28,467 research outputs found
Complexity Bounds for Ordinal-Based Termination
`What more than its truth do we know if we have a proof of a theorem in a
given formal system?' We examine Kreisel's question in the particular context
of program termination proofs, with an eye to deriving complexity bounds on
program running times.
Our main tool for this are length function theorems, which provide complexity
bounds on the use of well quasi orders. We illustrate how to prove such
theorems in the simple yet until now untreated case of ordinals. We show how to
apply this new theorem to derive complexity bounds on programs when they are
proven to terminate thanks to a ranking function into some ordinal.Comment: Invited talk at the 8th International Workshop on Reachability
Problems (RP 2014, 22-24 September 2014, Oxford
Multigraphs without large bonds are wqo by contraction
We show that the class of multigraphs with at most connected components
and bonds of size at most is well-quasi-ordered by edge contraction for all
positive integers . (A bond is a minimal non-empty edge cut.) We also
characterize canonical antichains for this relation and show that they are
fundamental
On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders
We investigate the ordinal invariants height, length, and width of well quasi
orders (WQO), with particular emphasis on width, an invariant of interest for
the larger class of orders with finite antichain condition (FAC). We show that
the width in the class of FAC orders is completely determined by the width in
the class of WQOs, in the sense that if we know how to calculate the width of
any WQO then we have a procedure to calculate the width of any given FAC order.
We show how the width of WQO orders obtained via some classical constructions
can sometimes be computed in a compositional way. In particular, this allows
proving that every ordinal can be obtained as the width of some WQO poset. One
of the difficult questions is to give a complete formula for the width of
Cartesian products of WQOs. Even the width of the product of two ordinals is
only known through a complex recursive formula. Although we have not given a
complete answer to this question we have advanced the state of knowledge by
considering some more complex special cases and in particular by calculating
the width of certain products containing three factors. In the course of
writing the paper we have discovered that some of the relevant literature was
written on cross-purposes and some of the notions re-discovered several times.
Therefore we also use the occasion to give a unified presentation of the known
results
On better-quasi-ordering classes of partial orders
We provide a method of constructing better-quasi-orders by generalising a
technique for constructing operator algebras that was developed by Pouzet. We
then generalise the notion of -scattered to partial orders, and use our
method to prove that the class of -scattered partial orders is
better-quasi-ordered under embeddability. This generalises theorems of Laver,
Corominas and Thomass\'{e} regarding -scattered linear orders and
trees, countable forests and N-free partial orders respectively. In particular,
a class of countable partial orders is better-quasi-ordered whenever the class
of indecomposable subsets of its members satisfies a natural strengthening of
better-quasi-order.Comment: v1: 45 pages, 8 figures; v2: 44 pages, 11 figures, minor corrections,
fixed typos, new figures and some notational changes to improve clarity; v3:
45 pages, 12 figures, changed the way the paper is structured to improve
clarity and provide examples earlier o
A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets
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
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