On the Complexity of Unary Tiling-Recognizable Picture Languages

IFIP International Federation for Information Processing, 2008

International Journal of Recent Technology and Engineering (IJRTE), 2019

In two dimensions, tiling the plane plays a vital role. Many picture generative models were attempted to tile the three dimensional space. A. Dharani et al. [6] introduced a new theoretical picture generative model to tile a three dimensional space using tetrahedral tile in two different ways namely Sequential Space Filling Grammar (SSFG) and Parallel Space Filling Grammar (PSFG). Local and recognizable tetrahedral picture languages are introduced in this paper and some of its properties are studied.

We investigate power series on pictures which map pictures to elements of a semiring and provide an extension of two-dimensional languages to a quantitative setting. We will assign weights to devices, ranging from tiling systems and domino systems to picture automata. We prove that, for commutative semirings, the behaviors of weighted picture automata are precisely alphabetic projections of series defined by rational operations and also coincide with the families of projections of tile-local and projections of hv-local series. Furthermore, we establish properties of unambiguous picture languages by characterizations in terms of unambiguous rational operations with injective projections and devices such as unambiguous (quadrapolic) picture automata or unambiguous domino systems.

This paper deals with descriptive complexity of picture languages of any dimension by syntactical fragments of existential second-order logic. - We uniformly generalize to any dimension the characterization by Giammarresi et al. \cite{GRST96} of the class of \emph{recognizable} picture languages in existential monadic second-order logic. - We state several logical characterizations of the class of picture languages recognized in linear time on nondeterministic cellular automata of any dimension. They are the first machine-independent characterizations of complexity classes of cellular automata. Our characterizations are essentially deduced from normalization results we prove for first-order and existential second-order logics over pictures. They are obtained in a general and uniform framework that allows to extend them to other "regular" structures. Finally, we describe some hierarchy results that show the optimality of our logical characterizations and delineate their lim...

We introduce two new models of two-dimensional computing devices called sgraffito automaton and restarting tiling automaton. The models fulfill two conditions. Firstly, they do not exceed the power of finite-state automata when working over one-dimensional inputs. And secondly, they allow a clear design of important computations. We study properties of the induced families of picture languages. We show that they include strictly REC and that restarting tiling automata are able to simulate sgraffito automata. An interesting family is settled by deterministic sgraffito automata. It can be seen as an alternative to DREC.

Italian Conference on Theoretical Computer Science, 2009

We present a model of automaton for picture language recognition, called Wang automaton, which is based on labeled Wang tiles. Wang automata combine features of both online tessellation acceptors and 4-way automata: as in online tessellation acceptors, computation assigns states to each picture position; as in 4-way automata, the input head visits the picture moving from one pixel to an adjacent one, according to some scanning strategy. Wang automata recognize the class REC, i.e. they are equivalent to tiling systems or online tessellation acceptors, and hence strictly more powerful than 4-way automata. We also introduce a natural notion of determinism for Wang automata, and study the resulting class, extending the more traditional approach of diagonal-based determinism, used e.g. by deterministic tiling systems. In particular, we prove that the concept of row (or column) ambiguity defines the class of languages recognized by Wang automata directed by boustrophedonic scanning strategies.

Annals of Combinatorics, 2009

We consider some problems concerning two relevant classes of twodimensional languages, i.e. the tiling recognizable languages, and the local languages, recently introduced by Giammarresi and Restivo and already extensively studied. We show that various classes of convex and column-convex polyominoes can be naturally represented as twodimensional words of tiling recognizable languages. Moreover we investigate the nature of the generating function of a tiling recognizable language, providing evidence that such a generating function need not be D-finite.

2013

Finite automaton with one counter (CA) is a fundamental model in automata theory. It has been widely examined from different point of views since sixties. One recent significant result, for example, is that the equivalence problem of deterministic one-way CAs is NLcomplete [Stanislav Böhm, Stefan Göller, Petr Jancar. STOC 2013: 131-140]. In the case of unary languages, on the other hand, we know little about the computational power of CAs. Since one-way nondeterministic pushdown automata, a generalization of one-way nondeterministic CAs, cannot recognize any nonregular unary language, it is interesting to focus on one-way alternating CAs (1ACAs) and two-way CAs (2CAs). Up to our knowledge, the only known unary non-regular languages recognized by 1ACAs and 2CAs are formed by the strings with exponential lengths.

Discrete Mathematics & Theoretical Computer Science, 2010

We present a model of automaton for picture language recognition, called Wang automaton, which is based on labeled Wang tiles. Wang automata combine features of both online tessellation acceptors and 4-way automata: as in online tessellation acceptors, computation assigns states to each picture position; as in 4-way automata, the input head visits the picture moving from one pixel to an adjacent one, according to some scanning strategy. Wang automata recognize the class REC, i.e. they are equivalent to tiling systems or online tessellation acceptors, and hence strictly more powerful than 4-way automata. We also introduce a natural notion of determinism for Wang automata, and study the resulting class, extending the more traditional approach of diagonal-based determinism, used e.g. by deterministic tiling systems. In particular, we prove that the concept of row (or column) ambiguity defines the class of languages recognized by Wang automata directed by boustrophedonic scanning strategies.

Pattern Recognition, 1999

Two different types of models called puzzle languages and recognizable picture languages describing digitized pictures in a two-dimensional plane have been introduced in the 1990s. We review here these models, reporting the main results already proved. We also give a few new results.