# String-to-String Interpretations with Polynomial-Size Output

## TL;DR

This paper characterizes string-to-string MSO interpretations as exactly the polyregular functions, showing they are closed under composition and have polynomial output size, connecting logical and automata-theoretic perspectives.

## Contribution

It establishes that string-to-string MSO interpretations are equivalent to polyregular functions and are closed under composition, bridging logic and automata theory.

## Key findings

- String-to-string MSO interpretations are exactly polyregular functions.
- Polyregular functions are recognized by pebble transducers.
- MSO interpretations are closed under composition.

## Abstract

String-to-string MSO interpretations are like Courcelle's MSO transductions, except that a single output position can be represented using a tuple of input positions instead of just a single input position. In particular, the output length is polynomial in the input length, as opposed to MSO transductions, which have output of linear length. We show that string-to-string MSO interpretations are exactly the polyregular functions. The latter class has various characterizations, one of which is that it consists of the string-to-string functions recognized by pebble transducers. Our main result implies the surprising fact that string-to-string MSO interpretations are closed under composition.