“…There have been recurrent attempts to generalize the notions of regularity from languages to functions, leading to classes of various expressiveness, such as 1 Mealy Machines (Mealy) [35], sequential functions (Seq) [44], rational functions (Rat) [20], regular functions (Reg) [22], and polyregular functions (Poly) [7]. For all of these models, an aperiodic counterpart can be defined (respectively, AMealy, ASeq, ARat, AReg, SF), lifting the correspondence between counter-free automata, star-free languages and aperiodic monoids to the functional setting [27,4,9,14,6,7,13]. The inclusions between these classes of functions are all known to be strict, and we depicted in Figure 1 the status of the membership problem associated to these strict inclusions (that is, given a function f , decide if f can be computed by a function of a proper subclass).…”