The MATRIX PACKING DOWN problem asks to find a row permutation of a given (0,1)-matrix in such a way that the total sum of the first non-zero column indexes is maximized. We study the computational complexity of this problem. We prove that the MATRIX PACKING DOWN problem is NP-complete even when restricted to zero trace symmetric (0,1)-matrices or to (0,1)-matrices with at most two 1's per column. Also, as intermediate results, we introduce several new simple graph layout problems which are proved to be NP-complete.

We introduce doubly-ranked (DR) monoids in order to study picturecodes. We show that a DR-monoid is free iff it is pictoriallystable. This allows us to associate with a set C of pictures apicture code B(C) which is the basis of the least DR-monoidincluding C.A weak version of the defect theorem for pictures is established.A characterization of picture codes through picture series isalso given.

In the infinite Post Correspondence Problem an instance (h,g)consists of two morphisms h and g, and the problem is todetermine whether or not there exists an infinite word ωsuch that h(ω) = g(ω). This problem was shown to beundecidable by Ruohonen (1985) in general. Recently Blondel and Canterini (Theory Comput. Syst.36(2003) 231–245) showed that this problem is undecidable for domainalphabets of size 105. Here we give a proof that the infinite PostCorrespondence Problem is undecidable for instances where themorphisms have domains of 9 letters. The proof uses a recentresult of Matiyasevich and Sénizergues and a modification of aresult of Claus.

In secret sharing, different access structures have different difficulty degrees for acceding to the secret. We give a numerical measure of how easy or how difficult is to recover the secret, depending only on the structure itself and not on the particular scheme used for realizing it. We derive some consequences.

Arithmetical complexity of a sequence is the number of words of length n that can be extracted from it according to arithmetic progressions. We study uniformly recurrent words of low arithmetical complexity and describe the family of such words having lowest complexity.

In this note we consider the longest word, which has periods p1,...,pn , and does not have the period gcd(p1,...,pn ).The length of such a word can be established by a simple algorithm. We give a short and natural way to prove that the algorithm is correct. We also give a new proof that the maximal word is a palindrome.

There exists a bijection between one-stack sortable permutations (permutations which avoid the pattern (231)) and rooted plane trees. We define an edit distance between permutations which is consistent with the standard edit distance between trees. This one-to-one correspondence yields a polynomial algorithm for the subpermutation problem for (231) pattern-avoiding permutations.Moreover, we obtain the generating function of the edit distance between ordered unlabeled trees and some special ones.For the general case we show that the mean edit distance between a rooted plane tree and all other rooted plane trees is at least n/ln(n).Some results can be extended to labeled trees considering coloredDyck paths or, equivalently, colored one-stack sortable permutations.

We consider conversions of regular expressions into k-realtimefinite state automata, i.e., automata in which the number ofconsecutive uses of ε-transitions, along any computation path,is bounded by a fixed constant k. For 2-realtime automata,i.e., for automata that cannot change the state, without readingan input symbol, more than two times in a row, we show that theconversion of a regular expression into such an automaton producesonly O(n) states, O(nlogn) ε-transitions, and O(n)alphabet-transitions. We also show how to easily transform these2-realtime machines into 1-realtime automata, still with onlyO(nlogn) edges. These results contrast with the known lowerbound Ω(n(logn)2 / loglogn), holding for 0-realtimeautomata, i.e., for automata with no ε-transitions.