aa r X i v : . [ m a t h . L O ] D ec Leaf Management
Jeffry L. HirstDecember 23, 2018
Abstract
Finding the set of leaves for an unbounded tree is a nontrivialprocess in both the Weihrauch and reverse mathematics settings. De-spite this, many combinatorial principles for trees are equivalent totheir restrictions to trees with leaf sets. For example, let d WF denotethe problem of choosing which trees in a sequence are well-founded,and let PK denote the problem of finding the perfect kernel of a tree.Let d WF L and PK L denote the restrictions of these principles to treeswith leaf sets. Then d WF , d WF L , PK , and PK L are all equivalent toΠ − CA over RCA , and all strongly Weihrauch equivalent. Introduction
The first section of this paper shows that for unbounded trees, finding leafsets is a nontrivial process. The second section describes an algorithm fortransforming trees into trees with leaf sets in such a way that propertiesrelated to infinite paths and perfect subtrees are preserved. The main equiv-alence results are presented in this section. The paper closes with a sectioncontaining an application to hypergraphs, where the use of a combinatorialprinciple restricted to sequences of trees with leaf sets is central to the proofof a Weihrauch equivalence.All relevant background information on reverse mathematics can be foundin Simpson’s text [7]. For background on Weihrauch analysis, see the workof Brattka, Gherardi, and Pauly [3].
Leaf sets
In second order arithmetic settings, a tree is encoded by a set of finite se-quences of natural numbers that is closed under initial subsequences. For1ny finite sequence σ , let | σ | denote the length of σ . A leaf in a tree is asequence that has no extensions in the tree. For a tree T , let leaf ( T ) denotethe set of leaves of T . A function b : N → N is a bounding function for T iffor every σ ∈ T and for every i < | σ | , σ ( i ) ≤ b ( i ). If a tree T has a boundingfunction, little set comprehension is required to calculate leaf ( T ). Proposition 1. ( RCA ) If b is a bounding function for the tree T , then leaf ( T ) exists.Proof. Working in
RCA , the sequence σ is a leaf of T if and only if σ ∈ T andfor each j ≤ b ( | σ | + 1) we have σ ⌢ j / ∈ T . Thus, the set leaf ( T ) is computableusing T and b as parameters, and exists by recursive comprehension.Let bleaf denote the Weihrauch problem that accepts a tree T and abounding function b as inputs and outputs the set leaf ( T ). For a boundedtree, the preceding proof describes a process for computing the leaf set. Con-sequently, bleaf is at the lowest level of the strong Weihrauch hierarchy, asstated in the following proposition. Proposition 2. bleaf ≡ sW . Finding leaf sets for trees without bounding functions is nontrivial. Theformulation of
LPO parallelized in the next proposition is the one precedingTheorem 6.7 of Brattka, Gherardi, and Pauly [3], and the Boolean negationof Definition 2.6 of Brattka and Gherardi [2].
Proposition 3. ( RCA ) The following are equivalent. (1)
ACA . (2) For every tree T , the set leaf ( T ) exists. (3) d LPO : If h p i i i ∈ N is a sequence of sequences of natural numbers, thenthere is a function z : N → { , } such that for each i , z ( i ) = 1 if andonly if ∃ n ( p i ( n ) = 0) .Proof. We will work in
RCA throughout the proof. To see that (1) implies(2), note that σ ∈ leaf ( T ) if and only if σ ∈ T and ∀ j ( σ ⌢ j / ∈ T ). Thusarithmetical comprehension proves the existence of leaf ( T ).To see that (2) implies (3), assume (2) and let h p i i i ∈ N be an instance of d LPO . Consider the tree T constructed from h p i i i ∈ N as follows. Every finite2equence of ones is in T . For each n ∈ N the sequence σ n consisting of n + 1ones followed by a zero is in T . The sequence σ ⌢n j is in T if and only if p n ( j ) = 0 and ∀ i < j ( p n ( i ) = 0). The set of sequences T exists by recursivecomprehension and is a tree because it is closed under initial segments. Apply(2) to find leaf ( T ). The function z : N → { , } defined by z ( n ) = 1 if andonly if σ n / ∈ leaf ( T ) exists by recursive comprehension and is a solution ofthe instance of d LPO .To complete the proof, by Lemma III.1.3 of Simpson [7], it suffices touse (3) to find the range of an injection. Let f : N → N be an injection.By recursive comprehension, we can find the sequence h p i i i ∈ N defined by p i ( n ) = 0 if f ( n ) = i , and p i ( n ) = 1 otherwise. Apply (3) to find z such that z ( i ) = 1 if and only if ∃ n ( p i ( n ) = 0). Then the set { i | z ( i ) = 1 } is the rangeof f , is computable from z , and so exists by recursive comprehension.By virtue of our circuitous reverse mathematics treatment, we can easilyprove the related Weihrauch reducibility result. Let leaf denote the problemthat accepts a tree T as input and outputs the leaf set leaf ( T ). Proposition 4. leaf ≡ sW d LPO .Proof.
The proof that (2) implies (3) for Proposition 3 also shows that d LPO ≤ sW leaf . To prove the reverse relation, fix T and let h σ i i i ∈ N be anenumeration of the sequences in T . For each i , let p i ( j ) = 0 if σ ⌢i j ∈ T andlet p i ( j ) = 1 otherwise. If z is a solution to this instance of d LPO , then it isalso a characteristic function for leaf ( T ). Transforming trees
As shown in the previous section, finding the leaf set for an arbitrary tree is anontrivial process in both the reverse mathematics and Weihrauch settings.However, in many cases it is possible to uniformly transform trees into treeswith leaf sets while preserving many Weihrauch equivalences and equivalencetheorems of reverse mathematics. The transformation can be defined usingthe following operations on finite sequences. For every σ ∈ N < N , let σ + 1denote the sequence with exactly the same length as σ such that for all n < | σ | , ( σ +1)( n ) = σ ( n )+1. For example, h , , i +1 = h , , i . Similarly,define σ · σ · n ) = σ ( n ) ·
1. Our main tree transformation is T ∗ as defined in the following theorem. Na¨ıvely, T ∗ is created by adding 1to every node of T and attaching a leaf labeled 0 to each positive node.3 emma 5. ( RCA ) Suppose T ⊂ N < N is a tree. The following are also trees: T − = { σ · | σ ∈ T } , T + = { σ + 1 | σ ∈ T } , and T ∗ = T + ∪ { σ ⌢ | σ ∈ T + } . Furthermore, given a sequence h T i i i ∈ N , we can find the sequence h T ∗ i , leaf ( T ∗ i ) i .Proof. Working in
RCA , it is easy to use recursive comprehension to provethe existence of the sets T − , T + , and T ∗ . The initial segments of the shifts σ · σ + 1 are the shifts of initial segments of σ , so T − and T + are trees.A proper initial segment of a sequence in the set { σ ⌢ | σ ∈ T + } is an initialsegment of an element of T + , so T ∗ is also a tree.To complete the proof, suppose h T i i i ∈ N is a sequence of trees. For each i , σ ∈ T ∗ i if and only if the last element of σ is positive and σ · ∈ T i , or if σ = τ ⌢ τ · ∈ T i . Thus recursive comprehension implies that h T ∗ i i i ∈ N exists. For each i , the sequence σ ∈ leaf ( T ∗ i ) if and only if σ ∈ T ∗ i and the lastentry of σ is 0. Thus RCA can prove that the sequence of pairs h T ∗ i , leaf ( T ∗ i ) i exists.The trees T and T ∗ share many properties. Information about paths andsubtrees of one can be uniformly transformed to information about the other.As described in Simpson [7, Definition I.6.6], a subtree S if T is perfect ifevery sequence in T has incompatible extensions in T . The perfect kernel of T is the union of all the perfect subtrees of T . Theorem 6. ( RCA ) A tree T and the transform T ∗ satisfy the following. (1) T is well-founded if and only if T ∗ is well-founded. (2) T has at most one path if and only if T ∗ has at most one path. (3) S is a perfect subtree of T if and only if S + is a perfect subtree of T ∗ . (4) K is the perfect kernel of T if and only if K + is the perfect kernel of T ∗ .Proof. Each part follows from the fact that the map taking σ to σ + 1 isa bijection between the paths of T and those of T ∗ and also between theperfect subtrees of T and those of T ∗ .The next two theorems list familiar equivalences for tree statements thatcontinue to hold when restricted to trees with leaf sets. For both proofs, thecentral tool is the transformation from T to T ∗ . In the following theorem,4he labels used for the combinatorial principles are consistent with those usedfor the associated Weihrauch problems by Kihara, Marcone, and Pauly [6]. Theorem 7. ( RCA ) The following are equivalent. (1)
ATR . (2) The Σ separation principle: For any Σ formulas ϕ ( n ) and ϕ ( n ) containing no free occurrences of Z , if ¬∃ n ( ϕ ( n ) ∧ ϕ ( n )) , then ∃ Z ∀ n (( ϕ ( n ) → n ∈ Z ) ∧ ( ϕ ( n ) → n / ∈ Z )) . (3) Σ − SEP : If h T ,i i i ∈ N and h T ,i i i ∈ N are sequences of trees such that foreach i , at most one of T ,i and T ,i has an infinite path, then there is aset Z such that for all n , T ,n has an infinite path implies n ∈ Z and T ,n has an infinite path implies n / ∈ Z . (4) Σ − SEP L : Item (3) for h T ,i , leaf ( T ,i ) i i ∈ N and h T ,i , leaf ( T ,i ) i i ∈ N , se-quences of trees with leaf sets. (5) Σ − CA − : If h T i i i ∈ N is a sequence of trees each with at most one infinitepath, then there is a set Z such that for all n , n ∈ Z if and only if T n has an infinite path. (6) Σ − CA − L : Item 5 for sequences of trees with leaf sets. (7)
PTT : If T has uncountably many paths then T has a non-empty perfectsubtree. (8) PTT L : Item 7 for trees with leaf sets.Proof.
The equivalence of (1) and (2) is Theorem V.5.1 of Simpson [7]. Theexistence of an infinite path in a tree can be written as a Σ formula, so (2)implies (3). To prove the converse, use a bootstrapping argument, proving ACA from (3) by creating a sequence of pairs of linear trees that compute therange of an injection. Then use ACA and (3) to derive (2) by an applicationof Lemma 3.14 of Friedman and Hirst [5]. The equivalence of (5) and (1) isTheorem V.5.2 of Simpson [7], and the equivalence of (7) and (1) is TheoremV.5.5 of Simpson [7]. Item (4) is a restriction of (3), so (3) implies (4) trivially.The converse is an immediate consequence of Theorem 6. Similarly, (5) and(6) are equivalent, as are (7) and (8).5o avoid confusion with the subsystem Π − CA , in following theoremwe use d WF as a label for the combinatorial principle denoted by Π − CA inthe article of Kihara, Marcone, and Pauly [6]. Note that d WF is the infiniteparallelization of the the principle WF , that takes a tree as an input andoutputs a 1 if the tree is well-founded and a 0 otherwise. Theorem 8. ( RCA ) The following are equivalent: (1) Π − CA : If ϕ ( n ) is a Π formula, then there is a set Z such that forall n , n ∈ Z if and only if ϕ ( n ) . (2) d WF : If h T i i i ∈ N is a sequence of trees, then there is a set Z such thatfor all n , n ∈ Z if and only if T has no infinite path. (3) d WF L : Item (2) for sequences of trees with leaf sets. (4) PK : Every tree has a perfect kernel. (5) PK L : Item (4) for trees with leaf sets.Proof.
The equivalence of (1) and (2) is Theorem VI.1.1 of Simpson [7]. Theequivalence of (1) and (4) is Theorem VI.1.3 of Simpson [7]. The restriction(3) follows trivially from (2), and the converse follows immediately fromTheorem 6. By a similar argument, (4) and (5) are equivalent.We now turn to the Weihrauch analogs of the preceding results. Themain tool is the computability theoretic version of Lemma 5.
Lemma 9.
There is a uniformly computable map from trees T to T − , andinvertible uniformly computable maps from T to T + and T ∗ . Also, there is acomputable functional mapping sequences of trees h T i i i ∈ N to h T ∗ i , leaf ( T ∗ i ) i .Proof. The processes described at the beginning of the section are uniformlycomputable, and for T + and T ∗ , uniformly computably invertible. Leaf setsare uniformly computable for trees of the form T ∗ .The following Weihrauch analog of Theorem 7 is based on the results ofKihara, Marcone, and Pauly [6]. Theorem 10.
PTT ≡ sW PTT L < W Σ − SEP . Also, the following princi-ples are strongly Weihrauch equivalent: Σ − SEP , Σ − SEP L , Σ − CA − , and Σ − CA − L . roof. The equivalences between the statements and the versions restrictedto trees with leaf sets follow from Lemma 9 and Theorem 6. The equivalenceof Σ − SEP and Σ − CA − is included in Theorem 3.11 of Kihara, Marcone,and Pauly [6], while PTT < sW Σ − SEP follows from their Corollary 3.7,Theorem 3.11, and Proposition 6.4 [6].We close the section with the Weihrauch analog of Theorem 8.
Theorem 11. WF ≡ sW WF L . Also, the following four principles are stronglyWeihrauch equivalent: d WF , d WF L , PK , and PK L .Proof. The equivalences WF ≡ sW WF L , d WF ≡ sW d WF L , and PK ≡ sW PK L allfollow immediately from Lemma 9 and Theorem 6. It suffices to show that d WF ≡ sW PK .To see that d WF ≤ sW PK , let h T i i i ∈ N be a sequence of trees, the input for d WF . For sequences σ and τ with | σ | = | τ | , let σ ∗ τ denote the sequenceconsisting of alternating entries of σ and τ . Thus for σ and τ of length n + 1, σ ∗ τ = h σ (0) , τ (0) , . . . , σ ( n ) , τ ( n ) i . Define the tree T by including thefollowing sequences for each i ∈ N : • h i i ∈ T for each i ∈ N , and • if σ ∈ T i , | σ | = n , and τ is a binary sequence of length n , then h i i ⌢ ( σ ∗ τ ) ∈ T and the initial segment of h i i ⌢ ( σ ∗ τ ) omitting the lastelement is also in T .The tree T is uniformly computable from the sequence h T i i i ∈ N . If T i hasan infinite path p , then for every binary sequence τ , all initial segments of h i i ⌢ ( p ∗ τ ) are in T . In this case, there is a perfect subtree of T above h i i ,so h i i is in the perfect kernel of T . If T i is well-founded, then the subtree ofextensions of h i i in T is also well-founded, so no perfect subtree of T contains h i i . Thus, if K is a perfect kernel for T , then T i is well-founded if and onlyif h i i ∈ K . Summarizing, Z = { i | h i i ∈ K } is the desired output for d WF .To see that PK ≤ sW d WF , let T be an input tree for PK . In the following,we freely conflate finite sequences with their natural number codes. For eachfinite sequence σ ∈ T , define T σ as follows: • h σ i ∈ T σ , and 7 if h σ, . . . , h τ , . . . , τ m ii ∈ T σ , and for each i ≤ m , τ ⌢i e i, and τ ⌢i e i, areincompatible extensions of τ i in T , then h σ, . . . , h τ , . . . , τ m i , h τ ⌢ e , , τ ⌢ e , , . . . , τ ⌢m e m, , τ ⌢m e m, ii ∈ T σ . The sequence h T σ i σ ∈ T (which can be viewed as h T i i i ∈ N ) is uniformly com-putable from T . For each σ ∈ T , T σ has an infinite path if and only if σ iscontained in a perfect subtree of T . Let Z be a solution of d WF for h T σ i σ ∈ T .Then Z = { σ ∈ T | T σ is well-founded } , and K = { σ ∈ T | σ / ∈ Z } is theperfect kernel of T . An application
This section presents a Weihrauch analysis closely related to Theorem 6 ofDavis, Hirst, Pardo, and Ransom [4]. A hypergraph H = ( V, E ) consists ofa set of vertices V = { v , v , . . . } and a set of edges E = { e , e , . . . } , whereeach edge in E is a set of vertices. For hypergraphs, an edge can be a setof any cardinality. If every edge of a hypergraph has cardinality exactly 2,then H is a graph. A k -coloring of a hypergraph H = ( V, E ) is a function f : V → k . A k -coloring is called proper if every edge with at least twovertices contains vertices of different colors. Let HPC ( k ) be the problem thataccepts a hypergraph H as input, outputs 1 if H has a proper k -coloring,and outputs 0 otherwise.The second part of the proof of the following theorem applies a leaf man-agement result from the preceding section. Theorem 12.
For all k ≥ , HPC ( k ) ≡ sW WF .Proof. To see that
HPC ( k ) ≤ sW WF , let H = ( V, E ) be a hypergraph inputfor
HPC ( k ). In the following, we freely conflate vertices and finite collectionsof vertices with their integer codes. Build a tree T by including sequences σ = h σ , σ , . . . , σ m i satisfying the following conditions for each i ≤ m . • If i = 2 j , then σ i is a set of two vertices in edge e j , or σ i is a code for ∅ and e j does not contain a pair of vertices in the list { v , . . . , v m } . • If i = 2 j + 1, then σ i < k . We view this as a color for v j . • The partial coloring of H given by the odd entries of σ uses distinctcolors on the pairs of vertices listed in the even entries.8he odd entries of any infinite path in T encode a proper k -coloring of H .Also, any proper k -coloring of H can be used to define an infinite paththrough T . (If H has no edges of cardinality less than 2, an infinite path canbe uniformly computed from any proper coloring, but this is not necessaryfor the current argument.) Thus, HPC ( k ) is 1 for H if and only if WF is 0for T .By Theorem 11, WF ≡ sW WF L , so to complete the proof it suffices toshow that WF L ≤ sW HPC ( k ). We will prove this for k = 2 and indicate howto modify the argument for larger values of k .Let T be an input for WF L , that is, a tree with a leaf set. Emulating theconstruction from the proof of Theorem 6 of Davis et al. [4], define a hyper-graph H as follows. The vertices of H include the five vertices { a , a , b , b , s } plus two vertices labeled σ and σ for each sequence σ ∈ T . The edges of H consist of • ( a , a ), ( a , s ), ( b , b ), and ( b , s ), • ( σ , σ ) for every nonempty σ ∈ T , • ( σ , s ) if σ is a leaf of T , • E σ = { σ } ∪ { τ | τ ∈ T ∧ ∃ n τ = σ ⌢ n } if σ ∈ T is not a leaf, and • E = { a , b } ∪ { σ | σ ∈ T ∧ | σ | = 1 } . H is uniformly computable from T and its leaf set. Note that the leaf set isused in the third and fourth bullets. H has a proper 2-coloring if and only if T has an infinite path. For details, see the proof of Theorem 6 of [4]. Thus WF L ≤ sW HPC (2). To prove the reduction for larger values of k , modifythe construction by adding a complete subgraph on k − H andconnecting each of its vertices to every vertex of H with an edge consistingof two vertices.Parallelization yields the Weihrauch analog of part of the reverse mathe-matical Theorem 6 of Davis et al. [4]. Corollary 13.
For k ≥ , d WF ≡ sW d HPC ( k ) .Proof. Fix k . By Theorem 12, WF ≡ sW HPC ( k ), so by Proposition 3.6 part(3) of Brattka, Gherardi, and Pauly [3], d WF ≡ sW d HPC ( k )The arguments used in Theorem 6 of [4] can be used to extend Theorem12 and Corollary 13 to conflict-free colorings.9 cknowledgements A talk related to this paper was presented at Dagstuhl Seminar 18361, orga-nized by Vasco Brattka, Damir Dzhafarov, Alberto Marcone, and Arno Pauly,and held September 2-7 of 2018 at Schloss Dagstuhl, the Leibniz-Zentrumf¨ur Informatik [1]. The author’s travel to the seminar was supported by aBoard of Trustees travel grant from Appalachian State University.
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