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Register a new userAn Application of Computable Distributions to the Semantics of Probabilistic Programs
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Added by
Daniel Huang
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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In this chapter, we explore how (Type-2) computable distributions can be used to give both (algorithmic) sampling and distributional semantics to probabilistic programs with continuous distributions. Towards this end, we sketch an encoding of computable distributions in a fragment of Haskell and show how topological domains can be used to model the resulting PCF-like language. We also examine the implications that a (Type-2) computable semantics has for implementing conditioning. We hope to draw out the connection between an approach based on (Type-2) computability and ordinary programming throughout the chapter as well as highlight the relation with constructive mathematics (via realizability).
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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 $Pi^0_2$-complete. 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 method of computing lower bounds of termination probability, and AST verification.
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