Step-Indexed Logical Relations for Probability

Bizjak, Aleš; Birkedal, Lars

doi:10.1007/978-3-662-46678-0_18

Cited by 38 publications

(37 citation statements)

References 22 publications

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“…One line of work has focused on operationally-based techniques for reasoning about contextual equivalence of programs. The methods are based on probabilistic bisimulations [24,25] or on logical relations [26]. Most of these approaches have been developed for languages with discrete distributions, but recently there has also been work on languages with continuous distributions [27,28].…”

Section: Related Workmentioning

confidence: 99%

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

Aguirre

Barthe

Birkedal

et al. 2018

Lecture Notes in Computer Science

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We extend the simply-typed guarded λ-calculus with discrete probabilities and endow it with a program logic for reasoning about relational properties of guarded probabilistic computations. This provides a framework for programming and reasoning about infinite stochastic processes like Markov chains. We demonstrate the logic sound by interpreting its judgements in the topos of trees and by using probabilistic couplings for the semantics of relational assertions over distributions on discrete types. The program logic is designed to support syntax-directed proofs in the style of relational refinement types, but retains the expressiveness of higher-order logic extended with discrete distributions, and the ability to reason relationally about expressions that have different types or syntactic structure. In addition, our proof system leverages a well-known theorem from the coupling literature to justify better proof rules for relational reasoning about probabilistic expressions. We illustrate these benefits with a broad range of examples that were beyond the scope of previous systems, including shift couplings and lump couplings between random walks.The goal of this paper is to develop a programming and reasoning framework for probabilistic computations over infinite objects, such as Markov chains. Although programming and reasoning frameworks for infinite objects and probabilistic computations are well-understood in isolation, their combination is challenging. In particular, one must develop a proof system that is powerful enough for proving interesting properties of probabilistic computations over infinite objects, and practical enough to support effective verification of these properties.Modelling probabilistic infinite objects A first challenge is to model probabilistic infinite objects. We focus on the case of Markov chains, due to its importance. A (discrete-time) Markov chain is a sequence of random variables {X i } over some fixed type T satisfying some independence property. Thus, the straightforward way of modelling a Markov chain is as a stream of distributions over T . Going back to the simple example outlined above, it is natural to think about this kind of discrete-time Markov chain as characterized by the sequence of positions {p i } i∈N , which in turn can be described as an infinite set indexed by the natural numbers. This suggests that a natural way to model such a Markov chain is to use streams in which each element is produced probabilistically from the previous one. However, there are some downsides to this representation. First of all, it requires explicit reasoning about probabilistic dependency, since X i+1 depends on X i . Also, we might be interested in global properties of the executions of the Markov chain, such as "The probability of passing through the initial state infinitely many times is 1". These properties are naturally expressed as properties of the whole stream. For these reasons, we want to represent Markov chains as distributions over streams. Seemingly, one downside of this repre...

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Section: Related Workmentioning

confidence: 99%

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

Aguirre

Barthe

Birkedal

et al. 2018

Lecture Notes in Computer Science

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“…There has been previous work on contextual equivalence for probabilistic languages with only discrete random variables. In particular, Bizjak and Birkedal [2015] define a step-indexed, biorthogonal logical relation whose structure is similar to ours, except that they sum where we integrate, and they use the probability of termination as the basic observation whereas we compare measures. Others have applied bisimulation techniques [Crubillé and Lago 2014;Sangiorgi and Vignudelli 2016] to languages with discrete choice; Ehrhard et al [2014] have constructed fully abstract models for PCF with discrete probabilistic choice using probabilistic coherence spaces.…”

Section: From Let-style To Direct-stylementioning

confidence: 99%

Contextual equivalence for a probabilistic language with continuous random variables and recursion

Wand

Culpepper

Giannakopoulos

et al. 2018

Proc. ACM Program. Lang.

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We present a complete reasoning principle for contextual equivalence in an untyped probabilistic language. The language includes continuous (real-valued) random variables, conditionals, and scoring. It also includes recursion, since the standard call-by-value fixpoint combinator is expressible.We demonstrate the usability of our characterization by proving several equivalence schemas, including familiar facts from lambda calculus as well as results specific to probabilistic programming. In particular, we use it to prove that reordering the random draws in a probabilistic program preserves contextual equivalence. This allows us to show, for example, that (let x = e 1 in let = e 2 in e 0 ) = ctx (let = e 2 in let x = e 1 in e 0 ) (provided x does not occur free in e 2 and does not occur free in e 1 ) despite the fact that e 1 and e 2 may have sampling and scoring effects.

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“…We see no strong obstacles in applying any of these to a typed version of our calculus, but it is beyond the scope of this work. Another topic for future work are methodologies for equivalence checking in the style of logical relations or bisimilarity, which have been recently shown to work well in discrete probabilistic calculi [3].…”

Section: Related Workmentioning

confidence: 99%

A lambda-calculus foundation for universal probabilistic programming

et al. 2016

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We develop the operational semantics of an untyped probabilistic λ-calculus with continuous distributions, and both hard and soft constraints, as a foundation for universal probabilistic programming languages such as CHURCH, ANGLICAN, and VEN-TURE. Our first contribution is to adapt the classic operational semantics of λ-calculus to a continuous setting via creating a measure space on terms and defining step-indexed approximations. We prove equivalence of big-step and small-step formulations of this distribution-based semantics. To move closer to inference techniques, we also define the sampling-based semantics of a term as a function from a trace of random samples to a value. We show that the distribution induced by integration over the space of traces equals the distribution-based semantics. Our second contribution is to formalize the implementation technique of trace Markov chain Monte Carlo (MCMC) for our calculus and to show its correctness. A key step is defining sufficient conditions for the distribution induced by trace MCMC to converge to the distribution-based semantics. To the best of our knowledge, this is the first rigorous correctness proof for trace MCMC for a higher-order functional language, or for a language with soft constraints.

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Step-Indexed Logical Relations for Probability

Cited by 38 publications

References 22 publications

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

Contextual equivalence for a probabilistic language with continuous random variables and recursion

A lambda-calculus foundation for universal probabilistic programming

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