Project 7: Abstract Vector Space Axiom Testbench

Build a verification tool that decides whether a candidate structure is a vector space and produces minimal counterexamples when it is not.

Quick Reference

Attribute Value
Difficulty Advanced (Level 3)
Time Estimate 1 week
Language Python (alternatives: Rust, Haskell)
Prerequisites Sets/functions, proof basics, linear combinations
Key Topics Vector space axioms, subspaces, basis, dimension

1. Learning Objectives

By completing this project, you will:

  1. Formalize all vector-space axioms as explicit testable contracts.
  2. Distinguish true subspaces from “looks-like” subsets that fail closure.
  3. Build counterexample generators that expose the first failed axiom.
  4. Connect abstract definitions to real data types (polynomials, signals, tuples).
  5. Document assumptions (field, operation definitions, domain closure) clearly.

2. Theoretical Foundation

2.1 Core Concepts

  • Vector space as structure: A set plus operations satisfying all axioms.
  • Subspace test: Nonempty + closure under addition and scalar multiplication.
  • Basis and dimension: Independence + spanning, independent of coordinates.
  • Counterexample as proof tool: One failed witness invalidates a universal claim.

2.2 Why This Matters

Most engineering bugs in math-heavy code are contract bugs, not arithmetic bugs. Teams often assume a set is “linear enough” and then apply linear methods that silently fail. This project trains you to encode algebraic preconditions directly into tooling.

2.3 Historical Context / Background

Linear algebra moved from coordinate manipulation to axiomatic structure during the 19th-20th century, which enabled the subject to apply to functions, operators, and abstract spaces far beyond geometric intuition.

2.4 Common Misconceptions

  • “Contains zero” means “subspace”.
  • Finite testing alone can prove universal claims.
  • Basis is unique for a space.

3. Project Specification

3.1 What You Will Build

A CLI tool that takes a structure definition file and reports:

  • VECTOR_SPACE or NOT_VECTOR_SPACE
  • Failed axiom name (if any)
  • Minimal witness tuple showing failure
  • Optional subspace classification relative to a parent space

3.2 Functional Requirements

  1. Accept candidate set/operations via JSON or declarative config.
  2. Check all vector-space axioms under selected field.
  3. Generate reproducible counterexamples with deterministic seeds.
  4. Support built-in fixtures: R^n, polynomial spaces, constrained subsets.
  5. Emit machine-readable and human-readable reports.

3.3 Non-Functional Requirements

  • Correctness: No false pass on fixture negatives.
  • Explainability: Every failure includes witness and axiom text.
  • Reproducibility: Same seed yields same witness.

3.4 Example Usage / Output

$ python axiom_testbench.py --spec specs/p2_over_r.json
[INFO] Structure: P2 over R
[PASS] additive closure
[PASS] scalar closure
[PASS] additive identity/inverse
[PASS] distributive/associative rules
[RESULT] VECTOR_SPACE

3.5 Real World Outcome

You end with a reusable validator that can be pointed at new symbolic structures before downstream algorithms are applied. In practice this prevents invalid assumptions from contaminating optimization, decomposition, or solver modules.

4. Solution Architecture

4.1 High-Level Design

Spec Parser -> Axiom Engine -> Witness Search -> Reporter

4.2 Key Components

Component Responsibility Key Decisions
Parser Load field/set/operations Strict schema validation
Axiom Engine Evaluate each axiom Explicit quantifier handling
Witness Search Find failing tuples Deterministic + adversarial mode
Reporter Human/JSON output Include exact failing expression

4.3 Data Structures

StructureSpec:
  field
  universe
  add_op
  scalar_op
  generators

4.4 Algorithm Overview

  1. Load spec and normalize operation signatures.
  2. Run deterministic fixture checks.
  3. Evaluate axioms in fixed order.
  4. On failure, shrink witness.
  5. Emit final classification.

5. Implementation Guide

5.1 Development Environment Setup

Create virtual environment
Install symbolic + test dependencies
Run fixture smoke suite

5.2 Project Structure

P07/
  specs/
  engine/
  witnesses/
  tests/
  reports/

5.3 The Core Question You’re Answering

“What is the minimum algebraic contract required before calling something a vector space?”

5.4 Concepts You Must Understand First

  1. Vector-space axioms
  2. Closure and quantifiers
  3. Counterexample-based reasoning

5.5 Questions to Guide Your Design

  1. Which axioms are hardest to falsify automatically?
  2. How do you avoid false confidence from random testing?

5.6 Thinking Exercise

Design three candidate sets that each fail a different axiom. Predict witness shape before running tool.

5.7 The Interview Questions They’ll Ask

  1. Why is closure under scalar multiplication essential?
  2. Why is one counterexample enough to reject a claim?
  3. Can dimension be defined without choosing coordinates?
  4. How do field assumptions alter outcomes?
  5. Why do algebraic contracts matter in production ML pipelines?

5.8 Hints in Layers

  • Hint 1: Start with fixed finite fixtures.
  • Hint 2: Add adversarial structured witnesses, not only random tuples.
  • Hint 3: Keep witness minimization separate from discovery.
  • Hint 4: Emit algebraic expression traces for failed checks.

5.9 Books That Will Help

Topic Book Chapter
Axioms and structure Axler Ch. 1
Proof/counterexample method Hefferon Intro proofs
Symbolic tool design SymPy docs Algebra modules

5.10 Implementation Phases

  1. Spec format + parser
  2. Axiom checks with fixtures
  3. Witness search and minimization
  4. Reporting and docs

5.11 Key Implementation Decisions

  • Prefer explicit theorem-language labels for each check.
  • Keep arithmetic backend swappable (exact vs floating-point).

6. Testing Strategy

6.1 Test Categories

  • Known positive structures
  • Known negative structures
  • Regression tests for witness determinism

6.2 Critical Test Cases

  1. R^n passes all axioms.
  2. “first coordinate fixed at 1” fails scalar closure.
  3. Polynomial odd/even subsets classified correctly.

6.3 Test Data

  • Curated symbolic fixtures plus generated random candidates.

7. Common Pitfalls & Debugging

  • Symptom: Tool passes invalid structures.
    • Why: Insufficient quantifier coverage.
    • Fix: Add symbolic branch with exact witnesses.
    • Quick test: Run known-fail fixture suite.
  • Symptom: Non-reproducible failures.
    • Why: Seed not propagated through witness search.
    • Fix: Global deterministic seed contract.
    • Quick test: Repeat same run 5 times.

8. Extensions & Challenges

  • Add finite-field support (GF(p)) and compare outcomes.
  • Add basis finder and dimension estimator for passing spaces.
  • Add auto-generated proof sketch text from witness outcomes.

9. Real-World Connections

  • Validating linear assumptions in feature-engineering pipelines.
  • Ensuring symbolic optimizers only consume valid linear structures.
  • Teaching tooling for proof-first math education platforms.

10. Resources

11. Self-Assessment Checklist

  • I can explain each vector-space axiom in plain language.
  • My tool emits minimal reproducible counterexamples.
  • I can classify at least 10 structures correctly.
  • I can explain why each failing structure fails.

12. Submission / Completion Criteria

  • CLI + fixtures + tests committed.
  • One report with three pass and three fail structures.
  • One reflection note mapping bugs to violated axioms.

This guide expands Project 7 from LINEAR_ALGEBRA_LEARNING_PROJECTS.md.