Development via Refinement in Probabilistic B — Foundation and Case Study

Rigorous Development of Complex Fault-Tolerant Systems

2006

Formal proofs of functional correctness and rigorous analyses of fault tolerance have, traditionally, been separate processes. In the former a programming logic (proof) or computational model (model checking) is used to establish that all the system's behaviours satisfy some (specification) criteria. In the latter, techniques derived from engineering are used to determine quantitative properties such as probability of failure (given failure of some component) or expected performance (an average measure of execution time, for example). To combine the formality and the rigour requires a quantitative approach within which functional correctness can be embedded. Programming logics for probability are capable in principle of doing so, and in this article we illustrate the use of the probabilistic guarded-command language (pGCL) and its logic for that purpose. We take self-stabilisation as an example of fault tolerance, and present program-logical techniques for determining, on the one hand, that termination occurs with probability one and, on the other, the the expected time to termination is bounded above by some value. An interesting technical novelty required for this is the recognition of both "angelic" and "demonic" refinement, reflecting our simultaneous interest in both upper-and lower bounds.

Section: Discussionmentioning

confidence: 99%

Programming-Logic Analysis of Fault Tolerance: Expected Performance of Self-stabilisation

Morgan

Rigorous Development of Complex Fault-Tolerant Systems

2006

“…Thus in this case only -an exactly fair coin-the refinement is accepted. 27 8.8 The non-existence of fair coins…”

Section: The Importance Of Fair Coinsmentioning

confidence: 99%

Security, Probability and Nearly Fair Coins in the Cryptographers’ Café

FM 2009: Formal Methods

Meinicke

Morgan

2009

Self Cite

Security and probability are both artefacts that we hope to bring increasingly within the reach of refinement-based Formal Methods; although we have worked on them separately, in the past, the goal has always been to bring them together. In this report we describe our ongoing work in that direction: we relate it to a well known problem in security, Chaum's Dining Cryptographers, where the various criteria of correctness that might apply to it expose precisely the issues we have found to be significant in our efforts to deal with security, probability and abstraction all at once. Taking our conviction into this unfamiliar and demanding territory, that abstraction and refinement are the key tools of software development, has turned out to be an exciting challenge.

“…Whilst the algebraic proof below ensures that the specification (7) is met, when presented with a concrete system (such as Fig.3) the hypotheses must be verified directly using the probabilistic semantics Fig.1, to complete the link between the concrete protocol and the abstract algebraic reasoning. The advantage of using this kind of verification rather than building a concrete model of the distributed protocol (in general involving many more than just 3 processes) is that all quantitative reasoning at the concrete level becomes entirely localised, and in some cases can even be automated [8].…”

Section: Fig 4 a Voting Run With Adversarial Choicementioning

confidence: 99%

“…More generally, approaches for carrying out such small intricate proofs can be done using quantitative program logic described elsewhere [17]. Whilst that logic is not optimised for checking full refinements, in some special cases there do exist practical verification rules for probabilistic refinement checking [8]. Now we have algebraic properties at Fig.3, we are able to prove entirely within the algebra pKA the refinement given by (7) and we note that crucially the proof does not rely on detailed arithemetic calculation at al.…”

Section: Fig 4 a Voting Run With Adversarial Choicementioning

confidence: 99%

Using probabilistic Kleene algebra pKA for protocol verification

The Journal of Logic and Algebraic Programming

Gonzalía

Cohen

et al. 2008

We propose a method for verification of probabilistic distributed systems in which a variation of Kozen's Kleene Algebra with Tests [11] is used to take account of the well-known interaction of probability and "adversarial" scheduling [17]. We describe pKA, a probabilistic Kleene-style algebra, based on a widely accepted model of probabilistic/demonic computation [7,25,17]. Our technical aim is to express probabilistic versions of Cohen's separation theorems[4]. Separation theorems simplify reasoning about distributed systems, where with purely algebraic reasoning they can reduce complicated interleaving behaviour to "separated" behaviours each of which can be analysed on its own. Until now that has not been possible for probabilistic distributed systems. We present two case studies. The first treats a simple voting mechanism in the algebraic style, and the second-based on Rabin's Mutual exclusion with bounded waiting [12]-is one where verification problems have already occurred: the original presentation was later shown to have subtle flaws [24]. It motivates our interest in algebras, where assumptions relating probability and secrecy are clearly exposed and, in some cases, can be given simple characterisations in spite of their intricacy. Finally we show how the algebraic proofs for these theorems can be automated using a modification of Kozen and Aboul-Hosn's KAT-ML [3].