Stochastic invariants for probabilistic termination

Abstract

Termination is one of the basic liveness properties, and we study the termination problem for probabilistic programs with real-valued variables. Previous works focused on the qualitative problem that asks whether an input program terminates with probability 1 (almost-sure termination). A powerful approach for this qualitative problem is the notion of ranking supermartingales with respect to a given set of invariants. The quantitative problem (probabilistic termination) asks for bounds on the termination probability, and this problem has not been addressed yet. A fundamental and conceptual drawback of the existing approaches to address probabilistic termination is that even though the supermartingales consider the probabilistic behaviour of the programs, the invariants are obtained completely ignoring the probabilistic aspect (i.e., the invariants are obtained considering all behaviours with no information about the probability).

Computing Expected Runtimes for Constant Probability Programs

“…Approaches based on supermartingales (e.g., [1,5,11,13,14,18]) use mappings similar to rdw P in order to infer a realvalued term which over-approximates the expected runtime. However, in the following (non-trivial) theorem we show that our transformation is not only an over-or under-approximation, but the termination behavior and the expected runtime of P and P rdw are identical.…”

“…Moreover, one could prove that µ P < 0 implies PAST by showing that x is a ranking supermartingale of the program [5,11,14,18]. That the program is not PAST if µ P ≥ 0 and not AST if µ P > 0 could be proved by showing that −x is a µ P -repulsing supermartingale [13].…”

“…Related Work. Many techniques to analyze (P)AST have been developed, which mostly rely on ranking supermartingales, e.g., [1,5,11,13,14,18,20,28,30]. Indeed, several of these works (e.g., [1,5,18,20]) present complete criteria for (P)AST, although (P)AST is undecidable.…”

“…For P race from Ex. 1 and a start configuration where the tortoise is 10 steps ahead of the hare (e.g., x 0 = (11, 1)), the prefix run (11, 1), (12, 1), (13,6) has the probability pr Prace (11,1) ( (11, 1), (12, 1), (13, 6) ) = δ (11,1),(11,1) · p (12,1)−(11,1) (11, 1) · p (13,6)−(12,1) (12, 1) = p (1,0) (11, 1) · p (1,5) (12, 1) = 6 11 · 1 22 = 3 121 . So we take into account whether the prefix run starts with x 0 = (11, 1) and multiply the probability to get from x = (11, 1) to x = (12, 1) with the probability to get from x = (12, 1) to x = (13, 6).…”

“…Example 45 (Probability Measure for P race ). Cyl Z 2 ( (11, 1), (12, 1), (13, 6) ) consists of all runs that start with (11, 1), (12, 1), (13,6). If the initial value is x 0 = (11, 1), then the probability that a run is in Cyl Z 2 ( (11, 1), (12, 1), (13,6) ) is P Prace (11,1) (Cyl Z 2 ( (11, 1), (12, 1), (13, 6) )) = pr Prace (11,1) ( (11, 1), (12, 1), (13, 6) ) = 3 121 .…”