on the number of provable formulas - iac.rm.cnr.itmarco/aila2005.pdf · aila 2005, 10 { 13 febbraio...

IntroductionCounting

The heuristic method

On the number of provable formulas

Marco Pedicini

Istituto per le Applicazioni del Calcolo ”M. Picone”, CNRRome – Italy

mailto:[email protected]

work in collaboration withQuintijn Puite (Utrecht)

AILA 2005, 10 – 13 Febbraio 2005

M. Pedicini On the number of provable formulas



Proofs and graphsProbabilistic correctness

Combinatorics in logic

Objects in logic have combinatorical nature:

formulas,

are intrisically discrete.







formulas,

sequents,








formulas,

sequents,

proofs.








formulas,

sequents,

proofs.


LL adds further difficulties by considering proof nets as first class objects.







formulas,

sequents,

proofs.


LL adds further difficulties by considering proof nets as first class objects.Combinatorics rise up at some extents as to the geometrical nature ofthese objects, and we find geometry in their graph representation.





Superposition

Variants of proof nets where superpositions of proofs have beenconsidered:

in separability a la Bohm for linear logic;





Superposition



in extending Girard’s ludics with exponential connectives;





Superposition




in the investigation of the ”true” notion of proof net;





Superposition





and also inspired by quantum computing.





Superposition





and also inspired by quantum computing.

In this work, we start to study the problem by elementary combinatorics,and we start from the following consideration:

a superposition of proof nets is given by a formal sum of weighted graphs





Formula tree

Consider a formula as the syntactic tree by forgetting non-geometricalinformation (connectives and atoms)

3

21

If fact we may chose different admissible decorations (rules) on nodes.





Where probabilities come out

With two possible connectives

&

or ⊗ we may assign to each node theconnective given by a weighted combination of them







&


&

&

&







&


&

&⊗







&


& &

⊗







&


&

⊗

⊗







&


⊗

&

⊗







&


⊗⊗

⊗







&


⊗

&

⊗

&

&

⊗

Node i is

&

with weight pi and with weight (1 − pi ) it is a ⊗.







&


⊗

&

⊗

&

&

⊗

Node i is

&

with weight pi and with weight (1 − pi ) it is a ⊗.In order to give them the nature of proof we need to make interact itwith axiom links







&


⊗

&

⊗

&

&

⊗

Node i is

&

with weight pi and with weight (1 − pi ) it is a ⊗.In order to give them the nature of proof we need to make interact itwith axiom links only in a few cases we will obtain proof nets, inparticular depending on weights we can establish the probability to becorrect for a given weighted tree.





By excluding symmetric cases we obtain that there are three possibleaxiom wirings

⊗

&

⊗

&

&

⊗






⊗

&

⊗

&

&

⊗

When do they yield a correct proof net ?






⊗

&

⊗

&

&

⊗


red/green wirings

In the first two cases it is a proof with probability(1 − p1)p2p3 + p1(1 − p2)p3






⊗

&

⊗

&

&

⊗


red/green wirings

In the first two cases it is a proof with probability(1 − p1)p2p3 + p1(1 − p2)p3

blue wiring

In the last case it is correct with probabilityp1p2(1 − p3).





Probabilistic correctness

Let us draw the picture of the global correctness of this object (p1 = p2) :

P(p1, p3) =1

3[2(1 − p1)p2p3 + 2p1(1 − p2)p3 + p1p2(1 − p3)]

00.2

0.40.6

0.8

1 0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

00.2

0.40.6

0.8

1

Is it possible to extends this process to the general case ?




Counting proof nets

As a preliminary step, we considered the problem of counting in howmany ways we may combine all possible axiom wirings with all possibleformula trees.




Counting proof nets


Easy way

For a given n generate all the structures by computer, and return theirnumber T (n);




Counting proof nets


Easy way

For a given n generate all the structures by computer, and return theirnumber T (n);cross fingers and submit the sequence

T (1), T (2), T (3), . . . , T (N)

to Sloane’s On-Line Encyclopedia of Integer Sequences(http://www.research.att.com/~ njas/sequences/)

...




Counting proof nets


Easy way

For a given n generate all the structures by computer, and return theirnumber T (n);cross fingers and submit the sequence

T (1), T (2), T (3), . . . , T (N)

to Sloane’s On-Line Encyclopedia of Integer Sequences(http://www.research.att.com/~ njas/sequences/)

...

Note that N is usually quite small, depending on RAM and CPU of yourPC.




Number of provable formulas (in MLL)

n

number of provable formulasT (n, 1)/(2nn!)

123456789

10

117

88282 725

11 556 5902 173 613 962

517 553 880 484149 714 681 114 349

51 094 054 734 001 49420 126 763 226 141 651 806

But Sloane cannot help us...”I am sorry, but the terms 1, 17, 882 do not match anything in thetable.”




Number of provable formulas (in MLL)

n


123456789

10

117

88282 725

11 556 5902 173 613 962

517 553 880 484149 714 681 114 349

51 094 054 734 001 49420 126 763 226 141 651 806

But Sloane cannot help us...”I am sorry, but the terms 1, 17, 882 do not match anything in thetable.”

Question:

Is it possible to have a formula computing T (n) ?

In facts, it is a non trivial task:

Unfortunately, we didn’t obtain a formula expressing the number ofprovable formulas for a given number of axiom links, but...M. Pedicini On the number of provable formulas



First StepSecond Step5th Step

Algebraization of proof generation

In order to make more effective the generation of proofs we introduce aformal structure of vector space.Let P

+ and P− be two isomorphic copies of the countably-dimensional

vector space

Rω :=

{

n∑

i=1

ciei

∣

∣

∣

∣

∣

ci ∈ R, n ∈ N

}

.

Proofs as vectors

The basic elements of P+ will be written [i ]+ (standing for a general

positive proof with i conclusions),Elements of P

− will be written [i ]− (standing for a general negative proofwith i conclusions).





MLL (negatively focalized)

Let us call P(n, c) (resp. Q(n, c)) the number of positive (resp. negative)MLLnf-proofs with n different given axioms and c conclusions.

Theorem

The functions P : N → P+ and Q : N → P

− satisfy the following

recursive definition.

P(1) = [2]+

P(n) =n−1∑

n′=1

(

n

n′

)

π(Q(n′)Q(n − n′)) (n > 1)

Q(n) = ν(P(n)) (n ≥ 1)





Let us define

P(n) :=∑

c≥1 P(n, c)[c]+ and Q(n) :=∑

c≥1 Q(n, c)[c]−.

We turn P− into an algebra

Definition

Let us define an (associative, commutative) multiplication (denoted byjuxtaposition)

P−× P

−→ P

− [i ]−[j ]− = [i + j − 1]−.

Intuitively corresponding to applying a ⊗-rule and remembering the mainformula:

A⊥ ` Γ B⊥ ` ∆

(A ⊗ B)⊥ ` Γ, ∆





Linear maps on the vector space

Definition

The de-focalization map π : P− → P

+ is defined by [i ]− 7→ [i ]+

It corresponds to forgetting the distinguished formula A⊥ ` Γ` A, Γ

The following map corresponds to the application of all possiblegeneralized

&-rules:

Definition

The map ν : P+ → P

− is defined by

[i ]+ 7→

i∑

q=1

(

i

q

)

q!Cq−1[i − q + 1]−.

remember we have to count all the possible

&

-trees we can attach to apositive sequent of length i .





Forgetting the formal structure

We have to generate elements of the formal vector space in order tocompute the coefficient of the element [1]−.The following statement is obtained by solving the previous recurringdefinition:

Theorem

P(1, 1) = 0

P(1, 2) = 1

P(n, c) =

n−1X

n′=1

“ n

n′

”

c−1X

c′=0n′−((n+1)−c)≤c′≤n′

Q(n′, 1 + c′)Q(n − n

′, 1 + ((c − 1) − c′))

(n > 1; 1 ≤ c ≤ n + 1)

Q(n, c) =

1+((n+1)−c)X

q=1

“q + (c − 1)

q

”

q!Cq−1P(n, q + (c − 1))

(n ≥ 1; 1 ≤ c ≤ n + 1)





Table for MLLnf

Upperbound for provable formulas.

By using the previous formula it is possible to determine by onlynumerical computations the following values:

nnumber of provable formulas

T (n, 1)/(2nn!)number of MLLnf-derivations

Q(n, 1)/(2nn!)123456789

10

117

88282 725

11 556 5902 173 613 962

517 553 880 484149 714 681 114 349

51 094 054 734 001 49420 126 763 226 141 651 806

117

1 174174 213

43 508 18616 093 558 826

8 162 702 679 8525 394 878 462 002 605

4 482 152 731 426 496 0504 558 136 970 068 451 778 302





Main result and conclusion

Similar construction, analogous to step 1, can be carried on for MLLsc,MLLcf, MLLpn;It is difficult to obtain step 2 for MLLpn.In fact, we have proved a heuristic result by using property of theexpectation function:

n

heuristic approximationT ′(n, 1)/(2nn!)


number of MLLnf-derivationsQ(n, 1)/(2nn!)

123456789

10

1.017.0

810.567 180.8

8 097 633.21 292 177 393.4

257 683 716 149.861 774 586 215 171.7

17 316 387 694 269 184.15 559 590 039 485 795 1xx.x

117

88282 725

11 556 5902 173 613 962

517 553 880 484149 714 681 114 349

51 094 054 734 001 49420 126 763 226 141 651 806

117

1 174174 213

43 508 18616 093 558 826

8 162 702 679 8525 394 878 462 002 605

4 482 152 731 426 496 0504 558 136 970 068 451 778 302


on the number of provable formulas - iac.rm.cnr.itmarco/aila2005.pdf · aila 2005, 10 { 13 febbraio...

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