Non-recursive trade-offs between two-dimensional automata and grammars

•Succinctness of descriptional systems for picture languages is studied.•Basic models of two-dimensional automata and grammars are considered.•It is shown that non-recursive trade-offs are a common phenomenon.•The results are based on the ability of the models to simulate Turing machines.

We study succinctness of descriptional systems for picture languages. Basic models of two-dimensional finite automata and generalizations of context-free grammars are considered. They include the four-way automaton of Blum and Hewitt, the two-dimensional online tessellation automaton of Inoue and Nakamura and the context-free Kolam grammar of Siromoney et al. We show that non-recursive trade-offs between the systems are very common. Basically, each separation result proving that one system describes a picture language which cannot be described by another system can usually be turned into a non-recursive trade-off result between the systems. These findings are strongly based on the ability of the systems to simulate Turing machines.