Efficient Timed Reachability Analysis using Clock Difference Diagrams

Abstract

One of the major problems in applying automatic verication tools to industrial-size systems is the excessive amount of memory required during the state-space exploration of a<br />model. In the setting of real-time, this problem of state-explosion requires extra attention as information must be kept not only on the discrete control structure but also on the values of continuous clock variables. In this paper, we present Clock Dierence Diagrams, CDD's, a BDD-like data-structure for<br />representing and eectively manipulating certain non-convex subsets of the Euclidean space, notably those encountered during verication of timed automata. A version of the real-time verication tool Uppaal using CDD's as a compact datastructure<br />for storing explored symbolic states has been implemented. Our experimental results demonstrate signicant space-savings: for 8 industrial examples, the savings are between 46%<br />and 99% with moderate increase in runtime. We further report on how the symbolic state-space exploration itself may be carried out using CDD's.

Symbolic Verification of Complex Real-time Systems with Clock-restriction Diagram

“…Nevertheless, in average, CRD with zones in reduced form (reduced CRD in short) is much more space-efficient than previous data-structures [1,7,15,16]. For example, given X = {x 1 , x 2 , x 3 } and C A = 10, in figure 1(b),(c),(d), we have the representations of zone {0 -x 2 < -3, x 1 -x 3 ≤ 1, x 2 -x1 < -4} in reduced CRD ( figure 1 (b)), in CRD with zones in closure form ( figure 1(c)), and in CDD with zones in closure form (called tightened form in [7]) without FALSE terminal vertex ( figure 1(d)). It is easy to see that as the number of clocks increases, reduced CRD will perform better and better.…”

“…To reduce information duplication in state-space representation, one solution is to convert all zones to their closure form [10,7] (called shortest-path closure in [14]), which is the set of all pairwise clock difference constraints derived from the all-pair shortest-path distances, and only store their closure form. Such conversion is expensive and, as we shall illustrate in figure 1(b)-(d), incurs large space consumption with data-structures like CDD and CRD.…”

“…Second, the default value of a variable in CRD is interpreted as ( <, ∞ ) while it is interpreted as (-∞, ∞ ) in CDD. To reduce information duplication in representations of state-spaces, CDD has to be transformed to closure form [10,7] which records the shortest-path distances between all pairs of clocks and is very space-inefficient.…”

“…It has been the general opinion that DBM (difference-bounded matrix) [10] is perhaps the best data-structure for the representation and manipulation of zones. For a long time, BDD-like structures [1,7,15,16] have not performed as well as the popular DBM (difference-bounded matrix) [10] which is a 2-dimensional matrix and nothing BDD-like. But recently, there comes a new BDD-like data-structure, called CRD (Clock-Restriction Diagram) [17], which has shown efficiency in experiments (see appendix 6).…”

“…Most modern model-checkers for real-time systems are built around some symbolic manipulation procedures [12] of zones implemented in data-structures like DBM [10], NDD [1], CDD [7], RED [15], or CRD [16]. A zone means a behaviorally equivalent state subspace of a timed automaton and is symbolically represented by a set of difference constraints between clock pairs.…”