A Robust Measure on FDFAs Following Duo-Normalized Acceptance

TL;DR

The paper introduces duo-normalized acceptance for FDFAs, showing it can yield exponential succinctness over normalized FDFAs while preserving non-deterministic logspace for Boolean operations and classical decisions. It then defines the diameter measure to align FDFA structure with Wagner hierarchy positions, proving that computing this measure for duo-normalized FDFAs is PSPACE-complete (contrast with NL for normalized FDFAs). By connecting to natural colors and reliable words, the authors establish semantic relationships between chain behavior and Wagner levels, and present the Colorful FDFA as a canonical model for succinctness comparisons. Overall, the work bridges FDFA formalisms with the Wagner hierarchy, highlighting practical implications for ω-regular language verification and learning, while delineating fundamental complexity trade-offs.

Abstract

Families of DFAs (FDFAs) are a computational model recognizing $ω$-regular languages. They were introduced in the quest of finding a Myhill-Nerode theorem for $ω$-regular languages, and obtaining learning algorithms. FDFAs have been shown to have good qualities in terms of the resources required for computing Boolean operations on them (complementation, union, and intersection) and answering decision problems (emptiness and equivalence); all can be done in non-deterministic logspace. In this paper we study FDFAs with a new type of acceptance condition, duo-normalization, that generalizes the traditional normalization acceptance type. We show that duo-normalized FDFAs are advantageous to normalized FDFAs in terms of succinctness as they can be exponentially smaller. Fortunately this added succinctness doesn't come at the cost of increasing the complexity of Boolean operations and decision problems -- they can still be preformed in non-deterministic logspace. An important measure of the complexity of an $ω$-regular language, is its position in the Wagner hierarchy. It is based on the inclusion measure of Muller automata and for the common $ω$-automata there exist algorithms computing their position. We develop a similarly robust measure for duo-normalized (and normalized) FDFAs, which we term the diameter measure. We show that the diameter measure corresponds one-to-one to the position on the Wagner hierarchy. We show that computing it for duo-normalized FDFAs is PSPACE-complete, while it can be done in non-deterministic logspace for traditional FDFAs.

Figures (5)

• Figure 1: Left, Middle: A DMA $\mathcal{M}$ and a DPA $\mathcal{D}$ for the language $L_1=L_{\infty aa \wedge \neg \infty bb}$. Right: The Wagner Hierarchy. An arrow from $\mathbb{C}$ to $\mathbb{D}$ says that $\mathbb{C} \subsetneq \mathbb{D}$. Note that many other strict inclusions follow by transitivity.
• Figure 2: Two FDFAs $\mathcal{F}_1=(\mathcal{Q},\{\mathcal{P}_\epsilon,\mathcal{P}_b\})$ and $\mathcal{F}_2=(\mathcal{Q},\{\mathcal{P}'_\epsilon,\mathcal{P}_b\})$ for the language $(\Sigma^*b)^\omega\cup(bb)^*a^\omega$ using normalized and duo-normalized acceptances, respectively.
• Figure 3: Left, Middle: Two FDFAs $\mathcal{F}^{\textsc{s}}=(\mathcal{Q},\{\mathcal{P}^{\textsc{s}}_\epsilon\})$ and $\mathcal{F}^{\text{\tiny{\SixFlowerPetalDotted}}}=(\mathcal{Q},\{\mathcal{P}^{\text{\tiny{\SixFlowerPetalDotted}}}_\epsilon\})$ for the language $L_{\infty aa \wedge \neg \infty bb}$ where $\mathcal{Q}$ is a one-state leading automaton. $\mathcal{F}^\textsc{s}$ uses normalized acceptance, $\mathcal{F}^\text{\tiny{\SixFlowerPetalDotted}}$ uses duo-normalized acceptance. Right: The progress DFA $\mathcal{P}_\epsilon$ for an FDFA accepting $\infty aa$ that uses duo-normalization and a one-state leading automaton.
• Figure 4: Three progress DFAs $\mathcal{P}_{}^{\textsc{m}}$, $\mathcal{P}_{}^{\text{\tiny{\SixFlowerPetalDotted}}}$ and $\mathcal{P}_{}^{\textsc{s}}$.
• Figure 5: Left: The inclusions among these classes of FDFAs (in black), as well as the placement of the canonical FDFAs in these classes (in blue). The letters p,s,r,l,c abbreviate periodic, syntactic, recurrent, limit, and colorful, resp. Right: Picture summarizing succinctness results on proper FDFAs. A double-line (resp. one-line) arrow form $\textsc{c}$ to $\textsc{d}$ indicates that $\textsc{c}$ can be exponentially (resp. quadratically) more succinct than $\textsc{d}$.

Theorems & Definitions (36)

• Example 1
• Theorem 2: Robustness of the inclusion measures Wagner75
• Example 3
• Example 4
• Definition 5: $\omega$-words decomposition wrt an FDFA
• Definition 6: Exact, Normalized, and Duo-Normalized acceptance
• Claim 6
• Example 7
• Theorem 8
• Definition 9: persistent decomposition wrt an FDFA