Visibly Recursive Automata

Abstract

As an alternative to visibly pushdown automata, we introduce visibly recursive automata (VRAs), composed of a set of classical automata that can call each other. VRAs are a strict extension of so-called systems of procedural automata, a model proposed by Frohme and Steffen. We study the complexity of standard language-theoretic operations and classical decision problems for VRAs. Since the class of deterministic VRAs forms a strict subclass in terms of expressiveness, we propose a (weaker) notion that does not restrict expressive power and which we call codeterminism. Codeterminism comes with many desirable algorithmic properties that we demonstrate by using it, e.g., as a stepping stone towards implementing complementation of VRAs.

Figures (7)

• Figure 1: Example of a VRA and a recursive run of it.
• Figure 2: Illustration of the semantics of a VRA on $cwr \in \Sigma_{\mathit{call}} \cdot \mathit{WM}(\widetilde{\Sigma}) \cdot \Sigma_{\mathit{ret}}$, with $q_i \in I^J$, $q_f\in F^J$ and $f(J)=\langle c,r \rangle$.
• Figure 3: Set of recursive language s of three automata $\mathcal{A}^{J_1}$, $\mathcal{A}^{J_2}$ and $\mathcal{A}^{J_3}$, and the corresponding set of languages of the automata $\mathcal{B^J}$, for all $\mathcal{J}\subseteq \{J_1,J_2,J_3\}$.
• Figure 4: A deterministic VRA with no equivalent SPA.
• Figure 5: A two-state deterministic VPA.

Theorems & Definitions (50)

• definition 1: Finite automaton
• definition 2: Well-matched word
• definition 3: Procedural alphabet
• definition 4: Visibly recursive automaton
• proposition 1
• proposition 2
• proof : Sketch
• theorem 1: Equivalence of VRAs and VPAs
• definition 5: Codeterministic VRA
• definition 6: Complete VRA