A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets

We characterize the importance of resources (like counters, channels, or alphabets) when measuring the expressiveness of Well-Structured Transition Systems (WSTS). We establish, for usual classes of well partial orders, the equivalence between the existence of order reflections (non-monotonic order embeddings) and the simulations with respect to coverability languages. We show that the non-existence of order reflections can be proved by the computation of order types. This allows us to extend the current classification of WSTS, in particular solving some open problems, and to unify the existing proofs.

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“…A more recent and difficult result, by Weiermann [9], provides us with the order type of multisets. These results are summed up here: [7][8][9].…”

Section: Definition 5 (Hessenberg 1906)mentioning

confidence: 98%

“…This leads us to the proposition that we use to separate many classes of WSTS (originally found in [9]): Proposition 6. (See [9].)…”

Section: How To Prove the Non-existence Of Reflections?mentioning

confidence: 99%

Ordinal theory for expressiveness of well-structured transition systems

Bonnet

Finkel

Haddad

et al. 2013

Information and Computation

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“…A proof for Theorem 5 can be found in [16]. However, the proof contains a small error for some exceptional cases.…”

Section: Bounds For the Maximal Order Types Of Multisets And Finite Smentioning

confidence: 99%

Well-partial-orderings and the big Veblen number

2014

Self Cite

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In this article we characterize a countable ordinal known as the big Veblen number in terms of natural well-partially ordered tree-like structures. To this end, we consider generalized trees where the immediate subtrees are grouped in pairs with address-like objects. Motivated by natural ordering properties, extracted from the standard notations for the big Veblen number, we investigate different choices for embeddability relations on the generalized trees. We observe that for addresses using one finite sequence only, the embeddability coincides with the classical tree embeddability, but in this article we are interested in more general situations (transfinite addresses and wellpartially ordered addresses). We prove that the maximal order type of some of these new embeddability relations hit precisely the big Veblen ordinal ϑ Ω Ω . Somewhat surprisingly, changing a little bit the well-partially ordered addresses (going from multisets to finite sequences), the maximal order type hits an ordinal which exceeds the big Veblen number by far, namely ϑ Ω Ω Ω . Our results contribute to the research program (originally initiated by Diana Schmidt) on classifying properties of natural well-orderings in terms of order-theoretic properties of the functions generating the orderings.

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“…De Jongh and Parikh [11], and Schmidt [17] have shown a way to compose order types with the disjoint union, the cartesian product, and the Higman ordering. A more recent and difficult result, by Weiermann [18], provides us with the order type of multisets. These results are summed up here: Proposition 7.…”

Section: Definition 5 (Hessenberg 1906 [11]) the Natural Addition mentioning

confidence: 99%

“…We refer the interested reader to [11] and [18] for the complete formulas. With these general results we can obtain many strict relations between wpo.…”

Section: Definition 5 (Hessenberg 1906 [11]) the Natural Addition mentioning

confidence: 99%

Ordinal Theory for Expressiveness of Well Structured Transition Systems

Bonnet

Finkel

Haddad

et al. 2011

Foundations of Software Science and Computational Structures

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Abstract. To the best of our knowledge, we characterize for the first time the importance of resources (counters, channels, alphabets) when measuring expressiveness of WSTS. We establish, for usual classes of wpos, the equivalence between the existence of order reflections (nonmonotonic order embeddings) and the simulations with respect to coverability languages. We show that the non-existence of order reflections can be proved by the computation of order types. This allows us to solve some open problems and to unify the existing proofs of the WSTS classification.

show abstract

A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets

Abstract: Abstract. We give a sharpening of a recent result of Aschenbrenner and Pong about the maximal order type of the term ordering on the finite multisets over a wpo. Moreover we discuss an approach to compute maximal order types of well-partial orders which are related to tree embeddings.

Cited by 16 publications

References 18 publications

Ordinal theory for expressiveness of well-structured transition systems

Ordinal theory for expressiveness of well-structured transition systems

Well-partial-orderings and the big Veblen number

Ordinal Theory for Expressiveness of Well Structured Transition Systems

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