L-Convex Polyominoes are Recognizable in Real Time by 2D Cellular Automata

A polyomino is said to be L-convex if any two of its cells are connected by a 4-connected inner path that changes direction at most once. The 2-dimensional language representing such polyominoes has been recently proved to be recognizable by tiling systems by S. Brocchi, A. Frosini, R. Pinzani and S. Rinaldi. In an attempt to compare recognition power of tiling systems and cellular automata, we have proved that this language can be recognized by 2-dimensional cellular automata working on the von Neumann neighborhood in real time. Although the construction uses a characterization of L-convex polyominoes that is similar to the one used for tiling systems, the real time constraint which has no equivalent in terms of tilings requires the use of techniques that are specific to cellular automata

On the Hierarchy of Block Deterministic Languages

A regular language is $k$-lookahead deterministic (resp. $k$-block deterministic) if it is specified by a $k$-lookahead deterministic (resp. $k$-block deterministic) regular expression. These two subclasses of regular languages have been respectively introduced by Han and Wood ($k$-lookahead determinism) and by Giammarresi et al. ($k$-block determinism) as a possible extension of one-unambiguous languages defined and characterized by Br\"uggemann-Klein and Wood. In this paper, we study the hierarchy and the inclusion links of these families. We first show that each $k$-block deterministic language is the alphabetic image of some one-unambiguous language. Moreover, we show that the conversion from a minimal DFA of a $k$-block deterministic regular language to a $k$-block deterministic automaton not only requires state elimination, and that the proof given by Han and Wood of a proper hierarchy in $k$-block deterministic languages based on this result is erroneous. Despite these results, we show by giving a parameterized family that there is a proper hierarchy in $k$-block deterministic regular languages. We also prove that there is a proper hierarchy in $k$-lookahead deterministic regular languages by studying particular properties of unary regular expressions. Finally, using our valid results, we confirm that the family of $k$-block deterministic regular languages is strictly included into the one of $k$-lookahead deterministic regular languages by showing that any $k$-block deterministic unary language is one-unambiguous

Solving 2D-pattern matching with networks of picture processors

We propose a solution based on networks of picture processors to the problem of picture pattern matching. The network solving the problem can be informally described as follows: it consists of two subnetworks, one of them extracts simultaneously all subpictures of the same size from the input picture and sends them to the second subnetwork. The second subnetwork checks whether any of the received pictures is identical to the pattern. We present an efficient solution based on networks with evolutionary processors only, for patterns with at most three rows or columns. Afterwards, we present a solution based on networks containing both evolutionary and hiding processors running in O(n+m+kl+k) computational (processing and communication) steps, where the input picture and the pattern are of size (n,m) and (k,l), respectively

Computing languages by (bounded) local sets

We introduce the definition of local structures as description of computations to recognize strings and characterize families of Chomsky's hierarchy in terms of projection of frontiers of local sets of structures. Then we consider particular grid structures we call bounded-grids and study the corresponding family of string languages by proving some closure properties and giving several examples

Exploring inside Tiling Recognizable Picture Languages to Find Deterministic subclasses

Tiling recognizable two-dimensional languages, also known as REC, generalize recognizable string languages to two dimensions and share with them several theoretical properties. Nevertheless family REC is not closed under complementation and this implies that it is intrinsically non-deterministic. We consider different notions of unambiguity and determinism and the corresponding REC subclasses: they define a hierarchy inside REC. We show that some definitions of unambiguity are equivalent to particular notions of determinism and therefore the corresponding classes have linear parsing algorithms and are closed under complementation