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Transforming Curves on Surfaces
, 1999
"... We describe an optimal algorithm to decide if one closed curve on a triangulated 2manifold can be continuously transformed to another, i.e., if they are homotopic. Suppose C 1 and C 2 are two closed curves on a surface M of genus g. Further, suppose T is a triangulation of M of size n such that C ..."
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We describe an optimal algorithm to decide if one closed curve on a triangulated 2manifold can be continuously transformed to another, i.e., if they are homotopic. Suppose C 1 and C 2 are two closed curves on a surface M of genus g. Further, suppose T is a triangulation of M of size n such that C 1 and C 2 are represented as edgevertex sequences of lengths k 1 and k 2 in T , respectively. Then, our algorithm decides if C 1 and C 2 are homotopic in O(n+k 1 +k 2 ) time and space, provided g 6= 2 if M orientable, and g 6= 3; 4 if M is nonorientable. This as well implies an optimal algorithm to decide if a closed curve on a surface can be continuously contracted to a point. Except for three low genus cases, our algorithm completes an investigation into the computational complexity of two classical problems for surfaces posed by the mathematician Max Dehn at the beginning of this century. The novelty of our approach is in the application of methods from modern combinatorial group theory...
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
"... ..."
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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Cited by 5 (0 self)
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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
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 ...
LIMITATIONS ON OUR UNDERSTANDING OF THE BEHAVIOR OF SIMPLIFIED PHYSICAL SYSTEMS
, 2008
"... Results going back to Turing and Gödel provide us with limitations on our ability to algorithmically decide the truth or falsity of mathematical assertions in a number of important mathematical contexts. Here we adapt some of this earlier work to very simplified mathematical models of discrete dete ..."
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Results going back to Turing and Gödel provide us with limitations on our ability to algorithmically decide the truth or falsity of mathematical assertions in a number of important mathematical contexts. Here we adapt some of this earlier work to very simplified mathematical models of discrete deterministic physical systems involving a few moving bodies (twelve point masses) in potentially infinite one dimensional space. There are two kinds of such limiting results that must be carefully distinguished. Results of the first kind state the nonexistence of any algorithm for determining whether any statement among a given set of statements is true or false. Results of the second kind are much deeper and present much greater challenges. They point to specific statements A, where we can neither prove nor refute A using accepted principles of mathematical reasoning. We give a brief survey of these limiting results. These include limiting results of the first kind: from number theory, group theory, and topology, in mathematics, and from idealized computing devices in theoretical computer science. We present a new limiting result of the first kind for simplified physical systems. We conjecture some related limiting results of the second kind, for simplified physical systems.
EFFICIENT COMPUTATION IN GROUPS AND SIMPLICIAL COMPLEXES BY
"... Abstract. Using HNN extensions of the BooneBritton group, a group E is obtained which simulates Turing machine computation in linear space and cubic time. Space in E is measured by the length of words, and time by the number of substitutions of defining relators and conjugations by generators requi ..."
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Abstract. Using HNN extensions of the BooneBritton group, a group E is obtained which simulates Turing machine computation in linear space and cubic time. Space in E is measured by the length of words, and time by the number of substitutions of defining relators and conjugations by generators required to convert one word to another. The space bound is used to derive a PSPACEcomplete problem for a topological model of computation previously used to characterize NPcompleteness and REcompleteness. Introduction. The ability of mathematical systems to simulate computation has often been used to prove unsolvability results. The first, and most instructive, example was Post's simulation of Turing machines by finitely presented semigroups [10]. For each deterministic Turing machine M, Post constructs a semigroup T(M) on generators we shall call qa, sb, where a and A range over certain finite sets. An
1 LIMITATIONS ON OUR UNDERSTANDING OF THE BEHAVIOR OF SIMPLIFIED PHYSICAL SYSTEMS
, 2008
"... Abstract. Results going back to Turing and Gödel provide us with limitations on our ability to algorithmically decide the truth or falsity of mathematical assertions in a number of important mathematical contexts. Here we adapt some of this earlier work to very simplified mathematical models of disc ..."
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Abstract. Results going back to Turing and Gödel provide us with limitations on our ability to algorithmically decide the truth or falsity of mathematical assertions in a number of important mathematical contexts. Here we adapt some of this earlier work to very simplified mathematical models of discrete deterministic physical systems involving a few moving bodies (twelve point masses) in potentially infinite one dimensional space. There are two kinds of such limiting results that must be carefully distinguished. Results of the first kind state the nonexistence of any algorithm for determining whether any statement among a given set of statements is true or false. Results of the second kind are much deeper and present much greater challenges. They point to specific statements A, where we can neither prove nor refute A using accepted principles of mathematical reasoning. We give a brief survey of these limiting results. These include limiting results of the first kind: from number theory, group theory, and topology, in mathematics, and from idealized computing devices in theoretical computer science. We present a new limiting result of the first kind for simplified physical systems. We conjecture some related limiting results of the second kind, for simplified physical systems.