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Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms
, 2002
"... We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices an ..."
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We recall that the calculation of homology with integer coecients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which in general are sparse. We provide a review of several algorithms for the calculation of Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices. Then we describe alternative approaches to the calculation of simplicial homology. The last section then describes motivating examples and actual experiments with the GAP package that was implemented by the authors. These examples also include as an example of other homology theories some calculations of Lie algebra homology.
Algorithmic problems in groups, semigroups and inverse semigroups
 Semigroups, Formal Languages and Groups
, 1995
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Parallel algorithms for group word problems. Doctoral Dissertation
, 1993
"... quality and form for publication on microfilm: ..."
Questions in Computable Algebra and Combinatorics
, 1999
"... this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these ..."
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this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25
Groups Presented by Finite TwoMonadic ChurchRosser Thue Systems
 Transactions of the American Mathematical Society
, 1986
"... Abstract. It is shown that a group G can be defined by a monoidpresentation of the form (2; 7"), where T is a finite twomonadic ChurchRosser Thue system over 2, if and only if G is isomorphic to the free product of a finitely generated free group with a finite number of finite groups. Introductio ..."
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Abstract. It is shown that a group G can be defined by a monoidpresentation of the form (2; 7"), where T is a finite twomonadic ChurchRosser Thue system over 2, if and only if G is isomorphic to the free product of a finitely generated free group with a finite number of finite groups. Introduction. In 1911 M. Dehn formulated three fundamental problems for groups given by presentations of the form (2; L), where 2 is some set of generators, 2 is a disjoint copy of 2, and L ç (2 U 2) * is a set of defining relators [12]. One of these problems is the word problem, which can be stated as follows: Let (2; L) be a group presentation. Given a word w e (2 U 2) * decide in a finite number of steps
Decision problems in group theory
 Proc. London Math. Soc
, 1982
"... At the 1976 Oxford Conference, Aanderaa introduced a new class of machines which he called F machines (later renamed as modular machines). Using these he gave two remarkably short and easy examples of finitely presented groups with unsolvable word problem. Both of these examples, together with an ex ..."
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At the 1976 Oxford Conference, Aanderaa introduced a new class of machines which he called F machines (later renamed as modular machines). Using these he gave two remarkably short and easy examples of finitely presented groups with unsolvable word problem. Both of these examples, together with an exposition of modular
Asymptotic cyclic expansion and bridge groups of formal proofs
 JOURNAL OF ALGEBRA
, 2001
"... Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify th ..."
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Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs.
The Computational Complexity Of TorsionFreeness Of Finitely Presented Groups
"... . We determine the complexity of torsionfreeness of finitely presented groups in Kleene's arithmetical hierarchy as \Pi 0 2 complete. This implies in particular that there is no effective listing of all torsionfree finitely presented groups, or of all nontorsionfree finitely presented groups. ..."
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. We determine the complexity of torsionfreeness of finitely presented groups in Kleene's arithmetical hierarchy as \Pi 0 2 complete. This implies in particular that there is no effective listing of all torsionfree finitely presented groups, or of all nontorsionfree finitely presented groups. 0. Introduction. One way of describing a group G is to give its presentation, i.e., to write G as G = hx i (i 2 I) j Ri (where fx i j i 2 Ig is a set of "generators" and R (the set of "relators") is a set of words in fx i ; x \Gamma1 i j i 2 Ig such that G ¸ = F=H where F is the free group generated by fx i j i 2 Ig and H is the normal subgroup of F generated by R. If we can find a free group F of finite rank and a finite set of relators R, then we call G a finitely presented group. Groups arising in applications, such as fundamental groups in topology, often are given naturally via their presentations. Unfortunately, a finite presentation does not yield very good information about ...
COMPLEXITY CLASSES OF EQUIVALENCE PROBLEMS REVISITED
, 2009
"... To determine if two given lists of numbers are the same set, we would sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms for equivalences arise in graph isomorphism and its variants, and the equa ..."
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To determine if two given lists of numbers are the same set, we would sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms for equivalences arise in graph isomorphism and its variants, and the equality of permutation groups given by generators. To determine if two given graphs are cospectral (have the same eigenvalues), however, we compute their characteristic polynomials and see if they are the same; the characteristic polynomial is a complete invariant for the equivalence relation of cospectrality. This is weaker than a canonical form, and it is not known whether a polynomialtime canonical form for cospectrality exists. Note that it is a priori possible for an equivalence relation to be decidable in polynomial time without either a complete invariant or canonical form. Blass and Gurevich (“Equivalence relations, invariants, and normal forms, I and II”, 1984) ask whether these conditions on equivalence relations – having an FP canonical form, having an FP complete invariant, and simply being in P – are in fact different. They showed that this question requires nonrelativizing techniques to resolve. Here we extend their results, and give new connections to probabilistic and quantum computation. We denote