Piecewise Testable Languages and Nondeterministic Automata

A regular language is k-piecewise testable if it is a finite boolean combination of languages of the form Sigma^* a_1 Sigma^* ... Sigma^* a_n Sigma^*, where a_i in Sigma and 0 = 0, it is an NL-complete problem to decide whether the language L(A) is piecewise testable and, for k >= 4, it is coNP-complete to decide whether the language L(A) is k-piecewise testable. It is known that the depth of the minimal DFA serves as an upper bound on k. Namely, if L(A) is piecewise testable, then it is k-piecewise testable for k equal to the depth of A. In this paper, we show that some form of nondeterminism does not violate this upper bound result. Specifically, we define a class of NFAs, called ptNFAs, that recognize piecewise testable languages and show that the depth of a ptNFA provides an (up to exponentially better) upper bound on k than the minimal DFA. We provide an application of our result, discuss the relationship between k-piecewise testability and the depth of NFAs, and study the complexity of k-piecewise testability for ptNFAs

On the Complexity of Universality for Partially Ordered NFAs

International audiencePartially ordered nondeterminsitic finite automata (poNFAs) are NFAs whose transition relation induces a partial order on states, i.e., for which cycles occur only in the form of self-loops on a single state. A poNFA is universal if it accepts all words over its input alphabet. Deciding universality is PSpace-complete for poNFAs, and we show that this remains true even when restricting to a fixed alphabet. This is nontrivial since standard encodings of alphabet symbols in, e.g., binary can turn self-loops into longer cycles. A lower coNP-complete complexity bound can be obtained if we require that all self-loops in the poNFA are deterministic, in the sense that the symbol read in the loop cannot occur in any other transition from that state. We find that such restricted poNFAs (rpoNFAs) characterise the class of R-trivial languages, and we establish the complexity of deciding if the language of an NFA is R-trivial. Nevertheless, the limitation to fixed alphabets turns out to be essential even in the restricted case: deciding universality of rpoNFAs with unbounded alphabets is PSpace-complete. Our results also prove the complexity of the inclusion and equivalence problems, since universality provides the lower bound, while the upper bound is mostly known or proved in the paper

On the Complexity of Universality for Partially Ordered NFAs

Partially ordered nondeterminsitic finite automata (poNFAs) are NFAs whose transition relation induces a partial order on states, i.e., for which cycles occur only in the form of self-loops on a single state. A poNFA is universal if it accepts all words over its input alphabet. Deciding universality is PSpace-complete for poNFAs, and we show that this remains true even when restricting to a fixed alphabet. This is nontrivial since standard encodings of alphabet symbols in, e.g., binary can turn self-loops into longer cycles. A lower coNP-complete complexity bound can be obtained if we require that all self-loops in the poNFA are deterministic, in the sense that the symbol read in the loop cannot occur in any other transition from that state. We find that such restricted poNFAs (rpoNFAs) characterise the class of R-trivial languages, and we establish the complexity of deciding if the language of an NFA is R-trivial. Nevertheless, the limitation to fixed alphabets turns out to be essential even in the restricted case: deciding universality of rpoNFAs with unbounded alphabets is PSPACE-complete. Our results also prove the complexity of the inclusion and equivalence problems, since universality provides the lower bound, while the upper bound is mostly known or proved in the paper

Partially Ordered Automata and Piecewise Testability

Partially ordered automata are automata where the transition relation induces a partial order on states. The expressive power of partially ordered automata is closely related to the expressivity of fragments of first-order logic on finite words or, equivalently, to the language classes of the levels of the Straubing-Th\'erien hierarchy. Several fragments (levels) have been intensively investigated under various names. For instance, the fragment of first-order formulae with a single existential block of quantifiers in prenex normal form is known as piecewise testable languages or $J$-trivial languages. These languages are characterized by confluent partially ordered DFAs or by complete, confluent, and self-loop-deterministic partially ordered NFAs (ptNFAs for short). In this paper, we study the complexity of basic questions for several types of partially ordered automata on finite words; namely, the questions of inclusion, equivalence, and ($k$-)piecewise testability. The lower-bound complexity boils down to the complexity of universality. The universality problem asks whether a system recognizes all words over its alphabet. For ptNFAs, the complexity of universality decreases if the alphabet is fixed, but it is open if the alphabet may grow with the number of states. We show that deciding universality for general ptNFAs is as hard as for general NFAs. Our proof is a novel and nontrivial extension of our recent construction for self-loop-deterministic partially ordered NFAs, a model strictly more expressive than ptNFAs. We provide a comprehensive picture of the complexities of the problems of inclusion, equivalence, and ($k$-)piecewise testability for the considered types of automata