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Term Coding and Dispersion: A Perfect-vs-Rate Complexity Dichotomy for Information Flow

Søren Riis

TL;DR

This work introduces Term Coding and a dispersion subclass to study how optimising local function interpretations affects global configurations. It establishes a polynomial-time computable dispersion exponent $D(\mathbf t)$ via a max-flow construction, while showing that exact perfect-dispersion thresholds are undecidable for $r\ge3$. A normalization-diversification pipeline reduces problems to flow-based representations, enabling a guessing-game bridge that yields a tight sandwich bound between dispersion and guessing numbers. The results delineate a sharp complexity dichotomy: asymptotic rate questions are tractable, whereas exact finite surjectivity is, in general, undecidable, highlighting a fundamental divide between rate and exact solvability in deterministic information flow. The framework and tools developed here lay groundwork for analyzing extremal combinatorics and network coding problems within a unified, graph-entropy-inspired formalism.

Abstract

We introduce a new framework term coding for extremal problems in discrete mathematics and information flow, where one chooses interpretations of function symbols so as to maximise the number of satisfying assignments of a finite system of term equations. We then focus on dispersion, the special case in which the system defines a term map $Θ^\mathcal I:\A^k\to\A^r$ and the objective is the size of its image. Writing $n:=|\A|$, we show that the maximum dispersion is $Θ(n^D)$ for an integer exponent $D$ equal to the guessing number of an associated directed graph, and we give a polynomial-time algorithm to compute $D$. In contrast, deciding whether \emph{perfect dispersion} ever occurs (i.e.\ whether $\Disp_n(\mathbf t)=n^r$ for some finite $n\ge 2$) is undecidable once $r\ge 3$, even though the corresponding asymptotic rate-threshold questions are polynomial-time decidable.

Term Coding and Dispersion: A Perfect-vs-Rate Complexity Dichotomy for Information Flow

TL;DR

This work introduces Term Coding and a dispersion subclass to study how optimising local function interpretations affects global configurations. It establishes a polynomial-time computable dispersion exponent via a max-flow construction, while showing that exact perfect-dispersion thresholds are undecidable for . A normalization-diversification pipeline reduces problems to flow-based representations, enabling a guessing-game bridge that yields a tight sandwich bound between dispersion and guessing numbers. The results delineate a sharp complexity dichotomy: asymptotic rate questions are tractable, whereas exact finite surjectivity is, in general, undecidable, highlighting a fundamental divide between rate and exact solvability in deterministic information flow. The framework and tools developed here lay groundwork for analyzing extremal combinatorics and network coding problems within a unified, graph-entropy-inspired formalism.

Abstract

We introduce a new framework term coding for extremal problems in discrete mathematics and information flow, where one chooses interpretations of function symbols so as to maximise the number of satisfying assignments of a finite system of term equations. We then focus on dispersion, the special case in which the system defines a term map and the objective is the size of its image. Writing , we show that the maximum dispersion is for an integer exponent equal to the guessing number of an associated directed graph, and we give a polynomial-time algorithm to compute . In contrast, deciding whether \emph{perfect dispersion} ever occurs (i.e.\ whether for some finite ) is undecidable once , even though the corresponding asymptotic rate-threshold questions are polynomial-time decidable.
Paper Structure (42 sections, 18 theorems, 44 equations, 2 figures, 1 algorithm)

This paper contains 42 sections, 18 theorems, 44 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Keywords.
  2. Introduction
  3. The computational problem
  4. A structural invariant: dependency graphs and guessing numbers
  5. The complexity dichotomy for dispersion
  6. Main contributions
  7. Relation to the max-flow/min-cut dispersion line
  8. Companion paper and scope
  9. Scope and self-contained presentation
  10. Roadmap
  11. Notation quick reference
  12. Term coding instances
  13. Terms and instances
  14. Entropy normalisation (intuition)
  15. Preprocessing: normalisation, quotienting, and diversification
  16. ...and 27 more sections

Key Result

Proposition 3.1

For every term-coding instance $\Gamma$ there exists an instance $\Gamma^{\mathrm{nf}}$ over the same signature whose set of variables is $X\cup Z$ (where $X$ are the original variables and $Z$ are fresh auxiliary variables) such that: In particular, $S_n(\Gamma)=S_n(\Gamma^{\mathrm{nf}})$ for all $n$.

Figures (2)

  • Figure 1: Dispersion as term coding via decoder symbols (Lemma \ref{['lem:disp-embed']}). Choosing $\mathbf h$ selects one preimage for each output $\mathbf y$ in the image of $\mathbf t$.
  • Figure 2: The simplified term graph for the diamond gadget. Cutting the four unit-capacity edges into $t$ separates the inputs from the outputs, giving a cut of capacity $4$.

Theorems & Definitions (64)

  • Remark 1.1: Two levels of variables
  • Example 1.1: A toy index-coding instance
  • Remark 1.2: Combinatorial graded feasibility (omitted here)
  • Example 1.2: A single-function diamond gadget (the dichotomy in microcosm)
  • Definition 2.1: Term-coding instance
  • Definition 2.2: Syntactic size of an instance
  • Remark 2.1: Why we fix an explicit size measure
  • Definition 2.3: Interpretation, solutions, and maximum code size
  • Remark 3.1: Effectiveness and size blow-up
  • Proposition 3.1: Flattening to normal form
  • ...and 54 more