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37
InductiveDataType Systems
, 2002
"... In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schem ..."
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Cited by 779 (24 self)
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In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.
The Isomorphism Problem for Toral Relatively Hyperbolic Groups
"... We provide a solution to the isomorphism problem for torsionfree relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually ..."
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Cited by 23 (7 self)
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We provide a solution to the isomorphism problem for torsionfree relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsionfree hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic nmanifolds, for n ≥ 3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsionfree relatively hyperbolic group with abelian parabolics is
Percolation On Fuchsian Groups
, 1996
"... . It is shown that, for site percolation on the Cayley graph of a cocompact Fuchsian group of genus 2, infinitely many infinite connected clusters exist almost surely for certain values of the parameter p = Pfsite is openg. In such cases, the set of limit points at 1 of an infinite cluster is show ..."
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Cited by 18 (4 self)
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. It is shown that, for site percolation on the Cayley graph of a cocompact Fuchsian group of genus 2, infinitely many infinite connected clusters exist almost surely for certain values of the parameter p = Pfsite is openg. In such cases, the set of limit points at 1 of an infinite cluster is shown to be a perfect, nowhere dense set of Lebesgue measure 0. These results are also shown to hold for a class of hyperbolic triangle groups. 1. Introduction Percolation on a "Euclidean" graph, such as the standard integer lattice Z d , exhibits a single threshold probability p c , above which infinite clusters exist with probability 1 and below which they exist with probability 0. In the percolation regime p ? p c the infinite cluster is unique [2]. The purpose of this paper is to show that percolation on a "noneuclidean" graph may exhibit several threshold probabilities, and in particular that for some values of p infinitely many infinite clusters may coexist, while for other values of p...
On Presentations of Algebraic Structures
 in Complexity, Logic and Recursion Theory
, 1995
"... This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicat ..."
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Cited by 17 (6 self)
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This paper is an expanded version of an part of a series of invited lectures given by the author during May 1995 in Siena, Italy to the COLORET II conference. This work is partially supported by Victoria University IGC and the Marsden Fund for Basic Science under grant VIC509. This paper is dedicated to the memory of my friend and teacher Chris Ash who contributed so much to effective structure theory and who left us far too young early in 1995
String rewriting and Gröbner bases  a general approach to monoid and group rings
 Proceedings of the Workshop on Symbolic Rewriting Techniques, Monte Verita
, 1995
"... The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The tech ..."
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Cited by 15 (5 self)
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The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Grobner bases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some noncommutative cases. Several results on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced for monoid rings having a finite convergent presentation by a semiThue system. For certain presentations, including free groups and contextfree groups, the existence of finite Grobner bases for finitely generated right ideals is shown and a procedure to com...
On the rational subset problem for groups
 Journal of Algebra
"... We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through free products amalgamated over finite subgroups and HNN extensi ..."
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Cited by 13 (9 self)
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We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through free products amalgamated over finite subgroups and HNN extensions with finite associated subgroups. We provide a simple proof of a result of Grunschlag showing that the decidability of this problem is a virtual property. We prove further that the problem is decidable for a direct product of a group G with a monoid M if and only if membership is uniformly decidable for Gautomaton subsets of M. It follows that a direct product of a free group with any abelian group or commutative monoid has decidable rational subset membership. © 2006 Elsevier Inc. All rights reserved.
ComputabilityTheoretic and ProofTheoretic Aspects of Partial and Linear Orderings
 Israel Journal of mathematics
"... Szpilrajn's Theorem states that any partial order P = hS; <P i has a linear extension L = hS; <L i. This is a central result in the theory of partial orderings, allowing one to de ne, for instance, the dimension of a partial ordering. It is now natural to ask questions like \Does a we ..."
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Cited by 9 (0 self)
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Szpilrajn's Theorem states that any partial order P = hS; <P i has a linear extension L = hS; <L i. This is a central result in the theory of partial orderings, allowing one to de ne, for instance, the dimension of a partial ordering. It is now natural to ask questions like \Does a wellpartial ordering always have a wellordered linear extension?" Variations of Szpilrajn's Theorem state, for various (but not for all) linear order types , that if P does not contain a subchain of order type , then we can choose L so that L also does not contain a subchain of order type . In particular, a wellpartial ordering always has a wellordered extension.
Asymptotic density and computably enumerable sets (tentative title), in preparation
"... Abstract. We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this re ..."
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Cited by 5 (3 self)
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Abstract. We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in three different ways: (i) The degrees of such sets A are precisely the nonlow c.e. degrees. (ii) There is a c.e. set A of density 1 with no computable subset of nonzero density. (iii) There is a c.e. set A of density 1 such that every subset of A of density 1 is of high degree. We also study the extent to which c.e. sets A can be approximated by their computable subsets B in the sense that A \ B has small density. There is a very close connection between the computational complexity of a set and the arithmetical complexity of its density and we characterize the lower densities, upper densities and densities of both computable and computably enumerable sets. We also study the notion of “computable at density r ” where r is a real in the unit interval. Finally, we study connections between density and classical smallness notions such as immunity, hyperimmunity, and cohesiveness. 1.