Benchmarking Linear Logic Proofs
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This document discusses benchmarking theorem proving in linear logic. It describes collecting theorems in linear logic to test various provers and logic variants. An initial collection of 61 intuitionistic logic theorems were translated into linear logic using 4 different translations, resulting in 244 test problems. The document also discusses generating theorems in linear logic programmatically using type inference to produce over 8 billion theorems for training neural networks as theorem provers. The goal is to better understand differences between logics and provers through benchmarking and experimentation.
Benchmarking Linear Logic Proofs
- 1. 1/33 Introduction Proving Theorems Generating Theorems Learning Theorems Benchmarking Linear Logic Proofs Valeria de Paiva Topos Institute, Berkeley, CA Visiting DI, PUC-RJ, RJ November, 2020 Valeria de Paiva ProofTheoryOnline2020
- 2. 2/33 Introduction Proving Theorems Generating Theorems Learning Theorems Thanks Tom and Anton for the invitation! Valeria de Paiva ProofTheoryOnline2020
- 3. 3/33 Introduction Proving Theorems Generating Theorems Learning Theorems Thank you friends for our continued collaboration! Valeria de Paiva ProofTheoryOnline2020
- 4. 4/33 Introduction Proving Theorems Generating Theorems Learning Theorems Based on The ILLTP Library for Intuitionistic Linear Logic, Linearity/TLLA 2018 (with Carlos Olarte, Elaine Pimentel and Gisele Reis) and Deriving Theorems in Implicational Linear Logic, Declaratively, ICLP 2020 and Training Neural Networks as Theorem Provers via the Curry-Howard Isomorphism, Computational Logic and Applications 2020 (with Paul Tarau) Valeria de Paiva ProofTheoryOnline2020
- 5. 5/33 Introduction Proving Theorems Generating Theorems Learning Theorems Motivations benchmark: “To measure the performance of (an item) relative to another similar item in an impartial scientific manner.” Benchmarks for theorem provers well-developed area of AI Since Logic Theorist (LT) 1956 Newell and Simon “the first artificial intelligence program”, proved 38 of the first 52 theorems of Principia Mathematica Valeria de Paiva ProofTheoryOnline2020
- 6. 6/33 Introduction Proving Theorems Generating Theorems Learning Theorems Motivation Many theorem provers: interactive and automatic Since 1993 TPTP: Thousands of Problems for Theorem Provers (http://www.tptp.org/) also CASC competition (CADE ATP System Competition) http://www.tptp.org/CASC/ not much for non-classical logics intuitionistic logic ILTP (http://www.iltp.de/) and some collections of modal problems including QMLTP (http://www.iltp.de/qmltp/) where is Linear Logic? Valeria de Paiva ProofTheoryOnline2020
- 7. 7/33 Introduction Proving Theorems Generating Theorems Learning Theorems Goals Collect problems/theorems in (a fragment of) Linear Logic Investigate variants of the logic Investigate variants of translations between logics Use provers/benchmarks/ML as tools for experiments in logic Logic as a Lab Science! Valeria de Paiva ProofTheoryOnline2020
- 8. 8/33 Introduction Proving Theorems Generating Theorems Learning Theorems Linear Logic: a tool for semantics A proof theoretic logic described by Girard in 1986. Basic idea: assumptions cannot be discarded or duplicated. They must be used exactly once – just like dollar bills (except when they’re marked by a modality “!”) Other approaches to accounting for logical resources before. Relevance Logic Great win for Linear Logic: Account for resources when you want to, otherwise fall back to traditional logic via translation A → B iff !A −◦ B Valeria de Paiva ProofTheoryOnline2020
- 9. 9/33 Introduction Proving Theorems Generating Theorems Learning Theorems Linear Logic In Linear Logic formulas denote resources. Resources are premises, assumptions and conclusions, as they are used in logical proofs. For example: $1 −◦ latte If I have a dollar, I can get a Latte $1 −◦ cappuccino If I have a dollar, I can get a Cappuccino $1 I have a dollar Can conclude either latte or cappuccino — But using my dollar and one of the premisses above, say $1 −◦ latte gives me a latte but the dollar is gone — Usual logic doesn’t pay attention to uses of premisses, A implies B and A gives me B but I still have A... Valeria de Paiva ProofTheoryOnline2020
- 10. 10/33 Introduction Proving Theorems Generating Theorems Learning Theorems Linear Implication and (Multiplicative) Conjunction Traditional implication: A, A → B B A, A → B A ∧ B Re-use A Linear implication: A, A −◦ B B A, A −◦ B A ⊗ B Cannot re-use A Traditional conjunction: A ∧ B A Discard B Linear conjunction: A ⊗ B A Cannot discard B Of course!: !A !A⊗!A Re-use !(A) ⊗ B B Discard Valeria de Paiva ProofTheoryOnline2020
- 11. 11/33 Introduction Proving Theorems Generating Theorems Learning Theorems Linear Logic Results Soundness and Completeness for coherence spaces and many other interesting models ”Have-your-cake-and-eat-it”theorem: Intuitionistic logic proves A → B iff Linear Logic proves !A −◦ B. A new graphical Natural Deduction system: proof nets A new style of proof systems: focused systems Some 30 years of limelight, especially in CS Valeria de Paiva ProofTheoryOnline2020
- 12. 12/33 Introduction Proving Theorems Generating Theorems Learning Theorems Why Bother with Benchmarks? Linear Logic has come of age useful in programming languages, game semantics, quantum physics, linguistics Several Provers? maybe should discuss adequacy or efficiency? Nah! Because it can help us understand the logic, where it differs or not from traditional systems Valeria de Paiva ProofTheoryOnline2020
- 13. 13/33 Introduction Proving Theorems Generating Theorems Learning Theorems (some) Linear Logic Provers LLTP (Maude) https://github.com/carlosolarte/Linear-Logic-Prover-in- Maude LL prover (http://bach.istc.kobe-u.ac.jp/llprover/) linTAP http://www.leancop.de/lintap/ Otten et al LL prover explorer, Lolli, etc YALLA (Coq) O. Laurent https://perso.ens-lyon.fr/olivier.laurent/yalla/ Valeria de Paiva ProofTheoryOnline2020
- 14. 14/33 Introduction Proving Theorems Generating Theorems Learning Theorems Choosing Logic Problems To be used as a standard a collection of problems should satisfy some criteria. at least the following: Formulae should able to distinguish different characteristics of the logical systems and provers (design choice points) important theorems and paradigmatic formulae should be present (how do we know?) Should be large enough so that we can do comparisons between different provers and systems (Not taken in consideration here) automatic comparison scripts and efficiency timings should be computed by third parties Valeria de Paiva ProofTheoryOnline2020
- 15. 15/33 Introduction Proving Theorems Generating Theorems Learning Theorems Design choices Classical or Intuitionistic Linear Logic? FILL? differences in provability is there a set of “principal”LL theorems? Easy place to find LL theorems: Intuitionistic Logic Use original theorem, everything provable in IL is provable in LL using Girard’s translation but hey, there are other (many) translations which one to choose and why? new use of computational provers and comparisons: to help to clarify the theory Valeria de Paiva ProofTheoryOnline2020
- 16. 16/33 Introduction Proving Theorems Generating Theorems Learning Theorems Our choices starting: Kleene “Introduction to Metamathematics”1952 a basic collection of intuitionistic theorems minimal set of intuitionistic theorems that a sound prover should be able to derive helpful to uncover bugs and sources of unsoundness (historical note: with Sara Kalvala “Linear Logic in Isabelle”, 1995. sequent calculus in Isabelle, deprecated now) Valeria de Paiva ProofTheoryOnline2020
- 17. 17/33 Introduction Proving Theorems Generating Theorems Learning Theorems Rudimentary Intuitionistic Logic Rudimentary fragment of Intuitionistic Logic, ie (→, ∧, ¬)-fragment of IL. Why this fragment? Intuitionistic disjunction poses some problems in LL. Additive or multiplicative disjunction? Each way leads to very different systems. Concentrate on easy cases first, hence “rudimentary”IL. This gives us some 61 theorems from Kleene’s book (next slide) problems in Giselle’s repo https://github.com/meta-logic/lltp Valeria de Paiva ProofTheoryOnline2020
- 18. 18/33 Introduction Proving Theorems Generating Theorems Learning Theorems Kleene Examples Valeria de Paiva ProofTheoryOnline2020
- 19. 19/33 Introduction Proving Theorems Generating Theorems Learning Theorems Translations? IL theorems are not necessarily theorems in LL. Need translations. Which translations? Four ‘translations’ 1 Girard: (A → B)G = !A −◦ B 2 “Lazy”(A → B)K = !(A −◦ B) 3 Liang/Miller’s 0-1 4 Forgetful = read → as −◦ Last is not provability preserving! First experiment: 61 theorems (IL) multiplied by 4 gives us 244 ’problems’ to check in Linear Logic Give me an automated theorem prover, please! Valeria de Paiva ProofTheoryOnline2020
- 20. 20/33 Introduction Proving Theorems Generating Theorems Learning Theorems Results Olarte implemented a basic prover for IL as well as for ILLF and LLF (all focused systems) specified in Rewriting Logic and implemented in Maude (Meseguer) proofs of the original IL sequent, together with the derivation trees of the corresponding ILL sequents, when provable 22 sequents are not provable in ILL Obtained a collection of basic tests for LL Extended this basic collection using (translations of) problems from ILTP and from reachability of Petri Nets Ended up with 4,494 formulas in our ILLTP library Some comparison of translations, but need more! Valeria de Paiva ProofTheoryOnline2020
- 21. 21/33 Introduction Proving Theorems Generating Theorems Learning Theorems Results Valeria de Paiva ProofTheoryOnline2020
- 22. 22/33 Introduction Proving Theorems Generating Theorems Learning Theorems What have we got? proposed a set of Linear Logic sequent tests, and have implemented provers for logics LJ, ILL and CLL. All three provers are focused-based As a first experiment we tried 4 translations and we can empirically compare times for them Can we see patterns on these timeouts and failures? Does this help when comparing translations? Would like to compare with other translations, e.g. LKtq Would like to have parallel/ensemble provers, perhaps how can we push this work forward? Valeria de Paiva ProofTheoryOnline2020
- 23. 23/33 Introduction Proving Theorems Generating Theorems Learning Theorems Changing Gears Lukewarm reception at FLoC2018: not enough interest in the benchmarks from theorem provers (mostly they’re interested in specific applications, like automating proofs in logic itself – several embeddings of LL into Coq) Co-authors wanted to do different things Tarau’s papers “Formula Transformers and Combinatorial Test Generators for Propositional Intuitionistic Theorem Provers”and ”Combinatorial Testing Framework for Intuitionistic Propositional Theorem Provers”(2019) Promised test generators and formula readers automatically converting the ‘human-made’ tests of the ILTP library to Prolog and Python form. Mentioned a testing framework focusing on the implicational fragment of intuitionistic logic I asked if we could do the same for Linear Logic Valeria de Paiva ProofTheoryOnline2020
- 24. 24/33 Introduction Proving Theorems Generating Theorems Learning Theorems Building theorems This is the work presented on ICLP2020 Main insight of work: instead of building formulas of a logic and trying to prove whether they are theorems or not, with given rules, build only the provable linear ones True by construction and provably so, using the Curry-Howard isomorphism how? generate all theorems of a given size in LL use a low polynomial type inference algorithm associating a type (when it exists) to a lambda term rely on the Curry-Howard isomorphism, to generate simply typed lambda terms in normal form Valeria de Paiva ProofTheoryOnline2020
- 25. 25/33 Introduction Proving Theorems Generating Theorems Learning Theorems What do we get? an implicational intuitionistic logic prover specialized for linear formulas a big dataset for training neural networks that prove intuitionistic theorems preliminary results: very high success rate with seq2seq encoded LSTM neural networks (not in the paper https://arxiv.org/pdf/2009.10241.pdf) intuition: use combinatorial generation of lambda terms + type inference (easy) to solve some type inhabitation problems (hard). Valeria de Paiva ProofTheoryOnline2020
- 26. 26/33 Introduction Proving Theorems Generating Theorems Learning Theorems How do we go about it? implicational linear formulas (Prolog code), built as: binary trees of size N, counted by Catalan numbers Catalan(N) labeled with variables derived from set partitions counted by Bell(N + 1) (see A289679 in OEIS) linear lambda terms (proof terms for the implicational formulas) linear skeleton Motzkin trees (binary-unary trees with constraints enforcing one-to-one mapping from variables to their lambda binders) closed linear lambda terms after a chain of refinements, we derive a compact and efficient generator for pairs of linear lambda terms in normal form and their types (which always exist as they are all typable!) Voil´a! almost 8 billion theorems in a few hours Valeria de Paiva ProofTheoryOnline2020
- 27. 27/33 Introduction Proving Theorems Generating Theorems Learning Theorems Want to machine learn tautologies? (Tarau’s Training Neural Networks as Theorem Provers via the Curry-Howard Isomorphism at CLA 2020 https://www.youtube.com/channel/UCk0-_Hgr_o3KbRQWdeoYIHA) Can we train neural networks to learn inference on an interesting logic? Yes! our logic is implicational intuitionistic linear logic. We need to derive an efficient algorithm requiring a low polynomial effort per generated theorem and its proof term. Outcomes: implicational intuitionistic linear logic prover a dataset for training neural networks high success rate seq2seq encoded LSTM neural nets open: can techniques extend to harder (e.g., PSPACE-complete) logics? Valeria de Paiva ProofTheoryOnline2020
- 28. 28/33 Introduction Proving Theorems Generating Theorems Learning Theorems Machine learning tautologies Several versions for Boolean logic, e.g. Can Neural Networks Understand Logical Entailment? Evans, Saxton, Amos, Kohli, and Grefenstette, 2018. arXiv:1802.08535 (Automated proof synthesis for propositional logic with deep neural networks, arxiv 1805.11799 for intuitionistic logic) not the same ML-problem implicational fragment of LL is decidable Curry-Howard holds: linear lambda-calculus corresponds to implicational only linear logic polynomial algorithms for generating its theorems are useful: when turned into test sets, combining tautologies and their proof terms can be useful for testing correctness and scalability of linear logic theorem provers when turned into datasets, they can be used for training deep learning networks focusing on neuro-symbolic computations Valeria de Paiva ProofTheoryOnline2020
- 29. 29/33 Introduction Proving Theorems Generating Theorems Learning Theorems Machine learning tautologies can use combinatorial generation of lambda terms + type inference (easy) to solve some type inhabitation problems (hard) GOOD NEWS: there’s a size-preserving bijection between linear lambda terms in normal form and their principal types! a proof follows immediately from a paper by Noam Zeilberger who attributes this observation to Grigori Mints the bijection is proven by exhibiting a reversible transformation of oriented edges in the tree describing the linear lambda term in normal form, into corresponding oriented edges in the tree describing the linear implicational formula, acting as its principal type obtained a generator for all theorems of implicational linear intuitionistic logic of a given size, as measured by the number of lollipops, without having to prove a single theorem! Valeria de Paiva ProofTheoryOnline2020
- 30. 30/33 Introduction Proving Theorems Generating Theorems Learning Theorems The datasets the dataset containing generated theorems and their proof-terms in prefix form (as well as their LaTeX tree representations marked as Prolog “%” comments) is available at http://www.cse.unt.edu/~tarau/datasets/lltaut/ it can be used for correctness, performance and scalability testing of linear logic theorem provers the <formula, proof-term> pairs in the dataset are usable to test deep-learning systems on theorem proving tasks also, formulas with non-theorems added for IPILL Valeria de Paiva ProofTheoryOnline2020
- 31. 31/33 Introduction Proving Theorems Generating Theorems Learning Theorems Can Neural Nets help Theorem Proving? We search for good frameworks for neuro-symbolic computing theorem provers are computation-intensive, sometimes Turing-complete search algorithms two ways neural networks can help: fine-tuning the search, by helping with the right choice at choice points used via an interface to solve low-level ‘perception’-intensive tasks can we simply replace the symbolic theorem prover given a large enough training dataset? do we want to? Valeria de Paiva ProofTheoryOnline2020
- 32. 32/33 Introduction Proving Theorems Generating Theorems Learning Theorems Evaluating the Performance of our Neural Nets as Theorem Provers in fact, our seq2seq LSTM recurrent neural network trained on encodings of theorems and their proof-terms performs unusually well the experiments with training the neural networks using the IPILL and IIPC theorem dataset are available at: https://github.com/ptarau/neuralgs the < formula, proof term > generators are available at: https://github.com/ptarau/TypesAndProofs the generated datasets are available at: http://www.cse.unt.edu/~tarau/datasets/ Valeria de Paiva ProofTheoryOnline2020
- 33. 33/33 Introduction Proving Theorems Generating Theorems Learning Theorems Conclusions Yeah, this was too fast for me too! But it works and it’s based on Mints’ and Zeiberger’ results for Intuitionistic Logic, which are simplified in the Linear implicational case. Future work is to see if we can extend it to Linear-non-Linear Logic. To see if we can learn many more theorems. THANKS! Valeria de Paiva ProofTheoryOnline2020