HOList: An Environment for Machine Learning of Higher-Order Theorem - PowerPoint PPT Presentation

HOList: An Environment for Machine Learning of Higher-Order Theorem Proving Kshitij Bansal, Christian Szegedy

Can we create a human level AI to reason about mathematics?

Can we create a human level AI to reason about mathematics? Without relying on informal human mathematics Relying on informal human mathematics ● No need for autoformalization (requires high ● Needs auto-formalization level of natural language understanding) ● Requires no formalization on user side ● Need to formalize the notion of ● Could learn the human notion of “ interestingness ”. “interestingness”. ● User needs to learn an “alien” language just ● Lot of training data to bootstrap from to communicate a theorem to it ● Can’t communicate its discoveries ● May be hard to bootstrap (little training data)

Vision of joint proving and auto-formalization Formal Reasoning (Neural) Language Agent Model Proof Assistant Informal Corpus Formal Corpus

Which Proof Assistant? ● Coq ● Lean ● Isabelle ● HOL4 ● HOL Light ● Mizar

Vision of joint proving and auto-formalization Formal Reasoning (Neural) Language Agent Model Proof Assistant Informal Corpus Formal Corpus

AITP'18 Trained model predicting tactic applications. - Theorems Proofs: tree of (goal, - Formal Corpus tactic) to (subgoals )

Formal Reasoning Agent Trained model predicting tactic applications. Proof Assistant - Theorems Proofs: tree of (goal, - Formal Corpus tactic) to (subgoals )

HOList An Environment for Machine Learning of Higher-Order Theorem Proving Formal Reasoning Agent APIs for ML researchers and Training data, theorem prover model, trained developers. checkpoints. Proof Assistant - Theorems (Benchmark) Later: Initial experiments, Formal Corpus results, discussion.

APIs for Theorem Prover Developers and ML Researchers (Proof) Assistant One goal/subgoal to prove One proof step: Tactic application, relevant premises Subgoals or *proved* Proof Search Ranking of tactics and premises Formal Reasoning Agent One goal/subgoal to prove Machine Learning

Proof Assistant Service RegisterTheorem ApplyTactic Register a new theorem Apply a tactic to a goal, for use as premise in potentially generating later proofs. new subgoals. ● Request: ● Request: ○ Theorem ○ Goal ○ Tactic ● Response: one of ● Response: one of ○ TheoremFingerprint ○ Subgoals ○ Error ○ Error

Proof Search Tree API ● Apply a tactic to any goal at any time. ● Controlled by any algorithm, e.g. neural algorithms. ● Automated merging of identical goals . ● On the fly tracking of: ○ Goals that are closed ○ Subgoals that can’t help closing the main goal ● Collects statistics (e.g. running time, error codes). ● Serialized as ProofLog.

Proof Search Tree

Proof Search ● Our prover: simple BFS Prover built on this tree API, with limits on branching. ○ max_top_suggestions (default: 20) ○ max_successful_branches (default: 2) ○ max_explored_nodes (default: 100) ○ max_theorem_parameters (we used: 16) ● Built on Tree API, easy to extend for more interesting proof search.

APIs for Theorem Prover Developers and ML Researchers (Proof) Assistant One goal/subgoal to prove One proof step: Tactic application, relevant premises Subgoals or *proved* Proof Search Ranking of tactics and premises Formal Reasoning Agent One goal/subgoal to prove Machine Learning

Machine Learning ● Predictions API integrating with the proof search. (Goal, Tactic ID) -> Score ○ (Goal, Premise) -> Score ○ ● Our models, experiments: more in the next talk.

APIs for Theorem Prover Developers and ML Researchers Assistant Proof Search Machine Learning - Manages the state of the Given: RegisterTheorem proof search tree. - Current goal ApplyTactic - Allows arbitrary nodes to be explored. Score: - Tactic applied - Premises used HOL-Light

Making available to researchers Benchmark Theorem Database Theorems Definitions Core 2,320 240 required for creating in-built tactics Complex 16,623 396 separated into training, validation, testing FlySpeck 10,519 1,563 for evaluating generalization

Making available to researchers Data Model ● ● Checkpoints of two-tower Proof Logs: architecture from imitation learning Synthetic proofs ○ and reinforcement learning. Human proofs ○ ● Sample training code. Proof Logs as TF Examples ● Features: ○ Goal (string) ■ Labels: ○ Tactic applied (int) ■ Premises used (string) ■

Making available to researchers Code Docker images HOL Light (with our modifications) HOL Light (server) http://github.com/ gcr.io/deepmath/hol-light brain-research/hol-light DeepHOL prover DeepHOL prover http://github.com/ gcr.io/deepmath/deephol tensorflow/deepmath

http://deephol.org Code is on GitHub. Training data, checkpoints, docker images also being made available.

HOList: An Environment for Machine Learning of Higher-Order Theorem - PowerPoint PPT Presentation

HOList: An Environment for Machine Learning of Higher-Order Theorem Proving Kshitij Bansal, Christian Szegedy Can we create a human level AI to reason about mathematics? Can we create a human level AI to reason about mathematics? Without

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