We investigate computational properties of propositional logics for dynamical systems. First, we consider logics for dynamic topological systems (W.f), fi, where W is a topological space and f a homeomorphism on W. The logics come with ‘modal’ operators interpreted by the topological closure and interior, and temporal operators interpreted along the orbits {w, f(w), f2 (w), ˙˙˙} of points w ε W. We show that for various classes of topological spaces the resulting logics are not recursively enumerable (and so not (...) recursively axiomatisable). This gives a ‘negative’ solution to a conjecture of Kremer and Mints. Second, we consider logics for dynamical systems (W, f), where W is a metric space and f and isometric function. The operators for topological interior/closure are replaced by distance operators of the form ‘everywhere/somewhere in the ball of radius a, ‘for a ε Q +. In contrast to the topological case, the resulting logic turns out to be decidable, but not in time bounded by any elementary function. (shrink)

We describe approximation algorithms for MAX SAT with performance ratios arbitrarily close to 1, in particular, when performance ratios exceed the limits of polynomial-time approximation. Namely, given a polynomial-time α-approximation algorithm , we construct an -approximation algorithm . The algorithm runs in time of the order ck, where k is the number of clauses in the input formula and c is a constant depending on α. Thus we estimate the cost of improving a performance ratio. Similar constructions for MAX 2SAT (...) and MAX 3SAT are also described. Taking known algorithms as , we obtain particular upper bounds on the running time of. (shrink)