Reachability in Parametric Interval Markov Chains using Constraints

Parametric Interval Markov Chains (pIMCs) are a specification formalism that extend Markov Chains (MCs) and Interval Markov Chains (IMCs) by taking into account imprecision in the transition probability values: transitions in pIMCs are labeled with parametric intervals of probabilities. In this work, we study the difference between pIMCs and other Markov Chain abstractions models and investigate the two usual semantics for IMCs: once-and-for-all and at-every-step. In particular, we prove that both semantics agree on the maximal/minimal reachability probabilities of a given IMC. We then investigate solutions to several parameter synthesis problems in the context of pIMCs -- consistency, qualitative reachability and quantitative reachability -- that rely on constraint encodings. Finally, we propose a prototype implementation of our constraint encodings with promising results

Qualitative Reachability for Open Interval Markov Chains

Interval Markov chains extend classical Markov chains with the possibility to describe transition probabilities using intervals, rather than exact values. While the standard formulation of interval Markov chains features closed intervals, previous work has considered also open interval Markov chains, in which the intervals can also be open or half-open. In this paper we focus on qualitative reachability problems for open interval Markov chains, which consider whether the optimal (maximum or minimum) probability with which a certain set of states can be reached is equal to 0 or 1. We present polynomial-time algorithms for these problems for both of the standard semantics of interval Markov chains. Our methods do not rely on the closure of open intervals, in contrast to previous approaches for open interval Markov chains, and can characterise situations in which probability 0 or 1 can be attained not exactly but arbitrarily closely.Comment: Full version of a paper published at RP 201

Abstract Interpretation for Probabilistic Termination of Biological Systems

In a previous paper the authors applied the Abstract Interpretation approach for approximating the probabilistic semantics of biological systems, modeled specifically using the Chemical Ground Form calculus. The methodology is based on the idea of representing a set of experiments, which differ only for the initial concentrations, by abstracting the multiplicity of reagents present in a solution, using intervals. In this paper, we refine the approach in order to address probabilistic termination properties. More in details, we introduce a refinement of the abstract LTS semantics and we abstract the probabilistic semantics using a variant of Interval Markov Chains. The abstract probabilistic model safely approximates a set of concrete experiments and reports conservative lower and upper bounds for probabilistic termination

Real-time and Probabilistic Temporal Logics: An Overview

Over the last two decades, there has been an extensive study on logical formalisms for specifying and verifying real-time systems. Temporal logics have been an important research subject within this direction. Although numerous logics have been introduced for the formal specification of real-time and complex systems, an up to date comprehensive analysis of these logics does not exist in the literature. In this paper we analyse real-time and probabilistic temporal logics which have been widely used in this field. We extrapolate the notions of decidability, axiomatizability, expressiveness, model checking, etc. for each logic analysed. We also provide a comparison of features of the temporal logics discussed

Probabilistic Interval Temporal Logic and Duration Calculus with Infinite Intervals: Complete Proof Systems

The paper presents probabilistic extensions of interval temporal logic (ITL) and duration calculus (DC) with infinite intervals and complete Hilbert-style proof systems for them. The completeness results are a strong completeness theorem for the system of probabilistic ITL with respect to an abstract semantics and a relative completeness theorem for the system of probabilistic DC with respect to real-time semantics. The proposed systems subsume probabilistic real-time DC as known from the literature. A correspondence between the proposed systems and a system of probabilistic interval temporal logic with finite intervals and expanding modalities is established too.Comment: 43 page