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Weighted Picture Automata and Weighted Logics

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Abstract

We investigate formal power series on pictures. These are functions that map pictures to elements of a semiring and provide an extension of two-dimensional languages to a quantitative setting. We establish a notion of a weighted MSO logics over pictures. The semantics of a weighted formula will be a picture series. We introduce weighted 2-dimensional on-line tessellation automata (W2OTA) and prove that for commutative semirings, the class of picture series defined by sentences of the weighted logics coincides with the family of picture series that are computable by W2OTA. Moreover, we show that the family of behaviors of W2OTA coincide precisely with the class of picture series characterized by weighted (quadrapolic) picture automata and consequently, the notion of weighted recognizability presented here is robust. However, the weighted structures can not be used to get better decidability properties than in the language case. For every commutative semiring, it is undecidable whether a given MSO formula has restricted structure or whether the semantics of a formula has empty support.

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  1. Institut für Informatik, Universität Leipzig, Johannisgasse 26, 04103, Leipzig, Germany

    Ina Fichtner

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  1. Ina Fichtner
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Correspondence to Ina Fichtner.

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Supported by the Ph.D. program 446/3 “Wissensrepräsentation” of the German Research Foundation.

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Fichtner, I. Weighted Picture Automata and Weighted Logics. Theory Comput Syst 48, 48–78 (2011). https://doi.org/10.1007/s00224-009-9225-3

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