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Edwardian Proofs as Futuristic Programs


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Keynote at Infinite Possibilities Conference, Baltimore, MA, March 2012.

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Edwardian Proofs as Futuristic Programs

  1. 1. + Edwardian Proofs as Futuristic Programs Valeria de Paiva Rearden Commerce, Foster City, CA, USA
  2. 2. + outline   Personal history   Unity behind disparity:   Categorical logic   Curry-Howard Correspondence   Learning to enjoy change
  3. 3. + Personal and Inspiring?...
  4. 4. + PUC—Rio de Janeiro Dept Maths, BA and MA (Algebra)
  5. 5. + University of Cambridge DPMMS 1984-88, Lab1989-1995
  6. 6. + University of Birmingham, UK Computer Science 1996-2000
  7. 7. Xerox PARC NLTT: Natural+ Language Theory and Technology October 2000-May 2008
  8. 8. + Adventures in Searchland PARC Forum   Valeria de Paiva, Cuil, Search Analyst   30 July 2009 4:00-5:00pm George E. Pake Auditorium, PARC, Palo Alto, CA map/ directions  934/adventures-in- searchland.html   Video and audio
  9. 9. + Launching a hot product…
  10. 10. + After the hype, the blogsphere….
  11. 11. + The reasons for Cuil   There is (too much) information on the web.   Cuil organizes the web so that you can find information that you didnt know you wanted..
  12. 12. + The reasons are still there…   Reports estimate we can see only 15% of the existing web.   Probing the web is mostly popularity based. Youre likely to see what others have seen before. But your seeing increases the popularity of what you saw, thereby reducing the pool of available stuff.   Vicious or virtuous circle? How to measure?   Eli Pariser, The Filter Bubble, book and TED talk,
  13. 13. + Rearden Commerce? What’s that? ...
  14. 14. Women in Engineering+ and diversity… Diwali celebration 2011
  15. 15. + Inspiration? Universidade de Cambridge DPMMS
  16. 16. + Unity Behind Disparity: Proofs are Programs?...   It’s no secret that the bulk of mathematics, as we know it today got crystallized in the last years of the 19th century, first years of the 20th century.   The shock is still being felt. Notices of the AMS Jan 2012: A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today Frank Quinn   Very controversial…   Interested today in two aspects: notions of Proofs and the birth of Algebra…
  17. 17. + The easy one: Edwardian Algebra?  (Abstract) Algebra: area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras generally...  Abstract (instead of elementary) algebra: turn of the 20th century, van der Waerden first textbook Modern Algebra, 1930.  Sources: linear algebra, permutation groups, diophantine equations,...
  18. 18. + Category Theory: 1945-now   Basic
concepts/ theories.
   Topological

   Morphisms


   Detractors
   The

  19. 19. + Categories: a picture
  20. 20. + Category Theory   Category: a collection of objects and of morphisms, satisfying obvious laws   Functors: the natural notion of morphism between categories   Natural transformations: the natural notion of morphisms between functors   Constructors: products, sums, limits, duals....   Adjunctions: an abstract version of equality   Does this relate to logic?
  21. 21. + Back to Proofs? Mathematical Logic divided in: Model theory, Proof theory, Recursion theory and Set theory Category theory used for Model theory and Proof theory Proof theory is Hilberts brainchild: proofs as objects of mathematical study Famous address in 1900 to ICM, 23 problems that shaped Mathematics “…in mathematics there is no ignorabimus,"
  22. 22. + Which Proofs?   Mathematics in turmoil in the turn of the century because of paradoxes e.g. Russell’s   Hilbert’s Program: Base all of mathematics in finitistic methods   Proving the consistency of Arithmetic the big quest   Read the graphic novel!!
  23. 23. + Edwardian Proofs? Frege  One of the founders of modern symbolic logic put forward the view that mathematics is reducible to logic.  Begriffsschrift, 1879  Was the first to write proofs using a collection of abstract symbols
  24. 24. + Proofs of the 20th century? Gentzen Proofs   To prove the consistency of Arithmetic Gentzen invented his systems of   NATURAL DEDUCTION how mathematicians think…   SEQUENT CALCULUS how he could formalize the thinking to obtain the main result he needed, his Hauptsatz. (1934)
  25. 25. + Church and lambda-calculus  Alonzo Church introduced the lambda calculus in 1932.  By 1936 Church had realized that lambda terms could be used to express every function that could ever be computed by a machine.  Instead of saying ‘the function f where f(x) = t’, he simply wrote λx. t.  The lambda calculus is an universal programming language.  The Curry-Howard correspondence led logicians and computer scientists to develop a cornucopia of new logics based on the correspondence between proofs and programs.
  26. 26. Curry-Howard correspondence?Basic idea:types are formulae/objects in appropriate category,terms are proofs/morphisms in same category, logical constructors are appropriate categorical constructions. Most important: Reduction is proof normalizationOutcome: Transfer results/tools from logic to CT to CS 
  27. 27. + Curry-Howard Correspondence Triangles Lambda- calculus Cartesian Intuitionistic 1965 1963 Closed Propositional Categories Logic
  28. 28. + Curry-Howard Correspondence Triangles Functional Programming Category Proof Theory TheoryA long time in the making: Curry analogy, Howard (1969), published 1981, book 1986 ..
  29. 29. Categorical Proof Theorymodels derivations/proofs, not whether theoremsare true or notProofs definitely first-class citizensHow?Uses extended Curry-Howard isomorphismWhy is it good? Modeling derivations useful inlinguistics, functional programming, compilers..Why is it important? Widespread use of logic/algebra in CS means new important problems tosolve with our favorite tools, as well as jobs formathematicians & logicians.Why so little impact on math itself?
  30. 30. + Hey, where are my futuristic programs? “Time and again, a logician motivated by logical concerns and a computer scientist motivated by practical concerns have discovered exactly the same type system, usually with the logician getting there first. Most modern functional languages use the Hindley-Milner type system, discovered by the logician Hindley in 1969 and re-discovered by the computer scientist Milner in 1978.”   Prof P. Wadler, University of Edinburg, “Proofs are Programs: 19th Century Logic and 21st Century Computing”, Avaya 2000
  31. 31. Successes/Opportunities ofCategorical Semantics+Models for the untyped lambda-calculus+Typed programming language & polymorphism+Dependent Type Theory+Operational Semantics & Full abstraction results+Game Semantics-Proof theory of Classical Logic/Modal logics-Effect full computation, mobile/grid computing, etc-Practical impact-Acceptance of methodology in math
  32. 32. + Languages & provers   StandardML Cornell   HOL   Haskell   Isabelle   O’Caml   Twelf, Elf,…   Scala   Coq   F#   TAL, TIL, PCC Warning: FP In the Real world…
  33. 33. + How do I get into this? The same picture...
  34. 34. + ...different logic Linear Logic, a proof theoretic logic described by Jean-Yves Girard in 1986. Basic idea: assumptions cannot be discarded or duplicated. They must be used exactly once—just like dollar bills... Other approaches to accounting for logical resources. Great win of Linear Logic: Account for resources when you want to, otherwise fall back on traditional logic, A=>B iff !A –o B
  35. 35. + My example: Dialectica categories Nice version of Curry Howard correspondence for Intuitionistic LINEAR LOGIC in 1987 Why is it nice? Godels Dialectica interpretation (1958)
  36. 36. + Dialectica categories   Goal: to be as constructive as possible while being able to interpret all of classical arithmetic   De Paiva 1987 „Dialectica Categories in AMS Boulder Meeting, vol 92: An internal categorical version of the Dialectica, modelling the main connective, implication
  37. 37. + What is the point?  First, the construction ends up as a model of Linear Logic, instead of constructive logic. Then it allows us to see where the assumptions in Godels argument are used  It justifies linear logic in terms of more traditional logic and conversely explains the more traditional work in terms of a modern (linear, resource conscious) decomposition of concerns.  Theorems(87/89): Dialectica categories provide models of linear logic as well as an internal representation of the dialectica interpretation. Modeling the exponential ! Is hard, first model to do it.
  38. 38. + Dialectica categories: Very pretty: models of Intuitionistic and classical linear logic and special connectives that allow us to get back to usual logic Extended it in a number of directions—a robust proof can be pushed in many ways. In CS as a model of Petri nets(> 2 phds), it has a non-commutative version for Lambek calculus (linguistics), it has been used as a model of state (Correa et al) and even of quantum groups. Also used for generic models of Linear Logic (with Schalk04) and for Linguistics Analysis of the syntax-semantics interface, the Glue Approach (Dalrymple, Lamping and Gupta).
  39. 39. + Bridges?Translation
  40. 40. + NOT TOY BRIDGES …Categorical Logic, DialecticaCategories and Their ApplicationsAutomated Theorem Proving andSemantics of Programming LanguagesLinear Functional Programming,Abstract Machines and Modal TypeTheoriesLogics for NLP, for Linguistic Inferenceand for Contexts in AICombining symbolic methods andanalytics to solve big data problems
  41. 41. + 20 years later   Extension to higher-order logic via topos theory, Bodil Biering phd thesis (2008),   The Copenhagen interpreatation   also connections between different functional interpretations (Oliva, Gernest and Trifonov),via Shirahatas work (TAC vol 17).
  42. 42. + Working in interdisciplinary areas is hard, but rewarding.Taking Stock   The frontier between logic, computing, linguistics and categories is a fun place to be.   Mathematics teaches you a way of thinking, more than specific theorems.   Barriers: proprietary software and unwillingness to `waste time’ on formalizations   Enablers: international scientific communities, open access, NLP and other open software, growing interaction between fields,…   Handsome payoff expected…   Fall in love with your ideas and enjoy talking to many about them...
  43. 43. + NASSLI 2012: Come to Texas for the fun… The Fifth North American Summer School of Logic, Language, and Information at the University of Texas at Austin Johan van Benthem University of Amsterdam / Stanford University Twenty interdisciplinary Logical Dynamics of Information and Interaction graduate-level courses, 90 minutes each, every day for five Craige Roberts Ohio State University days, on the UT campus, Questions in Discourse taught by leading international Noah Goodman Stanford University scholars, and crossing the interfaces of philosophy, Stochastic Lambda Calculus and its Applications in Semantics and Cognitive Science linguistics, computer science, psychology, statistics, and Mark Steedman University of Edinburgh logic. Combinatory Categorial Grammar: Theory and Practice Chris Potts Stanford University Extracting Social meaning and Sentiment June 18–22, 2012 L L I Registration: $175 (academic rate) / $400 (professional rate) Student scholarships available, see website for application instructions Accommodation provided for $70 / night (single) or $35 / night (double) Additional Courses: Special Events: Catherine Legg - University of Waikato Jonathan Ginzburg - University of Paris June 16–17: Bootcamp Session Possible Worlds: A Course in Metaphysics Robin Cooper - Göteborg University June 23–24: Texas Linguistic Society Conference (for Computer Scientists and Linguists) Type theory with records for natural language Special sessions on American Sign Language, Adam Lopez - Johns Hopkins University semantics Semantics, and Computational Linguistics Statistical Machine Translation Jeroen Groenendijk - University of Amsterdam Details regarding Call for Papers at Floris Roelofsen - University of Amsterdam June 23: Turing Centennial Symposium Eric Pacuit - Stanford University Social Choice Theory for Logicians Inquisitive semantics Valeria de Paiva - Rearden Commerce Shalom Lappin - King’s College London Ulrik Buchholtz - Stanford University Alternative Paradigms for Computational Semantics Introduction to Category Theory Tandy Warnow - University of Texas Adam Pease - Rearden Commerce Estimating phylogenetic trees in linguistics and Ontology Development and Application with biology SUMO Hans Kamp - University of Texas Ede Zimmermann - University of Frankfurt Mark Sainsbury - University of Texas Intensionality Vagueness and context Thomas Icard - Stanford University Steve Wechsler - University of Texas Surface Reasoning Eric McCready - Osaka University Workshop on Meaning as Use: Indexicality and Nina Gierasimczuk - University of Groningen Expressives (Speakers: Eric McCready, Steve Belief Revision Meets Formal Learning Theory Wechsler, Hans Kamp, Chris Potts, Pranav Anand, and Sarah Murray)NASSLLI is sponsored by the NSF (BCS 1019206), the UT College of Liberal Arts, and the UT Departments of Linguistics, Philosophy, and Psychology. Poster by Derya Kadipasaoglu ( and Christopher Brown
  44. 44. + Thanks!
  45. 45. + Some references   Godels Collected Works, eds Feferman and Dawson   Categories for the Working Mathematician Saunders MacLane   The Curry-Howard Correspondence ed de Groote, Galliers paper   Proofs and Types – Jean-Yves Girard 
  46. 46. + Projects, Forums: a small sample  CLiCS: Categorical Logic in Computer Science, I and II from Oct 89-Dec95, report:  APPSEM I and II: Apr 1998-Mar 2002 and Jan03-Dec05  LINEAR: May 1998-Apr2002  CMU/Penn Manifest Security  Grey project, ZAP(Princeton) 06-10  LOGOSPHERE: digital libraries/mathematical knowledge  TRUST – Stanford Security Lab
  47. 47. + Applications of categories in CS   semantics/design of programming languages: -  Lisp, ML, OCamL, Haskell, (Lua?), etc.   Compilers, proving optimizations in code, ...   categorical semantics of specification (work on Institutions, )   security, (CMU, Microsoft, etc)   concurrency…   Modeling databases (sketches?)   (at large) functional programming, language design (Type hackery – SPJ)   Formal methods and high assurance   interactive theorem proving (HOL, Isabelle, Coq, PVS, Twelf, HOLlight..)
  48. 48. + Dialectica categories
  49. 49. + Prospects?   Still want to see more work on Hopf algebras and dialectica categories: quantum mechanics anyone?   Same about state modeling and separation logic: a couple of ideas not written anywhere   Also about super power games and relationships between functional interpretations
  50. 50. + Why do we want this bridge?   Want our computer to understand us   Want to to ask questions   large-scale intelligent information access   Want to search for content on the web   exploiting the large amounts of textual data on the web   Want automatic, high quality translation into other languages   Want intelligent, grammatical summarization, etc…
  51. 51. + Blueprint for bridge between language and logic   translation compositional and principled,   meaning preserving, at least truth value preserving…   a reasonable fragment of all language   generic and specific texts   “logical forms” obtained are useful for reasoning. Questions:   which kind of logic on the target?   how do we know when we’re done?   how do we measure quality of results?
  52. 52. + What makes language so hard?   Problems with non-existence   Must deal with intensionality, “Negotiations prevented a use Contexts! See strike” “Entailment, Intensionality and   Bad idea: ∃x∃y Strike(x) and Text Understanding", Negotiations (y) and Condoravdi et al, 2003 prevented(x,y)   Ambiguity is rampant and   Could do existence predicates, pervasive! but entailments don’t work. Instead: Do concepts! see   PACK and/or CHOOSE! “Preventing Existence”, Crouch et al, FOIS2001   Problems with intensionality… Anecdotal evidence at least 453 sentences out of 1586 on 100 tips photocopier repair will have a propositional complement
  53. 53. +NOT ONLY A TOY PLEASE…Need to deal with huge amounts of textNeed to be robustNeed to move easily between genres, subject domains…Where’s that bridge? Where does it lead to?
  54. 54. + NLP: Symbolic or Data Driven?   Knowledge-based   Shallow, open-domain representations of representations meaning   Broad-coverage   Deep/logical representations   Fails “gracefully” allow high precision and   Lower precision and recall recall, but   Hard to compare systems   Typically on restricted domains   Too sensitive to form of sentences   Hard for users to read/ interpret output of the system   Very hard for systems to build up knowledgeObvious that we need a mixture,which one is the hard question..
  55. 55. + PARC Layered Architecture XLE/LFG P arsing KR M Inference Text apping Engine F-structure Transfer semantics AKR Sources Assertions Question Query Unified Lexicon LFG Term rewriting ECD XLE KR mapping rules Textual Inference MaxEnt models Factives,Deverbals logicsBasic idea: canonicalization of meanings