When analyzing probabilistic computations, a powerful approach is to first
find a martingale---an expression on the program variables whose expectation
remains invariant---and then apply the optional stopping theorem in order to
infer properties at termination time. One of the main challenges, then, is to
systematically find martingales.
We propose a novel procedure to synthesize martingale expressions from an
arbitrary initial expression. Contrary to state-of-the-art approaches, we do
not rely on constraint solving. Instead, we use a symbolic construction based
on Doob's decomposition. This procedure can produce very complex martingales,
expressed in terms of conditional expectations.
We show how to automatically generate and simplify these martingales, as well
as how to apply the optional stopping theorem to infer properties at
termination time. This last step typically involves some simplification steps,
and is usually done manually in current approaches. We implement our techniques
in a prototype tool and demonstrate our process on several classical examples.
Some of them go beyond the capability of current semi-automatic approaches
