Abstract
We exhibit a first-order definable picture language which we prove is not expressible by any star-free picture expression, i. e., it is not star-free. Thus first-order logic over pictures is strictly more powerful than star-free picture expressíons are. This is in sharp contrast with the situation with words: the well-known McNaughton-Papert theorem states that a word language is expressible by a first-order formula if and only if it is expressible by a star-free (word) expression.
The main ingredients of the non-expressibility result are a Fraïssé-style algebraic characterization of star freeness for picture languages and combinatorics on words.
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Wilke, T. (1997). Star-free picture expressions are strictly weaker than first-order logic. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds) Automata, Languages and Programming. ICALP 1997. Lecture Notes in Computer Science, vol 1256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63165-8_191
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DOI: https://doi.org/10.1007/3-540-63165-8_191
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