A Naturally-Colored Translation from LTL to Parity and COCOA

Chains of co-B\"uchi automata (COCOA) have recently been introduced as a new canonical representation of omega-regular languages. The co-B\"uchi 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. The only currently known translation procedure goes 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 with the help of the definition of the natural color of words, such complex constructions can be avoided, leading to a more direct translation from LTL to COCOA. In this paper, we describe a surprisingly simple yet optimal translation from LTL to COCOA and a variant of it that translates from LTL to deterministic parity automata. It constitutes a novel path for translating from LTL to DPW as the translation procedure does not use any of the aforementioned intricate constructions. Instead, our procedure relies on standard operations on weak alternating automata, Miyano/Hayashi's breakpoint construction, an augmented subset construction, and some simple graph algorithms. With weak alternating automata as starting point, the procedure can also be applied to specifications in linear dynamic logic. The translation procedure runs in asymptotically-optimal doubly-exponential time and computes automata of asymptotically optimal size.