Growth Rates of Geometric Grid Classes of Permutations

Geometric grid classes of permutations have proven to be key in investigations of classical permutation pattern classes. By considering the representation of gridded permutations as words in a trace monoid, we prove that every geometric grid class has a growth rate which is given by the square of the largest root of the matching polynomial of a related graph. As a consequence, we characterise the set of growth rates of geometric grid classes in terms of the spectral radii of trees, explore the influence of “cycle parity” on the growth rate, compare the growth rates of geometric grid classes against those of the corresponding monotone grid classes, and present new results concerning the effect of edge subdivision on the largest root of the matching polynomial.

Cubical Sets and Trace Monoid Actions

This paper is devoted to connections between trace monoids and cubical sets. We prove that the category of trace monoids is isomorphic to the category of generalized tori and it is a reflective subcategory of the category of cubical sets. Adjoint functors between the categories of cubical sets and trace monoid actions are constructed. These functors carry independence preserving morphisms in the independence preserving morphisms. This allows us to build adjoint functors between the category of weak asynchronous systems and the category of higher dimensional automata.

APPROXIMATING THE MEAN SPEEDUP IN TRACE MONOIDS

Traces can be viewed as parallel processes and the "mean speedup" of a trace monoid has been introduced as a measure of "intrinsic parallelism" of the monoid. We study the problem of computing the mean speedup under two conditions: uniform distribution on the words of given length and uniform distribution on the traces of given height. In the first case, we give an approximability result showing a probabilistic fully polynomial time approximation scheme, while, in the second case, we prove that the problem is NP-hard to approximate within n1-c for every ∊ > 0, unless NP = coR. A further consequence is the hardness of the problem of generating traces of a given height uniformly at random.

LAZARD'S ELIMINATION (IN TRACES) IS FINITE-STATE RECOGNIZABLE

We prove that the codes issued from the elimination of any sub-alphabet in a trace monoid are finite-state recognizable. This implies in particular that the transitive factorizations of the trace monoids are recognizable by (boolean) finite-state automata.