Two-sorted algebraic decompositions of Brookes's shared-state denotational semantics

2025-01-25

AI Summary

This paper addresses the consistency challenge arising from preemptive interleaving of operations in multi-user shared-resource scenarios. Methodologically, it introduces a *dual-sort algebraic effects* theory to reconstruct Brookes’s (1996) shared-state concurrent trace semantics purely algebraically. It models preemption hierarchically via “hold” (global lock acquisition) and “cede” (voluntary relinquishment) operators, explicitly encoding atomicity as lock acquire/release actions; combined with two-sorted equations, closure-pair operators, and strict semantics, it establishes an algebraic decomposition framework. Key contributions are: (1) a representation theorem characterizing free algebras as trace sets satisfying closure conditions; (2) exact recovery of Brookes’s original model when the hold layer is variable-free; and (3) hierarchical, compositional analysis of trace semantics—yielding the first algebraic, verifiable semantic foundation for concurrent shared-state systems.

Technology Category

• Knowledge Representation and Reasoning: Common-Sense Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We use a two sorted equational theory of algebraic effects to model concurrent shared state with preemptive interleaving, recovering Brookes's seminal 1996 trace-based model precisely. The decomposition allows us to analyse Brookes's model algebraically in terms of separate but interacting components. The multiple sorts partition terms into layers. We use two sorts: a"hold"sort for layers that disallow interleaving of environment memory accesses, analogous to holding a global lock on the memory; and a"cede"sort for the opposite. The algebraic signature comprises of independent interlocking components: two new operators that switch between these sorts, delimiting the atomic layers, thought of as acquiring and releasing the global lock; non-deterministic choice; and state-accessing operators. The axioms similarly divide cleanly: the delimiters behave as a closure pair; all operators are strict, and distribute over non-empty non-deterministic choice; and non-deterministic global state obeys Plotkin and Power's presentation of global state. Our representation theorem expresses the free algebras over a two-sorted family of variables as sets of traces with suitable closure conditions. When the held sort has no variables, we recover Brookes's trace semantics.

Problem

• Multi-user Resource Sharing
• State Operation Complexity
• Consistency and Correctness

Innovation

• Hold and Cede Operations
• Shared Resource Management
• Algebraic Decomposition Consistency