We discuss several two-dimensional generalizations of the familiar Lyndon-Schützenberger periodicity theorem for words. We consider the notion of primitive array (as one that cannot be expressed as the repetition of smaller arrays). We count the number of m×n arrays that are primitive. Finally, we show that one can test primitivity and compute the primitive root of an array in linear time.

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Site-directed insertion is an overlapping insertion operation that can be viewed as analogous to the overlap assembly or chop operations that concatenate strings by overlapping a suffix and a prefix of the argument strings. We consider decision problems and language equations involving site-directed insertion. By relying on the tools provided by semantic shuffle on trajectories we show that one variable equations involving site-directed insertion and regular constants can be solved. We consider also maximal and minimal variants of the site-directed insertion operation.

A two-dimensional finite automaton has a read-only input head that moves in four directions on a finite array of cells labelled by symbols of the input alphabet. A threeway two-dimensional automaton is prohibited from making upward moves, while a two-way two-dimensional automaton can only move downward and rightward.We show that the language emptiness problem for unary three-way nondeterministic two-dimensional automata is NP-complete, and is in P for general-alphabet twoway nondeterministic two-dimensional automata. We show that the language equivalence problem for two-way deterministic two-dimensional automata is decidable, while both the equivalence and universality problems for two-way nondeterministic twodimensional automata are undecidable. The deterministic case is the first known positive decidability result for the equivalence problem on two-dimensional automata over a general alphabet. We show that there exists a unary three-way deterministic twodimensional automaton with a nonregular column projection, and we show that the row projection of a unary three-way nondeterministic two-dimensional automaton is always regular.

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