2 Dta 2 Nta Nta

15 2DTA 2NTA NTA p MITL k U k DTA Figure 5: Relationship between various classes Theorem 5.2 The membership problem for 2NTA is solvable in PSPACE. Proof. Consider a 2NTA A, and let k be the magnitude of the largest constant appearing in a clock constraint in the transitions of A. Given a timed word w = (;) of length n, deene R w to be the set comprising of i + c and (i + j + c)=2 for 1 i; j n and ?(k + 1) c (k + 1). Let A be the 2NTA that behaves like A, except that A is required to choose a value from the set R w whenever a clock is nondeterministically reset. Then A accepts the timed word w ii A accepts w. Now we can construct a nondeterministic linear-bounded Turing machine A 0 that, given a description of A and a timed word w, simulates the behavior of A on w$. We conjecture that in general, an automaton cannot check a formula of MITL P k with less than k reversals. If this is the case, then the previous corollary can be strengthened to MITL P k MITL P k+1. In Section 3 we addressed expressiveness, closure, and decidability questions for two-way timed automata with a bounded number of reversals. The same questions can be asked for the unrestricted classes 2DTA and 2NTA. These classes are perhaps not useful for algorithmic veriication, because we will see that unbounded two-wayness is so powerful as to lead to undecidability even for the emptiness problem of 2DTA. To get an insight into the power of 2DTAs let us consider temporal logics with punctuality constraints. Recall that the logic MITL disallows punctuality constraints such as \every p symbol is followed by a q symbol after exactly 1 time unit," 2(p ! 3 =1 q): (=) We showed in 4] that allowing such formulas leads to the undecidability of the satissability problem. Let MITL = be the extension of MITL that allows singular intervals as time bounds on the temporal operators. We show that the models of any formula of MITL = can be deened by a 2DTA. For instance, it is easy to modify the automaton A 4 of Figure 4 so that it accepts precisely the models of the formula =. On the other hand, the models of = cannot be recognized …