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14
Equations and rewrite rules: a survey
 In Formal Language Theory: Perspectives and Open Problems
, 1980
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What's so special about Kruskal's Theorem AND THE ORDINAL Γ0? A SURVEY OF SOME RESULTS IN PROOF THEORY
 ANNALS OF PURE AND APPLIED LOGIC, 53 (1991), 199260
, 1991
"... 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 Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, an ..."
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Cited by 43 (3 self)
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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 Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Veblen hierarchies, some subsystems of secondorder logic, slowgrowing and fastgrowing 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 prooftheoretic point of view because they are not provable in relatively strong logical systems. They are examples of socalled “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 KnuthBendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γ0 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.
BranchWidth and WellQuasiOrdering in Matroids and Graphs
 J. COMBIN. THEORY SER. B
, 2001
"... We prove that a class of matroids representable over a fixed finite field and with bounded branchwidth is wellquasiordered under taking minors. With some extra work, the result implies Robertson and Seymour's result that graphs with bounded treewidth (or equivalently, bounded branchwidth) are w ..."
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Cited by 37 (9 self)
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We prove that a class of matroids representable over a fixed finite field and with bounded branchwidth is wellquasiordered under taking minors. With some extra work, the result implies Robertson and Seymour's result that graphs with bounded treewidth (or equivalently, bounded branchwidth) are wellquasiordered under taking minors. We will not only derive their result from our result on matroids, but will also use the main tools for a direct proof that graphs with bounded branchwidth are wellquasiordered under taking minors. This proof also provides a model for the proof of the result on matroids, with all specific matroid technicalities strippedoff. 1
Graph Minor Theory
 BULLETIN (NEW SERIES) OF THE AMERICAN MATHEMATICAL SOCIETY
, 2005
"... A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a farreaching ..."
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Cited by 17 (0 self)
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A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a farreaching generalization of this, conjectured by Wagner: If a class of graphs is minorclosed (i.e., it is closed under deleting and contracting edges), then it can be characterized by a finite number of excluded minors. The proof of this conjecture is based on a very general theorem about the structure of large graphs: If a minorclosed class of graphs does not contain all graphs, then every graph in it is glued together in a treelike fashion from graphs that can almost be embedded in a fixed surface. We describe the precise formulation of the main results and survey some of its applications to algorithmic and structural problems in graph theory.
On Growth Rates of Closed Permutation Classes
, 2003
"... A class of permutations is called closed if 2 implies 2 , where the relation is the natural containment of permutations. Let n be the set of all permutations of 1; 2; : : : ; n belonging to . We investigate the counting functions n 7! j n j of closed classes. Our main result says that if ..."
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Cited by 16 (0 self)
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A class of permutations is called closed if 2 implies 2 , where the relation is the natural containment of permutations. Let n be the set of all permutations of 1; 2; : : : ; n belonging to . We investigate the counting functions n 7! j n j of closed classes. Our main result says that if j n j < 2 for at least one n 1, then there is a unique k 1 such that F n;k j n j F n;k n holds for all n 1 with a constant c > 0. Here F n;k are the generalized Fibonacci numbers which grow like powers of the largest positive root of x 1. We characterize also the constant and the polynomial growth of closed permutation classes and give two more results on these.
Finite generation of symmetric ideals
 TRANS. AMER. MATH. SOC
, 2005
"... Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let SX be the group of permutations of X. The group SX acts on R in a natural way, and this in turn gives R the structure of a left module over the left group ring R ..."
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Cited by 14 (8 self)
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Let A be a commutative Noetherian ring, and let R = A[X] be the polynomial ring in an infinite collection X of indeterminates over A. Let SX be the group of permutations of X. The group SX acts on R in a natural way, and this in turn gives R the structure of a left module over the left group ring R[SX]. We prove that all ideals of R invariant under the action of SX are finitely generated as R[SX]modules. The proof involves introducing a certain wellquasiordering on monomials and developing a theory of Gröbner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finitedimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.
Wellquasiordering infinite graphs with forbidden finite planar minor
 Transactions of the American Mathematical Society
, 1990
"... Abstract. We prove that given any sequence G\, Gi,... of graphs, where G\ is finite planar and all other G, are possibly infinite, there are indices;', j such that i < j and G ¡ is isomorphic to a minor of Gj. This generalizes results of Robertson and Seymour to infinite graphs. The restriction on G ..."
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Cited by 8 (2 self)
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Abstract. We prove that given any sequence G\, Gi,... of graphs, where G\ is finite planar and all other G, are possibly infinite, there are indices;', j such that i < j and G ¡ is isomorphic to a minor of Gj. This generalizes results of Robertson and Seymour to infinite graphs. The restriction on G \ cannot be omitted by our earlier result. The proof is complex and makes use of an excluded minor theorem of Robertson and Seymour, its extension to infinite graphs, NashWilliams ' theory of betterquasiordering, especially his infinite tree theorem, and its extension to something we call treestructures over QOcategories, which includes infinitary version of a wellquasiordering theorem of Friedman. 1.
Combinatorial Aspects of DavenportSchinzel Sequences
 Discrete Math
, 1995
"... A finite sequence u = a 1 a 2 : : : a p of some symbols is contained in another sequence v = b 1 b 2 : : : b q if there is a subsequence b i 1 b i 2 : : : b i p of v which can be identified, after an injective renaming of symbols, with u. We say that u = a 1 a 2 : : : a p is kregular if i \Gamma ..."
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Cited by 7 (6 self)
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A finite sequence u = a 1 a 2 : : : a p of some symbols is contained in another sequence v = b 1 b 2 : : : b q if there is a subsequence b i 1 b i 2 : : : b i p of v which can be identified, after an injective renaming of symbols, with u. We say that u = a 1 a 2 : : : a p is kregular if i \Gamma j k whenever a i = a j ; i ? j. We denote further by juj the length p of u and by kuk the number of different symbols in u. In this expository paper we give a survey of combinatorial results concerning the containment relation. Many of them are from the author's PhD thesis with the same title. Extremal results concern the growth rate of the function Ex(u; n) = max jvj, the maximum is taken over all kukregular sequences v, kvk n, not containing u. This is a generalization of the case u = ababa : : : which leads to DavenportSchinzel sequences. Enumerative results deal with the numbers of ababfree and abbafree sequences. We mention a well quasiordering result and a tree generalization of our extremal function from sequences (=colored paths) to colored trees.
An objectoriented dynamic logic with updates
, 2004
"... With the goal of this thesis being to create a dynamic logic for objectoriented languages, ODL is developed along with a sound and relatively complete calculus. The dynamic logic contains only the absolute logical essentials of objectorientation, yet still allows a “natural ” representation of all ..."
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Cited by 5 (1 self)
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With the goal of this thesis being to create a dynamic logic for objectoriented languages, ODL is developed along with a sound and relatively complete calculus. The dynamic logic contains only the absolute logical essentials of objectorientation, yet still allows a “natural ” representation of all other features of common objectoriented programming languages. ODL is an extension of a dynamic logic for imperative While programs by function modification and dynamic type checks. A generalisation of substitutions, called updates, constitute the central technical device for dealing with object aliasing arising from function modification and for retaining a manageable calculus in practical application scenarios. Further, object enumerators realise object creation in a natural yet powerful way. Finally, completeness is proven relative to firstorder arithmetic. Along with the soundness result, this proof constitutes the central part of this thesis and even copes with states containing uncomputable functions.
On Growth Rates of Hereditary Permutation Classes
, 2002
"... A class of permutations \Pi is called hereditary if ae oe 2 \Pi implies 2 \Pi, where the relation ae is the natural containment of permutations. Let \Pi n be the set of all permutations of 1; 2; : : : ; n belonging to \Pi. We investigate the counting functions n 7! j\Pi n j of hereditary classe ..."
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Cited by 5 (1 self)
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A class of permutations \Pi is called hereditary if ae oe 2 \Pi implies 2 \Pi, where the relation ae is the natural containment of permutations. Let \Pi n be the set of all permutations of 1; 2; : : : ; n belonging to \Pi. We investigate the counting functions n 7! j\Pi n j of hereditary classes. Our main result says that if j\Pi n j ! 2 for at least one n 1, then there is a unique k 1 such that F n;k j\Pi n j F n;k \Delta n holds for all n 1 with a constant c ? 0. Here F n;k are the generalized Fibonacci numbers which grow like powers of the largest positive root of x \Gamma \Delta \Delta \Delta \Gamma 1. We characterize also the constant and the polynomial growth of hereditary permutation classes and give two more results on these.