Generalized Deterministic Languages and their Automata: A Characterization of Restricted Temporal Logic

Let TL X F TL F be the class of formulas of temporal logic, where as the only temporal operatorsX (next) andF (eventually) are allowed (onlyF is allowed, resp.). For a class TL X F let L be the class of -definable languages. Among others, characterizations of L TL X F and L TL F in terms of forbidden patterns in finite automata are known. Here we ask for every bound k on the number of nested uses of the next operator for the expressive power of the respective fragment of TL X F . Denote by TL X k F the class of formulas in TL X F with nesting depth k in the next operator. Obviously, L TL X F S k L TL X k F and L TL F L TL X F . We prove a levelwise characterization of the above syntactical hierarchy (1) in terms of a certain pattern S k (cf. Figure 6) that must not appear in the transition graph of deterministic finite automata, and (2) in terms of a formal language representation involving k-leftdeterministic languages. With (1) we generalize the known forbidden pattern for L TL F which is S and in (2) we carry over the notion of leftdeterministic languages to decompositions of words into consecutive blocks of length k . Our characterization enables us to give concise proofs of strictness and decidability results for the family of classes L TL X k F k . As a by-product, we reproof the known forbidden pattern characterization for L TL X F in a novel way and we learn, how the pattern S turns into the pattern for L TL X F as level k increases. All results hold dually for restricted past temporal logic, k-rightdeterministic languages and the pattern Sk (cf. Figure 5). We point out similarities and differences to languages having dot-depth one.