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Oneway permutations, computational asymmetry and distortion
 Online prepublication DOI: http://dx.doi.org/10.1016/j.jalgebra.2008.05.035 ) (Preprint: ArXiv http://arxiv.org/abs/0704.1569
, 2008
"... Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of oneway transformations. We introduce a computational asymmetry function that measures the amount of onewayness of permutations. We also introduce the wor ..."
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Cited by 5 (4 self)
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Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of oneway transformations. We introduce a computational asymmetry function that measures the amount of onewayness of permutations. We also introduce the wordlength asymmetry function for groups, which is an algebraic analogue of computational asymmetry. We relate boolean circuits to words in a Thompson monoid, over a fixed generating set, in such a way that circuit size is equal to wordlength. Moreover, boolean circuits have a representation in terms of elements of a Thompson group, in such a way that circuit size is polynomially equivalent to wordlength. We show that circuits built with gates that are not constrained to have fixedlength inputs and outputs, are at most quadratically more compact than circuits built from traditional gates (with fixedlength inputs and outputs). Finally, we show that the computational asymmetry function is closely related to certain distortion functions: The computational asymmetry function is polynomially equivalent to the distortion of the path length in Schreier graphs of certain Thompson groups, compared to the path length in Cayley graphs of certain Thompson monoids. We also show that the results of Razborov and others on monotone circuit complexity lead to exponential lower bounds on certain distortions. 1
Monoid generalizations of the Richard Thompson groups
 Mathematics ArXiv: math.GR/0704.0189
, 2007
"... The groups Gk,1 of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call Mk,1, and to inverse monoids, called Invk,1; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids Mk,1 have connections with c ..."
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Cited by 4 (4 self)
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The groups Gk,1 of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call Mk,1, and to inverse monoids, called Invk,1; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids Mk,1 have connections with circuit complexity (studied in another paper). Here we prove that Mk,1 and Invk,1 are congruencesimple for all k. Their Green relations J and D are characterized: Mk,1 and Invk,1 are J0simple, and they have k − 1 nonzero Dclasses. They are submonoids of the multiplicative part of the Cuntz algebra Ok. They are finitely generated, and their word problem over any finite generating set is in P. Their word problem is coNPcomplete over certain infinite generating sets. 1 ThompsonHigman monoids Since their introduction by Richard J. Thompson in the mid 1960s [25, 22, 26], the Thompson groups have had a great impact on infinite group theory. Graham Higman generalized the Thompson groups to an infinite family [17]. These groups and some of their subgroups have appeared in many contexts and have been widely studied; see for example [9, 5, 12, 7, 14, 15, 6, 8, 20]. The definition of the ThompsonHigman groups lends itself easily to generalizations to inverse
ALPS’07  Groups and Complexity
, 2007
"... The connection between groups and recursive (un)decidability has a long history, going back to the early 1900s. Also, various polynomialtime algorithms have been known in group theory for a long time. However the impact of more general computational complexity (e.g., NPcompleteness or PSpacecompl ..."
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The connection between groups and recursive (un)decidability has a long history, going back to the early 1900s. Also, various polynomialtime algorithms have been known in group theory for a long time. However the impact of more general computational complexity (e.g., NPcompleteness or PSpacecompleteness) has been relatively small and recent. These lectures review a sampling of older facts about algorithmic problems in group theory, and then present more recent results about the connection with complexity: isoperimetric functions and NP; Thompson groups, boolean circuits, and coNP; Thompson monoids and circuit complexity; Thompson groups, reversible computing, and #P; distortion of Thompson groups within Thompson monoids, and oneway permutations. We are especially interested in deep connections between computational complexity and group theory. By “connection ” we do not just mean analyzing the computational complexity of algorithms about groups. We are more interested in algebraic characterizations of complexity classes in terms of group theory, i.e., in finding a “mirror image” of all of complexity theory within group theory. Conversely, we are interested in the computational nature of concepts that appear at first purely algebraic.
The R and Lorders of the ThompsonHigman monoid Mk,1 and their complexity
, 812
"... We study the monoid generalization Mk,1 of the ThompsonHigman groups, and we characterize the R and the Lpreorder of Mk,1. Although Mk,1 has only one nonzero Jclass and k−1 nonzero Dclasses, the R and the Lpreorder are complicated; in particular,
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We study the monoid generalization Mk,1 of the ThompsonHigman groups, and we characterize the R and the Lpreorder of Mk,1. Although Mk,1 has only one nonzero Jclass and k−1 nonzero Dclasses, the R and the Lpreorder are complicated; in particular, <R is dense (even within an Lclass), and <L is dense (even within an Rclass). We study the computational complexity of the R and the Lpreorder. When inputs are given by words over a finite generating set of Mk,1, the R and the Lpreorder decision problems are in P. The main result of the paper is that over a “circuitlike ” generating set, the Rpreorder decision problem of Mk,1 is Π P 2complete, whereas the Lpreorder decision problem is coNPcomplete. We also prove related results about circuits: For combinational circuits, the surjectiveness problem is Π P 2complete, whereas the injectiveness problem is coNPcomplete. 1