Alexander Weinert

Easy to Win, Hard to Master: Optimal Strategies in Parity Games with Costs

The winning condition of a parity game with costs requires an arbitrary, butnfixed bound on the cost incurred between occurrences of odd colors and the nextnoccurrence of a larger even one. Such games quantitatively extend parity gamesnwhile retaining most of their attractive properties, i.e, determining thenwinner is in NP and co-NP and one player has positional winning strategies.n We show that the characteristics of parity games with costs are vastlyndifferent when asking for strategies realizing the minimal such bound: Thensolution problem becomes PSPACE-complete and exponential memory is bothnnecessary in general and always sufficient. Thus, solving and playing parityngames with costs optimally is harder than just winning them. Moreover, we shownthat the tradeoff between the memory size and the realized bound is gradual inngeneral. All these results hold true for both a unary and binary encoding ofncosts.n Moreover, we investigate Streett games with costs. Here, playing optimally isnas hard as winning, both in terms of complexity and memory.

Approximating Optimal Bounds in Prompt-LTL Realizability in Doubly-exponential Time

We consider the optimization variant of the realizability problem for Prompt Linear Temporal Logic, an extension of Linear Temporal Logic (LTL) by the prompt eventually operator whose scope is bounded by some parameter. In the realizability optimization problem, one is interested in computing the minimal such bound that allows to realize a given specification. It is known that this problem is solvable in triply-exponential time, but not whether it can be done in doubly-exponential time, i.e., whether it is just as hard as solving LTL realizability. nWe take a step towards resolving this problem by showing that the optimum can be approximated within a factor of two in doubly-exponential time. Also, we report on a proof-of-concept implementation of the algorithm based on bounded LTL synthesis, which computes the smallest implementation of a given specification. In our experiments, we observe a tradeoff between the size of the implementation and the bound it realizes. We investigate this tradeoff in the general case and prove upper bounds, which reduce the search space for the algorithm, and matching lower bounds.

Visibly Linear Dynamic Logic.

We introduce Visibly Linear Dynamic Logic (VLDL), which extends Linear Temporal Logic (LTL) by temporal operators that are guarded by visibly pushdown languages over finite words. In VLDL one can, e.g., express that a function resets a variable to its original value after its execution, even in the presence of an unbounded number of intermediate recursive calls. We prove that VLDL describes exactly the omega-visibly pushdown languages. Thus it is strictly more expressive than LTL and able to express recursive properties of programs with unbounded call stacks. n nThe main technical contribution of this work is a translation of VLDL into omega-visibly pushdown automata of exponential size via one-way alternating jumping automata. This translation yields exponential-time algorithms for satisfiability, validity, and model checking. We also show that visibly pushdown games with VLDL winning conditions are solvable in triply-exponential time.We prove all these problems to be complete for their respective complexity classes.