  In this work, we introduce the notion of decisional width of a finite
relational structure and the notion of decisional width of a regular class of
finite structures. Our main result states that given a first-order formula
{\psi} over a vocabulary {\tau}, and a finite automaton F over a suitable
alphabet B({\Sigma},w,{\tau}) representing a width-w regular-decisional class
of {\tau}-structures C, one can decide in time f({\tau},{\Sigma},{\psi},w)|F|
whether some {\tau}-structure in C satisfies {\psi}. Here, f is a function that
depends on the parameters {\tau},{\Sigma},{\psi},w, but not on the size of the
automaton F representing the class. Therefore, besides implying that the
first-order theory of any given regular-decisional class of finite structures
is decidable, it also implies that when the parameters {\tau}, {\psi}, {\Sigma}
and w are fixed, decidability can be achieved in linear time on the size of the
input automaton F. Building on the proof of our main result, we show that the
problem of counting satisfying assignments for a first-order logic formula in a
given structure A of width w is fixed-parameter tractable with respect to w,
and can be solved in quadratic time on the length of the input representation
of A.
