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Sequences for learning Path Semantics

Reading sequences for learning path semantics.

This is a collection of papers organized in a way to build up gradually understanding of path semantics.
Some papers are marked with stars (e.g. ★★★★★) to signify their significance.

Introduction

Cheat Sheets

Visualization

Reference material

Background

Notation

Examples of Path Semantics

Foundation

Normal paths are formalized in 5 different ways:

The proof in Lean 3 assumes satisfied models of total normal paths using semiconjugates. This is currently the best available model of normal paths in dependent types.

Each of these pictures can be used to reason about normal paths. However, there exists no exhausting set of axioms (this is currently believed to be an undecidable problem). For some theories, like Boolean functions, normal paths are proven to be sound by exhaustive proof search.

In the broader theory of logic, one can study Path Semantical Logic.

Path Semantical Logic

It is recommended to skip this section, unless you are interested in logic or the foundation of Path Semantics.

Path Semantical Logic uses the core axiom to group propositions into levels. A quality at level N propagates into quality at level N+1 (2021 standard order).

For an experimental implementations, see:

Notice that Avatar Logic is required to weaken IPL to PSI properly.

Some theorems, such as the Implication Theorem or Creation Theorem, holds in PSL but are not provable in PSI without LEM (Law of Excluded Middle). Since e.g. the Creation Theorem is absurd, this implies that PSI is the natural foundation of Path Semantics.

Papers:

Higher Order Operator Overloading

Higher Order Operator Overloading (HOOO) is a way to extend the semantics of normal expressions into expressions for higher order reasoning.

Discrete Path Semantics

Boolean Path Semantics

Permutative Path Semantics

Concepts in Path Semantics

Commutativity

Calculus

Probabilistic Path Semantics

Examples of Probabilistic Path Semantics

Avatar Extensions

Summary of Avatar Extensions

Foundation of Avatar Logic: Avatar Logic to Set Theory ★★★★★

Avatar Extensions is a technique of abstract generalization by exploiting symmetries inside "smaller" theories. For an experimental implementation of Avatar Logic, see Avalog. For visualization of Avatar Graphs, see Avatar-Graph.

Joker Calculus

For an implementation in Rust, see the Joker Calculus project.

Language Bias Hypotheses

Consciousness in Wolfram Models

Related to Homotopy Physics

Implementation: Avatar Hypergraph Rewriting

Conversations With GPT-3

Homotopy Physics

Homotopy Physics applies the theory of Avatar Extensions to Quantum Physics. Related to Consciousness in Wolfram Models.

Papers for the theoretical background to understand Homotopy Physics applied to experiments:

Proof Techniques

Sized Type Theory

MX Grammar for Dependent Types

Slot Lambda Calculus

Dual Number Monotonic Density

Cocyclic Graphs

Directional Set Algebra

Closed Natural Numbers

Satisfied Natural Numbers

Algexenotation

Asymmetric Velocity Logic

Answered Modal Logic

Semantics of Falsehood

Listing-Möbius Shifts

Theories

Artificial Intelligence

Moved to its own sequences page

Esoteric Path Semantics

Non-Deterministic Path Semantics