String-to-String Interpretations with Polynomial-Size Output

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.