Kleene automata and recursion theory
Following the introduction of a theoretical computational model on infinite objects, the ω-input Turing machine, we present a new type of infinite automata, the Kleene automata. We show it recognizes exactly the class of arithmetical ω-languages. Essentially, it is a proposisional automaton for which the transition relation is recursive and the interpretation of atomic formulas associated with each state is recursive. The acceptance conditions are built up hierarchically by adding to each level, the recursive disjonctions of negations of the previous level's formulas. The first level is a proposisional temporal logic restricted to the only one temporal operator next. We show the expressive power of this logic to be the class of recursive ω-languages.
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