Hall’s marriage theorem and subsystems of second order arithmetic Makoto Fujiwara Tohoku university

2011.11.18

Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

1 / 23

Contents of my master thesis Title : 1

Introduction and Preliminaries 1

.

. .

.

2

.

2 3

1

3

3

History of the Study for the Difficulty of Marriage Theorem Expanding Marriage Theorem Bounded Marriage Theorem with Bounding Function as Parameter

Reverse Mathematics of Kuratowski’s Theorem 1

.

Introduction Fundamental Concepts of Recursion Theory Subsystems of Second Order Arithmetic and Reverse Mathematics

Reverse Mathematics of Marriage Theorem 2

.

Reverse mathematics of marriage theorem and Kuratowski’s theorem

2

Formalization of Planar Graphs RCA0 ` Kratowski’s Theorem ↔ WKL0

Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

2 / 23

Contents of this talk

1

History of the Study for the Difficulty of Marriage Theorem

2

Reverse mathematics of Expanding Marriage Theorem

3

Reverse mathematics of Bounded Marriage Theorem with Bounding Function as Parameter

Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

3 / 23

Marriage Theorem A bipartite graph (B, G; R) is a 3-tuple where B and G are disjoint sets of vertices and R is a set of edges such that R ⊂ B × G. Let G = (B, G; R) is a bipartite graph. A solution of G is a injection M : B → G such that {(b, M(b))|b ∈ B} ⊂ R. For any X⊂fin B let NG (X) = {g ∈ G|∃b ∈ X((b, g) ∈ R)} and S G (X) = |NG (X)| − |X|.

Hall Condition : ∀X⊂fin B(S G (X) ≥ 0) Theorem (P. Hall, 1935) If G is a finite bipartite graph then there is a solution of G if it satisfies Hall condition. Theorem (M. Hall, 1948) If G is a bipartite graph such that for all b ∈ B, |NG (b)| < ∞ then there is a solution of G if it satisfies Hall condition. Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

4 / 23

Marriage Theorem A bipartite graph (B, G; R) is a 3-tuple where B and G are disjoint sets of vertices and R is a set of edges such that R ⊂ B × G. Let G = (B, G; R) is a bipartite graph. A solution of G is a injection M : B → G such that {(b, M(b))|b ∈ B} ⊂ R. For any X⊂fin B let NG (X) = {g ∈ G|∃b ∈ X((b, g) ∈ R)} and S G (X) = |NG (X)| − |X|.

Hall Condition : ∀X⊂fin B(S G (X) ≥ 0) Theorem (P. Hall, 1935) If G is a finite bipartite graph then there is a solution of G if it satisfies Hall condition. Theorem (M. Hall, 1948) If G is a bipartite graph such that for all b ∈ B, |NG (b)| < ∞ then there is a solution of G if it satisfies Hall condition. Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

4 / 23

Recursion Theoretic Analysis of Marriage Theorem Let G be a recursive baipartite graph such that for all b ∈ B, |NG (b)| < ∞ and G satisfies Hall condition. Then G has a solution by infinite marriage theorem. Now can we take the solution recursively? The answer is “no”. Theorem (A.Manaster and J.Rosenstein, 1972) There exists a recursive bipartite graph that satisfies Hall condition, but has no recursive solution. Can we modify this to have a recursive solution? A graph G = (V, E) is highly recursive if the function p : V → ω such that p(v) = |{v0 |(v, v0 ) ∈ E}| is recursive. Theorem (A.Manaster and J.Rosenstein, 1972) There exists a highly recursive bipartite graph that satisfies Hall condition, but has no recursive solution. Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

5 / 23

Recursion Theoretic Analysis of Marriage Theorem Let G be a recursive baipartite graph such that for all b ∈ B, |NG (b)| < ∞ and G satisfies Hall condition. Then G has a solution by infinite marriage theorem. Now can we take the solution recursively? The answer is “no”. Theorem (A.Manaster and J.Rosenstein, 1972) There exists a recursive bipartite graph that satisfies Hall condition, but has no recursive solution. Can we modify this to have a recursive solution? A graph G = (V, E) is highly recursive if the function p : V → ω such that p(v) = |{v0 |(v, v0 ) ∈ E}| is recursive. Theorem (A.Manaster and J.Rosenstein, 1972) There exists a highly recursive bipartite graph that satisfies Hall condition, but has no recursive solution. Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

5 / 23

Recursion Theoretic Analysis of Marriage Theorem Let G be a recursive baipartite graph such that for all b ∈ B, |NG (b)| < ∞ and G satisfies Hall condition. Then G has a solution by infinite marriage theorem. Now can we take the solution recursively? The answer is “no”. Theorem (A.Manaster and J.Rosenstein, 1972) There exists a recursive bipartite graph that satisfies Hall condition, but has no recursive solution. Can we modify this to have a recursive solution? A graph G = (V, E) is highly recursive if the function p : V → ω such that p(v) = |{v0 |(v, v0 ) ∈ E}| is recursive. Theorem (A.Manaster and J.Rosenstein, 1972) There exists a highly recursive bipartite graph that satisfies Hall condition, but has no recursive solution. Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

5 / 23

Recursion Theoretic Analysis of Marriage Theorem Let G be a recursive baipartite graph such that for all b ∈ B, |NG (b)| < ∞ and G satisfies Hall condition. Then G has a solution by infinite marriage theorem. Now can we take the solution recursively? The answer is “no”. Theorem (A.Manaster and J.Rosenstein, 1972) There exists a recursive bipartite graph that satisfies Hall condition, but has no recursive solution. Can we modify this to have a recursive solution? A graph G = (V, E) is highly recursive if the function p : V → ω such that p(v) = |{v0 |(v, v0 ) ∈ E}| is recursive. Theorem (A.Manaster and J.Rosenstein, 1972) There exists a highly recursive bipartite graph that satisfies Hall condition, but has no recursive solution. Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

5 / 23

Expanding Hall Condition : there is a function h s.t.

h(0) = 0 & ∀n∀X⊂fin B(|X| ≥ h(n) → S G (X) ≥ n)

Theorem (H.Kierstead, 1983) If G is highly recursive bipartite graph that satisfies expanding Hall condition with a recursive h, then G has a recursive solution. Theorem (H.Kierstead, 1983) There exists a highly recursive bipartite graph which satisfies expanding Hall condition but has no recursive solution.

Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

6 / 23

Kierstead also showed that a bipartite graph which satisfies expanding Hall condition has a solution even if there are boys who know infinite many girls. Theorem (H.Kierstead, 1983) If G is a bipartite graph then there is a solution of G if it satisfies Expanding Hall condition.

Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

7 / 23

Reverse Mathematics of Marriage Theorem FMT: If G = (B, G; R) is a bipartite graph such that |B| < ∞ then there is a solution of G if it satisfies Hall condition. MT: If G = (B, G; R) is a bipartite graph such that ∀b ∈ B ∃t ∀g((b, g) ∈ R → g < t) then there is a solution of G if it satisfies Hall condition. BMT: If G = (B, G; R) is a bipartite graph such that ∃p : B → N s.t.∀b, g((b, g) ∈ R → g < p(b)) then there is a solution of G if it satisfies Hall condition. Theorem (J.Hirst, 1987) RCA0 ` FMT RCA0 ` MT ↔ ACA0 RCA0 ` BMT ↔ WKL0 Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

8 / 23

Symmetric Hall Condition :

∀X⊂fin B(S G (X) ≥ 0) & ∀Y⊂finG(S G (Y) ≥ 0) A symmetric solution is a bijectve solution. MTsym : If G = (B, G; R) is a bipartite graph such that

∀v ∈ B∪G ∃t ∀g((v, v0 ) ∈ R → v0 < t)

then there is a symmetric solution of G if it satisfies symmetric Hall condition. BMTsym : If G = (B, G; R) is a bipartite graph such that ∃p : B ∪ G → N s.t.∀v, v0 ((v, v0 ) ∈ R → v < p(v0 )) then there is a symmetric solution of G if it satisfies symmetric Hall condition. Theorem (J.Hirst, 1987) RCA0 ` MTsym ↔ ACA0 RCA0 ` BMTsym ↔ WKL0 Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

9 / 23

Reverse mathematics of Expanding Marriage Theorem

Expanding Hall Condition : Hall condition & ∀n∃m∀X⊂fin B(|X| ≥ m → S G (X) ≥ n)

Strongly Expanding Hall Condition : Hall condition & ∃hB :B → N s.t. ∀n∀X⊂fin B(|X| ≥ hB (n) → S G (X) ≥ n)

B-locally finite : ∀b ∈ B ∃t ∀g((b, g) ∈ R → g < t) B-bounded : ∃p : B → N s.t.∀b, g((b, g) ∈ R → g < p(b))

Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

10 / 23

EMT: If G = (B, G; R) is a bipartite graph which satisfies expanding Hall condition and B-locally finite then there is a solution of G. ES MT: If G = (B, G; R) is a bipartite graph which satisfies strongly expanding Hall condition and B-locally finite then there is a solution of G. BEMT: If G = (B, G; R) is a bipartite graph which satisfies expanding Hall condition and B-bounded then there is a solution of G. BES MT: If G = (B, G; R) is a bipartite graph which satisfies strongly expanding Hall condition and B-bounded then there is a solution of G. XMTG : the statement XMT with the assumption G-locally finite XMTGB : the statement XMT with the assumption G-bounded Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

11 / 23

We got the following results. ACA0   

MT(Hirst) EMT ES MT G

EMT EMTGB WKL0   

ES MTG

BMT(Hirst) BEMT BES MT BEMTG BEMTGB

RCA0

BES MTG ES MTGB ? BES MTGB (Kierstead)

FMT(Hirst) Conjecture : RCA0 ` ES MTGB Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

12 / 23

EM∗ T is the statement proved by Kierstead. EM∗ T: If G = (B, G; R) is a bipartite graph which satisfies expanding Hall condition then there is a solution of G. ES M∗ T: If G = (B, G; R) is a bipartite graph which satisfies strongly expanding Hall condition then there is a solution of G. Theorem ACA0 ` EM∗ T The next figure is given by combining this result with the previous figure.

Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

13 / 23

ACA0   

WKL0   

MT(Hirst)

EM∗ T EMT EM∗ TG EMTG EM∗ TGB EMTGB

ES M∗ T ES MT ES M∗ TG ES MTG

BMT(Hirst) BEMT BES MT BEMTG BEMTGB

RCA0

BES MTG ES M∗ TGB ? ES MTGB ? BES MTGB (Kierstead)

FMT(Hirst) Conjecture : RCA0 + some strong induction ` ES M∗ TGB Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

14 / 23

Symmetrical expanding Hall condition : Expanding Hall condition for B & Expanding Hall condition for G

Half strongly expanding Hall condition : Symmetrical expanding Hall condition & strongly expanding Hall condition for one of B or G

Strongly symmetrical expanding Hall condition : Strongly expanding Hall condition for B & strongly expanding Hall condition for G

Symmetrically locally finite : B-locally finite & G-locally finite Half bounded : symmetrically locally finite & (one of B or G)-bounded

Symmetrically bounded : B-bounded & G-bounded Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

15 / 23

EMTsym : If G is a b.g. which satisfies sym. expanding Hall condition and sym. locally finite then there is a sym. solution of G. ES MTsym : If G is a b.g. which satisfies sym. strong expanding H.c. and sym. locally finite then there is a sym. solution of G. BEMTsym : If G is a b.g. which satisfies sym. expanding H.c. and sym. bounded then there is a sym. solution of G. BES MTsym : If G is a b.g. which satisfies sym. strongly expanding H.c. and sym. bounded then there is a sym. solution of G. BEHS MTsym : If G is a b.g. which satisfies half strongly expanding H.c. and sym. bounded then there is a sym. solution of G. HBES MTsym : If G is a b.g. which satisfies sym. strongly expanding H.c. and half bounded then there is a sym. solution of G. Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

16 / 23

We got the following results. ACA0   

MTsym (Hirst)

EM∗ Tsym EMTsym

· · ·· · · ··

· · ·· · · ·· · · ·· · · ··

· · ·· · · ·· ES MTsym

· · ··

HBES MTsym WKL0   

BMTsym (Hirst) BEMTsym BEHS MTsym

RCA0

BES MTsym FMTsym

EM∗ Tsym : If G is a b.g. which satisfies sym. expanding H.c. then there is a sym. solution of G. Theorem ACA0 ` EM∗ Tsym Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

17 / 23

We got the following results. ACA0   

MTsym (Hirst)

EM∗ Tsym EMTsym

· · ·· · · ··

· · ·· · · ·· · · ·· · · ··

· · ·· · · ·· ES MTsym

· · ··

HBES MTsym WKL0   

BMTsym (Hirst) BEMTsym BEHS MTsym

RCA0

BES MTsym FMTsym

EM∗ Tsym : If G is a b.g. which satisfies sym. expanding H.c. then there is a sym. solution of G. Theorem ACA0 ` EM∗ Tsym Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

17 / 23

We got the following results. ACA0   

MTsym (Hirst)

EM∗ Tsym EMTsym

· · ·· · · ··

· · ·· · · ·· · · ·· · · ··

· · ·· · · ·· ES MTsym

· · ··

HBES MTsym WKL0   

BMTsym (Hirst) BEMTsym BEHS MTsym

RCA0

BES MTsym FMTsym

EM∗ Tsym : If G is a b.g. which satisfies sym. expanding H.c. then there is a sym. solution of G. Theorem ACA0 ` EM∗ Tsym Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

17 / 23

Corollary. If G is highly recursive bipartite graph that satisfies symmetric expanding Hall condition with a recursive h, then G has a recursive symmetric solution.

∵) RCA0 ` BES MTsym . Corollary. There exists a highly recursive bipartite graph which satisfies symmetric expanding Hall condition but does not have a recursive symmetric solution.

∵) RCA0 ` BEMTsym → WKL0 .

Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

18 / 23

Reverse mathematics of Bounded Marriage Theorem with Bounding Function as Parameter Theorem (J.Hirst, 1987) RCA0 ` BMT ↔ WKL0 In our formal system Z2 , we fix p : N → N and consider the following statement : if G = (B = {bi |i ∈ N}, G = {gi |i ∈ N}; R) be a bipartite graph which satisfies Hall condition and ∀i, j((b(i), g( j)) ∈ R ⇒ j ≤ p(i)), then there is a solution of G. We regard the bounding function p as parameter and consider the bounded marriage theorem dependent on p. With no restriction of p, it is equivalent to WKL0 over RCA0 . p(x) ≡ x seems to be the most simple function which satisfy the situation since H.c. requires ∀x¬∀x0 ≤ x(p(x0 ) < x). It is easily founded that in this case, we can get the solution within RCA0 . Therefore, our interet is where is the boundary. Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

19 / 23

BMT[P]: If G = (B = {bi |i ∈ N}, G = {gi |i ∈ N}; R) be a bipartite graph which satisfies Hall condition and ∀i, j((b(i), g( j)) ∈ R ⇒ j ≤ p(i)) and P(p), then there is a solution of G. Theorem

¯ . RCA0 ` BMT[∀x(p(x) ≤ x + k)] Theorem RCA0 +Π02 -IND ` BMT[∃k∀x(p(x) ≤ x + k)].

Question. RCA0 ` BMT[∃k∀x(p(x) ≤ x + k)]? or not? Theorem RCA0 ` ∀p : N → N (BMT[∀t∃x(p(x) > x + t) ∧ ∀x, x0 (x < x0 → p(x) ≤ p(x0 ))] ↔ WKL0 ). Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

20 / 23

We can consider the symmetric type of this sort of problems. BMTsym [P]: If G = (B = {bi |i ∈ N}, G = {gi |i ∈ N}; R) be a bipartite graph which satisfies symmetric Hall condition and ∀i, j((b(i), g( j)) ∈ R ⇒ j ≤ p(i) ∧ i ≤ p( j)) and P(p), then there is a symmetric solution of G. Theorem

¯ . RCA0 ` BMTsym [∀x(p(x) ≤ x + k)] Theorem RCA0 +Π02 -IND ` BMTsym [∃k∀x(p(x) ≤ x + k)]. Theorem RCA0 ` ∀p : N → N (BMTsym [∀t∃x(p(x) > x + t) ∧ ∀x, x0 (x < x0 → p(x) ≤ p(x0 ))] ↔ WKL0 ). Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

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Bibliography 1

2

3

4

W. Gasarch, “A survey of recursive combinatorics”, Handbook of recursive mathematics, Vol. 2, Stud. Logic Found. Math., 139, Amsterdam: North-Holland(1998), pp. 1041–1176. A. Manaster and J. Rosenstein, Effective matchmaking (recursion theoretic aspects of a theorem of Philip Hall), Proc. Amer. Math. Soc.,25(1972), pp. 615–654 H. A. Kierstead, An Effecttive Version of Hall’s Theorem, American Mathematical Society, 88(1983), pp. 124–128. J. R. Hirst, Combinatrics in Subsystems of Second Order Arithmetic, Ph.D. thesis, Pennsylvania State University, 1987.

Thank you for your attention. Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

22 / 23

おまけ 村上くんが持ってきた 7 頭の象のパズル。このパズル、意外と簡 単には解けない。しかし修論に追われる我々はなるべく早くこ のパズルを解きたい。度重なる検証の結果、我々は像を 85 回回 転させるだけでこのパズルを解くことができた。(なかなかの時 間を要した。) では一般に n 頭の象のパズルを解くには像を何回 回転させなければならないのか？ 定理 (Omoto-Fujiwara-Hoshino, 2011)

n 頭の像のパズルを解くのに必要な手数 an は高々

{

a2m = a2m+1 =

2 m 3 (4 − 1) である。 1 m+1 (4 − 1) 3

未解決問題

n 頭の像のパズルを解くのに最低限必要な手数は？ Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

23 / 23

おまけ 村上くんが持ってきた 7 頭の象のパズル。このパズル、意外と簡 単には解けない。しかし修論に追われる我々はなるべく早くこ のパズルを解きたい。度重なる検証の結果、我々は像を 85 回回 転させるだけでこのパズルを解くことができた。(なかなかの時 間を要した。) では一般に n 頭の象のパズルを解くには像を何回 回転させなければならないのか？ 定理 (Omoto-Fujiwara-Hoshino, 2011)

n 頭の像のパズルを解くのに必要な手数 an は高々

{

a2m = a2m+1 =

2 m 3 (4 − 1) である。 1 m+1 (4 − 1) 3

未解決問題

n 頭の像のパズルを解くのに最低限必要な手数は？ Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

23 / 23

おまけ 村上くんが持ってきた 7 頭の象のパズル。このパズル、意外と簡 単には解けない。しかし修論に追われる我々はなるべく早くこ のパズルを解きたい。度重なる検証の結果、我々は像を 85 回回 転させるだけでこのパズルを解くことができた。(なかなかの時 間を要した。) では一般に n 頭の象のパズルを解くには像を何回 回転させなければならないのか？ 定理 (Omoto-Fujiwara-Hoshino, 2011)

n 頭の像のパズルを解くのに必要な手数 an は高々

{

a2m = a2m+1 =

2 m 3 (4 − 1) である。 1 m+1 (4 − 1) 3

未解決問題

n 頭の像のパズルを解くのに最低限必要な手数は？ Makoto Fujiwara (Tohoku university)

Hall’s marriage theorem and subsystems of second order arithmetic

2011.11.18

23 / 23