Timed Parity Games: Complexity and Robustness

We consider two-player games played in real time on game structures with clocks where the objectives of players are described using parity conditions. The games are \emph{concurrent} in that at each turn, both players independently propose a time delay and an action, and the action with the shorter delay is chosen. To prevent a player from winning by blocking time, we restrict each player to play strategies that ensure that the player cannot be responsible for causing a zeno run. First, we present an efficient reduction of these games to \emph{turn-based} (i.e., not concurrent) \emph{finite-state} (i.e., untimed) parity games. Our reduction improves the best known complexity for solving timed parity games. Moreover, the rich class of algorithms for classical parity games can now be applied to timed parity games. The states of the resulting game are based on clock regions of the original game, and the state space of the finite game is linear in the size of the region graph. Second, we consider two restricted classes of strategies for the player that represents the controller in a real-time synthesis problem, namely, \emph{limit-robust} and \emph{bounded-robust} winning strategies. Using a limit-robust winning strategy, the controller cannot choose an exact real-valued time delay but must allow for some nonzero jitter in each of its actions. If there is a given lower bound on the jitter, then the strategy is bounded-robust winning. We show that exact strategies are more powerful than limit-robust strategies, which are more powerful than bounded-robust winning strategies for any bound. For both kinds of robust strategies, we present efficient reductions to standard timed automaton games. These reductions provide algorithms for the synthesis of robust real-time controllers

Oceanic response to Pliensbachian and Toarcian magmatic events: Implications from an organic-rich basinal succession in the NW Tethys

The Bächental bituminous marls (Bächentaler Bitumenmergel) belonging to the Sachrang Member of the Lower Jurassic Middle Allgäu Formation were investigated using a multidisciplinary approach to determine environmental controls on the formation of organic-rich deposits in a semi-restricted basin of the NW Tethys during the Early Jurassic. The marls are subdivided into three units on the basis of mineralogical composition, source-rock parameters, redox conditions, salinity variations, and diagenetic processes. Redox proxies (e.g., pristane/phytane ratio; aryl isoprenoids; bioturbation; ternary plot of iron, total organic carbon, and sulphur) indicate varying suboxic to euxinic conditions during deposition of the Bächental section. Redox variations were mainly controlled by sea-level fluctuations with the tectonically complex bathymetry of the Bächental basin determining watermass exchange with the Tethys Ocean. Accordingly, strongest anoxia and highest total organic carbon content (up to 13%) occur in the middle part of the profile (upper tenuicostatum and lower falciferum zones), coincident with an increase in surface-water productivity during a period of relative sea-level lowstand that induced salinity stratification in a stagnant basin setting. This level corresponds to the time interval of the lower Toarcian oceanic anoxic event (T-OAE). However, the absence of the widely observed lower Toarcian negative carbon isotope excursion in the study section questions its unrestricted use as a global chemostratigraphic marker. Stratigraphic correlation of the thermally immature Bächental bituminous marls with the Posidonia Shale of SW Germany on the basis of C27/C29 sterane ratio profiles and ammonite data suggests that deposition of organic matter-rich sediments in isolated basins in the Alpine realm commenced earlier (late Pliensbachian margaritatus Zone) than in regionally proximal epicontinental seas (early Toarcian tenuicostatum Zone). The late Pliensbachian onset of reducing conditions in the Bächental basin coincided with an influx of volcaniclastic detritus that was possibly connected to complex rifting processes of the Alpine Tethys and with a globally observed eruption-induced extinction event. The level of maximum organic matter accumulation in the Bächental basin corresponds to the main eruptive phase of the Karoo-Ferrar large igneous province (LIP), confirming its massive impact on global climate and oceanic conditions during the Early Jurassic. The Bächental marl succession is thus a record of the complex interaction of global (i.e., LIP) and local (e.g., redox and salinity variations, basin morphology) factors that caused reducing conditions and organic matter enrichment in the Bächental basin. These developments resulted in highly inhomogeneous environmental conditions in semi-restricted basins of the NW Tethyan domain during late Pliensbachian and early Toarcian time