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We extend the simply-typed guarded $lambda$-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.
We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be
We introduce a method for proving almost sure termination in the context of lambda calculus with continuous random sampling and explicit recursion, based on ranking supermartingales. This result is extended in three ways. Antitone ranking functions h
We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be
We consider multi-agent systems where agents actions and beliefs are determined aleatorically, or by the throw of dice. This system consists of possible worlds that assign distributions to independent random variables, and agents who assign probabili
We introduce two extensions of the $lambda$-calculus with a probabilistic choice operator, $Lambda_oplus^{cbv}$ and $Lambda_oplus^{cbn}$, modeling respectively call-by-value and call-by-name probabilistic computation. We prove that both enjoys conflu