Ramsey Theorem for pairs as a classical principle in Intuitionistic - - PowerPoint PPT Presentation

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Ramsey Theorem for pairs as a classical principle in Intuitionistic Arithmetic.

Silvia Steila, joint work with Stefano Berardi

Universit` a degli studi di Torino

British Logic Colloquium, Leeds September 6th, 2013

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 1 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Σ0

n-LLPO

Lesser Limited Principle of Omniscience. For any parameter a ∀x, x′ (P(x, a) ∨ Q(x′, a)) = ⇒ ∀xP(x, a) ∨ ∀xQ(x, a). (P, Q ∈ Σ0

n−1)

Pigeonhole Principle for Π0

n

For any parameter a ∀x ∃z [z ≥ x ∧ (P(z, a) ∨ Q(z, a))] = ⇒ ∀x ∃z [z ≥ x ∧ P(z, a)] ∨ ∀x ∃z [z ≥ x ∧ Q(z, a)]. (P, Q ∈ Π0

n)

EMn Excluded Middle for Σ0

n formulas. For any parameter a

∃x P(x, a) ∨ ¬∃x P(x, a). (P ∈ Π0

n−1)

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 2 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

. . . EM2 Σ0

2-MARKOV

∆0

2-EM

Σ0

2 -LLPO

Π0

2-EM

EM1 Σ0

1-MARKOV

∆0

1-EM

Σ0

1 -LLPO

Π0

1-EM

EM0 Classical Logic HA Thesis: RT2

2 is equivalent

to Σ0

3-LLPO in HA.

The purpose of this work is to study, from the viewpoint of first order arithmetic (no set variables, the only sets are the arithmetical sets), Ramsey Theorem for pairs for recursive assignement of two colors in

• rder to find some principle of classical logic equivalent to it in HA.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 3 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

If X is a set, [X]2 = {Y ⊆ X | |Y | = 2}. We can think of [X]2 as the complete graph on X. We only consider arithmetically definable sets. RT2

2(Σ0 n). Ramsey Theorem for graphs and Σ0 n 2-colorings

For any coloring ca : [ω]2 → 2 with a parameter a, there exists an infinite subset X of ω homogeneous for the given coloring, i.e. [X]2 is painted with only one color. (ca ∈ Σ0

n).

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 4 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

In this work we formalize Ramsey Theorem for two colors, for pairs and for recursive colorings by the following schema: {∀a(B(., ca) infinite homogeneous black ∨ W (., ca) infinite homogeneous white ) | for some B, W arithmetical predicates }. Here c = {ca | a ∈ ω} denotes any recursive family of recursive assignment of two colors, black and white. We call this a disjunctive schema and we prove it if we prove some instance of it.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 5 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

First part: RT2

2 implies Σ0 3-LLPO We will prove that in the first order intuitionistic arithmetic, RT2

2(Σ0 0) is

equivalent to the classical principle Σ0

3-LLPO. By definition of disjunctive

schema, Ramsey Theorem for graphs and a recursive 2-coloring implies Σ0

3-LLPO if the following holds: for each P in Σ0 3-LLPO, there exist a

finite number of recursive family of recursive colorings ca,0, . . . , ca,j−1 such that, fixed any Wi(., ca,i) and Bi(., ca,i), if we assume {∀a(Wi(., ca,i) is inf. and hom. ∨ Bi(., ca,i) is inf. and hom. ) | i ∈ j} then we deduce P.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 6 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Proof sketch of RT2

2 implies Σ0 3-LLPO Lemma

1

RT2

2(Σ0 0) implies EM1;

2

EM1 implies that, for any family F = {s(n, ·) | n ∈ ω} of recursive monotone and bounded above sequences enumerated by a binary primitive recursive function s : ω × ω → ω, each sequence in F is stationary;

3

EM1 implies that, for any family G = {t(n, ·) | n ∈ ω} of recursive sequences enumerated by a binary primitive recursive function t : ω × ω → ω for which there are at most k values of x such that t(n, x) = t(n, x + 1), each sequence in G is stationary.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 7 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Proof sketch of RT2

2 implies Σ0 3-LLPO Let a be a parameter. We assume the hypothesis of Σ0

3-LLPO:

∀x, x′ (H0(x, a) ∨ H1(x′, a)), where H0(x, a) := ∃y ∀z P0(x, y, z, a) H1(x, a) := ∃y ∀z P1(x, y, z, a) for some P0, P1 primitive recursive predicates.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 8 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Proof sketch of RT2

2 implies Σ0 3-LLPO Our thesis is ∀x H0(x, a) ∨ ∀x H1(x, a) we define a recursive 2-coloring such that: if there are infinitely many white edges from x, then for all y ≤ x H0(y, a) holds; if there are infinitely many black edges from x, then for all y ≤ x H1(y, a) holds. Applying RT2

2(Σ0 0), there exists an infinite homogeneous set X. We prove

that if there is some infinite set X homogeneous in color c, then Hc(x, a) holds for infinitely many x. We obtain ∀x Hc(x, a).

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 9 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Second part: Σ0

3-LLPO implies RT2 2 Now we modify Jockusch proof of Ramsey Theorem in order to obtain a proof in HA of Σ0

3-LLPO =

⇒ RT2

2.

Given a coloring c : [ω]2 → 2 we say that X ⊆ ω defines a 1-coloring if for all x ∈ X, any two edges from x to some y, z ∈ X have the same color. If X is infinite and defines a 1-coloring, thanks to the Pigeonhole Principle we may define an infinite arithmetical subset Y of X whose points all have the same color. Y is homogeneous for c. So we need to find an infinite set that defines a 1-coloring.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 10 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Proof Sketch of Σ0

3-LLPO implies RT2 2 A tree T included in a graph ω defines a 1-coloring w.r.t. T if for all x ∈ T for any two proper descendants y, z of x in T, the edges from x to y, z have the same color. Assume there exists some infinite binary tree T defining a 1-coloring w.r.t. T. Then T has an infinite branch B by K¨

• nig’s Lemma. B defines

an infinite 1-coloring and so proves RT2

2.

Therefore a sufficient condition for RT2

2 is the existence of an infinite

binary tree defining a 1-coloring.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 11 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Proof Sketch of Σ0

3-LLPO implies RT2 2

Jockusch proof:

tree infinite branch homogeneous set K¨

• nig

PP(Π0

2)

Our work (Σ0

3-LLPO):

tree (Π0

1)

infinite branch (Π0

2)

(it is unique) color of homogeneous set E M

2

P P ( Π

1

)

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 12 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Proof Sketch of Σ0

3-LLPO implies RT2 2 Given any set T we may equip it by the following ancestor/descendant relation 0 ≺T 1, x ≺T y iff x ∈ T , y ∈ ω, x < y and ∀z(z ≺T x(c({z, x} = c{z, y}))). Definition (Inductive definition of the set T in HA) Define Tn by induction on n. If n = 0 then T0 = x0 := 0. For n + 1, if Chosen(xn+1, Tn), then Tn+1 = Tn ∪ {xn+1}. T =

• n∈ω

Tn. Chosen is some suitable arithmetical predicate.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 13 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Proof Sketch of Σ0

3-LLPO implies RT2 2 We proved (using a part of Σ0

3-LLPO) that

T is a Π0

1 binary tree,

T has a unique infinite branch r such that if T has infinitely many edges with color c, then r has infinitely many edges with color c. Moreover using Σ0

3-LLPO =

⇒ Pigeonhole Principle for Π0

1 predicates ,

we obtain that T has infinitely many edges of color c, so r has infinite many edges of color c; their smaller nodes define a monochromatic set for the original graph.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 14 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Our proof recursively defines two monochromatic sets ∆0

3, one of each

color, that can not be both finite, even if we can not decide which of these is the infinite one. In Jockusch proof he shows that one of the homogeneous sets is Π0

2,

while the second one in ∆0

• 3. In our proof we can see that both the

homogeneous sets are ∆0

3, since our construction is symmetric with

respect to the two colors.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 15 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Conclusions

Apparently, Σ0

3-LLPO is a principle of uncommon use, so we want to

remark the equivalence between Σ0

3-LLPO and two more common

principles: EM2 and DeMorgan(Σ0

3).

DeMorgan(Σ0

n) := ¬(P ∧ Q) =

⇒ ¬P ∨ ¬Q. (P, Q ∈ Σ0

n)

Theorem (in intuitionistic arithmetic HA) Σ0

n -LLPO ⇐

⇒ DeMorgan(Σ0

n) + EMn−1.

We can see that the most of the proof uses only EM2 and that DeMorgan(Σ0

3) (and so Σ0 3-LLPO) is used only in the last part.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 16 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

Questions and future developments

Is Σ0

3-LLPO equivalent to Ramsey Theorem for graphs and recursive

n-colorings for any n ≥ 2? May we generalize our results to Σ0

n-colorings and the classical

principle Σ0

n+3 -LLPO, for any n ≥ 0?

We hope to apply, in future works, the method called interactive realizability to understand and explain the computational content of the modified Jockusch proof, and prove (constructively) some consequences of Ramsey Theory. The interactive realizability is a realizability interpretation for first order classical arithmetic introduced in 2008 by Stefano Berardi and Ugo de’ Liguoro.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 17 / 18

Introduction RT2 2 implies Σ0 3-LLPO Σ0 3-LLPO implies RT2 2 Conclusions Bibliography

References

Akama, Berardi, Hayashi, Kohlenbach. An Arithmetical Hierarchy of the Law of Excluded Middle and Related Principles. LICS, 2004. Berardi, de’ Liguoro. A Calculus of Realizers for EM1 Arithmetic. Proceedings of CSL ’08, 2008. Jockusch. Ramsey’s Theorem and Recursion Theory. JSL, 1972. Ramsey. On a problem of formal logic. Proceedings London Mathematical Society, 1930.

Silvia Steila, joint work with Stefano Berardi Ramsey Theorem as a classical principle in HA 18 / 18