1. Stefano Berardi - Università di Torino http://www.di.unito.it/~stefano
Classical Logic as Limit Completion Workshop on Proof Theory and Algorithms 23 to 29 March 2003, Edinburgh
Stefano Berardi - Università di Torino

2. The text of this talk and some related papers may be found in the home page of the author:

3. Acknowledgements
We thank Prof. S. Hayashi for suggesting the use of limits in modelizing Classical Arithmetic.
We thank all S. Hayashi’s Proof Animation Group, and in particular Y. Akama, for many valuable suggestions and comments.
We owe the idea for the constructive content of Excluded Middle and the use of backtracking to Coquand Game interpretation.

4. The thesis of the Talk
Call N = {0, 1, 2, 3, …} the set of natural numbers.
The thesis of the Talk is:
“Classical Logic is equivalent to an intuitionistic (and informative) theory of some topogical completion N of N.”

5. An overview of the results
There is a purely intuitionistic model N of the set of arithmetical maps, which is a topological completion of N.
On the top of N, we may define an Intuitionistic Realization R of Classical Arithmetic, explicitely showing some constructive content for all classical proofs.
No proof manipulation is required: the interpretation is semantical, not syntactical.
We extract some constructive content from all proofs of existential statement, not just from proofs of simply existential statements.

6. Plan of the Talk § 1. The constructive content of Excluded Middle.
§ 2. An intuitionistic model of 02-maps.
§ 3. An intuitionistic model of Excluded Middle over 1-quantifier formulas.
§ 4. An intuitionistic model of the whole Classical Arithmetic.

7. §1. The constructive content of Excluded Middle
Call EM the Excluded Middle Schema AA, and EM-k its restriction to all A of degree k.
EM-1 is equivalent to
xN.P(x)  xN.P(x) (P(x) decidable)
Call T the set of instants of time. T is an inhabited unbounded partial ordering. Say, T =N.
The constructive content of EM-1 is a recursive process E, learning, with time, whether xN.P(x) is true or not.
Let us see how E works.

8. The process E Fix any instant of time tT.
Call a(t)={x1,…,xn} the set of xiN such that we computed the truth value of P(xi) before instant t.
Call E(t){False}+N the opinion of E, in the instant t, about the truth of xN.P(x).
E(t) is False if E thinks that xN.P(x) is false. E(t) is some xN if E thinks that P(x) holds.
E returns some xi a(t) such that P(xi) (such that the value of P(xi) is known), if any exists.
If none exists, E says: “xN.P(x) is false”.
How much is the opinion of E reliable ?

9. Unreliability of E The opinion of E is quite unreliable.
In the case P is false over a(t)={x1,…,xn}, and true for some xN-a(t), E thinks that xN.P(x) is false. This is wrong.
E changes its mind in the first instant some x such that P(x) is found. However, E could never find some x such that P(x), unless we sistematically compute P(x) for all xN.
If P is false for all xN, the opinion of E is correct. But we will never know for sure it is, because xN.P(x) is undecidable (in general).
So what? Is there any use of E(t)?

10. The use of E
In spite of the first impression, E(t) is all we need to know about xN.P(x)  xN.P(x).
Fix any instant of time tT, and any computation working out a verifiable conclusion under the assumption xN.P(x)  xN.P(x).
Up to the instant t, the computation either used the assumption xN.P(x); or it used the assumption xN.P(x), but not for all xN, only for a finite subset a(t)={x1,…,xn}N.
In fact, the computation worked under the assumption xN.P(x)  xa(t).P(x).
This is exactly the information provided by E(t).

11. Backtracking with E A price to pay to compute with E is backtracking.
Whenever E changes its mind about the truth of xN.P(x), everything we computed out of the value of E(t) must be discarded and recomputed again.

12. § 2. A model of 02-maps
A preliminary conclusion. We have a constructive content of EM-1, provided we accept, as individuals, not only integers but also all sequences s:TX indexed over time, with target some XN.
We consider only convergent sequences. s(t) is not be allowed to change its value infinitely many times. We identify s with its limit value.
The next step. If we want a purely intuitionistic model, we need an intuitionistic theory of convergent successions over integers. The existing one is classical.

13. Stationarity is not enough
Suppose for simplicity T=N (the instants of time are disposed along a single timeline 0, 1, 2, 3, …).
We develop an intutionistic theory of convergence for sequences s: NN.
s is stationary iff xN.yx. s(y)=s(x).
Classically, convergence is stationarity.
Intuitionistically, stationarity it does not work:
we cannot prove that E(t) is stationary.

14. An intuitionistic notion of convergence
Intutionistically, we define s convergent iff s satisfies the no-CounterExample interpretation of stationarity.
Classically, convergence is equivalent to stationarity.
Intuitionistically, convergence is equivalent to:
:NN rec.. xN. s constant over [x, (x)]
Now we can prove that E is convergent.
What it is the intuition behind “convergence” ?

15. The intuition behind convergence
We fix any effective upper limit (x) (depending on x) to the set of yx for which we will check s(y)=s(x) (the stationarity of s in x).
(x) represents the computational resources available to check if x is stationary.
Then, no matter what  is, we find some x looking like a stationarity point, with respect to the segment [x,(x)] we have time to check:
s(0)=21 s(1)=13 … [s(x)=7 s(x+1)=7…s((x))=7] s(…)=55

16. An intuitionistic notion of equality for succession
s, t: NN are stationary equal iff
xN.yx. s(y)=t(y)
Classically, equality between convergent successions is being stationary equal.
Intutionistically, we say that s~t iff s, t satisfies the no-CounterExample interpretation of stationary equality.
Classically, s~t is equivalent to stationary equality.

17. An intuitionistic notion of equality for succession
s~t is intuitionistically equivalent to:
:NN rec.. xN. s,t equal over [x, (x)]
This means: no matter what  is, we find some x looking like a stationarity point, with respect to the segment [x,(x)] we suppose having “time to check”:
s(0)=21 s(1)=13 … [s(x)=7 s(x+1)=8…s((x))=9] s(…)=55
t(0)=33 t(1)=60 … [t(x)=7 t(x+1)=8…t((x))=9] t(…)=88

18. f*(s)(t) = f(s(t))(t), for all tT
The structure N2
Define N2={convergent successions s:NN}/~
N2 is completion of N under convergent successions (similar to the definition of Real number out of Rational numbers).
Any xN may be identified with some constant succession x*:NN, defined by
x*(t) = x, for all tT
Any recursive map f:NN2, over xN, may be extended to a map f*:N2N2 over all s:NN, by the so-called syncronous application:
f*(s)(t) = f(s(t))(t), for all tT
The maps f* are by defin. the morphisms of N2.

19. N2 is an intuitionistic model of 02-maps
Maps f*:N2N2 include (simulations of) decision procedures for all simply existential predicates. Maps f*:N2N2 are closed under m-operator, whenever the resulting map is total.
N2 is an equational model for 02-maps.
Simply existential properties over N2 are all sets of the form
{xN2| yN2. f*(x,y) = 0}
for some morphism f*:N2N2
Lifting. A simply existential property holds for all xN2 iff it holds for all (images in N2 of) points nN.

20. Relating N2 and N Induction for equational statements holds in N2.
N and N2 are classically isomorphic, yet they are not recursively isomorphic.
Thus, N and N2 are not intuitionistically isomorphic
Intutionistically, N2 “looks larger than N”.

21. The main feature of N2 is Conservativity
In N2 we derive abstract statements s~t, about the identity of limits whose exact value, often, will never be known. We could think that results about N2 have nothing to do with N.
Instead, the conclusions we draw about N2 have consequences about N.
(Conservativity, or Density of N in N2) Any solution we find in N2, of some recursive equation f(x)=0 of N, may be effectively turned into some solution nN of the same equation.
Thus, abstract reasoning in N2 may be used to effectively solve concrete problems in N.

22. The main feature of N2: Conservativity
Conservativity has a remarkably simple proof.
(Proof of Conservativity) Fix any recursive map f:NNN2. Assume LN2 is a solution of f(x)=0 in N2. This means that f*(L)~0. By definition, for any recursive  we may effectively find some nN such that f(L(x))=0 for all x[n,(n)]. Set =id. Then f(L(x))=0 for all x[n,n]. Thus, we may effectively find some nN such that f(L(n))=0.

23. § 3. A model R2 for Intuitionistic Arithmetic + EM-1
Recall that EM-1 is Excluded Middle over 1-quantifier formulas.
We will extend N2 to a Realization Model R2 for Intuitionistic Arithmetic and EM-1.
Using the family of constants E we will realize EM-1.
There is a difference with Heyting Realizability: we realize a statement not in an absolute sense, but under a set of equational assumptions.

24. Relative Realization
The realization relation will be |=r:A, with ={a1~b1, …, an~bn} set of equations over N2.
The intended meaning is: “if all equations in  are true, then r realizes A”.
In this way, realization of atomic statement in N2 will be relative recursive (w.r.t. N2), rather than recursive. This is unavoidable because equality in N2 is not recursive.
|=r:A will stay for |=r:A or “r realizes A without assumptions”.

25. Some preliminaries about N2
Limit value. LN2 has limit nN iff L~n*.
The set {1,2}*. We define:
{1,2}* = {xN2| x:N{1,2} }
The elements of {1,2}* are succession with limit in {1,2}. We cannot decide if it is 1 or 2, though.
Truth in N2. We say that
(a1~b1, …, an~bn  a~b) is true in N2
iff the limit of truth value of (a1(t)=b1(t) …, an(t)=bn(t)a(t)=b(t)), for tN, is True.
Intuitionistically, this condition is stronger than just “a1~b1, …, an~bn implies a~b”.

26. The Realization Model R2
|=dummy:a~b iff
(a~b) is true in N2
|=<c,r1,r2>:A1A2 iff c{1,2}* and for i=1,2
,(c=i)|=ri:Ai
|=<r1,r2>:A1A2 iff for i=1,2
|=ri:Ai
|=f:A1A2 iff for all 
if |=s:A1 then |=f(s):A2
|=<c,s>:x.A(x) iff
|=s:A(c)
|=f:x.A(x) iff for all aN2
|=f(a):A(a)

27. The main feature of R2: Conservativity
R2 inherites Conservativity from N2.
(Conservativity) If P is a decidable statement of N, and x.P(x) is realizable in R2, then we may effectively find some nN such that P(n).
Thus, intuitionistic reasoning in R2 may be used to effectively solve concrete problems in N.
Yet, intuitionistic reasoning in R2 includes (better, it simulates) Excluded Middle for 1-quantifier statements!

28. Comparing a connective with its interpretation: , 
The interpretations , xN2 of , xN are intuitionistically weaker than the original , xN.
Intuitively, if we prove A1A2 in R2, we have some c {1,2}* , such that if the value of c is i, then Ai is true in R2. But we have no way of computing the value of c. In an intuitionistic proof of A1A2, we know which Ai is true in R2.
Intuitively, if we prove xN.A(x) in R2, we have some cN2 such that A(c) is true in R2, but we do not know the value of c. In an intuitionistic proof of xN.A(x), we know which A(i) is true.

29. Comparing a connective with its interpretation: , 
The interpretations , xN2 of , xN are intuitionistically equivalent to the original , xN.
In the case of , this claim requires a proof based over the density of N in N2.

30. Comparing a connective with its interpretation: ,
The interpretations , of , are intuitionistically stronger than the original ,.
Intuitively, if we prove A1  A2 in R2, we have a way of sending each proof of A1 in R2, under assumption , into a proof of A2 in R2, under assumption . In an intuitionistic proof, we consider only the case =.
In the same way, if prove A in R2 we know more than just the falsity of A in R2.

31. § 4. A model of the whole Classical Arithmetic
In the construction of R2, we used only Intuitionistic Arithmetical reasoning, plus two properties of N2:
Density of N in N2. Every recursive map f:NN2 may be extended in a unique way to a map f*:N2N2.
Conservativity of N2 w.r.t. N. Every simple existential predicate of N2 covering N covers the whole N2.

32. Extending R2 to a model R of Classical Arithmetic
Any model N of N satisfying Density and Conservativity may be extended to a Realization Model R of Intuitionistic Arithmetic.
This construction may be performed within Intuitionistic Arithmetic.
Fix any k=1,2,3,…,.
If N is also a model of 0k-maps, then R is a model of EM-k (Excluded Middle over degree k formulas).
If k=, then R models Classical Arithmetic.

33. Iterating Completion of N
For all k, we may define a model Nk of 0k-maps satisfying Density and Conservativity, then a model Rk of EM-k on the top of it.
We define N3 by interpreting in R3 the completion N2 of N.
This is possible because the construction of N2 requires only arithmetical reasoning.
More in general, we define Nk+1 by interpreting in Rk the completion N2 of N.
Then we define Rk+1 on the top of Nk+1.
We get N2, R2, N3, R3, …in this order.

34. A more direct definition of Nk
We may define a model Nk of 0k-maps satisfying Density and Conservativity directly.
Nk is a set of successions of successions … iterated k-1 times. N1 is N.
There is a purely combinatorial definition of convergence and equality for elements of Nk.
It may be found in the 2002 talk
“Classical Logic as Limit”, Section 4
in the author’s web site:

35. A direct definition of Nk
Intuitively, Nk consists of learning processes with “backtracking” of level k-1.
Level 1 backtracking is the possibily of making hypothesis, then discard it forever if we find some contradiction with data.
An example is the process EN2.
Level 2 backtracking is the possibily of making level 2 hypothesis over level 1 hypothesis (hypothesis over data).
When a level 2 hypothesis is discarded, it is not discarded forever. We may come back to it, and reconsider it again.

36. A Claim: Generalized Conservativity
Using the family Nk of models we may prove the following new result, generalizing a Conservativity result from literature:
Theorem: EM is conservative w.r.t. EM-k, for all statements x.P(x), with P of degree k.
This means: if P is a degree k predicate, and there is proof of x.P(x) using EM, then there is a proof of x.P(x) using only EM-k.
Conservativity of Classical w.r.t. Intuitionistic Arithmetic and x.P(x) statements, with P decidable, follows as a particular case, when k=0.

37. Related Papers
Classical Logic as Limit. An intuitionistic model of 02-maps using Parallel Computations. Submitted to I.C.. Available in:
An Intuitionistic Model of Classical Arithmetic and Arithmetical maps. Draft Version.

38. Stefano Berardi - Università di Torino
Learning Processes and Parallel Computations First APPSEM-II Workshop to 28 March 2003 Nottingham, United Kingdom
Stefano Berardi - Università di Torino

39. Reference
In this talk we introduce the following paper (submitted to I.C.):
Classical Logic as Limit. An intuitionistic model of 02-maps using Parallel Computations
The paper is in the proceedings of the workshop. The talk and the paper are also available in

40. Learning Processes and programming
The goal of the paper is to use Learning processes in order to simulate
non-recursive processes
inside real programming
in an intuitionistic, informative, semantic, and compositional way.

41. Using Parallel Computations
Using parallel computations we may define a model of learning processes which is more efficient, and more adherent to our intuition of what “learning” is.
In the next page, we introduce an example of process “learning” the truth value of a non-decidable statement. Then we will represent it using parallel computations.

42. A process E learning if xN.P(x)
We change our mind to xN.P(x) true
We never change our mind again
We deduce P(x6) is false
We check and we find P(x6) true
We deduce P(x5) is false
We check it indeed is
We deduce P(x4) is false
We check it indeed is
Start: we know P(x1), P(x2), P(x3) are false
We assume xN.P(x) is false

43. What a Learning processes does
A learning process makes some hypothesis coherent with the available data.
It starts a computation from such hypothesis.
During such computation it gathers new data.
Whenever some new datum contradicts its hypothesis, it produces a new hypothesis.
After producing a new hypothesis it restarts the computation.
Points 1-5 are repeated over and over again.

44. Convergence
Classically, a learning process process is convergent iff (like E in the previous page) it is stationary (it is constant from some stage on) in all possible computations.
The notion of convergence may be re-expressed intutionistically, by taking the no-counterexample interpretation of stationarity.

45. Learning processes are Parallel computations
The search for new data may be seen as the choice of a particular timeline in the space of the event.
Searches done by different processes are on different subsets of data, and they may be thought taking place in parallel.

46. Learning processes are non-Deterministic
A process may change its hypothesis when receiving some counterexample from some other process.
When many counterexamples are sent, only one of them is chosen, in a non-deterministic way.

47. Outline of the paper In the paper we define the notions of:
time (an unbounded partial ordering);
forking of timelines (to describe different possible computations);
interfering with a timeline, executing some actions (to describe a process);
forcing a timeline to satisfy a particular property.

48. an intuitionistic model of 02-maps using parallel computations
Outline of the results
Our processes have an intuitionistic notion of convergence and of equality.
We defined a notion of morphisms over processes.
Morphisms and convergent processes, quotiented up to process equality, are:
an intuitionistic model of 02-maps
using parallel computations

49. Conclusions
Learning allows to use simulations of non-recursive maps in real programming.
Parallel non-deterministic computations implement learning processes in a way which is more efficient, and more adherent to our intuition of what learning is.

50. Appendix: Event Structure
An Even Structure is any list <T, T, A, act> such that
T,T is any inhabited, unbounded recursive partial ordering. Elements of T denote instants of time.
A is a finite or infinite recursive subset of N, denoting a set of actions which may take place in any given instant.
act:T {finite subsets of A} is any recursive, weakly increasing map. act(t) denotes the finite set of actions which took place up to instant t.

51. Timelines and Strategies
A timeline is any recursive map :N T.
A strategy is any recursive map A: T{finite subsets of A} suggesting in any instant some action to do.
A timeline  follows what a strategy A suggests in a step i iff t executes the actions A(i) in some step j. That is, iff A(i)  act(j) for some j.
A team is any group of strategy changing with time, coded by a recursive map
F:T{finite sets of strategies}.

52. Forcing a property
A timeline  follows (the suggestions of) a strategy A iff  follows A in infinitely many steps.
A timeline  follows (the suggestions of) a team F iff for infinitely many i, if AF(i) then  follows A in i.
A strategy (team) forces a property P of timelines iff all timelines following the strategy (team) satisfy P.
A property P may be forced iff there is some team forcing it.

53. Learning maps and Equality
Fix two maps L, M:TN (any two successions indexed by time).
For any timeline :NT, we have L, M:N N. Define a property P() of timelines by “L is convergent (as succession over integers)”.
We say that L is convergent (that is is a learning map) iff there is a team F forcing the property:
“L is convergent”
We say that L, M are equal iff there is a team F forcing the property:
“L, M are convergent to the same limit”

54. f*:{learning maps}{learning maps}
Morphisms
A map
f*:{learning maps}{learning maps}
is a morphisms on learning maps iff there is some recursive map
f:N{learning maps}
such that, for all tT
f*(L)(t) = f(L(t))(t)