Max 4 live otomata
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Our distinguishing SA are small and employ a common nondeterministic gadget. , surprisingly leads to partly suboptimal, partly incomparable classes. Considering only the relative order of clock expiration times, as suggested by the first algorithm of Bryans et al.
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), but turn out to be suboptimal in the more realistic non-prophetic case. In particular, memoryless schedulers suffice in the complete-information setting (as is implicit in the method of Kwiatkowska et al.
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We find that SA distinguish most classes. achieving optimal reachability probabilities (Sect. Within each perspective, we define classes of schedulers whose views of the state and history are variously restricted (Sect. We use two perspectives on schedulers from the literature: the classic complete-information residual lifetimes semantics , where optimality is defined via history-dependent schedulers that see the entire current state, and non-prophetic schedulers that cannot observe the timing of future events. Understanding the capabilities of scheduler classes helps decide on the tradeoff between simplicity and the ability to attain optimal results. schedulers that need little information and limited memory, so as to be explainable and suitable for implementation on e.g. When it comes to planning problems, on the other hand, practitioners desire simple solutions, i.e. For example, Markov decision process (MDP) model checking works well because memoryless schedulers suffice for reachability, and the efficient time-bounded analysis of continuous-time MDP (CTMDP) exploits a relationship between two scheduler classes that are sufficiently simple, but on their own do not realise the desired extremal probabilities .
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#Max 4 live otomata verification
Our motivation is, on the one hand, that a clear understanding of scheduler classes is crucial to design verification algorithms. With this paper, we take on the underlying problem from a fundamental perspective: we investigate the power of, and relationships between, different classes of schedulers for SA. Without that restriction , error bounds and convergence guarantees are lost.Įvidently, the combination of nondeterminism and continuous probability distributions is a particularly challenging one. for a model closely related to SA, but restricted to bounded continuous distributions. The only approach that handles nondeterminism is the region-based approximation scheme of Kwiatkowska et al. , however they again exclude nondeterminism. Regeneration is central to the work of Ballarini et al. The latter forces regeneration on every edge, making it impossible to use clocks as memory between locations. propose two algorithms that require an a priori fixed scheduler, continuous bounded distributions, and that all active clocks be reset when a location is entered. restrict to phase-type or matrix-exponential distributions, such that nondeterminism cannot arise (as each edge is guarded by a single clock). Numerical verification algorithms exist for very limited subclasses of SA only: Buchholz et al. deterministic delays), urgent edges, and edges waiting on multiple clocks. The latter may arise from non-continuous distributions (e.g. We consider closed systems of stochastic automata (SA ), which extend GSMP and feature both generally distributed stochastic delays as well as discrete nondeterministic choices. If the nondeterminism is considered controllable, one may alternatively be interested in the planning problem of synthesising a scheduler that satisfies certain probability bounds. These are the supremum and infimum of the probabilities of the property in the purely stochastic systems induced by classes of schedulers (also called strategies, policies or adversaries) that resolve all nondeterminism. It is then possible to verify such systems by considering the extremal probabilities of a property. Various models and formalisms have thus been proposed to extend continuous-time stochastic processes with nondeterminism . Modelling such uncertainty with probability is convenient for simulation, but not always adequate . Nondeterminism arises through inherent concurrency of independent processes , but may also be deliberate underspecification. This has led to a number of discrete event simulation tools, such as those for networking , whose probabilistic semantics is founded on generalised semi-Markov processes (GSMP ). The need to analyse continuous-time stochastic models arises in many practical contexts, including critical infrastructures , railway engineering , space mission planning , and security .