Tachis: Higher-Order Separation Logic with Credits for Expected Costs

TL;DR

Tachis introduces a higher-order separation logic with probabilistic cost credits to reason about the expected costs of probabilistic programs, enabling flexible and expressive analysis of various algorithms.

Contribution

It presents a novel probabilistic cost credit system within separation logic, allowing for reasoning about expected costs and amortized analysis in probabilistic programs.

Findings

Proved upper bounds on expected costs of randomized quicksort, hash tables, and meldable heaps.

Mechanized proofs using Coq, Iris, and Coquelicot demonstrate the approach's effectiveness.

Supports multiple cost models including runtime and entropy usage.

Abstract

We present Tachis, a higher-order separation logic to reason about the expected cost of probabilistic programs. Inspired by the uses of time credits for reasoning about the running time of deterministic programs, we introduce a novel notion of probabilistic cost credit. Probabilistic cost credits are a separation logic resource that can be used to pay for the cost of operations in programs, and that can be distributed across all possible branches of sampling instructions according to their weight, thus enabling us to reason about expected cost. The representation of cost credits as separation logic resources gives Tachis a great deal of flexibility and expressivity. In particular, it permits reasoning about amortized expected cost by storing excess credits as potential into data structures to pay for future operations. Tachis further supports a range of cost models, including running time and entropy usage. We showcase the versatility of this approach by applying our techniques to prove upper bounds on the expected cost of a variety of probabilistic algorithms and data structures, including randomized quicksort, hash tables, and meldable heaps. All of our results have been mechanized using Coq, Iris, and the Coquelicot real analysis library.