[Paper Review] What are the fundamental structures of concurrency? We still don't know!

Samson Abramsky|arXiv (Cornell University)|Jan 20, 2014

Computability, Logic, AI Algorithms4 references5 citations

TL;DR

This paper argues that despite decades of progress in process calculi and concurrency theory, no fundamental, syntax-independent theory of concurrency has emerged. Drawing on Petri nets, physics, and geometry—especially through categorical and diagrammatic approaches—it contends that the field lacks a unified foundation akin to those in physics, and calls for deeper integration with quantum information and geometric structures to achieve a true theory of information dynamics.

ABSTRACT

Process algebra has been successful in many ways; but we don't yet see the lineaments of a fundamental theory. Some fleeting glimpses are sought from Petri Nets, physics and geometry.

Motivation & Objective

• To question why, despite the success of process calculi, no fundamental theory of concurrency has emerged.
• To argue that the profusion of process calculi reflects a lack of syntax-independent, intrinsic definitions of core concepts like causality and concurrency.
• To advocate for a deeper integration of concurrency theory with physics and geometry, inspired by discrete models and categorical frameworks.
• To explore whether insights from quantum information and geometric structures could provide the missing bedrock for a unified theory of concurrency.
• To position Petri nets and related formalisms as the most promising current candidates for foundational concurrency structures, despite their limitations.

Proposed method

• Analyzing the historical development of process calculi and their limitations in achieving a canonical, fundamental theory.
• Drawing analogies between process calculi and established scientific tool-kits (e.g., differential equations, Fourier transforms), emphasizing their practical utility over foundational completeness.
• Examining Petri nets and event structures as syntax-independent frameworks that offer intrinsic definitions of concurrency, causality, and process.
• Exploring the influence of physics—particularly relativity and quantum mechanics—on the conceptual foundations of concurrency, including causal sets and non-locality.
• Investigating categorical and diagrammatic formalisms (e.g., in quantum mechanics and geometry of interaction) as potential unifying frameworks.
• Highlighting connections between topological invariants (e.g., Jones polynomial) and concurrency structures, suggesting deep geometric underpinnings.

Experimental results

Research questions

• RQ1 Why has no single, fundamental theory of concurrency emerged despite the success of process calculi?
• RQ2 What are the intrinsic, syntax-independent definitions of core concurrency concepts such as causality, concurrency, and process?
• RQ3 How can insights from physics—especially discrete spacetime models and quantum entanglement—inform the foundations of concurrency theory?
• RQ4 Can geometric and categorical structures provide a unifying framework for concurrency, akin to those in modern theoretical physics?
• RQ5 What role might quantum information and computational biology play in shaping a deeper, more fundamental theory of information dynamics?

Key findings

• Process calculi, while practically powerful, lack a syntax-independent, foundational theory, leading to a proliferation of similar but distinct formalisms.
• Petri nets and event structures offer the most promising extant frameworks for intrinsic definitions of concurrency, though they remain incomplete.
• The influence of physics—particularly relativity and quantum mechanics—on Petri's original work suggests that concurrency theory may be inherently tied to physical principles.
• Categorical and diagrammatic approaches in quantum mechanics (e.g., in the work with Coecke) show strong formal parallels with the geometry of interaction and concurrency, suggesting deep structural connections.
• The absence of a 'Church's thesis for concurrency' or a notion of expressive completeness indicates a lack of consensus on what constitutes a complete or universal model.
• Geometry, especially through topological invariants and causal sets, may provide the missing mathematical language for a fundamental theory of concurrency.