From Proof Animation to LimitComputable Mathematics

1
From Proof Animation toLimit-Computable
Mathematics

• Susumu Hayashi
• Kobe University

PA workshop Kyoto 2001/01/10,11
2
This is a joint work with Yoji Akama, Tohoku
Univ. Hajime Ishihara, JAIST Shyun-ichi Kimura,
Hiroshima Univ. Ulrich Kohlenbach, BRICS Masahiro
Nakata, Kobe Univ. Mariko Yasugi, Kyoto Sangyo
Univ.
3
Towards Animation of Proofs testing proofs by
examples-, Hayashi, et al., TCS, in print.
Available from my home page.
X test of proofs Synonym Proof
animation Analogy to specification animation of
formal methods
4
Why Proof Animation?

• Its a contraposition of the developments of
  correct programs by program extraction.
• If something is wrong with the extracted program,
  then something is wrong with the original proof.
• Normally, the proofs tested are incomplete proofs
  under development.
• We can find bugs in goals, subgoals, definitions
  and strategies of proofs before we go far into
  the development of proofs.

5
What is test of proof? An Example
-- ASSUMPTION -- There is a bag. And some white
or black marbles are in it.
-- CONCLUSION -- All marbles in the bag are of
the same color.
This is wrong.
However, we prove it by mathematical induction!
6
Proof of the theorem

• Base case n1 is easy
• The induction step

• The theorem holds for group A and B, since they
  have only n marbles. All the marbles are of the
  same color, since they share an.

What is wrong?
7
Animation of the proof
Animating the proof by an applet Just click the
button.
8
Systematic proof animation

• The applet was written by hands.
• Automatic generation of such an applet from a
  proof is the ultimate goal of Proof Animation
  project.

• graphics animation library
• Generation of algorithm from proof by
  Curry-Howard

9
The program extracted from a formal proof of the
puzzle.
10
Proof animation of classical proofs

• Classical reasoning is used even in finite
  mathematics.
• Thus, classical proof execution principles such
  as lm-calculus and double negation
  translationA-translation must be used for proof
  animation.

11
Accountability of proof execution

• A proof execution principle with the following
  two criteria is said accountable
• computational contents (programs) associated to
  proofs are legible.
• association between proofs and programs is
  legible.

12
Accountability of proof execution is
indispensable for proof animation

• Finding bugs of a proof by its execution is
  understanding proofs by understanding the
  execution.
• Thus associated computational contents and the
  association must be legible as the case of applet
  for the puzzle.

13
Almost accountable interpretation Berardis
approximation theory

• Almost all proof execution methods for classical
  logic do not meet the criteria.
• But Berardis approximation interpretation for
  classical proofs meets the first criteria for
  some examples
• Minimal value of numerical functions ForAll f
  Nat -gtNat.Exists nNat. ForAll
  xNat.f(n) is smaller than or equal to f(x).
• an semi-algorithm is extracted from a classical
  proof of the theorem.

14
Berardis semi-algorithm extracted

• Regard the function f as a stream f(1), f(2),
  f(3),
• Have a box of a natural number.
• Put f(1) in the box.
• Compare the content of the box with the next
  element of the stream. If the new one is smaller
  than the number in your box, put the new one in
  the box.Repeat it infinitely.
• Caution this is a little bit incorrect argument

15
In what sense the semi-algorithm compute the
answer?

• The process does not stop.
• But your box will eventually contain the correct
  answer and then the content will never been
  changed.
• In this sense, this non-terminating process
  computes the right answer in the limit.
• You will have a right answer, but you will never
  know when you got it.

16
Second criteria?

• This explanation of Berardis algorithm is enough
  for the first criteria of accountable proof
  execution.
• But, unfortunately, it is not straight-forward to
  see how this explanation is obtained from his
  interpretation applied to the proof of the
  minimum value theorem. No second criteria.

17
A solution to the problem of accountable
classical proof execution

• There might be no accountable proof execution for
  all classical proofs.
• Thus, find a fragment F of classical mathematics
  such that
• proof execution for F is accountable.
• Enough mathematics can be done in F.

18
Learning Theory gives such a fragment Golds
argument

• Berardis argument is the same as the central
  idea of Algorithmic Learning Theory
• pointed out by Yamamoto
• Paulins related comment on LPO
• Golds theory of Limiting Recursion in JSL. 1965
  a seminal work of learning theories.

19
Limiting Recursive Function

• f(x) limn g(x,n), then f is called limiting
  recursive, when g is recursive.
• g(x,1), g(x,2), is guessing (learning) the
  value of f(x).
• g is called a guessing function of f
• Berardis semi-algorithm is a function guessing
  the minimum value of the function f.

20
LCM Limit-Computable Mathematics

• A fragment of classical mathematics whose
  BHK-interpretation is realized by limiting
  recursive functions rather than recursive
  functions.
• A formalization constructive formal theories
  enhanced with S02 DNE.

21
How this idea came out? Hilberts finite basis
theorem

• In his 1890 proof of the finite basis theorem, D.
  Hilbert used the same semi-algorithm as
  Berardis. And this was called theology by
  Gordan.
• It solved Gordans problem one of the first
  successes of transfinite mode of thought in
  modern algebra.
• I was studying the paper as an far origin of
  Hilbert program

22
Hilberts argument 1890,1897 (1)

• In his 1890 paper and 1897 lectures at
  Goettingen, Hilbert was arguing almost the same
  as Berardi did!!
• It is the point Gordan was against(a letter from
  Gordan to Klein, Feb.24,1890) It does not
  satisfy the requirements of recursive proofs. No,
  full and clear arrangement (Einteilung) of forms.

23
Hilberts argument 1890,1897 (2)

• Because, it was a proof by limit-argument!
• Hilbert was proving a version of finite basis
  theorem. In a modern terminology. Every ideal of
  homogeneous polynomials with many (but fixed)
  variables are finitely generated.
• But he formulated it by means of stream and argue
  as Berardi or Gold.

24
The stream formulation by Hilbert 1890,1897 (3)
25
The proof by limit-arguments 1890,1897 (4)

• He prove it by induction on the numbers of
  variables. The most impressive is the base case.
• A stream of 1-variable forms are c1xr1, c2xr2,.
• The basis is a single form crxr s.t. r is the
  minimum of r1, r2,
• He argued as follow. (From, David Hilbert,
  Theory of Algebraic Invariants, pp. 126-7,
  Cambridge Univ. Press)

26
The proof by limit-arguments 1890,1897 (5A)

• Let c1xr1 be the first form of the sequence with
  a coefficient different from zero.
• We then look for the next form in the sequence
  whose order is less then r1
• if there is no such form, we retain c1xr1.
• But

27
The proof by limit-arguments 1890,1897 (5B)

• But if there is one, say c2xr2, then we proceed
  to the next form in the sequence whose order is
  less than r2.
• If we continue in this manner, then we finally
  arrive at a form cixriFm in the sequence with
  the property that none of the subsequent forms
  have order less than ri.
• Every form is then divisible by Fm

28
To formal businesses!

• I will explain formal developments of LCM and
  its realizability by the paper distributed on
  white boards.
• The paper Limiting first order realizabiliy
  interpretation, by Nakata and Hayashi
• Warning! The paper was just submitted for
  publication.Not yet accepted. The running head
  is a dummy.

29
Semi-classical principles

• S0n-LEM (Law of Excluded Middle) Exists
  x.A or not Exists x.A for P0n-1-formula A.
• P0n-LEM ForAll x.A or not ForAll x.A
  for S0n-1-formula A.
• S0n-DNE (Double Negation Elimination) (not
  not Exists x.A) implies Exists x.A for
  P0n-1-formula A.

30
Why these principles?

• Limiting recursive functions are equivalent to
  D02-functions by Shoenfields limit lemma
• a set is D0n1-set iff its characteristic
  function g(x) is defined as g(x)limt1
  limtnf(t1,, tn,x), where f is primitive
  recursive.

31
Relations to existing principles

• S01-LEM is LPO (limited principles of
  omniscience) without function variables.
• P01-LEM is weak LPO without function variables.
• S01-DNE is Markovs principle for recursive
  predicates.
• Note They are not equivalent to LPO and WLPO
  since they do not have function variables. No
  repetition!

32
The hierarchy of the semi-classical principles up
to n2
S02DNE
S01DNE
The two starred ? are difficult and due to U.
Kohlenbach.
HA -
33
Limiting realizability and S02-DNE

• HALHA S02-DNE (HA with Limit)
• Kleene-realizability with limiting recursive
  functions realizes HAL.
• This gives an accountable semi-classical proof
  execution method.
• Then next criteria enough mathematics?

34
An example of LCM Hilberts invariant theory

• In his 1890 proof of the finite basis theorem, D.
  Hilbert used the same semi-algorithm as
  Berardis. And this was called theology by
  Gordan.
• It solved Gordans problem one of the first
  successes of transfinite mode of thought in
  modern algebra.
• His proof is formalizable in HAL(f).

35
Analysis?

• The classical theorems provable only in
  approximated forms in Bishop constructive
  mathematics seem provable in LCM in exact forms,
  i.e., without using approximation.
• Examples Hahn-Banach theorem, ergodic theorem,
  minimum value theorem, etc.etc.

36
LCM will be the fragment!

• Proof animation by LCM
• LCM has an accountable computational
  interpretation by learning processes or
  approximation.
• Yet to be shown enough for many mathematics, but
  very promising.
• Its interesting by its own sake and is related
  to many other areas.

37
LCM results and conjectures I

• Many theorems of 19th and early 20th centuries
  math. will be provable in LCM including
  Hahn-Banach, etc.
• Even many statements in such math will be P03.
  (Berardi)
• Formal theories for such a higher order LCM are
  necessary.

38
LCM results and conjectures II

• Almost all abstract computable calculi will be
  closed under limiting-construction.
• Two positive answers
• w-BRFT (Nakata Hayashi)
• PCA (Akama)
• Yet to know for type theories
• Practical implication e.g. Lim(Coq) is directly
  coded in Coq. No change of Coq kernel for proof
  animation.

39
LCM results and conjectures III

• Such a limiting-construction will be explained by
  some internal construction in W-valued sets,
  where W is the complete Heyting algebra of
  co-finite subsets of w. Akamas construction.
• A relation to finitely presented categories?
• Markovs rule for P0n-formulas (admissible rule
  of S0n1-DNE) holds for HAS0n-LEM (conjectured
  by Berardi for n1 case and proved by Hayashi).

40
LCM results and conjectures IV

• Some computability theories over reals will be
  related to LCM.
• computable functions of BSS theory is in a sense
  limiting recursive.
• Yasugis argument of computability of Gaussian
  function can be explained by limit process.
• The other way around, LCM will be explained by
  these theories alike.

41
A future work calculus of limiting processes

• Each limiting recursive function has only one
  Limit.
• Hilberts proof suggests that each instance of
  LEM corresponds to a limit-process generating a
  stream of guesses.
• Since some LEMs are used in the proof, the
  limit computation associated to Hilberts proof
  will be understood as a net of communicating
  limit-processes.

42
A future work calculus of limiting processes
(continued)

• It is easy to have a clear and intuitive picture
  of such a communication in mind just by looking
  at Hilberts proof.
• We need a formal calculus for such a
  communication and its implementation for
  practical proof animation of Hilberts proof and
  others.

43
Other future works

• reverse mathematics of transcendency
• the proper logic of LCM
• relations to
• learning theory and recursion theorydegrees of
  unsolvability, Boolean hierarchy, etc.
• numerical analysis
• computer algebra (Sturmfels work)

44
Other future works (continued)

• Etc. etc.
• I will make their list and put it at my homepage
  in a month
• http//alan.scitec.kobe-u.ac.jp/hayashi