12,306 research outputs found
Isomorphism versus commensurability for a class of finitely presented groups
We construct a class of finitely presented groups where the isomorphism problem is solvable but the commensurability problem is unsolvable. Conversely, we construct a class of finitely presented groups within which the commensurability problem is solvable but the isomorphism problem is unsolvable. These are first examples of such a contrastive complexity behaviour with respect to the isomorphism problem
A recursive presentation for Mihailova's subgroup
We give an explicit recursive presentation for Mihailova's subgroup of
corresponding to a finite, concise and Peiffer aspherical
presentation . This partially answers a
question of R.I. Grigorchuk, [8, Problem 4.14]. As a corollary, we construct a
finitely generated recursively presented orbit undecidable subgroup of
.Comment: 9 page
Search and witness problems in group theory
Decision problems are problems of the following nature: given a property P
and an object O, find out whether or not the object O has the property P. On
the other hand, witness problems are: given a property P and an object O with
the property P, find a proof of the fact that O indeed has the property P. On
the third hand(?!), search problems are of the following nature: given a
property P and an object O with the property P, find something "material"
establishing the property P; for example, given two conjugate elements of a
group, find a conjugator. In this survey our focus is on various search
problems in group theory, including the word search problem, the subgroup
membership search problem, the conjugacy search problem, and others
Exploiting Anonymity in Approximate Linear Programming: Scaling to Large Multiagent MDPs (Extended Version)
Many exact and approximate solution methods for Markov Decision Processes
(MDPs) attempt to exploit structure in the problem and are based on
factorization of the value function. Especially multiagent settings, however,
are known to suffer from an exponential increase in value component sizes as
interactions become denser, meaning that approximation architectures are
restricted in the problem sizes and types they can handle. We present an
approach to mitigate this limitation for certain types of multiagent systems,
exploiting a property that can be thought of as "anonymous influence" in the
factored MDP. Anonymous influence summarizes joint variable effects efficiently
whenever the explicit representation of variable identity in the problem can be
avoided. We show how representational benefits from anonymity translate into
computational efficiencies, both for general variable elimination in a factor
graph but in particular also for the approximate linear programming solution to
factored MDPs. The latter allows to scale linear programming to factored MDPs
that were previously unsolvable. Our results are shown for the control of a
stochastic disease process over a densely connected graph with 50 nodes and 25
agents.Comment: Extended version of AAAI 2016 pape
Some undecidability results for asynchronous transducers and the Brin-Thompson group 2V
Using a result of Kari and Ollinger, we prove that the torsion problem for
elements of the Brin-Thompson group 2V is undecidable. As a result, we show
that there does not exist an algorithm to determine whether an element of the
rational group R of Grigorchuk, Nekrashevich, and Sushchanskii has finite
order. A modification of the construction gives other undecidability results
about the dynamics of the action of elements of 2V on Cantor Space.
Arzhantseva, Lafont, and Minasyanin prove in 2012 that there exists a finitely
presented group with solvable word problem and unsolvable torsion problem. To
our knowledge, 2V furnishes the first concrete example of such a group, and
gives an example of a direct undecidability result in the extended family of R.
Thompson type groups.Comment: 16 pages, 3 figure
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