Weighted Two-Dimensional Finite Automata

Two-dimensional finite automata (\(\textsf {2D}\text {-}\textsf {FA}\)) are a natural generalization of finite automata to two-dimension and used to recognize picture languages. In order to study quantitative aspects of computations of \(\textsf {2D}\text {-}\textsf {FA}\), we introduce weighted two-dimensional finite automata (\(\textsf {W2D}\text {-}\textsf {FA}\)), which can represent functions from some input alphabet into a semiring. In this work, we investigate some basic properties of these functions like upper bounds and closure properties. First, we prove that the value of such a function is bounded by \(2^{O(n^2)}\). Then, we will see that this upper bound is actually sharp, and a deterministic \(\textsf {W2D}\text {-}\textsf {FA}\) of a restricted type already can compute a function that reaches this bound. Finally, we study the closure properties of the classes of functions that are computed by \(\textsf {W2D}\text {-}\textsf {FA}\) of various types under some rational operations, e.g., sum, Hadamard product, vertical (horizontal) multiplication, and scalar multiplication.