Real Functions computed by Parametric Weighted Finite Automata

Weighted finite automata (WFA) are an extension of finite automata that can be used to compute real functions and parametric WFA (PWFA) are a multidimensional variant of WFA. This paper shows that each PWFA computable set is computable by a PWFA that has an alphabet cardinality of two. Further we show that real polynomials with solely rational coefficients can be computed by PWFA that do not use non-rational numbers. We study a class of PWFA that behaves like WFA on mirrored input words and obtain the result that the only smooth real functions definable by PWFA in this family are polynomials.