artificial intelligence in theorem proving
play

Artificial Intelligence in Theorem Proving Cezary Kaliszyk VTSA - PowerPoint PPT Presentation

Apr 13, 2023 •199 likes •1.18k views

Artificial Intelligence in Theorem Proving Cezary Kaliszyk VTSA Overview Last Lecture theorem proving problems premise selection deep learning for theorem proving state estimation Today automated reasoning learning in classical ATPs

  1. Artificial Intelligence in Theorem Proving Cezary Kaliszyk VTSA

  2. Overview Last Lecture theorem proving problems premise selection deep learning for theorem proving state estimation Today automated reasoning learning in classical ATPs learning for tableaux reinforcement learning in TP longer proofs Cezary Kaliszyk Artificial Intelligence in Theorem Proving 2 / 72

  3. What about ATPs Proof by contradiction Assume that the conjecture does not hold Derive that axioms and negated conjecture imply ⊥ Saturation Convert problem to CNF Enumerate the consequences of the available clauses Goal: get to the empty clause Redundancies Simplify or eliminate some clauses (contract) Cezary Kaliszyk Artificial Intelligence in Theorem Proving 3 / 72

  4. Calculus Resolution Cezary Kaliszyk Artificial Intelligence in Theorem Proving 4 / 72

  5. Calculus Ordered Resolution A σ strictly maximal wrt C σ and B maximal wrt D σ . Cezary Kaliszyk Artificial Intelligence in Theorem Proving 4 / 72

  6. Calculus Ordered Resolution A σ strictly maximal wrt C σ and B maximal wrt D σ . Equality axioms? Cezary Kaliszyk Artificial Intelligence in Theorem Proving 4 / 72

  7. Calculus Ordered Resolution A σ strictly maximal wrt C σ and B maximal wrt D σ . Equality axioms? Ordered Paramodulation Cezary Kaliszyk Artificial Intelligence in Theorem Proving 4 / 72

  8. Calculus Ordered Resolution A σ strictly maximal wrt C σ and B maximal wrt D σ . Equality axioms? Ordered Paramodulation ( s = t ) σ and L [ s ′ ] σ ′ maximal in their clauses. Cezary Kaliszyk Artificial Intelligence in Theorem Proving 4 / 72

  9. Completion Cezary Kaliszyk Artificial Intelligence in Theorem Proving 5 / 72

  10. Superposition Calculus Basis of E, Vampire, Spass, Prover9, ≈ Metis Cezary Kaliszyk Artificial Intelligence in Theorem Proving 6 / 72

  11. Beyond the Calculus Tautology Deletion a ∨ b ∨ ¬ a ∨ d Subsumption (forward and backward) e.g. E uses Feature Vector Indexing 0 {C3} {C3} 0 2 0 {C3, C4} {C4} {C4} 0 1 0 1 {C1, C2, C3, C4} {C2} {C2} {C2} 2 1 0 {C1} {C1} {C1} Cezary Kaliszyk Artificial Intelligence in Theorem Proving 7 / 72

  12. Still... fof(6, axiom,![X1]:![X2]:![X4]:gg(X1,sup_sup(X1,X2,X4)),file(’i/f/1/goal_138__Q_Restricted_Rewriting.qrstep fof(32, axiom,![X1]:![X2]:gg(set(product_prod(X1,X1)),transitive_rtrancl(X1,X2)),file(’i/f/1/goal_138__Q_Re fof(55, axiom,![X1]:![X19]:![X20]:(member(product_prod(X1,X1),X19,X20)=>member(product_prod(X1,X1),X19,tran fof(68, axiom,![X1]:![X5]:![X3]:![X36]:![X20]:![X37]:![X16]:(ord_less_eq(set(product_prod(X1,X3)),X36,X20)= fof(70, axiom,![X1]:![X20]:transitive_rtrancl(X1,transitive_rtrancl(X1,X20))=transitive_rtrancl(X1,X20),fil fof(74, axiom,![X1]:![X24]:![X34]:![X33]:((~(member(X1,X24,X34))=>member(X1,X24,X33))=>member(X1,X24,sup_su fof(78, axiom,![X1]:![X11]:![X13]:transitive_rtrancl(X1,sup_sup(set(product_prod(X1,X1)),transitive_rtrancl fof(79, axiom,![X1]:![X22]:![X39]:(member(X1,X22,collect(X1,X39))<=>pp(aa(X1,bool,X39,X22))),file(’i/f/1/go fof(85, axiom,![X1]:(semilattice_sup(X1)=>![X23]:![X24]:![X22]:(ord_less_eq(X1,sup_sup(X1,X23,X24),X22)<=>( fof(86, axiom,![X1]:![X11]:relcomp(X1,X1,X1,transitive_rtrancl(X1,X11),transitive_rtrancl(X1,X11))=transiti fof(98, axiom,![X1]:![X33]:![X34]:(gg(set(X1),X34)=>(ord_less_eq(set(X1),X33,X34)<=>sup_sup(set(X1),X33,X34 fof(99, axiom,![X1]:![X33]:![X34]:ord_less_eq(set(X1),X33,sup_sup(set(X1),X33,X34)),file(’i/f/1/goal_138__Q fof(100, axiom,![X3]:![X1]:supteq(X1,X3)=sup_sup(set(product_prod(term(X1,X3),term(X1,X3))),supt(X1,X3),id( fof(102, axiom,![X1]:![X34]:![X33]:ord_less_eq(set(X1),X34,sup_sup(set(X1),X33,X34)),file(’i/f/1/goal_138__ fof(103, axiom,![X1]:![X33]:![X18]:![X34]:(ord_less_eq(set(X1),X33,X18)=>(ord_less_eq(set(X1),X34,X18)=>ord fof(109, axiom,![X1]:![X34]:![X33]:(gg(set(X1),X33)=>(ord_less_eq(set(X1),X34,X33)=>sup_sup(set(X1),X33,X34 fof(114, axiom,![X1]:![X33]:![X18]:![X34]:![X48]:(ord_less_eq(set(X1),X33,X18)=>(ord_less_eq(set(X1),X34,X4 fof(116, axiom,![X1]:![X33]:ord_less_eq(set(X1),X33,X33),file(’i/f/1/goal_138__Q_Restricted_Rewriting.qrste fof(125, axiom,![X1]:![X24]:![X33]:![X34]:(member(X1,X24,X33)=>(~(member(X1,X24,X34))=>member(X1,X24,minus_ fof(127, axiom,![X1]:![X24]:![X33]:![X34]:(member(X1,X24,minus_minus(set(X1),X33,X34))=>~((member(X1,X24,X3 fof(131, axiom,![X1]:![X33]:(gg(set(X1),X33)=>collect(X1,aTP_Lamp_a(set(X1),fun(X1,bool),X33))=X33),file(’i fof(134, axiom,![X1]:(order(X1)=>![X35]:![X49]:((gg(X1,X35)&gg(X1,X49))=>(ord_less_eq(X1,X35,X49)=>(ord_les fof(136, axiom,![X1]:(preorder(X1)=>![X35]:![X49]:![X50]:(ord_less_eq(X1,X35,X49)=>(ord_less_eq(X1,X49,X50) fof(143, axiom,![X1]:![X33]:![X34]:(ord_less_eq(set(X1),X33,X34)<=>![X52]:(gg(X1,X52)=>(member(X1,X52,X33)= fof(160, axiom,![X1]:![X39]:![X35]:![X33]:(pp(aa(X1,bool,X39,X35))=>(member(X1,X35,X33)=>?[X30]:(gg(X1,X30) fof(171, axiom,![X1]:![X65]:![X66]:(pp(aa(X1,bool,aTP_Lamp_a(set(X1),fun(X1,bool),X65),X66))<=>member(X1,X6 fof(186, axiom,![X67]:semilattice_sup(set(X67)),file(’i/f/1/goal_138__Q_Restricted_Rewriting.qrsteps_comp_s Cezary Kaliszyk Artificial Intelligence in Theorem Proving 8 / 72 fof(187, axiom,![X67]:preorder(set(X67)),file(’i/f/1/goal_138__Q_Restricted_Rewriting.qrsteps_comp_supteq_s

  13. Still the search space is huge: What can we learn? What has been learned CASC: Strategies AIM: Hints Hammers: Premises What can be chosen in Superposition calculus Term ordering (Negative) literal selection Clause selection Cezary Kaliszyk Artificial Intelligence in Theorem Proving 9 / 72

  14. E-Prover given-clause loop Most important choice: unprocessed clause selection [ Schulz 2015 ] Cezary Kaliszyk Artificial Intelligence in Theorem Proving 10 / 72

  15. Learning for E: Data Collection Mizar top-level theorems [ Urban 2006 ] Encoded in FOF 32,521 Mizar theorems with ≥ 1 proof training-validation split (90%-10%) replay with one strategy Collect all CNF intermediate steps and unprocessed clauses when proof is found Cezary Kaliszyk Artificial Intelligence in Theorem Proving 11 / 72

  16. Deep Network Architectures Logistic loss Fully Connected Max Pooling (1 node) Fully Connected Conv 5 (1024) + ReLU (1024 nodes) Conv 5 (1024) + ReLU Concatenate Conv 5 (1024) + ReLU Negated conjecture Clause Embedder embedder Input token embeddings Negated conjecture Clause tokens tokens Overall network Convolutional Embedding Cezary Kaliszyk Non-dilated and dilated convolutions Artificial Intelligence in Theorem Proving 12 / 72

  17. Recursive Neural Networks Curried representation of first-order statements Separate nodes for apply , or , and , not Layer weights learned jointly for the same formula Embeddings of symbols learned with rest of network Tree-RNN and Tree-LSTM models 1 1 Relation to graphs Cezary Kaliszyk Artificial Intelligence in Theorem Proving 13 / 72

  18. Model accuracy Model Embedding Size Accuracy: 50-50% split Tree-RNN-256 × 2 256 77.5% Tree-RNN-512 × 1 256 78.1% Tree-LSTM-256 × 2 256 77.0% Tree-LSTM-256 × 3 256 77.0% Tree-LSTM-512 × 2 256 77.9% CNN-1024 × 3 256 80.3% ⋆ CNN-1024 × 3 256 78.7% CNN-1024 × 3 512 79.7% CNN-1024 × 3 1024 79.8% WaveNet-256 × 3 × 7 256 79.9% ⋆ WaveNet-256 × 3 × 7 256 79.9% WaveNet-1024 × 3 × 7 1024 81.0% WaveNet-640 × 3 × 7(20%) 640 81.5 % ⋆ WaveNet-640 × 3 × 7(20%) 640 79.9% ⋆ = train on unprocessed clauses as negative examples Cezary Kaliszyk Artificial Intelligence in Theorem Proving 14 / 72

  19. Improving Proof Search inside E Overview Select one Using a deep neural network Unprocessed Processed Clauses Clauses Superposition Problem Deep neural network evaluation is slow Slower than combining selected clause with all processed clauses 2 2 State of 2016 Cezary Kaliszyk Artificial Intelligence in Theorem Proving 15 / 72

  20. Hybrid heuristic Optimizations for performance Batching Combining TF with auto 100% 100% Pure CNN Auto Hybrid CNN WaveNet 640* 80% 80% Pure CNN; Auto WaveNet 256 Hyrbid CNN; Auto WaveNet 256* Percent unproved Percent unproved WaveNet 640 60% 60% CNN CNN* 40% 40% 20% 20% 0% 0% 10 2 10 3 10 4 10 5 10 2 10 3 10 4 10 5 Processed clause limit Processed clause limit Cezary Kaliszyk Artificial Intelligence in Theorem Proving 16 / 72

  21. Harder Mizar top-level statements Model DeepMath 1 DeepMath 2 Union of 1 and 2 Auto 578 581 674 ⋆ WaveNet 640 644 612 767 ⋆ WaveNet 256 692 712 864 WaveNet 640 629 685 997 ⋆ CNN 905 812 1,057 CNN 839 935 1,101 Total (unique) 1,451 1,458 1,712 Overall proved 7 . 4% of the harder statements Cezary Kaliszyk Artificial Intelligence in Theorem Proving 17 / 72

  22. Harder Mizar top-level statements Model DeepMath 1 DeepMath 2 Union of 1 and 2 Auto 578 581 674 ⋆ WaveNet 640 644 612 767 ⋆ WaveNet 256 692 712 864 WaveNet 640 629 685 997 ⋆ CNN 905 812 1,057 CNN 839 935 1,101 Total (unique) 1,451 1,458 1,712 Overall proved 7 . 4% of the harder statements Batching and hybrid necessary Model accuracy unsatisfactory Cezary Kaliszyk Artificial Intelligence in Theorem Proving 17 / 72

  23. ENIGMA [ Jakubuv,Urban 2017 ] Cezary Kaliszyk Artificial Intelligence in Theorem Proving 18 / 72

  24. ENIGMA [ Jakubuv,Urban 2017 ] Evaluation on AIM E’s auto-schedule: 261 Single best strategy: 239 Cezary Kaliszyk Artificial Intelligence in Theorem Proving 18 / 72

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Artificial Intelligence in Theorem Proving Cezary Kaliszyk VTSA 2019 Computer Theorem Proving

Artificial Intelligence in Theorem Proving Cezary Kaliszyk VTSA 2019 Computer Theorem Proving

Artificial Intelligence in Theorem Proving Cezary Kaliszyk VTSA 2019 Computer Theorem Proving Computer used to automate reasoning in a logic Traditionally part of artificial intelligence (not machine learning) Field of research since the

1.25k views • 101 slides
ECE 4524 Artificial Intelligence and Engineering Applications Lecture 10: Theorem Proving in

ECE 4524 Artificial Intelligence and Engineering Applications Lecture 10: Theorem Proving in

ECE 4524 Artificial Intelligence and Engineering Applications Lecture 10: Theorem Proving in Propositional Logic Reading: AIAMA 7.5 Todays Schedule: Inference and Proofs Conjunctive Normal Form Resolution Forward and Backward

658 views • 16 slides
On Theorem Proving for Program Checking Historical perspective and recent developments Maria

On Theorem Proving for Program Checking Historical perspective and recent developments Maria

Outline Introduction Where is theorem proving in program checking Inside theorem proving Big and little engines together: a new theorem proving style Decision procedures with speculative inferences Current and future challenges On Theorem

944 views • 69 slides
Visual theorem proving with the Incredible Proof Machine The idea Theorem Proving without

Visual theorem proving with the Incredible Proof Machine The idea Theorem Proving without

Interactive Theorem Proving, Nancy August 22, 2016 Joachim Breitner Karlsruhe Institute of Technology University of Pennsylvania, Philadelphia Visual theorem proving with the Incredible Proof Machine The idea Theorem Proving without Syntax

579 views • 43 slides
Deep Prolog: End-to-end Differentiable Proving in Knowledge Bases Tim Rockt aschel University

Deep Prolog: End-to-end Differentiable Proving in Knowledge Bases Tim Rockt aschel University

Deep Prolog: End-to-end Differentiable Proving in Knowledge Bases Tim Rockt aschel University College London Computer Science 2nd Conference on Artificial Intelligence and Theorem Proving 26th of March 2017 Overview Machine Learning

714 views • 40 slides
Artificial intelligence Artificial Intelligence is the science of PHILOSOPHY OF ARTIFICIAL

Artificial intelligence Artificial Intelligence is the science of PHILOSOPHY OF ARTIFICIAL

Artificial intelligence Artificial Intelligence is the science of PHILOSOPHY OF ARTIFICIAL making machines do things that would INTELLIGENCE require intelligence if done by men Marvin Minsky Prof.Dr John-Jules Meyer Weak AI

730 views • 8 slides
What is Artificial Intelligence? . . . Exactly what the computer provides is the ability not to be

What is Artificial Intelligence? . . . Exactly what the computer provides is the ability not to be

What is Intelligence? What is Artificial Intelligence? Sessions 1 &amp; 2 Introduction to AI; Planning &amp; Search CSE 592 Applications of Artificial Intelligence Henry Kautz Winter 2003 What is Artificial Intelligence? . . . Exactly what

552 views • 17 slides
Automated Theorem Proving 1/4: Introduction and Propositional Theorem Proving A.L. Lamprecht

Automated Theorem Proving 1/4: Introduction and Propositional Theorem Proving A.L. Lamprecht

Automated Theorem Proving 1/4: Introduction and Propositional Theorem Proving A.L. Lamprecht Course Program Semantics and Verfication 2020, Utrecht University September 21, 2020 Lecture Notes Automated Reasoning by Gerard A.W. Vreeswijk.

609 views • 40 slides
mdmax at School Learning Selection in Equational Reasoning Sarah Winkler University of

mdmax at School Learning Selection in Equational Reasoning Sarah Winkler University of

mdmax at School Learning Selection in Equational Reasoning Sarah Winkler University of Innsbruck 4th Conference on Artificial Intelligence and Theorem Proving 10 April 2019 1 Equational Theorem Proving 0 + x x ( x ) + x 0 x + (

761 views • 58 slides
Automated Theorem Proving 2/4: First-Order Theorem Proving A.L. Lamprecht Course Program

Automated Theorem Proving 2/4: First-Order Theorem Proving A.L. Lamprecht Course Program

Automated Theorem Proving 2/4: First-Order Theorem Proving A.L. Lamprecht Course Program Semantics and Verfication 2020, Utrecht University September 23, 2020 Lecture Notes Automated Reasoning by Gerard A.W. Vreeswijk. Available for

1.03k views • 54 slides
Theorem-Proving Environments Nathan Ng CSC2547: Learning to Search Theorem Proving What is a

Theorem-Proving Environments Nathan Ng CSC2547: Learning to Search Theorem Proving What is a

Theorem-Proving Environments Nathan Ng CSC2547: Learning to Search Theorem Proving What is a theorem? Statement proven based on basis of previously established statements Premise: If I attend UofT, I am a student Premise: I attend

375 views • 25 slides
Applying Formal Verification to Reflective Reasoning R. Kumar 1 B. Fallenstein 2 1 Data61, CSIRO

Applying Formal Verification to Reflective Reasoning R. Kumar 1 B. Fallenstein 2 1 Data61, CSIRO

Applying Formal Verification to Reflective Reasoning R. Kumar 1 B. Fallenstein 2 1 Data61, CSIRO and UNSW ramana@intelligence.org 2 Machine Intelligence Research Institute benya@intelligence.org Artificial Intelligence for Theorem Proving,

1.43k views • 47 slides
AI Artificial Intelligence Definition artificial intelligence / rd

AI Artificial Intelligence Definition artificial intelligence / rd

20191008 AI Artificial Intelligence Definition artificial intelligence / rd fiSH l in tel j ns/ noun the theory and development of computer systems able to perform tasks that normally require human

1.15k views • 28 slides
Artificial Intelligence Artificial Intelligence Artificial Intelligence Study and design of

Artificial Intelligence Artificial Intelligence Artificial Intelligence Study and design of

Artificial Intelligence Artificial Intelligence Artificial Intelligence Study and design of machines that can think like a human being Attempt to understand human intelligence Artificial Intelligence But how do you define intelligence?

516 views • 15 slides
1.1 What is AI? 1. What is Artificial Intelligence? 2. AI Past and Present 3. Rational

1.1 What is AI? 1. What is Artificial Intelligence? 2. AI Past and Present 3. Rational

Foundations of Artificial Intelligence February 17, 2020 1. Introduction: What is Artificial Intelligence? 1.1 What is AI? Foundations of Artificial Intelligence 1. Introduction: What is Artificial Intelligence? 1.2 Acting Humanly 1.3

623 views • 6 slides
Health Warning Principles, Techniques, Applications Theorem Proving is addictive Gerwin Klein

Health Warning Principles, Techniques, Applications Theorem Proving is addictive Gerwin Klein

W HAT YOU WILL LEARN how to use a theorem prover background, how it works how to prove and specify NICTA Advanced Course Slide 1 Slide 3 Theorem Proving Health Warning Principles, Techniques, Applications Theorem Proving is

262 views • 9 slides
Mesoscale and submesoscale variability and biogeochemical interactions Annalisa Bracco Georgia

Mesoscale and submesoscale variability and biogeochemical interactions Annalisa Bracco Georgia

Mesoscale and submesoscale variability and biogeochemical interactions Annalisa Bracco Georgia Institute of Technology School of Earth and Atmospheric Sciences Part 1 Mesoscale (eddy) dynamics and biogeochemical processes Outline n

472 views • 31 slides
Algorithms in Nature SOP &amp; MIS Maximal Independent Set (MIS) A fundamental problem in

Algorithms in Nature SOP &amp; MIS Maximal Independent Set (MIS) A fundamental problem in

Algorithms in Nature SOP &amp; MIS Maximal Independent Set (MIS) A fundamental problem in distributed computing: - Often required to establish wireless networks - Used for routing messages, grouping sensors etc. For a set of nodes

749 views • 29 slides
Nutrition and Bone: Protein, Sodium Chloride, and Other Nutrients Deborah E. Sellmeyer, MD

Nutrition and Bone: Protein, Sodium Chloride, and Other Nutrients Deborah E. Sellmeyer, MD

Nutrition and Bone: Protein, Sodium Chloride, and Other Nutrients Deborah E. Sellmeyer, MD Associate Professor of Medicine Johns Hopkins School of Medicine Acid Base Balance and Skeletal Health Acid precursors: sulfur containing amino acids

502 views • 13 slides
Bellerophon: Tactical Theorem Proving for Hybrid Systems Nathan Fulton , Stefan Mitsch, Brandon

Bellerophon: Tactical Theorem Proving for Hybrid Systems Nathan Fulton , Stefan Mitsch, Brandon

Bellerophon: Tactical Theorem Proving for Hybrid Systems Nathan Fulton , Stefan Mitsch, Brandon Bohrer, Andr Platzer Carnegie Mellon University Cyber-Physical Systems Cyber-Physical Systems combine computation and control. Hybrid Systems

964 views • 68 slides
Proving Uniformity and Independence by Self-Composition and Coupling Gilles Barthe Thomas

Proving Uniformity and Independence by Self-Composition and Coupling Gilles Barthe Thomas

Proving Uniformity and Independence by Self-Composition and Coupling Gilles Barthe Thomas Espitau Benjamin Grgoire Justin Hsu* Pierre-Yves Strub 1 A puzzle A random walk on a cycle Start at position s { 0 , 1 , . . . , n 1 }

755 views • 57 slides
Proving the Viability of RDF as a Universal Healthcare Exchange Language David Booth, Ph.D.

Proving the Viability of RDF as a Universal Healthcare Exchange Language David Booth, Ph.D.

- DRAFT - Proving the Viability of RDF as a Universal Healthcare Exchange Language David Booth, Ph.D. Latest version of these slides: http://dbooth.org/2014/proving/ See also associated paper Imagine a world in which all healthcare systems

1.39k views • 138 slides
Data Refinement: model-oriented proof methods and their comparison Willem-Paul de Roever

Data Refinement: model-oriented proof methods and their comparison Willem-Paul de Roever

Data Refinement: model-oriented proof methods and their comparison Willem-Paul de Roever University of Kiel, Germany SYNCHRON 2003 Marseille-Luminy, France December 15, 2003 1 Overview Refinement Data refinement Simulation

964 views • 58 slides
! Proving!(or!Attacking)!Lost!Profits! Damages!in!Trade!Secrets!and!Other!Cases! !

! Proving!(or!Attacking)!Lost!Profits! Damages!in!Trade!Secrets!and!Other!Cases! !

! Proving!(or!Attacking)!Lost!Profits! Damages!in!Trade!Secrets!and!Other!Cases! ! Texas!Bar!CLE!!Live!Webcast! ! ! ! ! Kelly!Leonard! ! ! Strasburger*&amp;*Price* ! Moderator:!Zach!Wolfe! ! ! Fleckman*&amp;*McGlynn !

515 views • 16 slides

More recommend