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  0 1 -functions, i.e. recursive functions. In the same sense, LCM is a mathematics based on  0 2 -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  0 2 -functions? (1)  0 2 -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 for details of Proof Animation.

Why  0 2 -functions? (2) The  0 2 -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 f be a function form Nat to Nat. Then, there is n : Nat such that f(n) is the smallest value among f(0), f(1), f(2),… Nat : the set of natural numbers

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 f as a stream f(0), f(1), f(2), … Have a box of a natural number. We denote the content of the box by x.

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 x will 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: lim n → ∞ f(n) = x Thus, E. M. Gold (1965, J.S.L.30) called it “x is computable in the limit”

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 guesses of 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)=lim a1 lim a2 …. lim an f(a 1,a 2,…,a n,x) for a recursive function f, if and only if, g is a  0 n+1 -function. In this sense, “single limit” is the jump A’ :  0 n →  0 n+1 in recursion theory.

Logic based on limit-computable functions (1) As the  0 1 -functions are the recursive functions,  0 n -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  0 n -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 single jump  0 n →  0 n+1

Formal semantics of LCM (1) 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) Good Kripke or forcing style semantics and categorical semantics are longed for.

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

 0 n - and  0 n -formulas  0 n -and  0 n -formulas are defined as the usual prenex normal forms. Thus,  0 3 -formula is Exists x.ForAll y.Exists z.A A definition not restricted to prenex form is possible but omitted here for simplicity.

Semi-classical principles (LEM)  0 n -LEM (Law of Excluded Middle): A or not A for  0 n -formula A. Similarly for  0 n -LEM  0 n -LEM (A ↔ B) → A or not A for  0 n -formula A and  0 n -formula B

Semi-classical principles (DNE)  0 n -DNE (Double Negation Elimination): (not not A) → A for  0 n -formula A.  0 n -DNE is defined similarly Note:  0 1 -DNE is Markov’s principle for recursive predicates.

Some examples  0 1 -LEM ForAll x.A or not ForAll x.A  0 1 -LEM Exists x.A or not Exists x.A  0 2 -DNE not not Exists x.ForAll y.A → Exists x.ForAll y.A  0 3 -LEM Exists x.ForAll y.Exists z.A or not Exists x.ForAll y.Exists z.A

 0 n – DNE  0 n – LEM  0 n-1 – LEM  0 n – LEM  0 n+1 – DNE  0 n – LEM Hierarchy of semi-classical principles (1) The arrows indicate derivability in HA

Important Remark (1) If we allow function parameters in recursive formulas, then the hierarchy collapses with the help of the full principle of function definition ForAll 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:  0 1 -LEM,  0 1 -LEM,  0 1 -DNE,  0 2 -DNE In some semantics the followings also hold:  0 2 -LEM,  0 2 -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  0 n –DNE is not derivable from the scheme  0 n –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  0 1 –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 B for A, B:  0 1 -formulas WKL is constructively equivalent to LLPO plus the bounded countable choice for  0 1 -formulas.

The strength of WKL  0 1 –LEM derives WKL with a help of a function definition principle for  0 1 –graphs. In contrast, WKL cannot constructively derive  0 1 –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  0 1 -LEM is proved by three different proofs: monotone functional interpretation (Kohlenbach) Standard realizability plus low degree model of WKL 0 (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