Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Term Coding: An Entropic Framework for Extremal Combinatorics and the Guessing--Number Sandwich Theorem

Søren Riis

TL;DR

Term Coding converts universal algebraic identities into graded extremal problems by maximizing the code size $S_n(\Gamma)$ over interpretations, producing a normalised exponent $R(\Gamma,n)$ and a limiting exponent $α(Γ)$ tied to graph-entropy via guessing numbers. A central result, the guessing-number sandwich, bounds $S_n(Γ)$ between entropy-based quantities derived from a dependency graph (and its labelled variant), yielding $S_n(Γ)=n^{α(Γ)+o(1)}$. The framework is demonstrated through case studies including Steiner-type identities, self-orthogonal Latin squares, and network-coding-like constraints, with non-integer exponents explained by polymatroid/entropy geometry. This approach provides quantitative measures of proximity to satisfiability, bridging universal algebra with information theory and offering concrete bounds, constructions, and insights for extremal combinatorics and dispersion-like problems.

Abstract

Term Coding asks: given a finite system of term identities $Γ$ in $v$ variables, how large can its solution set be on an $n$--element alphabet, when we are free to choose the interpretations of the function symbols? This turns familiar existence problems for quasigroups, designs, and related objects into quantitative extremal questions. We prove a guessing-number sandwich theorem that connects term coding to graph guessing numbers (graph entropy). After explicit normalisation and diversification reductions, every instance yields a canonical directed dependency structure with guessing number $α$ such that the maximum code size satisfies $\log_n \Sn(Γ)=α+o(1)$ (equivalently, $\Sn(Γ)=n^{α+o(1)}$), and $α$ can be bounded or computed using entropy and polymatroid methods. We illustrate the framework with examples from extremal combinatorics (Steiner-type identities, self-orthogonal Latin squares) and from information-flow / network-coding style constraints (including a five-cycle instance with fractional exponent and small storage/relay maps).

Term Coding: An Entropic Framework for Extremal Combinatorics and the Guessing--Number Sandwich Theorem

TL;DR

Term Coding converts universal algebraic identities into graded extremal problems by maximizing the code size over interpretations, producing a normalised exponent and a limiting exponent tied to graph-entropy via guessing numbers. A central result, the guessing-number sandwich, bounds between entropy-based quantities derived from a dependency graph (and its labelled variant), yielding . The framework is demonstrated through case studies including Steiner-type identities, self-orthogonal Latin squares, and network-coding-like constraints, with non-integer exponents explained by polymatroid/entropy geometry. This approach provides quantitative measures of proximity to satisfiability, bridging universal algebra with information theory and offering concrete bounds, constructions, and insights for extremal combinatorics and dispersion-like problems.

Abstract

Term Coding asks: given a finite system of term identities in variables, how large can its solution set be on an --element alphabet, when we are free to choose the interpretations of the function symbols? This turns familiar existence problems for quasigroups, designs, and related objects into quantitative extremal questions. We prove a guessing-number sandwich theorem that connects term coding to graph guessing numbers (graph entropy). After explicit normalisation and diversification reductions, every instance yields a canonical directed dependency structure with guessing number such that the maximum code size satisfies (equivalently, ), and can be bounded or computed using entropy and polymatroid methods. We illustrate the framework with examples from extremal combinatorics (Steiner-type identities, self-orthogonal Latin squares) and from information-flow / network-coding style constraints (including a five-cycle instance with fractional exponent and small storage/relay maps).
Paper Structure (49 sections, 19 theorems, 97 equations, 1 figure, 4 tables)

This paper contains 49 sections, 19 theorems, 97 equations, 1 figure, 4 tables.

Table of Contents

  1. Keywords.
  2. Introduction
  3. From universal identities to graded extremal problems
  4. A first example: Steiner triple systems and "near misses"
  5. Scope and companion papers
  6. The computational engine: dependency graphs and guessing numbers
  7. Non-integer exponents and the "entropic envelope"
  8. Organisation
  9. Term Coding instances
  10. Terms and equations
  11. Interpretations and codes
  12. Two guiding classes of instances
  13. Universal identities.
  14. Local constraints with global optimisation.
  15. Normalisation
  16. ...and 34 more sections

Key Result

Proposition 3.2

For every Term Coding instance $\Gamma$ there is a normal-form instance $\Gamma^{\mathrm{nf}}$ obtained by introducing auxiliary variables such that for every $n$ and every interpretation $\mathcal{I}$ on an $n$--element set $A$, the restriction map $A^{V(\Gamma^{\mathrm{nf}})}\to A^{V(\Gamma)}$ ind and hence $S_n(\Gamma^{\mathrm{nf}})=S_n(\Gamma)$ for all $n$. Moreover, $\Gamma^{\mathrm{nf}}$ can

Figures (1)

  • Figure 1: The dependency graph of the diversified $C_5$ instance \ref{['eq:c5-div']}: the bidirected 5-cycle on $\{x,y,z,\alpha,\beta\}$.

Theorems & Definitions (63)

  • Remark 1.1: Table \ref{['tab:sts-refined']}
  • Definition 2.1: Term Coding instance
  • Definition 2.2: Code of an interpretation
  • Remark 2.3: Normalised exponent
  • Definition 3.1: Normal form
  • Proposition 3.2: Normalisation
  • proof
  • Remark 3.3: Term-DAG normalisation
  • Definition 3.4: Term-DAG / no-collision property
  • Lemma 3.5: Quotienting variable equalities
  • ...and 53 more