Weighted Two-Dimensional Finite Automata

Abstract

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.

This work is supported by the Fundamental Research Funds for the Central Universities under Grant GK201903094.