Algebraic Representations of Regular Array Languages

We consider formal sequences of d-dimensional vectors and symbols representing d-dimensional arrays and introduce the operations of catenation as well as iterated catenation of these formal sequences (d-dimensional formal arrays). Together with the usual set union, these operations allow us to define d-dimensional regular array expressions and thus to develop an algebraic representation of regular array languages generated by specific d-dimensional array grammars. In that way, specific infinite regular array languages allow for a finite representation as regular array expressions. Whereas, in general, it is undecidable whether the array language generated by a given regular array grammar is empty, finite or infinite, for these specific regular array grammars, these questions are decidable.

[1]  Arto Salomaa,et al.  Formal languages , 1973, Computer science classics.

[2]  Arto Salomaa,et al.  Semirings, Automata, Languages , 1985, EATCS Monographs on Theoretical Computer Science.

[3]  Rudolf Freund.  Control Mechanisms on #-Context-Free Array Grammars , 1994, Mathematical Aspects of Natural and Formal Languages.

[4]  Rudolf Freund,et al.  One-dimensional matrix array grammars , 1993, J. Inf. Process. Cybern..

[5]  Rudolf Freund,et al.  The Generative Power of d-Dimensional #-Context-Free Array Grammars , 1998, MCU.

[6]  Yasunori Yamamoto,et al.  The complexity of some decision problems about two-dimensional array grammars , 1983, Inf. Sci..

[7]  Ivan Hal Sudborough,et al.  Complexity and Decidability for Chain Code Picture Languages , 1985, Theor. Comput. Sci..

[8]  Patrick Shen-Pei Wang,et al.  Some New Results on Isotonic Array Grammars , 1980, Inf. Process. Lett..

[9]  C. Cook,et al.  A chomsky hierarchy of isotonic array grammars and languages , 1978 .

[10]  Yasunori Yamamoto,et al.  Context-sensitivity of Two-Dimensional Regular Array Grammars , 1989, Int. J. Pattern Recognit. Artif. Intell..

[11]  Gheorghe Păun,et al.  Mathematical Aspects of Natural and Formal Languages , 1994, World scientific series in computer science.

[12]  Ivan Hal Sudborough,et al.  The Membership and Equivalence Problems for Picture Languages , 1987, Theor. Comput. Sci..