Determinising Parity Automata

Parity word automata and their determinisation play an important role in automata and game theory. We discuss a determinisation procedure for nondeterministic parity automata through deterministic Rabin to deterministic parity automata. We prove that the intermediate determinisation to Rabin automata is optimal. We show that the resulting determinisation to parity automata is optimal up to a small constant. Moreover, the lower bound refers to the more liberal Streett acceptance. We thus show that determinisation to Streett would not lead to better bounds than determinisation to parity. As a side-result, this optimality extends to the determinisation of B\"uchi automata

Solving parity games in big steps

This article proposes a new algorithm that improves the complexity bound for solving parity games. Our approach combines McNaughton's iterated fixed point algorithm with a preprocessing step, which is called prior to every recursive call. The preprocessing uses ranking functions similar to Jurdzifiski's, but with a restricted co-domain, to determine all winning regions smaller than a predefined parameter. The combination of the preprocessing step with the recursive call guarantees that McNaughton's algorithm proceeds in big steps, whose size is bounded from below by the chosen parameter. Higher parameters lead to smaller call trees, but they also result in an expensive preprocessing step. An optimal parameter balances the cost of the recursive call and the preprocessing step, resulting in an improvement of the known upper bound for solving parity games from O (m (2n/c)(1/2c))to approximately O (m (6e(1) ((6) over bar) n/c(2))(1/3c) )

Time and Parallelizability Results for Parity Games with Bounded Tree and DAG Width

Parity games are a much researched class of games in NP intersect CoNP that are not known to be in P. Consequently, researchers have considered specialised algorithms for the case where certain graph parameters are small. In this paper, we study parity games on graphs with bounded treewidth, and graphs with bounded DAG width. We show that parity games with bounded DAG width can be solved in O(n^(k+3) k^(k + 2) (d + 1)^(3k + 2)) time, where n, k, and d are the size, treewidth, and number of priorities in the parity game. This is an improvement over the previous best algorithm, given by Berwanger et al., which runs in n^O(k^2) time. We also show that, if a tree decomposition is provided, then parity games with bounded treewidth can be solved in O(n k^(k + 5) (d + 1)^(3k + 5)) time. This improves over previous best algorithm, given by Obdrzalek, which runs in O(n d^(2(k+1)^2)) time. Our techniques can also be adapted to show that the problem of solving parity games with bounded treewidth lies in the complexity class NC^2, which is the class of problems that can be efficiently parallelized. This is in stark contrast to the general parity game problem, which is known to be P-hard, and thus unlikely to be contained in NC

Software Synthesis is Hard -- and Simple

While the components of distributed hardware systems can reasonably be assumed to be synchronised, this is not the case for the components of distributed software systems. This has a strong impact on the class of synthesis problems for which decision procedures exist: While there is a rich family of distributed systems, including pipelines, chains, and rings, for which the realisability and synthesis problem is decidable if the system components are composed synchronously, it is well known that the asynchronous synthesis problem is only decidable for monolithic systems. From a theoretical point of view, this renders distributed software synthesis undecidable, and one is tempted to conclude that synthesis of asynchronous systems, and hence of software, is much harder than the synthesis of synchronous systems. Taking a more practical approach, however, reveals that bounded synthesis, one of the most promising synthesis techniques, can easily be extended to asynchronous systems. This merits the hope that the promising results from bounded synthesis will carry over to asynchronous systems as well

Bounded Satisfiability for PCTL

While model checking PCTL for Markov chains is decidable in polynomial-time, the decidability of PCTL satisfiability, as well as its finite model property, are long standing open problems. While general satisfiability is an intriguing challenge from a purely theoretical point of view, we argue that general solutions would not be of interest to practitioners: such solutions could be too big to be implementable or even infinite. Inspired by bounded synthesis techniques, we turn to the more applied problem of seeking models of a bounded size: we restrict our search to implementable -- and therefore reasonably simple -- models. We propose a procedure to decide whether or not a given PCTL formula has an implementable model by reducing it to an SMT problem. We have implemented our techniques and found that they can be applied to the practical problem of sanity checking -- a procedure that allows a system designer to check whether their formula has an unexpectedly small model