连续分布函数程序的概率终止

Raven Beutner, Luke Ong
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引用次数: 6

摘要

研究了具有递归、随机条件和连续分布抽样的高阶概率泛函规划的终止问题。关于具有连续分布的程序的终止概率的推理是困难的,因为终止执行的枚举不能提供任何非平凡的界限。我们提出了一种新的基于区间轨迹的运算语义,它相对于标准的基于采样的语义是健全和完备的,其中(可数)枚举可以提供任意紧的下界。因此,我们首次证明了具有连续分布的程序的决定几乎确定终止(AST)为Π20-complete(对于CbN)。我们还根据交集类型系统提供了语义的组合表示。在第二部分中,我们提出了一种证明非仿射程序的AST的方法,即递归程序,在递归体的求值过程中,可以从不同的调用点进行多次递归调用(对一阶函数)。与确定性语言不同,递归调用站点的数量对终止概率有直接影响。我们的框架支持一个证明系统,该系统可以对远远超出现有方法范围的程序进行AST验证。我们已经构造了我们的方法的原型实现,用于计算终止概率的下界,以及AST验证。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On probabilistic termination of functional programs with continuous distributions
We study termination of higher-order probabilistic functional programs with recursion, stochastic conditioning and sampling from continuous distributions. Reasoning about the termination probability of programs with continuous distributions is hard, because the enumeration of terminating executions cannot provide any non-trivial bounds. We present a new operational semantics based on traces of intervals, which is sound and complete with respect to the standard sampling-based semantics, in which (countable) enumeration can provide arbitrarily tight lower bounds. Consequently we obtain the first proof that deciding almost-sure termination (AST) for programs with continuous distributions is Π20-complete (for CbN). We also provide a compositional representation of our semantics in terms of an intersection type system. In the second part, we present a method of proving AST for non-affine programs, i.e., recursive programs that can, during the evaluation of the recursive body, make multiple recursive calls (of a first-order function) from distinct call sites. Unlike in a deterministic language, the number of recursion call sites has direct consequences on the termination probability. Our framework supports a proof system that can verify AST for programs that are well beyond the scope of existing methods. We have constructed prototype implementations of our methods for computing lower bounds on the termination probability, and AST verification.
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