deletions (all of which will be called G-moves), ( 2 ) insertions of kk-l or h -l k , which are called k-insertions, ( 3 ) deletions of kk" or k-'k, which are called k-deletions, and (4) replacing a-*kca6 by k ' , where u E A, 6 and E are f l , which is called a contraction. .…”

Section: Since U M Comes By X-insertions and R-insertions Froin U Itmentioning

confidence: 99%

“…Once again, the proof can be slightly simplified using modular machines m in [2] or [8]. Once again, the proof can be slightly simplified using modular machines m in [2] or [8].…”

Section: T H E O R E M 13 If H O ( M ) Has Complezity At Least 3 Thmentioning

confidence: 99%

Seperating the intrinsic complexity and the derivational complexity of the word problem for finitely presented groups

Cohen

Madlener

Otto

1993

Mathematical Logic Qtrly

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A pseudc-natural algorithm for the word problem of a finitely presented group is an algorithm which not only tells us whether or not a word w equals 1 in the group but also gives a derivation of 1 from w when w eqiials 1. In [13], [14] Madlener and Otto show that, if we measure complexity of a primitive recursive algorithm by its level in the Grzegorczyk hierarchy. there are groups in which a pseudenatural algorithm is arbitrarily more complicated than an algorithm which simply solves the word problem. In a given group the lowest. degree of complexity that can be realised by a pseudc-nat.ura1 algorithin is essentially t.he derivational complexity of that group. Thus the result separates the derivat.iona1 complexity of the word problem of a finitely presented group from it.s int.rinsic complexity. The proof given in [13] involves the construction of a finitely presented group G from a Turing machine T such that the intrinsic complexity of the word problem for G reflects the complexity of the halting problem of T, while the derivational complexity of the word problem for G reflects the runtime complexity of T. The proof of one of the crucial leminas in [13] is only sketched, and part of the purpose of this paper is to give the full details of this proof. We will also obtain a variant of their proof, using modular machines rather than Turing machines. As for several other results, this simplifies proofs considerably. MSC: 03D40, 20F10.Keywords: Word problem for groups, Complexity of word problems. the

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Modular Machines and The Higman-Clapham-Valiev Embedding Theorem

Cited by 11 publications

References 4 publications

Algorithmic Problems in Varieties

Algorithmic Problems in Varieties

A Finitely Presented Group With Almost Solvable Conjugacy Problem

Seperating the intrinsic complexity and the derivational complexity of the word problem for finitely presented groups

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