Joseph B. Kruskal. Well-quasi-ordering, the Tree Theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, May 1960.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on the first that is even smaller than the first, causing a contradiction so S must be well quasi ordered. This minimal bad sequence idea was attributed by Robertson and Seymour to Nash Williams [13] Theorem 4.1 is a labeled version of a classical theorem by Kruskal. Theorem 4. 2 (Kruskal [10]) The collection of finite rooted trees is well quasi ordered by topological containment. Of course there might be many ways of subdividing the edges of P to obtain Q. What we mean is that there is some way of subdividing the edges of P such that we obtain Q and OE P (v) OE Q (u) for each ....

J. B. Kruskal. Well-quasi-ordering, the tree theorem, and v'azsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1960.

....relation is then such that all infinite sequences are good. There are several other equivalent definitions of well binary relations and well quasi orders. Higman [20] used an alternate definition of well quasi orders in terms of the finite basis property (or finite generating set in [28]) Both definitions are equivalent by Theorem 2.1 in [20] A different (but also equivalent) definition of a wqo is(e.g. 30, 66] A quasi order V is a wqo iff for all quasi orders V which contain V (i.e. v V v ) v V v ) the corresponding strict partial order OE V is a wfo. This ....

....not really interesting and one can therefore drop this requirement, leading to the use of wbr s. Later on in Sections 5 and 6 we will actually develop wbr s which are not wqo s. An interesting wqo is the homeomorphic embedding relation Theta, which derives from results by Higman [20] and Kruskal [28]. It has been used in the context of term rewriting systems in [9, 10] and adapted for use in supercompilation [64] in [61] Its usefulness as a stop criterion for partial evaluation is also discussed and advocated in [43] Some complexity results can be found in [63] and [17] also summarised in ....

[Article contains additional citation context not shown here]

J. B. Kruskal. Well-quasi ordering, the tree theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1960.

.... Lemma on Trees In order to prove their result on well quasi ordering of graphs with bounded tree width, Robertson and Seymour [5] invoke a lemma on trees , which they prove in the same paper. The proof of this lemma on trees extends a simple proof by Nash Williams [4] of the result of Kruskal [3] that forests are well quasi ordered under taking minors, or actually, more strongly, by topological containment . We also use Robertson and Seymour s lemma on trees. To make this paper self contained, we include a proof of that lemma. We need some definitions. A rooted tree is a finite directed ....

J. Kruskal. Well-quasi-ordering, the tree theorem, and V'azsonyi's conjecture. Transactions of the American Mathematical Society 95 (1960) 210--225.

....the current branch is not admissible. De nition 5. Given a wqo we de ne whistle as follows: whistle (L; M i M is the farthest ancestor of L such that label(M) label(L) and whistle (L; fail if there is no such ancestor. A particularly useful wqo is the homeomorphic embedding [16, 20]. The following is the de nition from [37] which adapts the pure homeomorphic embedding from [7] by adding a rudimentary treatment of variables. De nition 6. The (pure) homeomorphic embedding relation on expressions is inductively de ned as follows (i.e. is the least relation satisfying ....

....is that A B i A can be obtained from B by striking out certain parts, or said another way, the structure of A reappears within B. We have f(a; b) p(f(g(a) b) but p(a) 6 p(b) and f(a; b) 6 p(f(a; c) f(c; b) The relation is a wqo on the set of expressions over a nite alphabet [16, 20]. We also de ne the weak homeomorphic embedding relation on expressions by replacing rule 1. of De nition 6 by: 1. t X for all variables X is weaker in the sense that less sequences are admissible (and hence it is a wqr; it is not a wqo as it is not transitive) but it will be useful in ....

J. B. Kruskal. Well-quasi ordering, the tree theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210-225, 1960.

....a recursive path order on term graphs by analogy with the well known order on terms [26, 27] and demonstrate its use for proving termination of ) coll . Our exposition is based on [91] where a class of simplification orders on term graphs is established by extending Kruskal s Tree Theorem [73] from trees to term graphs. Definition 6.9 (Top and immediate subgraphs) Let G be a term graph and e be the unique edge such that att G (e) root G v 1 : v n for some nodes v 1 ; v n . Then the top of G, denoted by top G , is the subgraph consisting of e and the nodes root G ; v 1 ; ....

Joseph B. Kruskal. Well-quasi-ordering, the Tree Theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1960.

.... g(x i 1 ; x i m ) with f an n ary function symbol in F , g an m ary function symbol in F , n m 0, f g, and 1 6 i 1 Delta Delta Delta i m 6 n whenever m 1. We abbreviate Emb (F ; to emb . We conclude this preliminary section by recalling the Tree Theorem of Kruskal [6]. A partial order on a set A is called a partial well order (PWO for short) if every partial order on A that extends (including itself) is well founded. Kruskal s Tree Theorem can be phrased as follows: emb is a PWO on T (F) whenever is a PWO on the signature F . A special case states that ....

J.B. Kruskal, Well-Quasi-Ordering, the Tree Theorem, and Vazsonyi 's Conjecture, Transactions of the American Mathematical Society, 95 (1960), pp. 210--225.

....a recursive path order on term graphs by analogy with the wellknown order on terms [26,27] and demonstrate its use for proving termination of ) coll . Our exposition is based on [93] where a class of simplification orders on term graphs is established by extending Kruskal s Tree Theorem [74] from trees to term graphs. 1.6. TERMINATION 35 Definition 1.6.9 (Top and immediate subgraphs) Let G be a term graph and e be the unique edge such that att G (e) root G v 1 : v n for some nodes v 1 ; v n . Then the top of G, denoted by top G , is the subgraph consisting of e and ....

Joseph B. Kruskal. Well-quasi-ordering, the Tree Theorem, and Vazsonyi 's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1960.

....is also monotonic and stable under substitutions, i.e. a reduction ordering , then it suffices to check that l r for every rule l r in the system. Monotonic orderings including the subterm relation are called simplification orderings, and its well foundedness follows from Kruskal s theorem [Kru60]. Inside this class, path orderings, and in particular the recursive path ordering 1 (RPO) Der82] have received a special attention (see [Der87, DJ90] Unfortunately, although these orderings are simple and easy to use, they turn out, in many cases, to be a weak termination proving tool, as ....

J.B. Kruskal. Well-quasi-ordering, the Tree Theorem, and Vazsonyi 's conjecture. Transactions of the American Mathematical Society, 95:210--225,1960.

....branch is not admissible. Definition 5. Given a wqo # we define whistle # as follows: whistle # (L, #) M i# M is the farthest ancestor of L such that label(M) # label(L) and whistle # (L, #) fail if there is no such ancestor. A particularly useful wqo is the homeomorphic embedding [19, 25]: Definition 6. The (pure) homeomorphic embedding relation # on expressions is inductively defined as follows (i.e. # is the least relation satisfying the rules) 1. X # Y for all variables X,Y 2. s # f(t 1 , t n ) if s # t i for some i 3. f(s 1 , s n ) # f(t 1 , t n ) ....

....is that A # B i# A can be obtained from B by striking out certain parts, or said another way, the structure of A reappears within B. We have f(a, b) #p(f(g(a) b) but p(a) # #p(b) and f(a, b) # #p(f(a) f(b) The relation # is a wqo on the set of expressions over a finite alphabet [19, 25]. We also define the weak homeomorphic embedding relation # on expressions by replacing rule 1. of Definition 6 by: 1. t #X for all variables X # is weaker in the sense that less sequences are admissible (and hence it is a wqr) but it will be useful in establishing exact relationships to ....

J. B. Kruskal. Well-quasi ordering, the tree theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1960.

....next lemma was first proved by Higman in [Hig52] and is thus referred to as Higman s Lemma. The elegant proof given in [Ros82] originates from Nash Williams (see [NW63] Lemma 3. 6 (Theorem 10.23 in [Ros82] If (T; is a wqo, then so is (T ; Next, we will prove Kruskal s Theorem (see [Kru60]) in the setting of term rewriting. We will follow the proof given by Nash Williams in [NW63] see also [DJ90, Der79, Gal91] To this end, we first define the following TRSs. Definition 3.7 Let F be a signature, V be a countable set of variables, and let (F [ V; be a quasi ordering. We define ....

J. Kruskal. Well-Quasi-Ordering, the Tree Theorem, and Vazsonyi's Conjecture. Transactions of the American Mathematical Society 95, pages 210--225, 1960.

....by deleting the conditions from R is terminating. Therefore, it suffices to show the assertion for an unconditional simplifying TRS R. It is well known that is a well quasi ordering 5 if F [V is finite (this is a consequence of Higman s Lemma [Hig52] a special case of Kruskal s Tree Theorem [Kru60]: since F [ V is well quasi ordered by j, T (F ; V) is well quasi ordered by the pure homeomorphic embedding relation OE emb , and emb is contained in , cf. Der82] However, we assume V to be denumerable, thus is not in general a well quasi ordering (variables are incomparable, hence ....

J. Kruskal. Well-Quasi-Ordering, the Tree Theorem, and Vazsonyi's Conjecture. Transactions of the American Mathematical Society 95, pages 210--225, 1960.

....Thus, t(fl) h(e 1 ; e n ) let x 1 =e 1 ; xn=en in h(x 1 ; xn ) t 0 (ffl) This concludes the proof. ut Next we prove that M ps maintains a finitary, continuous predicate. The following result, known as Kruskal s Tree Theorem, is due to Higman [19] and Kruskal [22]. Its classical proof is due to Nash Williams [28] Theorem 41. E H (V ) Theta) is a well quasi order, provided H is finite. Proof. Collapse all variables to one 0 ary operator and use the proof in [14] ut Corollary 42. The relation Theta is a well quasi order on E. 12 Since t(fl) is ....

J.B. Kruskal. Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1960.

....rigid equation, we prove an auxiliary statement. LEMMA 3.8. Let t 0 ; t 1 ; be an infinite sequence of terms in a finite signature all whose variables belong to a finite set. Then there are numbers i; j such that i j and the constraint t i t j is unsatisfiable. Proof. Following (Kruskal, 1960) we introduce a partial ordering on terms as the smallest reflexive and transitive relation satisfying 1. f(s 1 ; s n ) s i for all i 2 f1; ng; 2. if s t then r[s] r[t] 3 Suggested by G.Becher (private communication) main.tex Date: October 28, 1998; Time: 19:24; Version ....

....1. f(s 1 ; s n ) s i for all i 2 f1; ng; 2. if s t then r[s] r[t] 3 Suggested by G. Becher (private communication) main.tex Date: October 28, 1998; Time: 19:24; Version not used What You Always Wanted to Know About Rigid E Unification 9 By Kruskal s Tree Theorem (Kruskal, 1960) there exist i; j such that i j and t j t i . It is easy to see that the constraint t i t j is unsatisfiable. Similar statements have been proven in (Dershowitz, 1982; Plaisted, 1986) THEOREM 3.9 (Termination of BSE) For any constraint rigid equation R Delta C, there exists a finite ....

J. Kruskal. Well quasi ordering, the tree problem and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1960.

....relation is then such that all infinite sequences are good. There are several other equivalent definitions of well binary relations and well quasi orders. Higman [27] used an alternate definition of well quasi orders in terms of the finite basis property (or finite generating set in [36]) Both definitions are equivalent by Theorem 2.1 in [27] A different (but also equivalent) definition of a wqo is(e.g. 38, 77] A quasi order V is a wqo iff for all quasi orders V which contain V (i.e. v V v 0 ) v V v 0 ) the corresponding strict partial order OE V is a wfo. This ....

....have to generate wfo s) and one can therefore drop this requirement, leading to the use of wbr s. Later on in Sections 5 and 6 we will actually develop wbr s which are not wqo s. An interesting wqo is the homeomorphic embedding relation Theta, which derives from results by Higman [27] and Kruskal [36]. It has been used in the context of term rewriting systems in [14, 15] and adapted for use in supercompilation [75] in [72] Its usefulness as a stop criterion for partial evaluation is also discussed and advocated in [52] Some complexity results can be found in [74] and [24] also summarised ....

[Article contains additional citation context not shown here]

J. B. Kruskal. Well-quasi ordering, the tree theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1960.

....ffl ffl Delta Delta Delta is OE 2 (0) in the Veblen Feferman Schutte hierarchy [ Veblen, 1908; Feferman, 1968; Schmidt, 1976 ] Less natural orderings on (nonbinary) ordered trees correspond to much larger ordinals in that hierarchy. In particular, some orderings based on Kruskal s Tree Theorem [ Kruskal, 1960 ] correspond to the first impredicative ordinal, Gamma 0 , and even to larger ones [ Friedman, 19 ; Simpson, 1985; Smory nski, 1986; Dershowitz, 1987; Gallier, 1991 ] The significance of Gamma 0 for computer science is discussed in [ Gallier, 1991 ] 5 Conclusions It has been argued [ ....

J. B. Kruskal. Well-quasi-ordering, the Tree Theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, May 1960.

....R Delta C. Proof. Sketch) The main idea is the following. We note that if there is an infinite derivation, we have a solvable infinite set of constraints containing ft 0 t 1 ; t 1 t 2 ; g, such that t i 2 T ( Sigma ; X) for finite Sigma and X . This contradicts to Kruskal s theorem [41]. 2 Inequality constraints are not needed for soundness or completeness of our method. The pragmatics behind inequality constraints is to ensure that the search for solutions of a rigid equation is finite. In addition, the use of ordering constraints prunes the search space. To illustrate this ....

J. Kruskal. Well quasi ordering, the tree problem and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1960.

....Finally, we can present our concrete unfolding rule: Definition20. concrete unfolding rule) Unfold the left most selectable atom in each goal of the SLD tree under construction. If no atom is selectable, do not unfold any atom in the goal. The following theorem is a variant of Kruskal s theorem [18], see also [9] Theorem 21. For any infinite sequence A 0 ; A 1 ; for some 0 i j: A i Theta A j . The following corollary follows from Definitions 9, 19 and 20, and Theorem 21. Corollary 22. Let P be a program, G a goal, and U the unfolding rule in Definition 20. Then U (P; G) is a ....

J.B. Kruskal. Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1960.

....reduction path has both Omega and x:y strongly embedded in it (for some variable y other than x) Section 10 reviews open problems. 1.3. Related Work. The result in Section 4 is a translation into calculus of Kruskal s Tree Theorem due to Higman [8] and in a more general formulation to Kruskal [14]. The result has a beautiful proof due to Nash Williams [18] Our proof follows the idea of Nash William s proof but with some modifications to account for terms instead of finite trees. This involves both simplifications 1 In effect the context in reduction of Psi has been turned into an ....

....such that M i v M j . There are only finitely many terminals; otherwise they would form a bad sequence. Thus there is an N 2 IN such that for all n N there is an m n with M n v Mm . From this one easily defines by recursion an infinite ascending subsequence.2 4.5. Theorem (Higman [8] Kruskal [14]) v is a well quasi order on . Proof (Nash Williams [18] Suppose the theorem were false, i.e. there were a bad sequence. By Lemma 4.3 there is a bad sequence oe = M 0 ; M 1 ; such that for all n 2 IN there is no bad sequence whose first n 1 elements are (in this order) M 0 ; M 1 ; ....

J.B. Kruskal. Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1954.

....should be part of the definition. Moreover, it is easy to show that the class of simply terminating TRSs is not affected by imposing closure under substitutions. Dershowitz [3, 4] showed that every simply terminating TRS is terminating. The proof is based on the beautiful Tree Theorem of Kruskal [26]. Definition 4.2 An infinite sequence t 1 , t 2 , t 3 , of terms in T (F ; V) is selfembedding if there exist 1 6 i j such that t i E emb t j . Theorem 4.3 (Kruskal s Tree Theorem Finite Version) Every infinite sequence of ground terms is self embedding. We refrain from proving ....

J.B. Kruskal, Well-Quasi-Ordering, the Tree Theorem, and Vazsonyi's Conjecture, Transactions of the American Mathematical Society 95 (1960) 210--225.

.... Theta A. Finally, we can present our concrete unfolding rule: Definition 3.9. concrete unfolding rule) Repeatedly unfold the left most selectable atom in each leaf of the SLD tree under construction until no atom is selectable. The following theorem is an extension of Higman Kruskal s theorem ([28, 35], see also [17] proven in [51] Theorem 3.10. For any infinite sequence A 0 ; A 1 ; of atoms, for some 0 i j: A i Theta A j . And we obtain the following corollary from Definitions 3.6, 3.8 and 3.9, and Theorem 3.10. 23 Corollary 3.11. Let P be a program, G a goal, and U the ....

....Q and there exists an ordered subconjunction A 0 1 : A 0 n of Q 0 such that A i Theta A 0 i for all i. This relation Theta still satisfies Theorem 3. 10 (for sequences of conjunctions) This can be proven easily using the results of [51] combined with Higman Kruskal s theorem ([28, 35], see also [17] by considering as a functor of variable arity (i.e. an associative operator) To complete the definition of abstraction, it remains to be decided how to split a maximal connected subconjunction Q 0 deriving from some B 2 bodies(U(P; QL ) when it indeed embeds a goal Q on the ....

J.B. Kruskal. Well-quasi-ordering, the tree theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1960.

....in t into the nodes in s such that the function symbol labeling a node in t is less than or equivalent to (under ) the label of the corresponding node in s, and such that distinct edges in t map to disjoint paths of s. The following deep result is at the heart of our argument: Tree Theorem ( Kruskal, 1960 ] If is a well quasi ordering of a vocabulary F , then the embedding relation emb is a well quasi ordering of the terms T (F) It has a beautiful proof, due to [ Nash Williams, 1963 ] Proof. Note that, by the infinite version of Ramsey s Theorem, any infinite sequence of elements of a ....

J. B. Kruskal, Well-quasi-ordering, the Tree Theorem, and Vazsonyi's conjecture, Transactions of the American Mathematical Society 95, pp. 210-225 (May 1960).

....rigid equation, we prove an auxiliary statement. Lemma 3.8 Let t 0 ; t 1 ; be an infinite sequence of terms in a finite signature all whose variables belong to a finite set. Then there are numbers i; j such that i j and the constraint t i t j is unsatisfiable. Proof. Following Kruskal [42] we introduce a partial ordering on terms as the smallest reflexive and transitive relation satisfying 2 Suggested by G.Becher (private communication) 9 Section 3. Rigid basic superposition 1. f(s 1 ; s n ) s i for all i 2 f1; ng; 2. if s t then r[s] r[t] By Kruskal s ....

....a partial ordering on terms as the smallest reflexive and transitive relation satisfying 2 Suggested by G.Becher (private communication) 9 Section 3. Rigid basic superposition 1. f(s 1 ; s n ) s i for all i 2 f1; ng; 2. if s t then r[s] r[t] By Kruskal s Tree Theorem [42] there exist i; j such that i j and t j t i . It is easy to see that the constraint t i t j is unsatisfiable. 2 Similar statements have been proven by Dershowitz [16] and Plaisted [57] Theorem 3.9 (Termination of BSE) For any constraint rigid equation R Delta C, there exists a finite ....

J. Kruskal. Well quasi ordering, the tree problem and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, 1960.

....all terms s and variable arity function symbols f in F . Definition 2.8. A rewrite ordering is a simplification ordering if it enjoys the subterm property and the deletion property (the last one only for variable arity function symbols) When F is finite, is proved [Der82] using Kruskal s theorem [Kru60]) that any simplification ordering on T (F ; X ) is well founded (and hence, it is reduction ordering) Thus, when F is finite, simplification orderings can also be used for proving termination of rewrite systems. Theorem 2.2. Let F be finite. A rewrite system R(F ; X ; R) is terminating if all ....

Joseph B. Kruskal. Well-quasi-ordering, the Tree Theorem, and Vazsonyi 's conjecture. Transactions of the American Mathematical Society, 95:210--225, May 1960.

....any relation S on the set of natural numbers, if S has the fbp so does S . By this being the most general formulation of Higman s lemma we mean that it is the most general version regarding the requirements on S. But words can be generalized to trees, for instance, as stated by Kruskal theorem [Kru60]. The statement we proved in type theory is the following. THEOREM 2.6 (Higman s lemma) Given any relation S, if S has the fbp so does S . Even if the requirement that S is a relation on natural numbers is dropped, this formulation should not be considered to be more general than the one of ....

J. Kruskal. Well-Quasi-Ordering, the Tree Theorem, and Vazsonyi's Conjecture. Transactions of the American Mathematical Society, 95:210-- 225, 1960.

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Joseph B. Kruskal. Well-quasi-ordering, the Tree Theorem, and Vazsonyi's conjecture. Transactions of the American Mathematical Society, 95:210--225, May 1960.

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