Kruskal's Tree Theorem Research Articles

Generalized fusible numbers and their ordinals

Erickson defined the fusible numbers as a set F of reals generated by repeated application of the function x+y+12. Erickson, Nivasch, and Xu showed that F is well ordered, with order type ε0. They also investigated a recursively defined function M:R→R. They showed that the set of points of discontinuity of M is a subset of F of order type ε0. They also showed that, although M is a total function on R, the fact that the restriction of M to Q is total is not provable in first-order Peano arithmetic PA.In this paper we explore the problem (raised by Friedman) of whether similar approaches can yield well-ordered sets F of larger order types. As Friedman pointed out, Kruskal's tree theorem yields an upper bound of the small Veblen ordinal for the order type of any set generated in a similar way by repeated application of a monotone function g:Rn→R.The most straightforward generalization of x+y+12 to an n-ary function is the function x1+⋯+xn+1n. We show that this function generates a set Fn whose order type is just φn−1(0). For this, we develop recursively defined functions Mn:R→R naturally generalizing the function M.Furthermore, we prove that for any linear function g:Rn→R, the order type of the resulting F is at most φn−1(0).Finally, we show that there do exist continuous functions g:Rn→R for which the order types of the resulting sets F approach the small Veblen ordinal.

Kruskal's Tree Theorem for Acyclic Term Graphs

In this paper we study termination of term graph rewriting, where we restrict our attention to acyclic term graphs. Motivated by earlier work by Plump we aim at a definition of the notion of simplification order for acyclic term graphs. For this we adapt the homeomorphic embedding relation to term graphs. In contrast to earlier extensions, our notion is inspired by morphisms. Based on this, we establish a variant of Kruskal's Tree Theorem formulated for acyclic term graphs. In proof, we rely on the new notion of embedding and follow Nash-Williams' minimal bad sequence argument. Finally, we propose a variant of the lexicographic path order for acyclic term graphs.

Certified Kruskal's Tree Theorem

This article presents the first formalization of Kurskal's tree theorem in a proof assistant. The Isabelle/HOL development is along the lines of Nash-Williams' original minimal bad sequence argument for proving the tree theorem. Along the way, proofs of Dickson's lemma and Higman's lemma, as well as some technical details of the formalization are discussed.

On effective construction of the greatest solution of language inequality [formula omitted

In this paper, we consider effective constructions of the greatest solution of the language inequality XA⊆BX. It has been proved by Kunc in 2005 that the greatest solution of XA⊆BX is regular provided that B is regular, no matter what A is. However this proof is based on Kruskal's tree theorem, and does not provide any effective way to construct the greatest solution.We focus on this gap in this paper. We give an effective construction of the greatest solution for the following two cases:(i)A,B are regular and there exists k⩾1 such that pref(B)Ak∩B⩽kpref(B)=∅, where pref(B) is the set of prefixes of words in B,(ii)A,B are regular and B is a code with finite decoding delay.Our construction takes the point of view of games. As shown by Kunc in his regularity proof, the construction of the greatest solution can be reduced to the construction of the winning region of a two-player game. Our contribution is to show that the winning regions of the two-player game for the two cases can be constructed effectively.The main ingredient of the construction for the first case is a shrinking lemma for the words on which one of the players has a winning strategy. While the construction for the second case is based on the observation that the two-player game can be reduced to a two-player reachability game played on the transition graph of a one-counter machine.

Structure theorem for tournaments omitting N5

AbstractA finite tournament T is tight if the class of finite tournaments omitting T is well‐quasi‐ordered. We show here that a certain tournament N5 on five vertices is tight. This is one of the main steps in an exact classification of the tight tournaments, as explained in [10]; the third and final step is carried out in [11]. The proof involves an encoding of the indecomposable tournaments omitting N5 by a finite alphabet, followed by an application of Kruskal's Tree Theorem. This problem arises in model theory and in computational complexity in a more general form, which remains open: the problem is to give an effective criterion for a finite set {T1,…,Tk} of finite tournaments to be tight in the sense that the class of all finite tournaments omitting each of T1,…,Tk is well‐quasi‐ordered. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 165–192, 2003

Proofs as Graphs

This note introduces a method of representing and reasoning about the actions of a class of proof procedures. A graph-like structure, called a proof diagram, is introduced in which conclusions of inferences can be shared. A version of Kruskal's Tree Theorem is developed for these structures and from there a notion of minimal proof is introduced. The notion of minimal proof allows us to make a link between standard treatments of proof and proofs generated by mechanical theorem provers. We discuss various uses for the proof diagrams including the development of tactics by graph reduction and propose a graph based meta-language for theorem provers.

Finitely constrained classes of homogeneous directed graphs

AbstractGiven a finite relational language L is there an algorithm that, given two finite sets and of structures in the language, determines how many homogeneous L structures there are omitting every structure in and embedding every structure in ?For directed graphs this question reduces to: Is there an algorithm that, given a finite set of tournaments Γ, determines whether Γ, the class of finite tournaments omitting every tournament in Γ. is well-quasi-order?First, we give a nonconstructive proof of the existence of an algorithm for the case in which Γ consists of one tournament. Then we determine explicitly the set of tournaments each of which does not have an antichain omitting it. Two antichains are exhibited and a summary is given of two structure theorems which allow the application of Kruskal's Tree Theorem. Detailed proofs of these structure theorems will be given elsewhere.The case in which Γ consists of two tournaments is also discussed.

What's so special about Kruskal's theorem and the ordinal Γ o? A survey of some results in proof theory

This paper consists primarily of a survey of results of Harvey Friedman about some proof-theoretic aspects of various forms of Kruskal's tree theorem, and in particular the connection with the ordinal Γ 0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Verlen hierarchies, some subsystems of second-order logic, slow-growing and fast-growing hierarchies including Girard's result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the ‘tree theorem’, as well as a ‘finite miniaturization’ of Kruskal's theorem due to Harvey Friedman. These versions of Kruskal's theorem are remarkable from a proof-theoretic point of view because they are not provable in relatively strong logical systems. They are examples of so-called ‘natural independence phenomena’, which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal's tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of Knuth-Bendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γ o and Kruskal's theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence.

Orderings for term-rewriting systems

Methods of proving that a term-rewriting system terminates are presented. They are based on the intuitive notion of ‘simplification orderings’, orderings in which any term that is syntactically simpler than another is smaller than the other. As a consequence of Kruskal's Tree Theorem, any nonterminating system must be self-embedding in the sense that it allows for the derivation of some term from a simpler one; thus termination is guaranteed if every rule in the system is a reduction in some simplification ordering. Most of the orderings that have been used for proving termination are indeed simplication orderings; using this notion often allows for much easier proofs. A particularly useful class of simplification orderings, the ‘recursive path orderings’, is defined. Examples of the use of simplication orderings in termination proofs are given.