We develop an algorithm for computing bounded reachability probability for hybrid systems, i.e., the probability that the system reaches an unsafe region within a finite number of discrete transitions. In particular, we focus on hybrid systems with continuous dynamics given by solutions of nonlinear ordinary differential equations (with possibly nondeterministic initial conditions and parameters), and probabilistic behaviour given by initial parameters distributed as continuous (with possibly infinite support) and discrete random variables. Our approach is to define an appropriate relaxation of the (undecidable) reachability problem, so that it can be solved by $δ$-complete decision procedures. In particular, for systems with continuous random parameters only, we develop a validated integration procedure which computes an arbitrarily small interval that is guaranteed to contain the reachability probability. In the more general case of systems with both nondeterministic and probabilistic parameters, our procedure computes a guaranteed enclosure for the range of reachability probabilities. We have applied our approach to a number of nonlinear hybrid models and validated the results by comparison with Monte Carlo simulation.
Labeled transition systems are typically used to represent the behavior of nondeterministic processes, with labeled transitions defining a one-step state to-state reachability relation. This model has been recently made more general by modifying the transition relation in such a way that it associates with any source state and transition label a reachability distribution, i.e., a function mapping each possible target state to a value of some domain that expresses the degree of one-step reachability of that target state. In this extended abstract, we show how the resulting model, called ULTraS …
Interface theories are powerful frameworks supporting incremental and compositional design of systems through refinements and constructs for conjunction, and parallel composition. In this report we present a first Interface Theor -- |Modal Mixed Interfaces -- for systems exhibiting both non-determinism and randomness in their behaviour. The associated component model -- Mixed Markov Decision Processes -- is also novel and subsumes both ordinary Markov Decision Processes and Probabilistic Automata.
Two of the most studied extensions of trace and testing equivalences to nondeterministic and probabilistic processes induce distinctions that have been questioned and lack properties that are desirable. Probabilistic trace-distribution equivalence differentiates systems that can perform the same set of traces with the same probabilities, and is not a congruence for parallel composition. Probabilistic testing equivalence, which relies only on extremal success probabilities, is backward compatible with testing equivalences for restricted classes of processes, such as fully nondeterministic proce…
We present a spectrum of trace-based, testing, and bisimulation equivalences for nondeterministic and probabilistic processes whose activities are all observable. For every equivalence under study, we examine the discriminating power of three variants stemming from three approaches that differ for the way probabilities of events are compared when nondeterministic choices are resolved via deterministic schedulers. We show that the first approach - which compares two resolutions relatively to the probability distributions of all considered events - results in a fragment of the spectrum compatibl…
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 non…
The combination of nondeterminism and probability in concurrent systems lead to the development of several interpretations of process behavior. If we restrict our attention to linear properties only, we can identify three main approaches to trace and testing semantics: the trace distributions, the trace-by-trace and the extremal probabilities approaches. In this paper, we propose novel notions of behavioral metrics that are based on the three classic approaches above, and that can be used to measure the disparities in the linear behavior of processes wrt trace and testing semantics. We study t…
We explore language semantics for automata combining probabilistic and nondeterministic behavior. We first show that there are precisely two natural semantics for probabilistic automata with nondeterminism. For both choices, we show that these automata are strictly more expressive than deterministic probabilistic automata, and we prove that the problem of checking language equivalence is undecidable by reduction from the threshold problem. However, we provide a discounted metric that can be computed to arbitrarily high precision.
Backwards reachability is an efficient zone-based approach for model checking probabilistic timed automata w.r.t. PTCTL properties. Current implementations, however, are restricted to maximum probabilities of reachability properties. In this paper, we report on our new implementation of backwards reachability as part of the Modest Toolset. Its support for minimum and maximum probabilities of until formulas makes it the most general implementation available today. We compare its behaviour to the experimental results reported in the original papers presenting the backwards reachability technique…
Probabilistic timed automata are a suitable formalism to model systems with real-time, nondeterministic and probabilistic behaviour. We study two-player zero-sum games on such automata where the objective of the game is specified as the expected time to reach a target. The two players---called player Min and player Max---compete by proposing timed moves simultaneously and the move with a shorter delay is performed. The first player attempts to minimise the given objective while the second tries to maximise the objective. We observe that these games are not determined, and study decision proble…
The stochastic Boolean satisfiability (SSAT) problem has been introduced by Papadimitriou in 1985 when adding a probabilistic model of uncertainty to propositional satisfiability through randomized quantification. SSAT has many applications, among them probabilistic bounded model checking (PBMC) of symbolically represented Markov decision processes. This article identifies a notion of Craig interpolant for the SSAT framework and develops an algorithm for computing such interpolants based on a resolution calculus for SSAT. As a potential application area of this novel concept of Craig interpola…
We extend the theory of labeled Markov processes with internal nondeterminism, a fundamental concept for the further development of a process theory with abstraction on nondeterministic continuous probabilistic systems. We define nondeterministic labeled Markov processes (NLMP) and provide three definition of bisimulations: a bisimulation following a traditional characterization, a state based bisimulation tailored to our "measurable" non-determinism, and an event based bisimulation. We show the relation between them, including that the largest state bisimulation is also an event bisimulation.…
There is a scalability gap between probabilistic and non-probabilistic verification. Probabilistic model checking tools are based either on explicit engines or on (Multi-Terminal) Binary Decision Diagrams. These structures are complemented in areas of non-probabilistic verification by more scalable techniques, such as IC3. We present a symbolic probabilistic model checking algorithm based on IC3-like incremental construction of inductive clauses to partition the state space, interleaved with incremental construction of a system of linear inequalities. This paper compares our implementation to …
We evaluate numerically-precise Monte Carlo (MC), Quasi-Monte Carlo (QMC) and Randomised Quasi-Monte Carlo (RQMC) methods for computing probabilistic reachability in hybrid systems with random parameters. Computing reachability probability amounts to computing (multidimensional) integrals. In particular, we pay attention to QMC methods due to their theoretical benefits in convergence speed with respect to the MC method. The Koksma-Hlawka inequality is a standard result that bounds the approximation of an integral by QMC techniques. However, it is not useful in practice because it depends on th…
Computing reachability probabilities is a fundamental problem in the analysis of probabilistic programs. This paper aims at a comprehensive and comparative account on various martingale-based methods for over- and under-approximating reachability probabilities. Based on the existing works that stretch across different communities (formal verification, control theory, etc.), we offer a unifying account. In particular, we emphasize the role of order-theoretic fixed points---a classic topic in computer science---in the analysis of probabilistic programs. This leads us to two new martingale-based …
Given a Probabilistic Finite Automata (PFA), a set of states S, and an error threshold e > 0, our algorithm approximates the infimum probability (quantifying over all infinite words) that the automata reaches S. Our result contrasts with the known result that the approximation problem is undecidable if we consider the supremum instead of the infimum. Since we study the probability of reaching a set of states, instead of the probability of ending in an accepting state, our work is more related to model checking than to formal languages.
We revisit the symbolic verification of Markov chains with respect to finite horizon reachability properties. The prevalent approach iteratively computes step-bounded state reachability probabilities. By contrast, recent advances in probabilistic inference suggest symbolically representing all horizon-length paths through the Markov chain. We ask whether this perspective advances the state-of-the-art in probabilistic model checking. First, we formally describe both approaches in order to highlight their key differences. Then, using these insights we develop Rubicon, a tool that transpiles Pris…
This paper introduces Farkas certificates for lower and upper bounds on minimal and maximal reachability probabilities in Markov decision processes (MDP), which we derive using an MDP-variant of Farkas' Lemma. The set of all such certificates is shown to form a polytope whose points correspond to witnessing subsystems of the model and the property. Using this correspondence we can translate the problem of finding minimal witnesses to the problem of finding vertices with a maximal number of zeros. While computing such vertices is computationally hard in general, we derive new heuristics from ou…
The weakest pre-expectation calculus has been proved to be a mature theory to analyze quantitative properties of probabilistic and nondeterministic programs. We present an automatic method for proving quantitative linear properties on any denumerable state space using iterative backwards fixed point calculation in the general framework of abstract interpretation. In order to accomplish this task we present the technique of random variable abstraction (RVA) and we also postulate a sufficient condition to achieve exact fixed point computation in the abstract domain. The feasibility of our approa…
We study monads resulting from the combination of nondeterministic and probabilistic behaviour with the possibility of termination, which is essential in program semantics. Our main contributions are presentation results for the monads, providing equational reasoning tools for establishing equivalences and distances of programs.
We consider the problem of computing reachability probabilities: given a Markov chain, an initial state of the Markov chain, and a set of goal states of the Markov chain, what is the probability of reaching any of the goal states from the initial state? This problem can be reduced to solving a linear equation Ax = b for x, where A is a matrix and b is a vector. We consider two iterative methods to solve the linear equation: the Jacobi method and the biconjugate gradient stabilized (BiCGStab) method. For both methods, a sequential and a parallel version have been implemented. The parallel versi…
Basic Parallel Processes (BPPs) are a well-known subclass of Petri Nets. They are the simplest common model of concurrent programs that allows unbounded spawning of processes. In the probabilistic version of BPPs, every process generates other processes according to a probability distribution. We study the decidability and complexity of fundamental qualitative problems over probabilistic BPPs -- in particular reachability with probability 1 of different classes of target sets (e.g. upward-closed sets). Our results concern both the Markov-chain model, where processes are scheduled randomly, and…
The majority of existing probabilistic model checking case studies are based on well understood theoretical models and distributions. However, real-life probabilistic systems usually involve distribution parameters whose values are obtained by empirical measurements and thus are subject to small perturbations. In this paper, we consider perturbation analysis of reachability in the parametric models of these systems (i.e., parametric Markov chains) equipped with the norm of absolute distance. Our main contribution is a method to compute the asymptotic bounds in the form of condition numbers for…
In this paper we introduce imprecise probability for session types. More exactly, we use a probabilistic process calculus in which both nondeterministic external choice and probabilistic internal choice are considered. We propose the probabilistic multiparty session types able to codify the structure of the communications by using some imprecise probabilities given in terms of lower and upper probabilities. We prove that this new probabilistic typing system is sound, as well as several other results dealing with both classical and probabilistic properties. The approach is illustrated by a simp…
We propose a simple technique for verifying probabilistic models whose transition probabilities are parametric. The key is to replace parametric transitions by nondeterministic choices of extremal values. Analysing the resulting parameter-free model using off-the-shelf means yields (refinable) lower and upper bounds on probabilities of regions in the parameter space. The technique outperforms the existing analysis of parametric Markov chains by several orders of magnitude regarding both run-time and scalability. Its beauty is its applicability to various probabilistic models. It in particular …
Proving programs terminating is a fundamental computer science challenge. Recent research has produced powerful tools that can check a wide range of programs for termination. The analog for probabilistic programs, namely termination with probability one ("almost-sure termination"), is an equally important property for randomized algorithms and probabilistic protocols. We suggest a novel algorithm for proving almost-sure termination of probabilistic programs. Our algorithm exploits the power of state-of-the-art model checkers and termination provers for nonprobabilistic programs: it calls such …
Probabilistic behavior is omnipresent in computer controlled systems, in particular, so-called safety-critical hybrid systems, because of various reasons, like uncertain environments, or fundamental properties of nature. In this paper, we extend existing hybrid process algebra $ACP_{hs}^{srt}$ with probability without replacing nondeterministic choice operator. In view of some shortcomings in existing approximate probabilistic bisimulation, we relax the constrains and propose a novel approximate probabilistic bisimulation relation. After that, we present a performance evaluation language, CTRM…
Probabilistic transition system specifications (PTSSs) in the $nt μfθ/ ntμxθ$ format provide structural operational semantics for Segala-type systems that exhibit both probabilistic and nondeterministic behavior and guarantee that bisimilarity is a congruence for all operator defined in such format. Starting from the $nt μfθ/ ntμxθ$ format, we obtain restricted formats that guarantee that three coarser bisimulation equivalences are congruences. We focus on (i) Segala's variant of bisimulation that considers combined transitions, which we call here "convex bisimulation"; (ii) the bisimulation e…
