By Frank Drewes (auth.), Werner Kuich, George Rahonis (eds.)

This Festschrift quantity is released in honor of Symeon Bozapalidis at the celebration of his retirement after greater than 35 years of educating. the themes coated are: weighted automata over phrases and bushes, tree transducers, quantum automata, graphs, photographs and kinds of semigroups.

Since 1982 -- on the Aristotle collage of Thessaloniki -- Symeon's major pursuits were heavily hooked up with the algebraic foundations in laptop technology. particularly, he contributed to the improvement of the speculation of tree languages and sequence, the axiomatization of graphs, photograph thought, and fuzzy languages.

The quantity, which specializes in the study pursuits of Symeon, comprises 15 completely refereed invited papers, written by way of his colleagues, acquaintances, and scholars. lots of the papers have been provided on the Workshop on Algebraic Foundations in desktop technological know-how, held in Thessaloniki, Greece, in the course of November 7--8, 2011.

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M} be the sequence of all weights from im(μ). Then we have at most |W1 | · . . · |Wm | many sequences Û = π1 , . . , πm , such that πi ∈ Wi for all i ∈ {1, . . , m}. Next we choose some aperiodic word π ∈ {1, 2}ω . Due to the statement above, each sequence Û forms only a ﬁnite number of preﬁxes of π. Let k be the length of the longest preﬁx of π which a sequence Û can form. Consequently, no value W ∈ im( M ) contains a word from {1, 2}∗ that is a preﬁx of π of length greater than k. Let u ∈ {1, 2}∗ − be a preﬁx of π of length at least k + 1 and let t ∈ TΣ be a tree with ← u ∈ dom(t).

Clearly, L(GPCP ) = L(GPCP ), which proves the assertion for grammars with two rules. 6 Concluding Remarks We have deﬁned square-reﬁnement collage grammars and surveyed decidability and complexity results regarding these grammars or their special cases, namely (partial-) array collage grammars, which are closely related to Bozapalidis’ picture-reﬁnement grammars, and grid collage grammars. A result that has been excluded from the presentation in this paper, because its proof can be formulated much more conveniently in the tree-based setting, 26 F.

M with m ∈ IN be an arbitrary sequence of words from {1, 2}∗ . Then the set km Pm = π1k1 . . πm k1 , . . , km ∈ IN0 (where πi0 = ε) contains only a ﬁnite number of preﬁxes of π. This can be shown by an induction on m. We say, the sequence π1 , . . , πm forms a ﬁnite number of preﬁxes of π. 38 M. Droste et al. Now we assume there is a linear order < on K and let M = (Q, Σ, μ, F ) be a wta. Let W1 < . . < Wm with Wi ⊆ {1, 2}∗ for i ∈ {1, . . , m} be the sequence of all weights from im(μ).

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