Limit-Computable Mathematics and its Applications

1. Limit-Computable Mathematics and its Applications Susumu Hayashi & Yohji Akama Sep, 22, 2002 CSL’02, Edinburgh, Scotland, UK

2. LCM: Limit-Computable Mathematics • Constructive mathematics is a mathematics based on D01-functions, i.e. recursive functions. • In the same sense, LCM is a mathematics based on D02-functions.

3. The aim of the talk • The talk aims to present basic theoretical ideas of LCM and a little bit of the intended application as the motivation. • Thus, in this talk • THEORY • APPLICATION (Proof Animation) • although the original project was application oriented and still the motto is kept.

4. Why D02-functions? (1) • D02-functions are used as models of learning processes, and, in a sense, semi-computable. • The original and ultimate goal of LCM project is materialization of Proof Animation • Proof Animation is debugging of proofs. • See http://www.shayashi.jp/PALCM/ for details of Proof Animation.

5. Why D02-functions? (2) • The D02-functions are expected to be useful for Proof Animation as learning theoretic algorithms were useful in E. Shapiro’s Algorithmic Debugging of Prolog programs • Shapiro’s debugger debugged Prolog programs, i.e. axiom systems in Horn logic. • In a similar vein, an LCM proof animator is expected to debug axiom systems and proofs of LCM logic, which is at least a super set of predicate constructive logic.

6. An example of semi-computable learning process (1) • MNP (Minimal Number Principle):Let fbe a function form Nat to Nat. Then, thereis n : Natsuch thatf(n)is the smallest value among f(0), f(1), f(2),…Nat : the set of naturalnumbers

7. An example of semi-computable learning process (2) • Such an n is not Turing-computable from f. • However, the number n is obtained in finite time from f by a mechanical “computation”.

8. A limit-computation of n (1) • Regard the function fas a stream f(0), f(1), f(2), … • Have a box of a natural number. We denote the content of the box byx.

9. A limit-computation of n (2) • Initialize the box by setting x=0. • Compare f(x)with the next element of the stream, say f(n).If the new one is smaller than f(x), then put n in the box. Otherwise, keep the old value in the box. • Repeat the last step forever.

10. A limit-computation of n (3) • The process does not stop. But your box will eventually contain the correct answer and after then the content xwill never change. • In this sense, the non-terminating process “computes” the right answer in finite time. • You will have a right answer, but you will never know when you got it.

11. A limit-computation of n (4) • By regarding the set of natural numbers as a discreate topology space, the process “computing” x is understood as the limit: limn → ∞f(n) = x • Thus, E. M. Gold (1965, J.S.L.30) called it “x is computable in thelimit”

12. Limit computation as Learning process (1) • In computational learning theory initiated by Gold, the infinite series f(0), f(1), f(2),…is regarded as guessesof a learner to learn the limit value.

13. Limit computation as Learning process (2) • f is called a guessing function: • The learner is allowed to change his mind. A guessing function represents a history of his mind changes. • When the learner stops mind changes in finite time, it succeeded to learn the right value. Otherwise, it failed to learn.

14. Limit and recursive hierarchy • Shoenfield’s Limit Lemma • A function g is defined by g(x)=lima1 lima2 ….liman f(a1,a2,…,an,x)for a recursive function f, if and only if, g is a D0n+1-function. • In this sense, “single limit” is the jump A’: D0n →D0n+1 in recursion theory.

15. Logic based on limit-computable functions (1) • As the D01-functions are the recursive functions, D0n-functions may be regarded as a generalized domain of computable functions. • For example, they satisfy axioms of some abstract recursion theory, e.g. BRFT by Strong & Wagner.

16. Logic based on limit-computable functions (2) • Semantics of constructive mathematics is given by realizability interpretations and type theories based on recursive functions. • Thus, when recursive functions are replaced by D0n-functions, a new mathematics is created.

17. Logic based on limit-computable functions (3) • For n=2, it is a mathematics based on limit-computation or computational learning. It is LCM. • Note that limits in LCM are not nested. • We may regard LCM is a mathematics based on the singlejump D0n →D0n+1

18. Formal semantics of LCM (1) • Good Kripke or forcing style semantics and categorical semantics are longed for. • Existing formal semantics of LCM are given by limit-function spaces and realizability interpretations or some interpretations similar. • The first and simplest one is Kleene realizability with limit partial functions with partial recursive guessing functions (Nakata & Hayashi)

19. Formal semantics of LCM (2) • Learning theoretic limits must be extended to higher order functions to interpret logical implication and etcetras. Some extensions are necessary even for practical application reasons as well. • E.g. Nakata & Hayashi used “partial guessing functions”, which are rarely used in learning theory.

20. Formal semantics of LCM (3) • Combinations of different approaches to limit-functions plus different realizability interpretations (Kleene, modified, etc) make different semantics of LCM, e.g., • Nakata & Hayashi already mentioned • Akama & Hayashi: lim-CCC and modified realizability • Berardi: A limit semantics based on limits over directed sets.

21. What kind of logic hold? • Logical axioms and rules of LCM depend on these semantics just as modified realizability and Kleene realizability define different constructive logics. • However, they have common characteristics: semi-classical principles hold

22. S0n- and P0n-formulas • S0n-andP0n-formulas are defined as the usual prenex normal forms. • Thus, S03-formula isExists x.ForAll y.Exists z.A • A definition not restricted to prenex form is possible but omitted here for simplicity.

23. Semi-classical principles (LEM) • S0n-LEM　(Law of Excluded Middle):Aor not Afor S0n-formula　A. • Similarly for P0n-LEM • D0n-LEM(A ↔ B)→ Aor not Afor S0n-formula　Aand P0n-formula　B

24. Semi-classical principles (DNE) • S0n-DNE　(Double Negation Elimination):(not not A) → Afor S0n-formula　A. • P0n-DNE is defined similarly • Note: S01-DNE is Markov’s principle for recursive predicates.

25. Some examples • P01-LEM ForAll x.A or not ForAll x.A • S01-LEM Exists x.A or not Exists x.A • S02-DNE not notExists x.ForAll y.A →Exists x.ForAll y.A • S03-LEM Exists x.ForAll y.Exists z.A or not Exists x.ForAll y.Exists z.A

26. Hierarchy of semi-classical principles (1) S0n–LEM P0n–LEM S0n–DNE P0n+1–DNE The arrows indicate derivability in HA D0n–LEM S0n-1–LEM

27. Important Remark (1) • If we allow function parameters in recursive formulas, then the hierarchy collapses with the help of the full principle of function definitionForAll x.Exists!y.A(x,y) → Exists f.Forall x.A(x,f(x)) • Because of the combination of these two iterate applications of limits.

28. Important Remark (2) • We keep the function definition principle and forbid function parameters in recursive predicates. • We may introduce function parameters for recursive functions.

29. LCM semi-classical principles • In all of the known semantics of LCM, the followings hold:P01-LEM, S01-LEM, S01-DNE, P02-DNE • In some semantics the followings also hold:D02-LEM, S02-DNE • These are LCM-principles since interpretable by single limits. The principles beyond these need iterated limits, and so non-LCM.

30. Hierarchy of semi-classical principles (2) • The converse of arrows in the hierarchy of semi-classical principles are conjectured not to be derivable in HA. • If the scheme S0n–DNE is not derivable from the scheme P0n–LEM, then the conjecture is proved for the n-level. • The conjecture have been solved for n=1, 2 levels, which include all of the LCM semi-classical principles. It is still open for the higher levels.

31. What theorems are provable in LCM? (1) • Transfers from Reverse Mathematics: • If sets are identified with {0,1}-valued function, almost all theorems proved in systems of Reverse Mathematics can be transferred into LCM. • Since Reverse Math. covers large parts of mathematics, we can prove very many classical theorems in LCM almost automatically thanks to e.g. Simpson’s book.

32. A recent development in LCM • P01–LEM is the weakest LCM semi-classical principle considered. • Even below it, there is an interesting semi-classical principle and corresponding theorems. • It’s Weak Koenig Lemma (WKL): “any binary branching tree with infinite nodes has an infinite path”.

33. WKL and LLPO • Bishop’s LLPO:not not (A or B) → A or Bfor A, B: P01-formulas • WKL is constructively equivalent to LLPO plus the bounded countable choice for P01-formulas.

34. The strength of WKL • P01–LEM derives WKL with a help of a function definition principle for P01–graphs. • In contrast, WKL cannot constructively derive P01–LEM. • Thus, WKL is strictly weaker than LCM. • Still WKL is constructively equivalent to many mathematical theorems like Gödel’s completeness theorem for classical predicate logic, Heine-Borel theorem, etc. etc…

35. Three underivability proofs • The underivability of P01-LEM is proved by three different proofs: • monotone functional interpretation (Kohlenbach) • Standard realizability plus low degree model of WKL0 (Berardi, Hayashi, Yamazaki) • Lifschitz realizability (Hayashi)

36. Open problem • WKL seems to represent a class of non-deterministic or multi-valued computation. Monotone functional interpretation and Lifschitz realizability and seem to give their models. • On the other hand, Hayashi’s proof uses Jockush-Soare’s the low degree theorem and the usual realizability, i.e., usual computation. • The relationship between these two groups of proofs would be a relationship of forcing and generic construction. • Open problem:Find out exact relationship.

37. Collaborators • The results on hierarchy and calibration are obtained in our joint works with the following collaborators: S. Berardi, H. Ishihara, U.Kohlenbach, T. Yamazaki, M. Yasugi