  This paper is about certain string-to-string functions, called the
polyregular functions. These are like the regular string-to-string functions,
except that they can have polynomial (and not just linear) growth. The class
has four equivalent definitions: 1. deterministic two-way transducers with
pebbles; 2 the smallest class of string-to-string functions that is closed
under composition, contains all sequential functions as well as two extra
functions called squaring and iterated reverse 3. a fragment of the
lambda-calculus, which has a list type constructor and limited forms of
iteration such as map but not fold; 4. an imperative programming language,
which has for loops that range over input positions.
  The first definition comes from [milo2003typechecking], while the remaining
three are new to the author's best knowledge. The class of polyregular
functions contains known classes of string-to-string transducers, such as the
sequential, rational, or regular ones, but goes beyond them because of
super-linear growth. Polyregular functions have good algorithmic properties,
such as: - the output can be computed in linear time (in terms of combined
input and output size); - the inverse image of a regular word language is
(effectively) regular.
  We also identify a fragment of polyregular functions, called the first-order
polyregular functions, which has additional good properties, e.g. the output
can be computed by an AC0 circuit.

Formal Languages and Automata Theory