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### Limit-Computable Mathematics and its Applications

Susumu Hayashi & Yohji Akama

Sep, 22, 2002

CSL’02, Edinburgh, Scotland, UK

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.

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.

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.

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.

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

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”.

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.

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.

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.

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”

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.

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.

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.

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.

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.

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

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)

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.

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.

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

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.

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

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.

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

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

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.

Important Remark (2)

- We keep the function definition principle and forbid function parameters in recursive predicates.
- We may introduce function parameters for recursive functions.

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.

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.

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.

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”.

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.

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…

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)

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.

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

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