Categories and Types in Logic, Language, and Physics

Abstract

Multidimensional Dyck languages generalize the language of balanced brackets to alphabets of size > 2. Words contain each alphabet symbol with the same multiplicity. In addition, reading a word from left to right, there are always at least as many ai as a(i+1), where aj is the jth alphabet symbol in the lexicographic ordering. We compare the Dyck languages with MIX languages, where the multiplicity constraint is respected, but the order of the symbols is free. To understand the combinatorics of the Dyck languages, we study the bijection with standard Young tableaux of rectangular shape, and, for the three-dimensional case, with planar webs for combinatorial spider categories. We present a typelogical analysis of Dyck languages with an alphabet of size d in terms of a construction that aligns (d-1) grammars for the two-symbol case. © Springer-Verlag Berlin Heidelberg 2014.