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A Uniqueness Theorem for Distributed Computation under Physical Constraint

Zhiyuan Ren, Mingxuan Lu, Wenchi Cheng

TL;DR

This paper establishes a rigorous axiomatic system that formalizes these physical constraints and proves that for the broad class of computations admitting an idempotent merge operator, there exists a unique, optimal paradigm.

Abstract

Foundational models of computation often abstract away physical hardware limitations. However, in extreme environments like In-Network Computing (INC), these limitations become inviolable laws, creating an acute trilemma among communication efficiency, bounded memory, and robust scalability. Prevailing distributed paradigms, while powerful in their intended domains, were not designed for this stringent regime and thus face fundamental challenges. This paper demonstrates that resolving this trilemma requires a shift in perspective - from seeking engineering trade-offs to deriving solutions from logical necessity. We establish a rigorous axiomatic system that formalizes these physical constraints and prove that for the broad class of computations admitting an idempotent merge operator, there exists a unique, optimal paradigm. Any system satisfying these axioms must converge to a single normal form: Self-Describing Parallel Flows (SDPF), a purely data-centric model where stateless executors process flows that carry their own control logic. We further prove this unique paradigm is convergent, Turing-complete, and minimal. In the same way that the CAP theorem established a boundary for what is impossible in distributed state management, our work provides a constructive dual: a uniqueness theorem that reveals what is \textit{inevitable} for distributed computation flows under physical law.

A Uniqueness Theorem for Distributed Computation under Physical Constraint

TL;DR

This paper establishes a rigorous axiomatic system that formalizes these physical constraints and proves that for the broad class of computations admitting an idempotent merge operator, there exists a unique, optimal paradigm.

Abstract

Foundational models of computation often abstract away physical hardware limitations. However, in extreme environments like In-Network Computing (INC), these limitations become inviolable laws, creating an acute trilemma among communication efficiency, bounded memory, and robust scalability. Prevailing distributed paradigms, while powerful in their intended domains, were not designed for this stringent regime and thus face fundamental challenges. This paper demonstrates that resolving this trilemma requires a shift in perspective - from seeking engineering trade-offs to deriving solutions from logical necessity. We establish a rigorous axiomatic system that formalizes these physical constraints and prove that for the broad class of computations admitting an idempotent merge operator, there exists a unique, optimal paradigm. Any system satisfying these axioms must converge to a single normal form: Self-Describing Parallel Flows (SDPF), a purely data-centric model where stateless executors process flows that carry their own control logic. We further prove this unique paradigm is convergent, Turing-complete, and minimal. In the same way that the CAP theorem established a boundary for what is impossible in distributed state management, our work provides a constructive dual: a uniqueness theorem that reveals what is \textit{inevitable} for distributed computation flows under physical law.

Paper Structure

This paper contains 40 sections, 14 theorems, 3 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical Context and Related Models of Computation
  3. In-Network Computing Models and Systems
  4. Communication and I/O Lower Bounds
  5. Logical Convergence: CALM and Semilattices
  6. Operational Models and their Physical Infeasibility
  7. A Constructive Duality to Impossibility Theorems
  8. Summary of Theoretical Positioning
  9. Formal Model and Axioms
  10. Execution Model and Cost Metrics
  11. Axiomatic System
  12. The SDPF Paradigm
  13. Single-Read Reuse (S1)
  14. Stateless Micro-tasks (S2)
  15. Idempotent Merge (S3)
  16. ...and 25 more sections

Key Result

Proposition 1

Let the set of tiles be $\{\text{tile}_k\}$ with consumption counts $U_k$. If an algorithm allows some consumers to independently fetch the same tile from the source (without in-network reuse), then In particular, when some $U_k=\Omega(n)$, then $R_A=\Omega(n)$.

Figures (3)

  • Figure 1: A conceptual comparison between traditional process-centric models and the data-centric SDPF paradigm. (a) Process-Centric Model; (b) The SDPF Data-Centric Model. In SDPF, intelligence and control logic are embedded in the data flow, while executors are stateless and anonymous.
  • Figure 2: The constructive reduction proof of Theorem 5.5. Any paradigm claiming optimality under the axioms is progressively transformed by eliminating inefficiencies (T1-T4), inevitably converging to the unique SDPF normal form.
  • Figure 3: Simulating an SK-combinator reduction ($Sxyz \to xz(yz)$) using SDPF primitives. The transformation is guaranteed to be atomic and idempotent by the Combine operation and its unique rid.

Theorems & Definitions (37)

  • Definition 1: SDPF Normal Form
  • Proposition 1: Necessity of Single-Read Reuse
  • proof : Proof Sketch
  • Proposition 2: Necessity of Statelessness and Idempotency
  • proof : Proof Sketch
  • Proposition 3: Necessity of Barrier-less Asynchronous Scheduling
  • proof : Proof Sketch
  • Proposition 4: Necessity of Sliding Window
  • proof : Proof Sketch
  • Theorem 1: Uniqueness up to metric-equivalence
  • ...and 27 more