D. Giammarresi and A. Restivo. Two-dimensional finite state recognizability. Fundamenta Informaticae, 25(3-4):399--422, 1996. Special issue: Formal Language Theory.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Note that if the pictures of L have the same size, we have (L ; then the notation L is coherent with the fact that denotes the set of all possible pictures over the alphabet Sigma. 5 Closure of recognizable languages by tiling D. Giammarresi and A. Restivo have shown in [7] that the set of all pictures obtained by tiling by a finite set of polyominoes is recognizable. With the same idea, it is straightforward to see that the language obtained by tiling by a finite picture language is also recognizable. In this section we extend this result to tilings by a ....

D. Giammarresi and A. Restivo. Two-dimensional finite state recognizability. Fundamenta Informaticae, 25(3-4):399--422, 1996. Special issue: Formal Language Theory.

....allowed subtrees. Then, it can be proved that projections of frontiers of local sets of binary trees characterize context free languages. As third case, we take as structures rectangular grid graphs with frontiers equal to the last row. Local sets of grids are defined as local picture languages in [4, 5]. In [3, 7] is proved that context sensitive languages can be characterized as projections of frontiers of local picture languages. Notice that, when regular, context free or context sensitive languages are given by a local sets of structures plus a projection it is easy to get both a description ....

....Let w 0 2 L 0 be the string such that (w 0 ) w. From the above example, it is clear that #w 0 # is de facto the computation to accept string w 2 L and we will refer to it as a local computation for w. 3 The notion of local languages has been extended from string to picture languages in [4, 5]. A picture is essentially a two dimensional array of symbols from a finite alphabet. Given a picture p of size m Theta n, its bordered version is defined by adding symbols # all around the rectangle: the result is then a picture of size (m 2) Theta (n 2) whose first and last rows and ....

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D. Giammarresi and A. Restivo. Two-dimensional finite state recognizability. Fundamenta Informaticae. vol. 25, no. 3,4 , pages 399--422, 1996.

....in first order logic. In [6] it was shown that the first result essentially carries over to picture (or two dimensional ) languages in the following sense: a picture language is recognized by a tiling system if and only if it is definable in existential monadic second order logic (while in [5], see also [6] full monadic second order logic had been proven to be strictly more powerful) In this paper, we show that the second result does not carry over to picture languages. More precisely, we exhibit a simple, first order definable picture language, denoted L (see page 349) and show ....

....star free picture languages is strictly contained in the class of first order definable picture languages. This clarifies an interesting question about the fine structure of the class of all recognizable picture languages, which was brought up in [6] It should also be noted that by a result from [5], the class of first order definable picture languages is strictly contained in the class of all recognizable picture languages. As with star free word expressions, star free picture expressions are built from singleton sets using boolean combinations and concatenation. Of course, due to the ....

D. Giammarresi and A. Restivo. Two-dimensional finite state recognizability. Fundamenta Informaticae. To appear.

....pictures) as model theoretical structures and showed as recognizability corresponds to the notions of definability on existential monadic second order logic (cf. 38] This is coherent with the string language recognizability theory where Buchi s Theorem holds. In a recent proposal (cf. [13, 14]) a notion of recognizability of a set of pictures in terms of tiling systems is introduced. The underlying idea is to define recognizability by projection of local properties . Informally, recognition in a tiling system is defined in terms of a finite set of square pictures of side two which ....

....languages by borrowing and extending notation from the theory of one dimensional languages. Next, we will give formal definitions of concatenation operations between two dimensional strings (pictures) and two dimensional languages. The notations used can be mainly found in [21] or in [14]. Let Sigma be a finite alphabet. Definition 2.1 A two dimensional string (or a picture ) over Sigma is a two dimensional rectangular array 3 of elements of Sigma. The set of all two dimensional strings over Sigma is denoted by Sigma . A two dimensional language over Sigma is a subset ....

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D. Giammarresi and A. Restivo. Two-dimensional finite state recognizability. Fundamenta Informaticae. Special Issue: Formal Language Theory, vol. 25, no. 3,4 (1996), 399--422.

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