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Complexity of deep computations via topology of function spaces

Eduardo Dueñez, José Iovino, Tonatiuh Matos-Wiederhold, Luciano Salvetti, Franklin D. Tall

TL;DR

The paper develops a unified framework connecting topology of function spaces, model theory, and computational complexity to study the limits of deep computations. By modeling computations as elements of a space of types and introducing Compositional Computation Structures with an Extendibility Axiom, it derives precise criteria (NIP, Rosenthal compacta, PAC learnability) that separate tame (polynomial) from wild (exponential) regimes and characterizes when deep computations admit well-behaved limits. It provides concrete CCS examples (Newton's method, finite-precision thresholds, prefix tests) to illustrate the tameness or complexity of deep computations and shows that, under countable predicates, the deep computation closures are Rosenthal compacta, admitting PAC-type classifications via Todorčević’s trichotomy and the Argyros–Dodos–Kanellopoulos seven-minimal families. The probabilistic extension (Monte Carlo computability) and Talagrand stability results establish when deep computations have measurable, computable limits, enabling probabilistic analyses and opening avenues for stochastic or quantum computation models within a rigorous topological-model-theoretic framework.

Abstract

We use topological methods to study complexity of deep computations and limit computations. We use topology of function spaces, specifically, the classification Rosenthal compacta, to identify new complexity classes. We use the language of model theory, specifically, the concept of \emph{independence} from Shelah's classification theory, to translate between topology and computation. We use the theory of Rosenthal compacta to characterize approximablility of deep computations, both deterministically and probabilistically.

Complexity of deep computations via topology of function spaces

TL;DR

The paper develops a unified framework connecting topology of function spaces, model theory, and computational complexity to study the limits of deep computations. By modeling computations as elements of a space of types and introducing Compositional Computation Structures with an Extendibility Axiom, it derives precise criteria (NIP, Rosenthal compacta, PAC learnability) that separate tame (polynomial) from wild (exponential) regimes and characterizes when deep computations admit well-behaved limits. It provides concrete CCS examples (Newton's method, finite-precision thresholds, prefix tests) to illustrate the tameness or complexity of deep computations and shows that, under countable predicates, the deep computation closures are Rosenthal compacta, admitting PAC-type classifications via Todorčević’s trichotomy and the Argyros–Dodos–Kanellopoulos seven-minimal families. The probabilistic extension (Monte Carlo computability) and Talagrand stability results establish when deep computations have measurable, computable limits, enabling probabilistic analyses and opening avenues for stochastic or quantum computation models within a rigorous topological-model-theoretic framework.

Abstract

We use topological methods to study complexity of deep computations and limit computations. We use topology of function spaces, specifically, the classification Rosenthal compacta, to identify new complexity classes. We use the language of model theory, specifically, the concept of \emph{independence} from Shelah's classification theory, to translate between topology and computation. We use the theory of Rosenthal compacta to characterize approximablility of deep computations, both deterministically and probabilistically.
Paper Structure (18 sections, 24 theorems, 37 equations, 2 figures)

This paper contains 18 sections, 24 theorems, 37 equations, 2 figures.

Table of Contents

  1. General topological preliminaries: From continuity to Baire class 1
  2. From Rosenthal's dichotomy to the Bourgain-Fremlin-Talagrand dichotomy to Shelah's NIP
  3. NIP as a universal dividing line between polynomial and exponential complexity
  4. Rosenthal compacta
  5. The special case B_1(X, R^ P) with P countable.
  6. Compositional Computation Structures: A structural approach to floating-point computations
  7. Compositional Computation Structures
  8. Computability and definability of deep computations and the Extendibility Axiom
  9. Newton's method as a CCS
  10. Finite precision threshold classifiers as a CCS
  11. Finite precision prefix test
  12. Classifying deep computations
  13. NIP, Rosenthal compacta, and deep computations
  14. The Todorčević trichotomy and levels of PAC learnability
  15. The Argyros-Dodos-Kanellopoulos heptachotomy, and approximability of deep computation by minimal classes
  16. ...and 3 more sections

Key Result

Lemma 1.3

Let $X$ be a Polish space and $A\subseteq B_1(X)$ be pointwise bounded. The following are equivalent:

Figures (2)

  • Figure 1: Newton's method approximating $p(z)=z^3-2z+2$. Notice the regions of divergence.
  • Figure 2: Newton's method approximating $p(z)=z^3-1$.

Theorems & Definitions (56)

  • Lemma 1.3
  • proof
  • Theorem 1.4: Rosenthal's Dichotomy, Rosenthal:1974
  • Theorem 1.5: "The BFT Dichotomy". Bourgain-Fremlin-Talagrand BFT_1978_PCompactBaire
  • Definition 1.6
  • Proposition 1.7
  • proof
  • Theorem 1.8
  • proof
  • Definition 1.9
  • ...and 46 more