Alphabeta MathTabular prediction and control by interaction
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How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 2 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 2 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Combinatorics
Linear Algebra
Machine Learning Algorithms
Measure Theory
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Outer Measure and the Caratheodory Extension Theorem
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Radon Nikodym Theorem and Lebesgue Decomposition
Probability
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Lemma
Predictably weighted bounded martingale differences have a convergent series
Definition
Optimal action-value Bellman operator
Lemma
A finite MDP policy generates action-state-reward trajectories
Lemma
Coordinate-wise stochastic approximation converges to a contraction fixed point
Algorithm
Tabular Q-learning
Definition
Episodic returns, temporal-difference errors, and policy support
Lemma
The optimal action value is attained by a greedy stationary policy
Theorem
Tabular Q-learning converges under per-pair Robbins–Monro steps and full visitation
Lemma
Finite-episode importance sampling has the target-policy mean under support
Algorithm
Tabular TD(0) policy prediction
Algorithm
Tabular on-policy Sarsa control
Algorithm
First-visit Monte Carlo policy evaluation
Sources
- John N. Tsitsiklis, Asynchronous Stochastic Approximation and Q-Learning
- Richard S. Sutton and Andrew G. Barto, Reinforcement Learning: An Introduction, 2nd ed.
- Han-Dong Yang, Optimal Control and Reinforcement Learning, Chapter 2: Value-based Reinforcement Learning
- Leslie Pack Kaelbling, Michael L. Littman, and Anthony R. Cassandra, Planning and Acting in Partially Observable Stochastic Domains, Brown Technical Report CS-96-08
- Leslie Pack Kaelbling, Michael L. Littman, and Anthony R. Cassandra, Planning and Acting in Partially Observable Stochastic Domains, Brown CS-96-08