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Local generation of languages: the monotonic binary sequences

Mathieu Hoyrup

TL;DR

The paper studies local generation of the monotone binary language ${\mathsf{Mon}}_n$ using a communication-complex framework, showing that minimal generating complexes are interval-based and arise either from vertex insertions of smaller-length instances or from new constructions at particular lengths (notably yielding the ${K_2},{K_5},{K_7},{K_8}$ families). It develops a unifying inference system to discard non-generating complexes and proves both upper and lower bounds on interval sizes, establishing a near $3n/4$ proportion for the maximal interval length with exact results at several small $n$ values. The work connects discrete generation with combinatorial topology, offering a formal bridge to reformulations via chromatic labeled complexes and outlining open questions about complete classifications and asymptotic behavior. Overall, it advances understanding of how distributed, local generation constraints shape language generation in a topological setting.

Abstract

In a previous article, we have introduced the problem of local generation of languages, where the communication underlying the generation procedure is captured by a simplicial complex. We study in details this problem for the language of binary monotonic sequences. We prove general results and identify several classes of minimal simplicial complexes generating this language.

Local generation of languages: the monotonic binary sequences

TL;DR

The paper studies local generation of the monotone binary language using a communication-complex framework, showing that minimal generating complexes are interval-based and arise either from vertex insertions of smaller-length instances or from new constructions at particular lengths (notably yielding the families). It develops a unifying inference system to discard non-generating complexes and proves both upper and lower bounds on interval sizes, establishing a near proportion for the maximal interval length with exact results at several small values. The work connects discrete generation with combinatorial topology, offering a formal bridge to reformulations via chromatic labeled complexes and outlining open questions about complete classifications and asymptotic behavior. Overall, it advances understanding of how distributed, local generation constraints shape language generation in a topological setting.

Abstract

In a previous article, we have introduced the problem of local generation of languages, where the communication underlying the generation procedure is captured by a simplicial complex. We study in details this problem for the language of binary monotonic sequences. We prove general results and identify several classes of minimal simplicial complexes generating this language.
Paper Structure (41 sections, 41 theorems, 72 equations, 19 figures, 1 table)

This paper contains 41 sections, 41 theorems, 72 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview of the main results.
  3. Definitions
  4. Background
  5. Intervals
  6. Graphical representation of complexes
  7. Symmetries
  8. Examples of complexes generating Mon
  9. A complex generating Mon5
  10. Verification.
  11. A complex generating Mon7
  12. Verification.
  13. A complex generating Mon8
  14. Structural results
  15. Decomposition
  16. ...and 26 more sections

Key Result

Proposition 1

The minimal complexes generating $\mathsf{Mon}_n$ and containing several simplices of size $n-1$ are obtained from $K_2$ by vertex insertions, and have the form $\langle I_n\setminus \{a\},I_n\setminus \{b\}\rangle$ where $a,b\in I_n$ are distinct.

Figures (19)

  • Figure 1: $\mathbb{I}\mathrel{\angle}\mathbb{J}$
  • Figure 2: A graphical representation of $K_5$
  • Figure 3: The complex $K_5$
  • Figure 4: The visibility diagram of a function generating $\mathsf{Mon}_5$
  • Figure 5: The complex $K_7$
  • ...and 14 more figures

Theorems & Definitions (88)

  • Proposition : Proposition \ref{['prop_mono_2_simplices']}
  • Theorem : Theorem \ref{['thm_one_interval']}
  • Theorem : Theorem \ref{['thm_intervals']}
  • Theorem : Theorem \ref{['thm_mu']}
  • Definition 2.1: Language generation
  • Proposition 2.1: Canonical form
  • Proposition 2.2: Image of a language
  • Lemma 2.1
  • proof
  • Proposition 2.3: Symmetries
  • ...and 78 more