We present a logic for stating properties such as, “after a request for service there is at least a 98% probability that the service will be carried out within 2 seconds”. The logic extends the temporal logic CTL by Emerson, Clarke and Sistla with time and probabilities. Formulas are interpreted over discrete time Markov chains. We give algorithms for checking that a given Markov chain satisfies a formula in the logic. The algorithms require a polynomial number of arithmetic operations, in size of both the formula and the Markov chain. A simple example is included to illustrate the algorithms.
Probabilistic transition systems is a basic semantic model for description and analysis of e.g. reliability aspects of concurrent and distributed systems. We present a formalism for specifying probabilistic transition systems, or processes, which itself is based on transition systems. Roughly, a specification has the form of a transition system where transitions are labeled by sets of allowed probabilities. For instance, a suficiently reliable medium can be specified by a transition system where transitions that represent loss and delivery of messages are labeled by appropriate intervals of probabilities.We define a satisfaction relation between processes and specifications which generalizes probabilistic bisimulation equivalence as proposed by Larsen and Skou. We also propose two criteria for refinement between specifications. One stronger criterion is analogous to the definition of simulation between non-probabilistic processes. This criterion is relatively unproblematic to establish. We show that it is analogous to the extension from processes t o modal transition systems b y Larsen and Thomsen. Another weaker criterion views a specification as defining a set of probabilistic processes; refinement is then simply Containment between sets of processes. We present a complete method for verifying containment between specifications, which extends methods f o r deciding containment between finite automata or tree acceptors.
We present regular model checking, a framework for algorithmic verification of infinite-state systems with, e.g., queues, stacks, integers, or a parameterized linear topology. States are represented by strings over a finite alphabet and the transition relation by a regular length-preserving relation on strings. Major problems in the verification of parameterized and infinite-state systems are to compute the set of states that are reachable from some set of initial states, and to compute the transitive closure of the transition relation. We present two complementary techniques for these problems. One is a direct automata-theoretic construction, and the other is based on widening. Both techniques are incomplete in general, but we give sufficient conditions under which they work. We also present a method for verifying ω-regular properties of parameterized systems, by computation of the transitive closure of a transition relation.
We consider the veri cation of a particular class of in nite-state systems, namely systems consisting of nite-state processes that communicate via unbounded lossy FIFO channels. This class is able to model e.g. link protocols such as the Alternating Bit Protocol and HDLC. For this class of systems, we show that several interesting veri cation problems are decidable by giving algorithms for verifying (1) the reachability problem: is a nite set of global states reachable from some other global state of the system, (2) safety properties over traces formulated as regular sets of allowed nite traces, and (3) eventuality properties: do all computations of a system eventually reach a given set of states. We have used the algorithms to verify some idealized sliding-window protocols with reasonable time and space resources. Our results should be contrasted with the well-known fact that these problems are undecidable for systems with unbounded perfect FIFO channels.
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