Polyregular Functions Characterisations and Refutations

Regular functions are a well-studied robust class of string-to-string functions, one of whose characterisations is that they are exactly the functions recognisable by deterministic two-way transducers, that is, finite automata with output. This implies that the growth of a regular function-the function describing the output length in terms of the input length-is always linear. To go beyond linear growth, one can equip the two-way transducers with multiple reading heads (pebbles), the number of which then still constitutes a bound on the degree of the polynomial describing the growth. The functions recognised by these pebble automata are called polyregular. Over the past years, the properties of polyregular functions have been studied extensively. Just as for regular functions, various equivalent characterisations have been found, and variants of the corresponding models have been investigated. This paper gives an introduction to the realm of polyregular functions by discussing some of those characterisations, recent developments, and the parameters in the models that are linked to the growth. The second part presents simple constructions which show the asymmetry of the link between the growth degree and the number of heads. That is, in general, the growth degree of a polyregular function does not bound the minimum number of pebbles needed in an automaton to compute the function.