Abstract
A recognizable picture language is defined as the projection of a local picture language defined by a set of two-by-two tiles, i.e. by a strictly-locally-testable (SLT) language of order 2. The family of recognizable picture languages is also defined, using larger k by k tiles, \(k>2\), by the projection of the corresponding SLT language. A basic measure of the descriptive complexity of a picture language is given by the size of the SLT alphabet using two-by-two tiles, more precisely by the so-called alphabetic ratio of sizes: SLT-alphabet/picture-alphabet. We study how the alphabetic ratio changes moving from two to larger tile sizes, and we obtain the following result: any recognizable picture language over an alphabet of size n is the projection of an SLT language over an alphabet of size 2n. Moreover, two is the minimal alphabetic ratio possible in general. This result reproduces into two dimensions a similar property (known as Extended Medvedev’s theorem) of the regular word languages, concerning the minimal alphabetic ratio needed to define a language by means of a projection of an SLT word language.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.