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Equivalence and Conditional Independence in Atomic Sheaf Logic

Alex Simpson

Abstract

We propose a semantic foundation for logics for reasoning in settings that possess a distinction between equality of variables, a coarser equivalence of variables, and a notion of conditional independence between variables. We show that such relations can be modelled naturally in atomic sheaf toposes.

Equivalence and Conditional Independence in Atomic Sheaf Logic

Abstract

We propose a semantic foundation for logics for reasoning in settings that possess a distinction between equality of variables, a coarser equivalence of variables, and a notion of conditional independence between variables. We show that such relations can be modelled naturally in atomic sheaf toposes.
Paper Structure (10 sections, 35 theorems, 65 equations, 4 figures)

This paper contains 10 sections, 35 theorems, 65 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Multiteam semantics
  3. Atomic sheaves
  4. Atomic sheaf logic
  5. Atomic equivalence
  6. Independent pullbacks
  7. Atomic conditional independence
  8. Probability sheaves
  9. The Schanuel topos
  10. Discussion and related work

Key Result

Proposition 3.5

$\mathbb{S}\mathsf{ur}$ is coconfluent.

Figures (4)

  • Figure 1: Multiteam semantics of atomic formulas
  • Figure 2: Semantics of atomic sheaf logic
  • Figure 3: Axioms for equivalence
  • Figure 4: Axioms for conditional independence

Theorems & Definitions (68)

  • Definition 2.1: Equiextension for nondeterministic variables
  • Definition 2.2: Conditional independence for nondeterministic variables
  • Example 3.1: Representable presheaves
  • Example 3.2: Product presheaves
  • Example 3.3
  • Definition 3.4: Coconfluence
  • Proposition 3.5
  • proof
  • Definition 3.6: Invariant element
  • Definition 3.7: Descendent
  • ...and 58 more