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Limits of structures and Total NP Search Problems

Ondřej Ježil

TL;DR

A new proof of this unprovability is obtained and it is shown that $\textbf{WeakPigeon}$ is not many-one reducible to $\textbf{RetractionWeakPigeon}$ in the oracle setting.

Abstract

For an infinite class of finite graphs of unbounded size, we define a limit object, to be called a $\textit{wide limit}$, relative to some computationally restricted class of functions. The limit object is a first order Boolean-valued structure. The first order properties of the wide limit then reflect how a computationally restricted viewer "sees" a generic member of the class. The construction uses arithmetic forcing with random variables [Krajíček, Forcing with random variables and proof complexity 2011]. We give sufficient conditions for universal and existential sentences to be valid in the limit, provide several examples, and prove that such a limit object can then be expanded to a model of weak arithmetic. To illustrate the concept we give an example in which the wide limit relates to total NP search problems. In particular, we take the wide limit of all maps from $\{0,\dots,k-1\}$ to $\{0,\dots,\lfloor k/2\rfloor-1\}$ to obtain a model of $\forall \text{PV}_1(f)$ where the problem $\textbf{RetractionWeakPigeon}$ is total but $\textbf{WeakPigeon}$, the complete problem for $\textbf{PWPP}$, is not. Thus, we obtain a new proof of this unprovability and show it implies that $\textbf{WeakPigeon}$ is not many-one reducible to $\textbf{RetractionWeakPigeon}$ in the oracle setting.

Limits of structures and Total NP Search Problems

TL;DR

A new proof of this unprovability is obtained and it is shown that is not many-one reducible to in the oracle setting.

Abstract

For an infinite class of finite graphs of unbounded size, we define a limit object, to be called a , relative to some computationally restricted class of functions. The limit object is a first order Boolean-valued structure. The first order properties of the wide limit then reflect how a computationally restricted viewer "sees" a generic member of the class. The construction uses arithmetic forcing with random variables [Krajíček, Forcing with random variables and proof complexity 2011]. We give sufficient conditions for universal and existential sentences to be valid in the limit, provide several examples, and prove that such a limit object can then be expanded to a model of weak arithmetic. To illustrate the concept we give an example in which the wide limit relates to total NP search problems. In particular, we take the wide limit of all maps from to to obtain a model of where the problem is total but , the complete problem for , is not. Thus, we obtain a new proof of this unprovability and show it implies that is not many-one reducible to in the oracle setting.
Paper Structure (18 sections, 19 theorems, 73 equations)

This paper contains 18 sections, 19 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wide Limits
  4. Wide Limits for Shallow Decision Trees
  5. Sufficient conditions for validity of universal and existential sentences
  6. Expanding Wide Limits to Two-Sorted Arithmetic
  7. The structures $K(F,G)$
  8. Preservation of sentences concerning the wide limit
  9. Total NP search problems and Wide Limits
  10. Preliminaries on Total NP Search problems
  11. The wide limit of all instances of $\textsc{WeakPigeon}$
  12. Model for separation of $\textsc{RetractionWeakPigeon}$ and $\textsc{WeakPigeon}$
  13. Construction of the model
  14. Non-totality of $\textsc{WeakPigeon}$
  15. Totality of $\textsc{RetractionWeakPigeon}$
  16. ...and 3 more sections

Key Result

Lemma 2.1

There is a function $\text{st}:\mathbb{Q}^\mathcal{M}_{fin}\to \mathbb{R}$ assigning to each finite $\mathcal{M}$-rational a real number. The function $\text{st}$ is a ring homomorphism and the kernel of $\text{st}$ is exactly the ideal of infinitesimal numbers. When $q$ is a finite $\mathcal{M}$-ra

Theorems & Definitions (61)

  • Lemma 2.1: The existence of a standard part goldbring2014lecture
  • Theorem 2.2: goldbring2014lecture
  • Theorem 2.3: Bernoulli's inequality
  • Theorem 2.4: Exponential equality
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4: krajicek2010forcing
  • Definition 3.5
  • Definition 3.6: The wide limit
  • ...and 51 more