By Stephen L. Bloom, Zoltan Ésik, Werner Kuich (auth.), Symeon Bozapalidis, George Rahonis (eds.)

This publication constitutes the refereed lawsuits of the 3rd foreign convention on Algebraic Informatics, CAI 2009, held in Thessaloniki, Greece, in may possibly 2009.

The sixteen complete papers have been conscientiously reviewed and chosen from 25 submissions. The papers disguise subject matters similar to algebraic semantics on graph and timber, formal energy sequence, syntactic gadgets, algebraic photo processing, finite and endless computations, acceptors and transducers for strings, bushes, graphs arrays, and so on. choice difficulties, algebraic characterization of logical theories, strategy algebra, algebraic algorithms, algebraic coding thought, algebraic features of cryptography.

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**Extra resources for Algebraic Informatics: Third International Conference, CAI 2009, Thessaloniki, Greece, May 19-22, 2009, Proceedings**

**Example text**

Proposition 14. ([14]) L(CFKG) ⊂ L(PG). Namely, rules A → B rules: C of a CF Kolam grammar G in CNF are equivalent to RTG A→ ###### #BBC C# #BBC C# ###### and similarly an equivalent form can be stated for rules A → B C. This is compatible with the constraint of Pr˚usˇa grammars given in Remark 1 and so for each CF Kolam grammar there exists an equivalent Pr˚usˇa’s grammar. The inclusion is proper because the language of Example 1 cannot be generated by a CF Kolam grammar. The time complexity of picture recognition problem for CF Kolam grammars in CNF has been recently proved [19] to be O(m2 n2 (m + n)).

14]) A regional tile grammar (RTG) is a tile grammar (see Definition 14), in which every variable size rule A → ω is such that LOC(ω) is a regional language. We note that Example 1 is regional, while the picture language presented in Example 2 is not. For languages generated by regional tile grammars a parsing algorithm generalizing the CKY algorithm is given. A subpicture is conveniently identified by its subdomain as in original algorithm a substring is identified by the positions of its first and last characters.

We know from [39] that for every CFKG G, if L(G) does not contain the empty picture, there exists a CFKG G in Chomsky Normal Form, such that L(G) = L(G ). Also, the 40 A. Cherubini and M. Pradella classical algorithm to translate a string grammar into Chomsky Normal Form can be easily adapted to CFKGs. Example 4. The following Chomsky Normal Form grammar G defines the set of pictures such that each column is a palindrome: S → V S | A1 A2 | B1 B2 | a | b; V → A1 A2 | B1 B2 | a | b; A2 → V A1 | a; B2 → V B1 | b; A1 → a; B1 → b.