Monoids and Maximal Codes

In recent years codes that are not Uniquely Decipherable (UD) are been studied partitioning them in classes that localize the ambiguities of the code. A natural question is how we can extend the notion of maximality to codes that are not UD. In this paper we give an answer to this question. To do this we introduce a partial order in the set of submonoids of a monoid showing the existence, in this poset, of maximal elements that we call full monoids. Then a set of generators of a full monoid is, by definition, a maximal code. We show how this definition extends, in a natural way, the existing definition concerning UD codes and we find a characteristic property of a monoid generated by a maximal UD code.Comment: In Proceedings WORDS 2011, arXiv:1108.341

On the group of a rational maximal bifix code

We give necessary and sufficient conditions for the group of a rational maximal bifix code $Z$ to be isomorphic with the $F$-group of $Z\cap F$, when $F$ is recurrent and $Z\cap F$ is rational. The case where $F$ is uniformly recurrent, which is known to imply the finiteness of $Z\cap F$, receives special attention. The proofs are done by exploring the connections with the structure of the free profinite monoid over the alphabet of $F$

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

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