المفاهيم الأساسية
This paper presents the first sound and relatively complete proof rules for qualitative and quantitative termination of probabilistic programs with discrete probabilistic choice and demonic bounded nondeterminism.
الملخص
The paper considers an imperative programming model with variables ranging over rationals, a finite set of program locations, and a guarded transition relation between locations representing computational steps. The model includes primitives for probability distributions over available transitions, allowing for the expression of bounded nondeterministic and probabilistic choice.
The qualitative termination problem (Almost-Sure Termination, AST) asks if the program terminates almost surely, no matter how nondeterminism is resolved. The quantitative termination problem asks for bounds on the probability of termination.
The authors provide two different proof rules for AST:
The first rule uses a supermartingale function that is unbounded and non-increasing in expectation on "most" states, along with a variant function that certifies finite paths to the terminal state.
The second rule takes a more local view, requiring proofs of near termination (with some non-zero probability) from every reachable state. If these proofs together indicate a non-zero lower bound of termination across all states, a zero-one law implies almost-sure termination.
For quantitative termination, the authors build on the stochastic invariant technique of Chatterjee et al. [10], providing a sound and complete rule that requires stochastic invariants for each natural number n.
The key technical tool is the "unrolling lemma", which shows that if a program terminates with positive probability, then there is a finite upper bound on the length of the shortest terminal run. This lemma is used crucially in the proofs of completeness.
The authors show that their proof rules are sound and relatively complete with respect to the theory of arithmetic, and that many existing proof rules can be transformed into their system, indicating its applicability in practice.
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