Past, Present, and Infinite Future

I was supposed to deliver one of the speeches at Wolfgang Thomas’s retirement ceremony. Wolfgang had called me on the phone earlier and posed some questions about temporal logic, but I hadn’t had good answers at the time. What I decided to do at the ceremony was to take up the conversation again and show how it could have evolved if only I had put more effort into answering his questions. Here is the imaginary conversation with Wolfgang. The contributions are (1) the first direct translation from counter-free ω-automata into future temporal formulas, (2) a definition of bimachines for ω-words, (3) a translation from arbitrary temporal formulas (including both, future and past operators) into counter-free ω-bimachines, and (4) an automata-based proof of separation: every arbitrary temporal formula is equivalent to a boolean combination of pure future, present, and pure past formulas when interpreted in ω-words. 1998 ACM Subject Classification F.1.1 Models of Computation, F.1.2 Modes of Computation, F.4.1 Mathematical Logic, F.4.3 Formal Languages