Results 1 
9 of
9
ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
On Fraïssé’s conjecture for linear orders of finite Hausdorff rank
, 2007
"... We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is ϕ2(0), the first fixed point of the εfunction. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA + 0 + “ϕ2(0) is wellordered ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is ϕ2(0), the first fixed point of the εfunction. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA + 0 + “ϕ2(0) is wellordered” and, over RCA0, implies ACA ′ 0 + “ϕ2(0) is wellordered”.
THE PROFILE OF RELATIONS.
, 2007
"... Dedicated to Roland Fraïssé, at the occasion of his 86 th birthday. Abstract. Le profil d’une structure relationelle R est la fonction ϕR qui compte pour chaque entier n le nombre de ses sousstructures à n éléments, les sousstructures isomorphes étant identifiées. Dans cet exposé, je donne quelque ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Dedicated to Roland Fraïssé, at the occasion of his 86 th birthday. Abstract. Le profil d’une structure relationelle R est la fonction ϕR qui compte pour chaque entier n le nombre de ses sousstructures à n éléments, les sousstructures isomorphes étant identifiées. Dans cet exposé, je donne quelques exemples, notamment des exemples venant des groupes, et présente quelques faits frappants concernant le comportement des profils. J’indique le rôle joué par quelques notions de la théorie de l’ordre et de la combinatoire (eg belordre, algèbre ordonnée, théorème de Ramsey) dans l’étude du profil. Comme illustration, je montre que le profil d’une structure relationnelle R dont l’âge est inépuisable et de hauteur au plus ω(k + 1) satisfait l’inégalité ϕR(n) ≤ ` ´ n+k pour k tout entier n. Les recherches en cours suggèrent de voir le profil d’une
A Complex Example of a Simplifying Rewrite System
"... . For a string rewriting system, it is known that termination by a simplification ordering implies multiple recursive complexity. This theoretical upper bound is, however, far from having been reached by known examples of rewrite systems. All known methods used to establish termination by simplific ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
. For a string rewriting system, it is known that termination by a simplification ordering implies multiple recursive complexity. This theoretical upper bound is, however, far from having been reached by known examples of rewrite systems. All known methods used to establish termination by simplification yield a primitive recursive bound. Furthermore, the study of the order types of simplification orderings suggests that the recursive path ordering is, in a broad sense, a maximal simplification ordering. This would imply that simplifying string rewrite systems cannot go beyond primitive recursion. Contradicting this intuition, we construct here a simplifying string rewriting system whose complexity is not primitive recursive. This leads to a new lower bound for the complexity of simplifying string rewriting systems. Introduction Rewriting systems serve as a model of computation in many fields of theoretical computer science, for instance automated theorem proving, algebraic specificat...
Ackermannian and PrimitiveRecursive Bounds with Dickson’s Lemma
"... Abstract—Dickson’s Lemma is a simple yet powerful tool widely used in decidability proofs, especially when dealing with counters or related data structures in algorithmics, verification and modelchecking, constraint solving, logic, etc. While Dickson’s Lemma is wellknown, most computer scientists ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract—Dickson’s Lemma is a simple yet powerful tool widely used in decidability proofs, especially when dealing with counters or related data structures in algorithmics, verification and modelchecking, constraint solving, logic, etc. While Dickson’s Lemma is wellknown, most computer scientists are not aware of the complexity upper bounds that are entailed by its use. This is mainly because, on this issue, the existing literature is not very accessible. We propose a new analysis of the length of bad sequences over (N k, ≤), improving on earlier results and providing upper bounds that are essentially tight. This analysis is complemented by a “user guide ” explaining through practical examples how to easily derive complexity upper bounds from Dickson’s Lemma. I.
doi:10.1006/inco.2002.3160 A Characterisation of Multiply Recursive Functions with Higman’s Lemma
, 1999
"... We prove that string rewriting systems which reduce by Higman’s lemma exhaust the multiply recursive functions. This result provides a full characterisation of the expressiveness of Higman’s lemma when applied to rewriting theory. The underlying argument of our construction is to connect the order t ..."
Abstract
 Add to MetaCart
We prove that string rewriting systems which reduce by Higman’s lemma exhaust the multiply recursive functions. This result provides a full characterisation of the expressiveness of Higman’s lemma when applied to rewriting theory. The underlying argument of our construction is to connect the order type and the derivation length via the Hardy hierarchy. C ○ 2002 Elsevier Science (USA) 1.
Ackermannian and PrimitiveRecursive Bounds with Dickson’s Lemma ∗
, 2011
"... Dickson’s Lemma is a simple yet powerful tool widely used in decidability proofs, especially when dealing with counters or related data structures in algorithmics, verification and modelchecking, constraint solving, logic, etc. While Dickson’s Lemma is wellknown, most computer scientists are not a ..."
Abstract
 Add to MetaCart
Dickson’s Lemma is a simple yet powerful tool widely used in decidability proofs, especially when dealing with counters or related data structures in algorithmics, verification and modelchecking, constraint solving, logic, etc. While Dickson’s Lemma is wellknown, most computer scientists are not aware of the complexity upper bounds that are entailed by its use. This is mainly because, on this issue, the existing literature is not very accessible. We propose a new analysis of the length of bad sequences over (N k, ≤), improving on earlier results and providing upper bounds that are essentially tight. This analysis is complemented by a “user guide ” explaining through practical examples how to easily derive complexity upper bounds from Dickson’s Lemma. 1