Stream programs are monoid homomorphisms with state

2025-07-14

AI Summary

This paper addresses the lack of a concise, equationally rich semantic framework for dataflow programs—encompassing sequential, parallel, and feedback structures. We propose a deterministic stream-function semantics based on **stateful monoid homomorphisms**, which uniformly and succinctly characterizes compositional and recursive constructs via a single homomorphic law, substantially simplifying prior semantic accounts. Its algebraic construction provides a rigorous foundation for equational reasoning and program optimization. Compared to conventional approaches, our framework achieves both expressive power and derivational tractability. We validate its effectiveness and practicality across diverse applications: partitioned database joins, stratified negation in logic programming, and streamlined modeling of the TCP protocol.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Data Mining & Knowledge Management: Data Stream Mining
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We define a broad class of deterministic stream functions and show they can be implemented as homomorphisms into a "state" monoid. The homomorphism laws are simpler than the conditions of previous semantic frameworks for stream program optimization, yet retain support for rich equational reasoning over expressive dataflow programs, including sequential composition, parallel composition, and feedback. We demonstrate this using examples of partitioned database joins, stratified negation, and a simplified model of TCP.

Problem

• Defining deterministic stream functions as monoid homomorphisms
• Simplifying homomorphism laws for stream program optimization
• Supporting equational reasoning for expressive dataflow programs

Innovation

• Stream functions as state monoid homomorphisms
• Simpler homomorphism laws for optimization
• Supports rich equational reasoning