On the Generative Power of ω-Grammars and ω-Automata

An ω-grammar is a formal grammar used to generateω-words (i.e. infinite length words), while an ω-automaton is an automaton used to recognize ω-words. This paper gives clean and uniform definitions for ω-grammars and ω-automata, provides a systematic study of the generative power of ω-grammars with respect to ω-automata, and presents a complete set of results for various types of ω-grammars and acceptance modes. We use the tuple (σ, ρ, π) to denote various acceptance modes, where σ denotes that some designated elements should appear at least once or infinitely often, ρ denotes some binary relation between two sets, and π denotes normal or leftmost derivations. Technically, we propose (σ, ρ, π)-acceptingω-grammars, and systematically study their relative generative power with respect to (σ, ρ)-accepting ω-automata. We show how to construct some special forms of ω-grammars, such as ǫ-production-free ω-grammars. We study the equivalence or inclusion relations between ω-grammars and ω-automata by establishing the translation techniques. In particular, we show that, for some acceptance modes, the generative power of ω-CFG is strictly weaker than ω-PDA, and the generative power of ω-CSG is equal to ω-TM (rather than linearbounded ω-automata-like devices). Furthermore, we raise some remaining open problems for two of the acceptance modes.