Polyregular Functions

TL;DR

This paper introduces polyregular functions, a class of string-to-string functions with polynomial growth, characterized by multiple equivalent definitions, and explores their properties, algorithms, and subclasses.

Contribution

It provides four equivalent definitions of polyregular functions, including new characterizations via lambda-calculus and imperative programming, and analyzes their properties and subclasses.

Findings

Polyregular functions include known transducer classes but extend beyond them.

Output can be computed in linear time relative to input and output size.

Inverse images of regular languages remain regular within this class.

Abstract

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.