Monadic Second-Order Logic over Rectangular Pictures and Recognizability by Tiling Systems

A logical approach to locality in pictures languages

“…The normalization of the logic -reducing it to its "one first-order variable" fragment, explicitly expresses the local feature of MSO on pictures. The results and methods of Item 1 can be summed as follows: one exploits the homogeneous framework of logic for making simpler, more explicit -using only one first-order variable -and more uniform the proof and ideas of the main result of Giammarresi et al [23,22]; this allows us to generalize it to any dimension and, potentially, to other regular structures.…”

“…In a series of papers culminating in [23], Giammarresi et al proved that a 2-picture language is recognizable (i.e. is the projection of a local picture language) iff it is definable in existential monadic logic (EMSO).…”

“…[23,22,49,2]): on the pixel domain [1, n] d where the sets p −1 (s), s ∈ Σ, are encoded by unary relations (Q s ) s∈Σ and the underlying d-dimensional grid is encoded by d successor functions (see Definition 2.2);…”

“…It is a simple generalization to any dimension of the notion of hv-local 2-dimensional picture language of [41] (see also [22,20,21,23,24,2]). To define it formally, we need to mark the border of pictures.…”

Picture Languages: From Wang Tiles to 2D Grammars

“…In fact, a language is locally threshold testable iff it is definable by a first-order formula in the signature {S 1 , S 2 , {P a } a∈Σ } ( [32]), while is locally testable if and only if it is definable by a first-order formula in the signature {S 1 , S 2 , {P a } a∈Σ , lef t, right, top, bottom}, where lef t, right, top, bottom are unary predicates saying that a position is at the respective border [40].…”