Behavioural Preservation in Fault Tolerant Patterns

Abstract

In the development of critical systems it is common practise to make use of redundancy in order to achieve higher levels of reliability. There are well established design patterns that introduce redundancy and that are widely documented in the literature and adopted by the industry. However there have been few attempts to formally verify them with respect to behavioural preservation. In this work, we purpose an approach to specify such design patterns, called here fault tolerant patterns, using HOL. We use the theorem prover HOL4 to prove the compositionality and correctness of the fault tolerant patterns. We illustrate our approach by modelling three classical fault tolerant patterns: homogeneous redundancy, heterogeneous redundancy and triple modular redundancy. Our model takes into account that the original system (without redundancy) computes a certain function with some delay and is amenable to random failures. In order to prove that a fault tolerant pattern preserves the behaviour of its subsystems, we defined new notions of refinement. Systems engineers commonly accept the fact that fault tolerant patterns do not change the functionality of a system. However, this practise is not compatible with the classical refinement notions. Thus we defined axiomatic notions of refinement to prove that the formalised fault tolerant patterns preserve the behaviour of its subsystems. We also proved that our fault tolerant patterns are compositional in the sense that we can apply fault tolerant patterns consecutively and for an arbitrary number of times. The result of that is still a system whose delay, failure model and functionality can be systematically discovered (by proof) with almost no effort. In order to illustrate the usage of the patterns we applied the triple modular redundancy pattern to a simplified avionic Elevator Control System. We showed that once a fault tolerant pattern is verified, the application of it to a specific system and the proof of the behavioural preservation of the resulting system becomes trivial. This work has been done in collaboration with the Brazilian aircraft manufacturer Embraer.