favorite8Note that in each of these cases, we concentrate on the corresponding (stable) eigenspace, construct a separating semilinear invariant for this restriction of the problem, and extend it to the full space by allowing any value on other coordinates.
favorite72It is quite easy to prove that no semilinear invariant exists (except for the whole space R2 ) for this instance, whatever the value of y.
favorite4This gives rise to the semilinear invariant problem, where an instance is given by a set of square matrices A1 , .
favorite20We are interested in instances of the Monniaux Problem in which there are no guards, all transitions are affine (or equivalently linear, since affine transitions can be made linear by increasing the dimension of the ambient space by 1), and invariants are semilinear.
favorite1In fact, as observed by Monniaux , if one considers a domain of convex polyhedra having a uniformly bounded number of faces (therefore subsuming in particular the domains just described), then for any class of programs with polynomial transition relations and guards, the existence of separating invariants becomes decidable, as the problem can equivalently be phrased in the first-order theory of the reals.
favorite8Patricia Bouyer, Thomas Brihaye, and Nicolas Markey.
favorite3This function can also compute ValG (`, 1) for all ` and all games G (even with non-urgent locations) since time can not elapse anymore when the clock has valuation 1.
favorite5The key point of those algorithms is to reduce the problem to the computation of optimal values in a restricted family of PTGs called Simple Priced Timed Games (SPTGs for short), where the underlying automata contain no guard, no reset, and the play is forced to stop after one time unit.
favorite4This assumption was justified in [11, 7] by showing that, in the absence of non-Zeno assumption, the existence problem, that is to decide whether Min has a strategy guaranteeing to reach a target location with a cost below a given threshold, is indeed undecidable for PTGs with non-negative prices and three or more clocks.
favorite43We consider priced timed games with one clock and arbitrary (positive and negative) weights and show that, for an important subclass of theirs (the so-called simple priced timed games), one can compute, in exponential time, the optimal values that the players can achieve, with their associated optimal strategies.