State Complexity of Permutation and the Language Inclusion Problem up to Parikh Equivalence on Alphabetical Pattern Constraints and Partially Ordered NFAs

We investigate the state complexity of the permutation operation, or the commutative closure, on Alphabetical Pattern Constraints (APCs). This class corresponds to level [Formula: see text] of the Straubing-Thérien hierarchy and includes the finite, the piecewise testable, or [Formula: see text]-trivial, and the [Formula: see text]-trivial and [Formula: see text]-trivial languages. We give a sharp state complexity bound expressed in terms of the longest strings in the unary projection languages of an associated finite language. Additionally, for a subclass, we give sharp bounds expressed in terms of the size of a recognizing input automaton and the size of the alphabet. We also state a related state complexity bound for the commutative closure on finite languages. Lastly, we investigate the language inclusion, equivalence and universality problems on APCs up to permutational, or Parikh, equivalence. These problems are known to be [Formula: see text]-complete on APCs in general, even for fixed alphabets. We show them to be decidable in polynomial time for fixed alphabets if we only want to solve them up to Parikh equivalence. We also correct a mistake from the conference version in a bound on the size of recognizing automata for the commutative closure.