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Separation and Gluing of Explanations on Sites of Dynamical Systems

Paul Z. Wang

Abstract

We construct a Grothendieck site whose objects are Mealy machines over definable sets in an o-minimal structure and whose coverings are jointly surjective families of definable open immersions. On this site, we define presheaves of explanations -- systems equipped with an interpretable interface, parameterised by a ``judge.'' We prove that the behavioral presheaf (quotienting by observable output equivalence) is separated: a global explanation is determined by its local restrictions. We show that gluing fails in general -- locally consistent explanations need not assemble globally -- and give, for stateless explanatory systems of the restricted-interface presheaf, a necessary and sufficient topological condition for the sheaf property in terms of robust disconnection of fibers of the judge.

Separation and Gluing of Explanations on Sites of Dynamical Systems

Abstract

We construct a Grothendieck site whose objects are Mealy machines over definable sets in an o-minimal structure and whose coverings are jointly surjective families of definable open immersions. On this site, we define presheaves of explanations -- systems equipped with an interpretable interface, parameterised by a ``judge.'' We prove that the behavioral presheaf (quotienting by observable output equivalence) is separated: a global explanation is determined by its local restrictions. We show that gluing fails in general -- locally consistent explanations need not assemble globally -- and give, for stateless explanatory systems of the restricted-interface presheaf, a necessary and sufficient topological condition for the sheaf property in terms of robust disconnection of fibers of the judge.
Paper Structure (32 sections, 11 theorems, 9 equations, 2 figures)

This paper contains 32 sections, 11 theorems, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Systems, Sites, and the O-Minimal Setting
  4. Heterogeneous Mealy machines
  5. The o-minimal setting
  6. The site $\mathrm{Sys}_{\mathrm{oi}}$
  7. Judges and Presheaves of Explanations
  8. Judges and sections
  9. Equivalence relations and the presheaf hierarchy
  10. Separation
  11. The Gluing Landscape
  12. Gluing for $\mathcal{F}_j$
  13. Gluing fails for $\mathcal{F}_j^{\mathrm{beh}}$
  14. Positive results: connected and robustly disconnected fibers
  15. Summary of the gluing landscape
  16. ...and 17 more sections

Key Result

Proposition 2.7

$\mathrm{Sys}_{\mathrm{oi}}$ has pullbacks (computed componentwise as intersections of definable opens), and the coverings above define a Grothendieck pretopology MacLaneMoerdijk1994, making $(\mathrm{Sys}_{\mathrm{oi}}, \mathrm{Cov})$ a site.

Figures (2)

  • Figure 1: The $2$-state system of Counterexample \ref{['cex:ri-sep-fail']} (separation failure for $\mathcal{F}_j^{\mathrm{ri}}$). Edge labels: input/output. The input-splitting covering $\mathcal{S}|_{\{a\}}$, $\mathcal{S}|_{\{b\}}$ restricts each patch to a single input; an alternative explanation agrees on pure-input sequences but disagrees on the mixed sequence $(a,b)$.
  • Figure 2: The $4$-state system of Counterexample \ref{['cex:beh-gluing-fail']} (gluing failure for $\mathcal{F}_j^{\mathrm{beh}}$). Edge labels: input/output; the judge collapses inputs ($j_I : \{a,b\} \to \{\bullet\}$). The dashed and dotted boxes show the covering patches; the overlap is $\{s_1, s_2\}$. The obstruction: $s_0 \xrightarrow{b} s_2$ and $s_3 \xrightarrow{b} s_2$ force incompatible behavioral requirements on the after-state $s_2$.

Theorems & Definitions (42)

  • Definition 2.1
  • Definition 2.2: vandenDries1998
  • Example 2.3
  • Remark 2.4
  • Definition 2.5: Definable open immersion of systems
  • Definition 2.6: Coverings
  • Proposition 2.7
  • proof : Proof sketch
  • Theorem 2.8
  • Theorem 2.9
  • ...and 32 more