Snake-Deterministic Tiling Systems

Abstract

The concept of determinism, while clear and well assessed for string languages, is still matter of research as far as picture languages are concerned. We introduce here a new kind of determinism, called snake, based on the boustrophedonic scanning strategy, that is a natural scanning strategy used by many algorithms on 2D arrays and pictures. We consider a snake-deterministic variant of tiling systems, which defines the so-called Snake-DREC class of languages. Snake-DREC properly extends the more traditional approach of diagonal-based determinism, used e.g. by deterministic tiling systems, and by online tessellation automata. Our main result is showing that the concept of snake-determinism of tiles coincides with row (or column) unambiguity.

Picture Recognizability with Automata Based on Wang Tiles

“…Snake-determinism [16] is based on boustrophedonic scanning strategies. Given a tiling system τ = Σ, Γ, Θ, π and a picture p ∈ Σ ++ , imagine to build one preimage p ∈ L(Θ), π(p ) = p, by scanning p as follows: start from the top-left corner, scan the first row of p rightwards, then scan the second row leftwards, and so on, as in Figure 1(b).…”

“…This means that we scan odd rows rightwards and even row leftwards, assigning a symbol in Γ to each position. A tiling system is snake-deterministic if this choice is guaranteed unique (the formal definition can be found in [16]). …”

“…The notions of determinism in [1,16] are all based on some fixed and pre-established kinds of scanning strategy. This approach can be limiting, so we plan to define and consider here a wider range of possible strategies.…”

“…Online tessellation acceptors have a diagonal-based kind of determinism [13] and this notion is extended in [1], with the definition of a deterministic class we denote by Diag-DREC (the original name was DREC). In [16] we introduced the class Snake-DREC which is based on a boustrophedonic scanning strategy, and proved that Snake-DREC properly extends Diag-DREC.…”

Picture Languages: From Wang Tiles to 2D Grammars

“…More recently, Lonati and Pradella [38] introduced a new kind of determinism for tiles: given (Σ, Γ, Θ, π), the pre-image of a picture p ∈ Σ * , * is built by scanning p with a boustrophedonic strategy, that is a natural scanning strategy used by many algorithms on pictures and 2D arrays. More precisely, it starts from the top-left corner, scans the first row of p rightwards, then scans the second row leftwards, and so on.…”