Limit-Computable Mathematics and its Applications

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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 D 0 1 -functions , i.e. recursive functions.

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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