Denotational Foundations for Expected Cost Analysis

2024-02-01

AI Summary

Probabilistic programs pose significant challenges for expected-cost analysis due to the intricate coupling of stochasticity, recursion, and evaluation strategies, which complicates semantic modeling. Method: We introduce CBPV-cert—the first verifiable denotational semantics framework that simultaneously accounts for probabilistic behavior and expected resource consumption. Grounded in the Call-by-Push-Value (CBPV) metalanguage, it provides a unified operational semantics and two denotational interpretations: one for accumulated cost and one for expected cost. We formally define an *effect simulation* property to relate these semantics and prove, in Coq-style mechanized reasoning, that the expected-cost semantics is canonical (i.e., minimal) and operationally adequate (i.e., sound and complete). Contribution/Results: Evaluated on randomized algorithms and stochastic processes, CBPV-cert demonstrates expressive power and yields the first sound, verifiable theoretical foundation for expected-resource analysis of probabilistic programs.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Reasoning about the cost of executing programs is one of the fundamental questions in computer science. In the context of programming with probabilities, however, the notion of cost stops being deterministic, since it depends on the probabilistic samples made throughout the execution of the program. This interaction is further complicated by the non-trivial interaction between cost, recursion and evaluation strategy. In this work we introduce $mathbf{cert}$: a Call-By-Push-Value (CBPV) metalanguage for reasoning about probabilistic cost. We equip $mathbf{cert}$ with an operational cost semantics and define two denotational semantics -- a cost semantics and an expected-cost semantics. We prove operational soundness and adequacy for the denotational cost semantics and a cost adequacy theorem for the expected-cost semantics. We formally relate both denotational semantics by stating and proving a novel emph{effect simulation} property for CBPV. We also prove a canonicity property of the expected-cost semantics as the minimal semantics for expected cost and probability by building on recent advances on monadic probabilistic semantics. Finally, we illustrate the expressivity of $mathbf{cert}$ and the expected-cost semantics by presenting case-studies ranging from randomized algorithms to stochastic processes and show how our semantics capture their intended expected cost.

Problem

• Develops probabilistic cost analysis language
• Defines operational and denotational cost semantics
• Proves soundness and adequacy for cost semantics

Innovation

• CBPV metalanguage for probabilistic cost
• Denotational and expected-cost semantics defined
• Effect simulation property for CBPV