1. Kazushige Terui RIMS, Kyoto University

2. On 11/10/2015, IPSJ announced: Computer Shogi Project has been completed. What’s next?  Go, or … Computer Shomei (Proof)  Can computer compete with human mathematicians in theorem proving?  How similar is Shomei to Shogi?

4.  Being studied in Automated Theorem Proving  CASC competition every year  Fundamental difficulty  Linked to Interactive Theorem Proving  Dates back to Foundations of Mathematics  Formal proofs are also studied in Theory of Proofs and Programs in Theoretical Computer Science  Cf. Todai Robot Project (NII)

5. Translated by human Input to a prover Theorem. Proof …

6. Consider the sentences built from  polynomials with rational coefficients, ＝, ≦  “and” “or” “implies” “not”  “for every real number x …”  “there exists a real number x such that …” Then, any sentence can be  tranformed into one without “for all”, “there exists” （ quantifier elimination ）  decided true or not （ decidability ）  proved from the axioms of (ordered) real- closed fields if true （ completeness ）

7.  Theorem provers work so-so fine for:  Some specific domains (eg. RCF)  Finitely axiomatizable theories over 1 st order logic  Difficulty arises when the theory includes: “for every set X…” “for every function f…” “for every natural number n…” Identifying the limit: foundations of math

9. “Mathematics of mathematicians” (1870s ～）  computability theory  proof theory  model theory （ + set theory, nonclassical logics ） Outcomes  Computers （ Turing ） ⇒ Computer Science （ 1950’s ～）  Formal systems （ Frege, Russell, Hilbert ） ⇒ Computer Shomei

10.  A math theory expressed as a package of formal language, axioms and inference rules.  Better to think of it as “Proving capability of an imaginary mathematician”  Example: Peano Arithmetic (PA) can  calculate basic funcitons on natural numbers  use mathematical induction  deal with integers and rationals (indirectly)  NOT deal with infinite sets and real numbers

16. Proof completed!

17.  Draw a proof figure by patching axioms and inference rules.  If a proof figure (without assumptions) can be drawn, one says: “PA proves sentence A.”

18. PA can prove  elementary theorems in number theory  also some advanced theorems which require analysis or algebra ⇐ since PA conservatively extends to ACA0  perhaps Fermat’s last theorem too??? （ open problem ）  NOT all theorems （ Variants of Ramsey’s theorem, Kruskal’s theorem, etc. ）

20. Negative solution to Hilbert’s Entscheidungsproblem (1928) Theorem ： There is no mechanical procedure to determine whether a given sentence of PA is true or not. Theorem （ Church, Turing 1936 ）： There is no mechanical procedure to determine whether a given sentence of 1 st order logic is logically true or not.

21. “short” ： provided by a parameter n “feasible” ： in polynomial time p(n) Negative solution to Gödel’s Entscheidungsproblem （ 1956 ） Theorem ： There is no “feasible” procedure to determine if a given sentence has a “short” proof or not (unless P=NP).

22. Theorem （ Gödel 1931 ）： There is a true sentence which PA cannot prove. Example ： G := “PA cannot prove G” （ 1 st incompleteness ） CON := “PA is consistent” （ 2 nd incompleteness ） The same holds even when new symbols and axioms are added to PA recursively and consistently.

23.  Incompleteness theorem does NOT entail:  Computer is inferior to Human  Computer Shomei is impossible  In fact, theorem provers work so-so fine for finitely axiomatizable theories over 1st order logic.  Solution to Robins’ Conjecture (1997)  On the other hand, they do not work well for  Integers (induction axioms)  Sets (2nd order logic or comprehension axioms)

24.  In 1 st order logic, one quantifies over objects. ∀ x.A(x) : “for every object x, A(x) holds”  In 2 nd order logic, one quantifies over sets of objects too. ∀ P.A(P) : “for every set P, A(P) holds”

25. Completeness Theorem （ Gödel1930 ）： There is a (recursive) complete formal system for 1 st order logic which proves all logically true sentences. Incompleteness Theorem （ Cor. to Gödel1931 ）： There is NO (recursive) complete formal system for 2 nd order logic which proves all logically true sentences (in the standard semantics).

26. 2 nd order ： “guessing from infinite”is essential （ lack of subformula prop, decidable unification ） 1 st order unification is linear time 2 nd order unification is undecidable 1 st order ： can avoid “guessing from infinite” （ subformula prop., Skolemization, unification ） Big Deviation from Shogi !!!

27. From 1 st order to higher order!  Needed for large part of mathematics  Some provers exist, but …  New ideas required

29.  Study of logic and computation based on the slogan “proofs are programs.” Constructive math Proof theory Programming languages

31. Proof simplification is a computational process: Fundamental of proof theory （ Gentzen1934 ）： Proof simplification eventually terminates in 1 st order predicate logic. Theorem （ Gentzen1936,38 ）： Consistency of PA can be proved by transfinite induction up to ε 0.

32. LogicComputation SentenceType Proof figureProgram Proof simplificationProgram execution A implies BFunctions from A to B Mathematical InductionRecursive programming ConsistencyTermination

33. ∈ ∈ Invariant of proofs

34.  Not just symbolic, but also algebraic, dynamic and continuous.  Subject of mathematical study.  Does it help to Computer Shomei?

36.  Support writing rigorous proofs “A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail.” (Voevodsky 2014)  Developed under strong influence of the theory of proofs and programs Proof assistantTheoretical backgrdOrigin Isabelle/HOLSimple type theoryChurch 1940 AgdaDependent type theory Martin-Löf 1970’s CoqCalculus of construction Coquand 1985

37.  Mainly for industrial purposes  Verification of software/hardware  Spreading in theoretical computer science  Used by 23% of articles submitted to POPL2014  About to be used in pure mathematics  Four color theorem （ Coq, Gonthier 2005 ）  Feit-Thompson thm （ Coq, Gonthier et.al. 2012 ）  Kepler conjecture （ Isabelle/HOL Light, Hales et.al. 2014 ）

38.  Computer supports Human writing a rigorous proof.  Human supports Computer learning how to prove.

39.  1998 ： T. Hales et. al. proves Kepler’s conjecture.  2002 ： Referee report “it’s 99% correct, but…”  2003 ： Hales launches the FLYSPECK project.  2014 ： Project completed （ see the report 2015 ）  Proof assistants: Isabelle/HOL and HOL Light  International team of more than 30 people  Detected a small error in the original proof  Are proof assistants really correct ? ⇒ The core of HOL Light: just 400 lines in OCaml  Byproduct: a reusable large-scale library: thousands of definitions and lemmas

40. Machine Learning!

41.  If premises are selected by human, 56 ％ are automatically provable.  The rest 44 ％ should contain difficult lemmas.  Currently learning is restricted to premise selection （ +α ）  Can a computer learn proof tactics too?  “Cornerstones” (lemmas) should be provided by human.  Train the provers so that they work with less cornerstones.

42.  「コンピュータは数学者になれるのか？」  「数学者はコンピュータに慣れるのか？」 証明の一部自動化 証明データの提供

43.  Proofs in mathematics get longer and longer  Supports from computers will be essential  New trend in ATP (Urban et. al.): Theorem Proving from Large-Scale Libraries  From Big Data to Big Proofs (Scott 2014)

44. Big Proof (1973 – 2006) (Photo removed. Google the word “big proof”)

45.  Computer Shomei?  Not yet competes with human mathematicians  Boundary between 1 st and 2 nd orders  How to overcome?  Development in proof assistants  Win-win relationship between human and computer  Learning higher facilities (proof tactics)  Theory of proofs and programs (?)

46. Thank you for your attention! Comments appreciated: terui@kurims.kyoto-u.ac.jp