288 research outputs found
Real Computational Universality: The Word Problem for a class of groups with infinite presentation
The word problem for discrete groups is well-known to be undecidable by a
Turing Machine; more precisely, it is reducible both to and from and thus
equivalent to the discrete Halting Problem.
The present work introduces and studies a real extension of the word problem
for a certain class of groups which are presented as quotient groups of a free
group and a normal subgroup. Most important, the free group will be generated
by an uncountable set of generators with index running over certain sets of
real numbers. This allows to include many mathematically important groups which
are not captured in the framework of the classical word problem.
Our contribution extends computational group theory from the discrete to the
Blum-Shub-Smale (BSS) model of real number computation. We believe this to be
an interesting step towards applying BSS theory, in addition to semi-algebraic
geometry, also to further areas of mathematics.
The main result establishes the word problem for such groups to be not only
semi-decidable (and thus reducible FROM) but also reducible TO the Halting
Problem for such machines. It thus provides the first non-trivial example of a
problem COMPLETE, that is, computationally universal for this model.Comment: corrected Section 4.
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Computability Theory
Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödelâs Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science
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Computability Theory
Computability and computable enumerability are two of the fundamental notions of mathematics. Interest in effectiveness is already apparent in the famous Hilbert problems, in particular the second and tenth, and in early 20th century work of Dehn, initiating the study of word problems in group theory. The last decade has seen both completely new subareas develop as well as remarkable growth in two-way interactions between classical computability theory and areas of applications. There is also a great deal of work on algorithmic randomness, reverse mathematics, computable analysis, and in computable structure theory/computable model theory. The goal of this workshop is to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work
Counting edge-injective homomorphisms and matchings on restricted graph classes
We consider the -hard problem of counting all matchings with
exactly edges in a given input graph ; we prove that it remains
-hard on graphs that are line graphs or bipartite graphs
with degree on one side. In our proofs, we use that -matchings in line
graphs can be equivalently viewed as edge-injective homomorphisms from the
disjoint union of length- paths into (arbitrary) host graphs. Here, a
homomorphism from to is edge-injective if it maps any two distinct
edges of to distinct edges in . We show that edge-injective
homomorphisms from a pattern graph can be counted in polynomial time if
has bounded vertex-cover number after removing isolated edges. For hereditary
classes of pattern graphs, we complement this result: If the
graphs in have unbounded vertex-cover number even after deleting
isolated edges, then counting edge-injective homomorphisms with patterns from
is -hard. Our proofs rely on an edge-colored
variant of Holant problems and a delicate interpolation argument; both may be
of independent interest.Comment: 35 pages, 9 figure
Computable Categoricity of Trees of Finite Height
We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a ÎŁ03-condition. We show that all trees which are not computably categorical have computable dimension Ï. Finally, we prove that for every n â„ 1 in Ï, there exists a computable tree of finite height which is â0n+1-categorical but not â0n-categorical
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