Events in Property Patterns

A pattern-based approach to the presentation, codification and reuse of property specifications for finite-state verification was proposed by Dwyer and his collegues. The patterns enable non-experts to read and write formal specifications for realistic systems and facilitate easy conversion of specifications between formalisms, such as LTL, CTL, QRE. In this paper, we extend the pattern system with events - changes of values of variables in the context of LTL.Comment: 14 pages, 3 figure

Probably Safe or Live

This paper presents a formal characterisation of safety and liveness properties \`a la Alpern and Schneider for fully probabilistic systems. As for the classical setting, it is established that any (probabilistic tree) property is equivalent to a conjunction of a safety and liveness property. A simple algorithm is provided to obtain such property decomposition for flat probabilistic CTL (PCTL). A safe fragment of PCTL is identified that provides a sound and complete characterisation of safety properties. For liveness properties, we provide two PCTL fragments, a sound and a complete one. We show that safety properties only have finite counterexamples, whereas liveness properties have none. We compare our characterisation for qualitative properties with the one for branching time properties by Manolios and Trefler, and present sound and complete PCTL fragments for characterising the notions of strong safety and absolute liveness coined by Sistla

Generalized Strong Preservation by Abstract Interpretation

Standard abstract model checking relies on abstract Kripke structures which approximate concrete models by gluing together indistinguishable states, namely by a partition of the concrete state space. Strong preservation for a specification language L encodes the equivalence of concrete and abstract model checking of formulas in L. We show how abstract interpretation can be used to design abstract models that are more general than abstract Kripke structures. Accordingly, strong preservation is generalized to abstract interpretation-based models and precisely related to the concept of completeness in abstract interpretation. The problem of minimally refining an abstract model in order to make it strongly preserving for some language L can be formulated as a minimal domain refinement in abstract interpretation in order to get completeness w.r.t. the logical/temporal operators of L. It turns out that this refined strongly preserving abstract model always exists and can be characterized as a greatest fixed point. As a consequence, some well-known behavioural equivalences, like bisimulation, simulation and stuttering, and their corresponding partition refinement algorithms can be elegantly characterized in abstract interpretation as completeness properties and refinements

Generalizing the Paige-Tarjan Algorithm by Abstract Interpretation

The Paige and Tarjan algorithm (PT) for computing the coarsest refinement of a state partition which is a bisimulation on some Kripke structure is well known. It is also well known in model checking that bisimulation is equivalent to strong preservation of CTL, or, equivalently, of Hennessy-Milner logic. Drawing on these observations, we analyze the basic steps of the PT algorithm from an abstract interpretation perspective, which allows us to reason on strong preservation in the context of generic inductively defined (temporal) languages and of possibly non-partitioning abstract models specified by abstract interpretation. This leads us to design a generalized Paige-Tarjan algorithm, called GPT, for computing the minimal refinement of an abstract interpretation-based model that strongly preserves some given language. It turns out that PT is a straight instance of GPT on the domain of state partitions for the case of strong preservation of Hennessy-Milner logic. We provide a number of examples showing that GPT is of general use. We first show how a well-known efficient algorithm for computing stuttering equivalence can be viewed as a simple instance of GPT. We then instantiate GPT in order to design a new efficient algorithm for computing simulation equivalence that is competitive with the best available algorithms. Finally, we show how GPT allows to compute new strongly preserving abstract models by providing an efficient algorithm that computes the coarsest refinement of a given partition that strongly preserves the language generated by the reachability operator.Comment: Keywords: Abstract interpretation, abstract model checking, strong preservation, Paige-Tarjan algorithm, refinement algorith

On Relaxing Metric Information in Linear Temporal Logic

Metric LTL formulas rely on the next operator to encode time distances, whereas qualitative LTL formulas use only the until operator. This paper shows how to transform any metric LTL formula M into a qualitative formula Q, such that Q is satisfiable if and only if M is satisfiable over words with variability bounded with respect to the largest distances used in M (i.e., occurrences of next), but the size of Q is independent of such distances. Besides the theoretical interest, this result can help simplify the verification of systems with time-granularity heterogeneity, where large distances are required to express the coarse-grain dynamics in terms of fine-grain time units.Comment: Minor change

Branching Bisimilarity with Explicit Divergence

We consider the relational characterisation of branching bisimilarity with explicit divergence. We prove that it is an equivalence and that it coincides with the original definition of branching bisimilarity with explicit divergence in terms of coloured traces. We also establish a correspondence with several variants of an action-based modal logic with until- and divergence modalities

Regular cell complexes in total positivity

This paper proves a conjecture of Fomin and Shapiro that their combinatorial model for any Bruhat interval is a regular CW complex which is homeomorphic to a ball. The model consists of a stratified space which may be regarded as the link of an open cell intersected with a larger closed cell, all within the totally nonnegative part of the unipotent radical of an algebraic group. A parametrization due to Lusztig turns out to have all the requisite features to provide the attaching maps. A key ingredient is a new, readily verifiable criterion for which finite CW complexes are regular involving an interplay of topology with combinatorics.Comment: accepted to Inventiones Mathematicae; 60 pages; substantially revised from earlier version