Bernoulli measure on strings, and Thompson-Higman monoids

The Bernoulli measure on strings is used to define height functions for the dense R- and L-orders of the Thompson-Higman monoids M_{k,1}. The measure can also be used to characterize the D-relation of certain submonoids of M_{k,1}. The computational complexity of computing the Bernoulli measure of certain sets, and in particular, of computing the R- and L-height of an element of M_{k,1} is investigated.Comment: 27 pages

One-way permutations, computational asymmetry and distortion

Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of one-way transformations. We introduce a computational asymmetry function that measures the amount of one-wayness of permutations. We also introduce the word-length 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 word-length. 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 word-length. We show that circuits built with gates that are not constrained to have fixed-length inputs and outputs, are at most quadratically more compact than circuits built from traditional gates (with fixed-length 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.Comment: 33 page