A Naturally-Colored Translation from LTL to Parity and COCOA

Rüdiger Ehlers, Ayrat Khalimov

Abstract: Chains of co-Buechi automata (COCOA) have recently been introduced as a new canonical representation of omega-regular languages. The co-Buechi automata in a chain assign to each omega-word its natural color, which depends only on the language itself and not on its automaton representation. The automata in such a chain can be minimized in polynomial time and are good-for-games, making the representation attractive for verification and reactive synthesis applications. However, since in such applications, a specification is usually given in linear temporal logic (LTL), to make COCOA useful, the specification first has to be translated into such a chain of automata. Currently, the only known translation procedure involves a detour through deterministic parity automata (LTL to DPW to COCOA), where the first step neglects the natural colors and requires intricate constructions by Safra or Esparza et al. This observation raises the question whether, by leveraging the definition of the natural color of words, these complex constructions can be avoided, leading to a more direct translation from LTL to COCOA. This paper presents a surprisingly simple yet optimal translation from LTL to COCOA. Our procedure relies on standard operations on weak alternating automata, Miyano-Hayashi's breakpoint construction, an augmented subset construction, and simple graph algorithms. With weak alternating automata as a starting point, the procedure can also handle specifications in linear dynamic logic.

External IDs:dblp:journals/corr/abs-2410-01021