Download N 2 - Università di Torino

Classical Logic as
Limit Completion
Workshop on
Proof Theory and Algorithms
23 to 29 March 2003, Edinburgh
Stefano Berardi - Università di Torino
http://www.di.unito.it/~stefano
• The text of this talk and some related
papers may be found in the home page of
the author:
http://www.di.unito.it/~stefano
2
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.
3
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.”
4
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
5
proofs of simply existential statements.
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.
6
§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.
7
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”.
8
• How much is the opinion of E reliable ?
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).
9
• So what? Is there any use of E(t)?
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).10
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.
11
§ 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.
12
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.
13
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” ?
14
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
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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.
16
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
17
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
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• The maps f* are by defin. the morphisms of N2.
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 moperator, 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)19
points nN.
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”.
20
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
21
effectively solve concrete problems in N.
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.
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§ 3. A model R2 for Intuitionistic
Arithmetic + EM-1
• Recall that EM-1 is Excluded Middle over 1quantifier 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.
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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”.
24
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
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just “a1~b1, …, an~bn implies a~b”.
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)
26
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!
27
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.28
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.
29
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.
30
§ 4. A model of the whole
Classical Arithmetic
•
In the construction of R2, we used only
Intuitionistic Arithmetical reasoning, plus two
properties of N2:
1. Density of N in N2. Every recursive map
f:NN2 may be extended in a unique way to a
map f*:N2N2.
2. Conservativity of N2 w.r.t. N. Every simple
existential predicate of N2 covering N covers the
whole N2.
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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.
32
Iterating Completion of N
•
•
•
•
•
•
For all k, we may define a model Nk of 0kmaps 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.
33
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:
http://www.di.unito.it/~stefano
34
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
35
reconsider it again.
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
36
k=0.
Related Papers
• Classical Logic as Limit. An intuitionistic model
of 02-maps using Parallel Computations.
Submitted to I.C.. Available in:
http://www.di.unito.it/~stefano
• An Intuitionistic Model of Classical Arithmetic
and Arithmetical maps. Draft Version.
37
Learning Processes
and Parallel Computations
First APPSEM-II Workshop
26 to 28 March 2003
Nottingham, United Kingdom
Stefano Berardi - Università di Torino
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 http://www.di.unito.it/~stefano
39
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.
40
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.
41
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(x4) is false
We check it indeed is
We deduce P(x5) 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
What a Learning processes does
1. A learning process makes some hypothesis
coherent with the available data.
2. It starts a computation from such hypothesis.
3. During such computation it gathers new data.
4. Whenever some new datum contradicts its
hypothesis, it produces a new hypothesis.
5. After producing a new hypothesis it restarts
the computation.
6. Points 1-5 are repeated over and over again.
43
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 reexpressed intutionistically, by taking the
no-counterexample
interpretation
of
stationarity.
44
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.
45
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 nondeterministic way.
46
Outline of the paper
• In the paper we define the notions of:
1. time (an unbounded partial ordering);
2. forking of timelines (to describe different
possible computations);
3. interfering with a timeline, executing
some actions (to describe a process);
4. forcing a timeline to satisfy a particular
property.
47
Outline of the results
1. Our processes have an intuitionistic
notion of convergence and of equality.
2. We defined a notion of morphisms over
processes.
3. Morphisms and convergent processes,
quotiented up to process equality, are:
an intuitionistic model of 02-maps
using parallel computations
48
Conclusions
• Learning allows to use simulations of nonrecursive 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.
49
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. 50
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}.
51
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.
52
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”
53
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)
54