Verifying nondeterministic probabilistic channel systems against ω\omega-regular linear-time properties

Lossy channel systems (LCSs) are systems of finite state automata that communicate via unreliable unbounded fifo channels. In order to circumvent the undecidability of model checking for nondeterministic LCSs, probabilistic models have been introduced, where it can be decided whether a linear-time property holds almost surely. However, such fully probabilistic systems are not a faithful model of nondeterministic protocols. We study a hybrid model for LCSs where losses of messages are seen as faults occurring with some given probability, and where the internal behavior of the system remains nondeterministic. Thus the semantics is in terms of infinite-state Markov decision processes. The purpose of this article is to discuss the decidability of linear-time properties formalized by formulas of linear temporal logic (LTL). Our focus is on the qualitative setting where one asks, e.g., whether a LTL-formula holds almost surely or with zero probability (in case the formula describes the bad behaviors). Surprisingly, it turns out that -- in contrast to finite-state Markov decision processes -- the satisfaction relation for LTL formulas depends on the chosen type of schedulers that resolve the nondeterminism. While all variants of the qualitative LTL model checking problem for the full class of history-dependent schedulers are undecidable, the same questions for finite-memory scheduler can be solved algorithmically. However, the restriction to reachability properties and special kinds of recurrent reachability properties yields decidable verification problems for the full class of schedulers, which -- for this restricted class of properties -- are as powerful as finite-memory schedulers, or even a subclass of them.Comment: 39 page

Recurrence and Transience for Probabilistic Automata

In a context of $omega$-regular specifications for infinite execution sequences, the classical B"uchi condition, or repeated liveness condition, asks that an accepting state is visited infinitely often. In this paper, we show that in a probabilistic context it is relevant to strengthen this infinitely often condition. An execution path is now accepting if the emph{proportion} of time spent on an accepting state does not go to zero as the length of the path goes to infinity. We introduce associated notions of recurrence and transience for non-homogeneous finite Markov chains and study the computational complexity of the associated problems. As Probabilistic B"uchi Automata (PBA) have been an attempt to generalize B"uchi automata to a probabilistic context, we define a class of Constrained Probabilistic Automata with our new accepting condition on runs. The accepted language is defined by the requirement that the measure of the set of accepting runs is positive (probable semantics) or equals 1 (almost-sure semantics). In contrast to the PBA case, we prove that the emptiness problem for the language of a constrained probabilistic B"uchi automaton with the probable semantics is decidable

Composition of Stochastic Transition Systems Based on Spans and Couplings

Conventional approaches for parallel composition of stochastic systems relate probability measures of the individual components in terms of product measures. Such approaches rely on the assumption that components interact stochastically independent, which might be too rigid for modeling real world systems. In this paper, we introduce a parallel-composition operator for stochastic transition systems that is based on couplings of probability measures and does not impose any stochastic assumptions. When composing systems within our framework, the intended dependencies between components can be determined by providing so-called spans and span couplings. We present a congruence result for our operator with respect to a standard notion of bisimilarity and develop a general theory for spans, exploiting deep results from descriptive set theory. As an application of our general approach, we propose a model for stochastic hybrid systems called stochastic hybrid motion automata

On Skolem-Hardness and Saturation Points in Markov Decision Processes

The Skolem problem and the related Positivity problem for linear recurrence sequences are outstanding number-theoretic problems whose decidability has been open for many decades. In this paper, the inherent mathematical difficulty of a series of optimization problems on Markov decision processes (MDPs) is shown by a reduction from the Positivity problem to the associated decision problems which establishes that the problems are also at least as hard as the Skolem problem as an immediate consequence. The optimization problems under consideration are two non-classical variants of the stochastic shortest path problem (SSPP) in terms of expected partial or conditional accumulated weights, the optimization of the conditional value-at-risk for accumulated weights, and two problems addressing the long-run satisfaction of path properties, namely the optimization of long-run probabilities of regular co-safety properties and the model-checking problem of the logic frequency-LTL. To prove the Positivity- and hence Skolem-hardness for the latter two problems, a new auxiliary path measure, called weighted long-run frequency, is introduced and the Positivity-hardness of the corresponding decision problem is shown as an intermediate step. For the partial and conditional SSPP on MDPs with non-negative weights and for the optimization of long-run probabilities of constrained reachability properties (aU b), solutions are known that rely on the identification of a bound on the accumulated weight or the number of consecutive visits to certain sates, called a saturation point, from which on optimal schedulers behave memorylessly. In this paper, it is shown that also the optimization of the conditional value-at-risk for the classical SSPP and of weighted long-run frequencies on MDPs with non-negative weights can be solved in pseudo-polynomial time exploiting the existence of a saturation point. As a consequence, one obtains the decidability of the qualitative model-checking problem of a frequency-LTL formula that is not included in the fragments with known solutions

Weak bisimulation for fully probabilistic processes

Bisimulations that abstract from internal computation have proven to be useful for verification of compositionally defined transition systems. In the literature of probabilistic extensions of such transition systems, similar bisimulations are rare. In this paper, we introduce weak and branching bisimulation for fully probabilistic systems, transition systems where nondeterministic branching is replaced by probabilistic branching. In contrast to the nondeterministic case, both relations coincide. We give an algorithm to decide weak (and branching) bisimulation with a time complexity cubic in the number of states of the fully probabilistic system. This meets the worst case complexity for deciding branching bisimulation in the nondeterministic case. In addition, the relation is shown to be a congruence with respect to the operators of PLSCCS, a lazy synchronous probabilistic variant of CCS. We illustrate that due to these properties, weak bisimulation provides all the crucial ingredients for mechanised compositional veri�cation of probabilistic transition systems

Construction of a cms on a given cpo

In dealing with denotational semantics of programming languages partial orders resp. metric spaces have been used with great benefit in order to provide a meaning to recursive and repetitive constructs. This paper presents two methods to define a metric on a subset M of a cpo D such that M is a complete metric spaces and the metric semantics on M coincides with the cpo semantics on D when the same semantic operators are used. The first method is to add a 'length' on a cpo which means a function ρ : D → IN 0 ∪{∞} of increasing power. The second is based on the ideas of [9] and uses pseudo rank orderings, i.e. monotone sequences of monotone functions ϖn : D → D. We show that SFP domains can be characterized as special kinds of rank orderded cpo's. We also discuss the connection between the Lawson topology and the topology induced by the metric

On the definability of concurrency and communication : event structures versus pomset classes

In the context of communicating parallel process systems various paradigma for communication resp. synchronisation have been proposed. Two well-known theoretical models for communicating systems are CCS and TCSP. A variety of semantics has been proposed for these and similar languages which can be characterized by different criteria: true versus interleaving parallelism, linear versus branching time models, operational versus denotational approaches, choice of the mathematical discipline to handle recursion and domain equations. In recent years interest has shifted more and more towards semantics that model true parallelism. The most known are petri net semantics, event structure and pomset semantics. The present paper investigates the question whether the two closely related approaches of event structures and pomsets are equally suitable to provide semantics for language constructs as avaible in CCS or TCSP. Given the variety of approaches to semantic description comparative studies like the present one are importantas a guideline. They help us to decide which method suits which purpose. In addition, comparative studies enhance the better understanding of the language constructs, and finally comparative studies of semantics that yield consistency results strengthen our confidence in the correctness of each, of the semantics involved. The paper is divided into seven sections. Section 2 introdtices CCS and TCSP and discusses their communication mechanisms. Section3 introduces event structures. Section 4 defines pomset classes. Section 5 models the communication mechanisms of CCS and TCSP using event structures and section 6 discusses the problems that arise when pomset classes are used. Section 7 is the conclusion. The appendix contains some formal definitions

The connection between an event structure semantics and an operational semantics for TCSP

The relation between an operational interleaving semantics for TSCP based on a transition system and a compositional true concurrency semantics based on event structures is studied. In particular we extend the consistency result of U. Goltz and R. Loogen [Ann. Soc. Math. Pol., Ser. IV, Fundam. Inf. 14, 39-73 (1991; Zbl 0717.68028)] for TCSP processes without recursion to the general case. Thus, we obtain for every TCSP process P that its operational meaning O(P) and the interleaving behaviour O(M[[P]]) which is derived from the event structure M[[P]] associated with P are bisimilar. (aus: Zentralblatt MATH