279 research outputs found
A Comparison of Well-Quasi Orders on Trees
Well-quasi orders such as homeomorphic embedding are commonly used to ensure
termination of program analysis and program transformation, in particular
supercompilation.
We compare eight well-quasi orders on how discriminative they are and their
computational complexity. The studied well-quasi orders comprise two very
simple examples, two examples from literature on supercompilation and four new
proposed by the author.
We also discuss combining several well-quasi orders to get well-quasi orders
of higher discriminative power. This adds 19 more well-quasi orders to the
list.Comment: In Proceedings Festschrift for Dave Schmidt, arXiv:1309.455
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
Well quasi-orders and context-free grammars
Let G be a context-free grammar and let L be the language of all the words derived from any variable of G. We prove the following generalization of Higman's theorem: any division order on L is a well quasi-order on L. We also give applications of this result to some quasi-orders associated with unitary grammars. (C) 2004 Elsevier B.V. All rights reserved
Reverse mathematics, well-quasi-orders, and Noetherian spaces
A quasi-order Q induces two natural quasi-orders on P(Q) P(Q) , but if Q is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq (Proceedings of the 22nd Annual IEEE Symposium 4 on Logic in Computer Science (LICS’07), pp. 453–462, 2007) showed that moving from a well-quasi-order Q to the quasi-orders on P(Q) P(Q) preserves well-quasi-orderedness in a topological sense. Specifically, Goubault-Larrecq proved that the upper topologies of the induced quasi-orders on P(Q) P(Q) are Noetherian, which means that they contain no infinite strictly descending sequences of closed sets. We analyze various theorems of the form “if Q is a well-quasi-order then a certain topology on (a subset of) P(Q) P(Q) is Noetherian” in the style of reverse mathematics, proving that these theorems are equivalent to ACA0 over RCA0. To state these theorems in RCA0 we introduce a new framework for dealing with second-countable topological spaces
Applying G\"odel's Dialectica Interpretation to Obtain a Constructive Proof of Higman's Lemma
We use G\"odel's Dialectica interpretation to analyse Nash-Williams' elegant
but non-constructive "minimal bad sequence" proof of Higman's Lemma. The result
is a concise constructive proof of the lemma (for arbitrary decidable
well-quasi-orders) in which Nash-Williams' combinatorial idea is clearly
present, along with an explicit program for finding an embedded pair in
sequences of words.Comment: In Proceedings CL&C 2012, arXiv:1210.289
Homeomorphic Embedding for Online Termination of Symbolic Methods
Well-quasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of techniques for program analysis, specialisation, transformation, and verification. In this paper we survey and discuss this use of homeomorphic embedding and clarify the advantages of such an approach over one using well-founded orders. We also discuss various extensions of the homeomorphic embedding relation. We conclude with a study of homeomorphic embedding in the context of metaprogramming, presenting some new (positive and negative) results and open problems
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
Regular tree languages and quasi orders
Regular languages were characterized as sets closed with respect to monotone well-quasi orders. A similar result is proved here for tree languages. Moreover, families of quasi orders that correspond to positive varieties of tree languages and varieties of finite ordered algebras are characterized
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