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

This e-book constitutes the refereed lawsuits of the 3rd foreign convention on Algebraic Informatics, CAI 2009, held in Thessaloniki, Greece, in might 2009.

The sixteen complete papers have been rigorously reviewed and chosen from 25 submissions. The papers conceal themes corresponding to algebraic semantics on graph and bushes, formal energy sequence, syntactic items, algebraic photograph processing, finite and endless computations, acceptors and transducers for strings, bushes, graphs arrays, and so on. selection difficulties, algebraic characterization of logical theories, method algebra, algebraic algorithms, algebraic coding conception, algebraic features of cryptography.

**Read or Download Algebraic Informatics: Third International Conference, CAI 2009, Thessaloniki, Greece, May 19-22, 2009, Proceedings PDF**

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**Sample text**

Analogously can be stated a vertical iteration lemma. Another necessary condition for a language being in REC uses the notion of syntactic equivalence modulo a language L. For a language L ∈ Σ ∗,∗ two isometric pictures p, q are called syntactically equivalent modulo L (in symbols, p ≈L q) if for all x1 , x2 , y1 , y2 ∈ Σ ∗,∗ of suitable sizes, x1 (y1 p y2 ) x2 ∈ L if and only if x1 (y1 q y2 ) x2 ∈ L. The function fL (|p|row , |p|col ) gives the number of ≈L -equivalence classes in Σ ∗,∗ of size (|p|row , |p|col ).

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. Theorem 10. ([14]) The parsing problem for RTG has polynomial time complexity. Analyzing the algorithm, one derives that the complexity of parsing for a picture of size (n, m) is O(μm4 n4 ) where constant μ depends on parameters of the grammar. The property of having polynomial time complexity for picture recognition, together with the remark that pictures with palindromic rules are not in REC immediately give the following results: Proposition 12.

32 A. Cherubini and M. Pradella Lemma 3. ([40]) Let L ∈ REC over Σ. For each positive integer n let {Mn } be a sequence such that 1. Mn ⊆ Σ n,+ × Σ n,+ ; 2. ∀(p, q) ∈ Mn , p q ∈ L; 3. ∀(p, q), (p , q ) ∈ Mn , {p q , p q} L. Then |Mn | is 2O(n) . The question of the existence of some language not in REC for which the above lemma fails to prove the non recognizability was posed. The language of squares over {a, b} with as many a’s as b’s was proposed as candidate. However, from a result in [49], it follows that the above language is recognizable.