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- 1. Explaining the Kruskal’s Tree Theorem Dr M Benini, Dr R Bonacina Università degli Studi dell’Insubria Logic Seminars JAIST, May 12th, 2017
- 2. The theorem Theorem 1 (Kruskal) The collection T (A) of all the ﬁnite trees labelled over a well quasi order A ordered by homeomorphic embedding, is a well quasi order T(A) = 〈T (A);≤E 〉. The T1 ≤E T2 relation means that there is an embedding η of T1 into T2, i.e., a function which maps the nodes and the arcs of T1 into those of T2 and preserves the structure of the T1 tree. It is worth noticing that the statement is not precise, since the deﬁnitions of ‘tree’ and ‘preserving the structure’ are left implicit. ( 2 of 33 )
- 3. The theorem In fact, there is an ambiguous point in the statement: it is usually intended as speaking of trees as some special graphs, with the notion of embedding captured via the graph minor relation, while it is proved by using an inductive deﬁnition of trees, close to the usual one in Computer Science, with a natural but ad-hoc notion of embedding. So, we are speaking of two distinct theorems, and about their relation. Also, we have many proofs of one of them, while the other is usually relegated to a footnote in the end, or a quick hint, or an exercise, when mentioned. But it is the unproved one which is really used in Mathematics. ( 3 of 33 )
- 4. The theorem Deﬁnition 2 (Tree) Let A = 〈A;≤A〉 be a quasi order. A tree T is inductively deﬁned as 1. a single node, called its root; 2. given the trees T1,...,Tn, then a tree is the structure composed by a node, called the root and T1,...,Tn, called the immediate subtrees of the root. A labelled tree (T,l) over A is a tree T and a function l from its nodes to A. Deﬁnition 3 (Tree) A tree is a ﬁnite, acyclic and connected graph. A labelled tree (T,l) over the quasi order A = 〈A;≤A〉 is a labelled graph which is a tree. The two deﬁnitions are evidently diﬀerent: to distinguish them, we refer to the trees as for the former one as pointed trees. ( 4 of 33 )
- 5. The theorem The notion of embedding for the ﬁrst deﬁnition of tree is Deﬁnition 4 (Pointed minor) Let A = 〈A;≤A〉 be a well quasi order. Let (T,lT ) and (T ,lT ) be pointed trees. Then (T,lT ) ≤K (T ,lT ) if and only if one of the following conditions applies 1. there is an immediate subtree (S ,lT ) of (T ,lT ) such that (T,lT ) ≤K (S ,lT ); 2. calling rT and rT the roots of T and T respectively, lT (rT ) ≤A lT (rT ) and there is an injective map from the immediate subtrees (S,lT ) of (T,lT ) to those (S ,lT ) of (T ,lT ) such that (S,lT ) ≤K (S ,lT ). ( 5 of 33 )
- 6. Well quasi orders Deﬁnition 5 (Quasi order) A quasi order A = 〈A;≤〉 is a class A with a binary relation ≤ on A which is reﬂexive and transitive. If the relation is also anti-symmetric, A is a partial order. Given x,y ∈ A, x ≤ y means that x and y are not related by ≤; x is equivalent to y, x y, when x ≤ y and y ≤ x; x is incomparable with y, x y, when x ≤ y and y ≤ x. The notation x < y means x ≤ y and x y; x ≥ y is the same as y ≤ x; x > y stands for y < x. Intuitively, a quasi order is an order in which we admit two elements to be equivalent but not equal. ( 6 of 33 )
- 7. Well quasi orders Deﬁnition 6 (Descending chain) Let A = 〈A;≤〉 be a quasi order. Every sequence {xi ∈ A}i∈I, with I an ordinal, such that xi ≥ xj for every i < j is a descending chain. If a descending chain {xi }i∈I is such that xi > xj whenever i < j, then it is a proper descending chain. A (proper) descending is ﬁnite when the indexing ordinal I < ω. If every proper descending chain in A is ﬁnite, then the quasi order is said to be well founded. ( 7 of 33 )
- 8. Well quasi orders Deﬁnition 7 (Antichain) Let A = 〈A;≤〉 be a quasi order. Every sequence {xi ∈ A}i∈I, with I an ordinal, such that xi xj for every i = j is an antichain. An antichain is ﬁnite when the indexing ordinal I < ω. If every antichain in A is ﬁnite, then the quasi order is said to satisfy the ﬁnite antichain property or, simply, to have ﬁnite antichains. Deﬁnition 8 (Well quasi order) A well quasi order is a well founded quasi order having the ﬁnite antichain property. ( 8 of 33 )
- 9. Nash-Williams’s toolbox Deﬁnition 9 (Bad sequence) Let A = 〈A;≤〉 be a quasi order. An inﬁnite sequence {xi }i∈ω in A is bad if and only if xi ≤ xj whenever i < j. A bad sequence {xi }i∈ω is minimal in A when there is no bad sequence yi i∈ω such that, for some n ∈ ω, xi = yi when i < n and yn < xn. In fact, in the following, a generalised notion of ‘being minimal’ is used: a bad sequence {xi }i∈ω is minimal with respect to µ and r in A when for every bad sequence yi i∈ω such that, for some n ∈ ω, xi r yi when i < n , it holds that µ(yn) <W µ(xn). Here, µ: A → W is a function from A to some well founded quasi order 〈W ;≤W 〉 and r is a reﬂexive binary relation on A. ( 9 of 33 )
- 10. Nash-Williams’s toolbox Theorem 10 (Characterisation) Let A = 〈A;≤〉 be a quasi order. Then, the following are equivalent: 1. A is a well quasi order; 2. in every inﬁnite sequence {xi }i∈ω in A there exists an increasing pair xi ≤ xj for some i < j; 3. every sequence {xi ∈ A}i∈ω contains an increasing subsequence xnj j∈ω such that xni ≤ xnj for every i < j. 4. A does not contain any bad sequence. ( 10 of 33 )
- 11. Nash-Williams’s toolbox Fact 11 Let A = 〈A;≤〉 be a well quasi order. Then, for every quasi order A+ = A;≤+ such that ≤ ⊆ ≤+ , A+ is a well quasi order. Fact 12 Let A = 〈A;≤A〉 be a well quasi order. Then, every quasi order B = 〈B;≤B〉 with B ⊆ A and ≤B the restriction of ≤A to B, is a well quasi order. ( 11 of 33 )
- 12. Nash-Williams’s toolbox Proposition 13 Let 〈A;≤〉 be a well quasi order and let ≈ be an equivalence relation on A such that ≤≈ is a quasi ordering of A/≈, with [x]≈ ≤≈ [y]≈ if and only if there are x ∈ [x]≈ and y ∈ [y]≈ such that x ≤ y . Then A/≈;≤≈ is a well quasi order. ( 12 of 33 )
- 13. Nash-Williams’s toolbox Lemma 14 (Dickson) Assume A and B to be non empty sets. Then A = 〈A;≤A〉 and B = 〈B;≤B〉 are well quasi orders if and only if A×B = 〈A×B;≤×〉 is a well quasi order, with the ordering on the Cartesian product deﬁned by (x1,y1) ≤× (x2,y2) if and only if x1 ≤A x2 and y1 ≤B y2. ( 13 of 33 )
- 14. Nash-Williams’s toolbox Lemma 15 Let A = 〈A;≤A〉 be a quasi order which is not a well quasi order, and let 〈W ;≤〉 be a well founded quasi order. Also, let f : A → W be a function and r ⊆ A×A a reﬂexive relation. Then, there is a bad sequence {xi }i∈ω on A that is minimal with respect to f and r: for every n ∈ ω and for every bad sequence yi i∈ω on A such that xi r yi whenever i < n, f (yn) < f (xn). So, if A is a quasi order, but not a well quasi order, then it contains a bad sequence which is minimal with respect to some measure f and some comparison criterion r, normally =. ( 14 of 33 )
- 15. Nash-Williams’s toolbox Let B = 〈B;≤B〉 be a quasi order. Let 〈W ;≤〉 be a total well founded quasi order, and let µ: B → W be a function. Suppose B is not a well quasi order, then there is {Bi }i∈ω bad in B and minimal with respect to µ and = by Lemma 15. Let p ∈ ω and let ∆: Bi : i ≥ p → ℘ﬁn(B), the collection of all the ﬁnite subsets of B, be such that (∆1) for every i ∈ ω and for every x ∈ ∆(Bi ), x ≤B Bi ; (∆2) for every i ∈ ω and for every x ∈ ∆(Bi ), µ(x) < µ(Bi ). Proposition 16 Let D = 〈 i>p ∆(Bi );≤B〉. Then D is a well quasi order. ( 15 of 33 )
- 16. Nash-Williams’s toolbox Summarising, we want to prove that B = 〈B;≤B〉 is a well quasi order, and we know it is a quasi order. Suppose B is not a well quasi order. Then there is a minimal bad sequence {Bi }i∈ω with respect to some reasonable measure µ and =. Deﬁne a decomposition ∆ of the elements in the bad sequence. Then, the collection of the components forms a well quasi order. Form a sequence C from the components: by using well known results, e.g., Dickson’s Lemma, it is usually easy to deduce that C lies in a well quasi order. Then, C contains an increasing pair. So, each component of Bn is less than a component in Bm. Recombine the pieces, and it follows (!) that Bn ≤ Bm, contradicting the initial assumption. Q.E.D. ( 16 of 33 )
- 17. The proof Theorem 17 (Kruskal) Let R (A) be the collection of pointed trees over A = 〈A;≤A〉. If A is a well quasi order then R(A) = R (A);≤K is a well quasi order. Proof. (i) Suppose R(A) is not a well quasi order. Then, by Lemma 15 there is a bad sequence (Ti ,li ) i∈ω in R(A) minimising |E (_)|. Let (Ti ,li ) i∈I be the subsequence of (Ti ,li ) i∈ω composed by the pointed trees with no edges. Then they contain just a single node, the root ri , so li (ri ) i∈I is a sequence in A with no increasing pair. Thus, by Theorem 10 on the A well quasi order, I is ﬁnite, so p = maxI is deﬁned and (Ti ,li ) i>p is such that E (Ti ) > 0 and, in particular, there is an edge from the root to some node. → ( 17 of 33 )
- 18. The proof → Proof. (ii) For i > p, deﬁne ∆(Ti ,li ) as the set composed by the two connected components T1 i ,li , T2 i ,li obtained deleting some arc {ri ,xi } ∈ E (Ti ): each component is a pointed tree having one endpoint of {ri ,xi } as its root. We stipulate that the root of T1 i is ri and the root of T2 i is xi . Clearly, if Tj i ,li ∈ ∆(Ti ,li ), Tj i ,li ≤K (Ti ,li ) and E Tj i ,li < E (Ti ,li ) . So D = i>p ∆(Ti ,li );≤K is a well quasi order by Lemma 16. Thus, by Dickson’s Lemma 14, D×D is a well quasi order. Considering the sequence T1 i ,li , T2 i ,li i>p , by Theorem 10 there are m > n such that T1 n ,ln ≤K T1 m,lm , thus ln (rn) ≤A lm (rm), and T2 n ,ln ≤K T2 m,lm , and the endpoints of the arc deleted by ∆ are similarly preserved, so (Tn,ln) ≤K (Tm,lm), contradicting (Ti ,li ) i∈ω to be bad. ( 18 of 33 )
- 19. Pointed trees versus graphs Consider the following pair of incomparable trees, and decompose them as in the previous proof: but = The decomposition yields two pairs of subtrees which are identical as graphs but diﬀerent as pointed trees. Thus, extending the proof of Kruskal’s Theorem to trees seen as graphs is not immediate. ( 19 of 33 )
- 20. Trees as graphs Deﬁnition 18 (Graph) A graph G = 〈V ,E〉 is composed by a set V of nodes or vertices, and a set E of edges or arcs, which are unordered pairs of distinct nodes. Given a graph G, V (G) denotes the set of its nodes and E(G) denotes the set of its edges. A graph G is ﬁnite when V (G) is so. No loops The deﬁnition induces a criterion for equality Obvious notion of isomorphism ( 20 of 33 )
- 21. Trees as graphs Deﬁnition 19 (Subgraph) G is a subgraph of H, G ≤S H, if and only if there is η: V (G) → V (H) injective such that, for every x,y ∈ E(G), η(x),η(y) ∈ E(H). Deﬁnition 20 (Induced subgraph) Let A ⊆ V (H). Then the induced subgraph G of H by A is identiﬁed by V (G) = A and E(G) = x,y ∈ E(H): x,y ∈ A . The notion of subgraph deﬁnes an embedding on graphs: G ≤S H says that there is a map η, the embedding, that allows to retrieve an image of G inside H. ( 21 of 33 )
- 22. Trees as graphs Deﬁnition 21 (Path) Let G be a graph and let x,y ∈ V (G). A path p from x to y, p: x y, of length n ∈ N is a sequence vi ∈ V (G) 0≤i≤n such that (i) v0 = x, vn = y, (ii) for every 0 ≤ i < n, {vi ,vi+1} ∈ E(G), and (iii) for every 0 < i < j ≤ n, vi = vj. Deﬁnition 22 (Connected graph) A graph is connected when there is at least one path between every pair of nodes. ( 22 of 33 )
- 23. Trees as graphs Deﬁnition 23 (Minor) G is a minor of H, G ≤M H, if and only if there is an equivalence relation ∼ on V (H) whose equivalence classes induce connected subgraphs in H, and G ≤S H/∼, with V (H/∼) = V (H)/∼ and E(H/∼) = [x]∼ ,[y]∼ : x ∼ y and x,y ∈ E(H) . For the sake of brevity, an equivalence inducing connected subgraphs as above, is called a c-equivalence. Fact 24 Let G be the collection of all the ﬁnite graphs. Then 〈G;≤S〉 and 〈G;≤M〉 are partial orders. ( 23 of 33 )
- 24. The proof, part II Theorem 25 (Kruskal) Let T (A) be the collection of all the pointed trees labelled over A = 〈A;≤A〉. If A is a well quasi order, then T(A) = T (A);≤A M is a well quasi order. Proof. (i) Notice how (T,lT ) ≤K (T ,lT ) implies (T,lT ) ≤A M (T ,lT ). In fact, a simple induction on Deﬁnition 4 suﬃces to establish the result: initially W = 1. if (T,lT ) ≤K (T ,lT ) because (T,lT ) ≤K (S ,lT ) with S an immediate subtree of T , then W is updated by adding the collection of nodes in the subgraph of T induced by V (T )V (S ); → ( 24 of 33 )
- 25. The proof, part II → Proof. (ii) 2. if (T,lT ) ≤K (T ,lT ) because lT (rT ) ≤A lT (rT ) and there is ξ injective mapping the immediate subtrees of T to the immediate subtrees of T such that (S,lT ) ≤K ξ(S,lT ), then [rT ] is the union of W and the collection of nodes of the subgraph of T composed by the immediate subtrees of T not in the image of ξ. Then, inductively, the equivalence classes of the roots of the subtrees are constructed, restarting with W = . The equivalence classes [x] form a partition on V (T ), and thus a c-equivalence ∼ as it is immediate to verify; moreover, there is an evident injective function from V (T) to V (T )/∼ which maps the root of each subtree in T into the root of some subtree in T . Finally, labels are trivially preserved. Thus, since R(A) is a well quasi order by Proposition 17, also T(A) is a well quasi order by Fact 11. ( 25 of 33 )
- 26. An unsatisfactory theorem So, Kruskal’s Theorem on pointed trees is extended to Kruskal’s Theorem on trees as graphs. The key of the proof is that (T,lT ) ≤K (T ,lT ) implies (T,lT ) ≤A M (T ,lT ), i.e., ≤K ⊆ ≤A M. Since ≤A M extends ≤K, every bad sequence which happens to exist in the collection of trees as graphs, is bad also in the collection of pointed trees, for any choice of roots. The same result holds for any quasi order extending ≤K. So, what makes ≤A M special? Why is the statement using ≤A M referred to as a Theorem? Does it depends only because it is useful? The general answer in Mathematics is that something is useful because it has a ‘good’ structure. And this is the case also for Kruskal’s result. ( 26 of 33 )
- 27. An alternative proof Deﬁnition 26 (Node ordering) Let (T,l) be a pointed tree, with r ∈ V (T) its root. If x,y ∈ V (T) then x ≤T y when r y = (x y)◦(r x). It is worth remarking that r x has to be a path, so it cannot contain the same node twice, except for the endpoints. This fact imposes a direction to the edges: x ≤T y when there is a path x y which ‘goes only down’, thus y is ‘below’ x in the tree, or x is ‘closer’ than y to the root. ( 27 of 33 )
- 28. An alternative proof Deﬁnition 27 (Embedding via node ordering) If (T,lT ) and (T ,lT ) are two pointed trees with labels over the quasi order A = 〈A;≤A〉, then (T,lT ) ≤K (T ,lT ) when there is ξ: V (T) → V (T ) injective such that: if x ≤T y then ξ(x) ≤T ξ(y), ξ preserves the node ordering of T; lT (x) ≤A lT (ξ(x)) for each x ∈ V (T). Comparing with Deﬁnition 4, it immediately follows that Fact 28 ≤K=≤K. ( 28 of 33 )
- 29. An alternative proof Theorem 29 (Kruskal) Let R (A) be the collection of pointed trees over A = 〈A;≤A〉. If A is a well quasi order, then R (A) = R (A);≤K is a well quasi order. Proof. Following the proof of Proposition 17, consider a bad sequence (Ti ,li ) i∈ω in R (A) minimising E (_) , deﬁne ∆ as before, thus D = i∈ω ∆(Ti ,li );≤K is a well quasi order, and by Dickson’s Lemma 14, D×D is a well quasi order. Thus, by the same argument in Proposition 17, an injective ξ: V (Tn) → V (Tm) preserving the node ordering of Tn and its labels can be found, for some n < m, thus showing that Tn ≤KTm and contradicting (Ti ,li ) i∈ω to be bad. ( 29 of 33 )
- 30. An alternative proof Deﬁnition 30 Let R (A) the collection of pointed trees over A; deﬁne an equivalence relation ≈ on R (A) such that (T,lT ) ≈ (T ,lT ) if and only if V (T) = V (T ), E (T) = E (T ) and lT = lT , i.e., if the pointed trees diﬀer only by the choice of the root. Consider ≤≈ K, with [(T,lT )] ≤≈ K [(T ,lT )] if there are rT ∈ V (T), rT ∈ V (T ) such that (T,lT ) ≤K (T ,lT ) as pointed trees with roots rT and rT respectively. Fact 31 Each equivalence class [_]≈ denotes a non-pointed tree, that is, R (A)/ ≈ is isomorphic to T (A). ( 30 of 33 )
- 31. An alternative proof This suggests that also the order relations on R(A)/ ≈ and T (A), i.e., ≤≈ K and ≤A M, may be related. The following result shows the connection between the order relation on pointed trees and the graph minor. Proposition 32 If (T,lT ) and (T ,lT ) are trees, then [(T,lT )] ≤≈ K [(T ,lT )] if and only if (T,lT ) ≤A M (T ,lT ). Thus R(A)/≈ = R (A)/ ≈;≤≈ K is a quasi order. Kruskal’s Theorem follows because by Proposition 29 and Proposition 13 R(A)/≈ is a well quasi order, and by Proposition 32, R(A)/≈ ∼= T(A). ( 31 of 33 )
- 32. An alternative proof Proposition 32 really explains the Kruskal’s Theorem: the graph minor relation ≤A M is not some arbitrary extension of ≤K; rather, ≤A M is the relation obtained by forgetting the direction a choice of some root imposes on a tree. In other words, the collection of ﬁnite trees is a well quasi order with respect to ≤A M because each tree summarises a set of pointed trees, diﬀering only by the node which acts as a root, and, in turn, ≤A M summarises via the obvious quotient the relation ≤K which preserves the structure of trees and their roots, the last emphasised bit being what is abstracted away. ( 32 of 33 )
- 33. The end Questions? ( 33 of 33 )