14 research outputs found
Length of an intersection
A poset \bfp is well-partially ordered (WPO) if all its linear extensions
are well orders~; the supremum of ordered types of these linear extensions is
the {\em length}, \ell(\bfp) of \bfp. We prove that if the vertex set
of \bfp is infinite, of cardinality , and the ordering is the
intersection of finitely many partial orderings on , ,
then, letting \ell(X,\leq_i)=\kappa\multordby q_i+r_i, with ,
denote the euclidian division by (seen as an initial ordinal) of the
length of the corresponding poset~: \ell(\bfp)<
\kappa\multordby\bigotimes_{1\leq i\leq n}q_i+ \Big|\sum_{1\leq i\leq n}
r_i\Big|^+ where denotes the least initial ordinal greater
than the ordinal . This inequality is optimal (for ).Comment: 13 page
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
Model Theoretic Complexity of Automatic Structures
We study the complexity of automatic structures via well-established concepts
from both logic and model theory, including ordinal heights (of well-founded
relations), Scott ranks of structures, and Cantor-Bendixson ranks (of trees).
We prove the following results: 1) The ordinal height of any automatic well-
founded partial order is bounded by \omega^\omega ; 2) The ordinal heights of
automatic well-founded relations are unbounded below the first non-computable
ordinal; 3) For any computable ordinal there is an automatic structure of Scott
rank at least that ordinal. Moreover, there are automatic structures of Scott
rank the first non-computable ordinal and its successor; 4) For any computable
ordinal, there is an automatic successor tree of Cantor-Bendixson rank that
ordinal.Comment: 23 pages. Extended abstract appeared in Proceedings of TAMC '08, LNCS
4978 pp 514-52
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
Ackermannian and Primitive-Recursive Bounds with Dickson's Lemma
Dickson's Lemma is a simple yet powerful tool widely used in termination
proofs, especially when dealing with counters or related data structures.
However, most computer scientists do not know how to derive complexity upper
bounds from such termination proofs, and the existing literature is not very
helpful in these matters.
We propose a new analysis of the length of bad sequences over (N^k,\leq) and
explain how one may derive complexity upper bounds from termination proofs. Our
upper bounds improve earlier results and are essentially tight
Ranking Functions for Size-Change Termination II
Size-Change Termination is an increasingly-popular technique for verifying
program termination. These termination proofs are deduced from an abstract
representation of the program in the form of "size-change graphs".
We present algorithms that, for certain classes of size-change graphs, deduce
a global ranking function: an expression that ranks program states, and
decreases on every transition. A ranking function serves as a witness for a
termination proof, and is therefore interesting for program certification. The
particular form of the ranking expressions that represent SCT termination
proofs sheds light on the scope of the proof method. The complexity of the
expressions is also interesting, both practicaly and theoretically.
While deducing ranking functions from size-change graphs has already been
shown possible, the constructions in this paper are simpler and more
transparent than previously known. They improve the upper bound on the size of
the ranking expression from triply exponential down to singly exponential (for
certain classes of instances). We claim that this result is, in some sense,
optimal. To this end, we introduce a framework for lower bounds on the
complexity of ranking expressions and prove exponential lower bounds.Comment: 29 pages
Ordinal Measures of the Set of Finite Multisets
Well-partial orders, and the ordinal invariants used to measure them, are relevant in set theory, program verification, proof theory and many other areas of computer science and mathematics. In this article we focus on a common data structure in programming, finite multisets of some well partial order. There are two natural orders one can define on the set of finite multisets of a partial order: the multiset embedding and the multiset ordering. Though the maximal order type of these orders is already known, other ordinal invariants remain mostly unknown. Our main contributions are expressions to compute compositionally the width of the multiset embedding and the height of the multiset ordering. Furthermore, we provide a new ordinal invariant useful for characterizing the width of the multiset ordering
Fixed Points and Noetherian Topologies
This paper provides a canonical construction of a Noetherian least fixed
point topology. While such least fixed point are not Noetherian in general, we
prove that under a mild assumption, one can use a topological minimal bad
sequence argument to prove that they are. We then apply this fixed point
theorem to rebuild known Noetherian topologies with a uniform proof.
In the case of spaces that are defined inductively (such as finite words and
finite trees), we provide a uniform definition of a divisibility topology using
our fixed point theorem. We then prove that the divisibility topology is a
generalisation of the divisibility preorder introduced by Hasegawa in the case
of well-quasi-orders.Comment: 18 pages, 2 figure
Weak Bisimulation Approximants
Bisimilarity ⌠and weak bisimilarity â are canonical notions of equivalence between processes, which are defined co-inductively, but may be approached â and even reached â by their (transfinite) inductively-defined approximants âŒÎ± and âα. For arbitrary processes this approximation may need to climb arbitrarily high through the infinite ordinals before stabilising. In this paper we consider a simple yet well-studied process algebra, the Basic Parallel Processes (BPP), and investigate for this class of processes the minimal ordinal α such that â = âα. The main tool in our investigation is a novel proof of Dicksonâs Lemma. Unlike classical proofs, the proof we provide gives rise to a tight ordinal bound, of Ï n, on the order type of non-increasing sequences of n-tuples of natural numbers. With this we are able to reduce a long-standing bound on the approximation hierarchy for weak bisimilarity â over BPP, and show that â = âÏ Ï