Relative Timing Based Verification of Concurrent Systems

The thesis presents a new theory and methodology for the formal verification of safety properties in timed systems. The correct operation of such systems not only depends on a set of functional properties but also on certain assumptions about the delays of the components of the system and the response times of the environment in which the system operates. The verification of this type of systems typically involves several computationally hard problems. In particular, the combinatorial state explosion problem becomes exacerbated by the time dimension.

The theory that supports the proposed verification approach extends the conventional BDD-based symbolic methods to the verification of timed systems, modeled by means of timed transition systems. The theory is based on the relative timing paradigm, which instead of considering exact time differences in the occurrence of events, considers the effect of delays in terms of relative orderings between events. For example, in order to guarantee that a race is not propagated in a digital circuit, it is often sufficient to check that certain signal switches before another, instead of identifying the exact instants of time in which both signals switch. Moreover, the timing information does not need to computed for the overall system, but only locally for the part of the system involved in the proof or disproof of a given property. This is possible thanks to a crucial observation, that the set of executions of a transition system can be covered by a set of partial orders. As a consequence, only a subset of the events of the system is involved in the proof of a property and the timing analysis can be carried out very efficiently.

Conventional methods for the verification of timed systems rely on the computation of the exact timed state space of the system as the first step of the analysis. Although efficient techniques have been devised to overcome the complexity issue (e.g. difference bound matrices), symbolic methods cannot be easily applied. Thus, the combinatorial time-state explosion problem often limits the applicability of such methods to moderate-size systems.

Instead, the approach proposed in the thesis relies on an incremental refinement of the untimed state space of the system, so that timing information is incorporated as soon as it is needed. The timing information is derived by an efficient off-line timing analysis over small sets of events. The refined state space is captured under the model of lazy transition system, which allows an efficient representation of the timed domain using conventional symbolic methods. As a consequence, the approach can be potentially applied to bigger systems or to systems with more level of detail, than those that can be handled by similar methods for the verification of timed systems. Moreover, the incremental nature of the approach provides a good way to obtain at least partial results even on systems for which complete solutions could be too complex to compute.

A key feature of the proposed verification approach is that not only proves or disproves the correctness of a timed system. If the system is correct the set of relative timing relations used for the proof are provided. Such relations constitute a set of sufficient timing constraints that guarantee the correctness of the system. On the other hand, if the system is incorrect, a counterexample failure trace is provided. The most important aspect of all this feedback is that it can be used as valuable back-annotation information along the design process. This feature, which allows to bridge the gap between verification and design, constitutes another differential aspect of our verification approach when compared to other equivalent verification methods.

The verification approach has been fully implemented in an experimental CAV tool called TRANSYT. The tool can handle hierarchical and distributed modular systems which can inter-operate by a variety of communication mechanisms. TRANSYT has successfully proved its functionality as well as the validity of the overall verification approach, by verifying a number of timed asynchronous circuits with up to more than 10⁶ untimed states. The experiments cover, for example, the verification of: complex-gate decompositions in quasi-speed-independent asynchronous circuits, delay-reset domino circuits, pulse-based systems, circuits optimized for speed using timing assumptions, etc. Additionally, compositional verification methods have been combined with the basic verification approach in order to tackle the size/complexity issues involved in the verification of complex timed systems. Thus, abstractions, assume-guarantee reasoning and mathematical induction have been used to prove the correctness the IPCMOS architecture. It is a scalable pipelined architecture which is aimed to the interconnection of different clock zones in a system.

Thanks to the rather theoretical nature of the proposed verification approach, its potential applicability covers a wider range of systems than those cited above, such as: custom transistor-level circuits that exploit the technology limits for performance, complex digital structures where synchronization is a crucial issue (e.g. dynamic MOS), asynchronous and GALS-type systems, real-time systems, etc.