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96
Term Rewriting Systems
, 1992
"... Term Rewriting Systems play an important role in various areas, such as abstract data type specifications, implementations of functional programming languages and automated deduction. In this chapter we introduce several of the basic comcepts and facts for TRS's. Specifically, we discuss Abstract Re ..."
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Cited by 565 (16 self)
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Term Rewriting Systems play an important role in various areas, such as abstract data type specifications, implementations of functional programming languages and automated deduction. In this chapter we introduce several of the basic comcepts and facts for TRS's. Specifically, we discuss Abstract Reduction Systems
What's so special about Kruskal's Theorem AND THE ORDINAL Γ0? A SURVEY OF SOME RESULTS IN PROOF THEORY
 ANNALS OF PURE AND APPLIED LOGIC, 53 (1991), 199260
, 1991
"... This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, an ..."
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Cited by 43 (3 self)
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This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Kruskal’s tree theorem, and in particular the connection with the ordinal Γ0. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Veblen hierarchies, some subsystems of secondorder logic, slowgrowing and fastgrowing hierarchies including Girard’s result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the “tree theorem”, as well as a “finite miniaturization ” of Kruskal’s theorem due to Harvey Friedman. These versions of Kruskal’s theorem are remarkable from a prooftheoretic point of view because they are not provable in relatively strong logical systems. They are examples of socalled “natural independence phenomena”, which are considered by most logicians as more natural than the metamathematical incompleteness results first discovered by Gödel. Kruskal’s tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of KnuthBendix completion procedures. There is also a close connection between a certain infinite countable ordinal called Γ0 and Kruskal’s theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence.
Conjunctive Partial Deduction: Foundations, Control, Algorithms, and Experiments
 J. LOGIC PROGRAMMING
, 1999
"... ..."
BranchWidth and WellQuasiOrdering in Matroids and Graphs
 J. COMBIN. THEORY SER. B
, 2001
"... We prove that a class of matroids representable over a fixed finite field and with bounded branchwidth is wellquasiordered under taking minors. With some extra work, the result implies Robertson and Seymour's result that graphs with bounded treewidth (or equivalently, bounded branchwidth) are w ..."
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Cited by 38 (9 self)
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We prove that a class of matroids representable over a fixed finite field and with bounded branchwidth is wellquasiordered under taking minors. With some extra work, the result implies Robertson and Seymour's result that graphs with bounded treewidth (or equivalently, bounded branchwidth) are wellquasiordered under taking minors. We will not only derive their result from our result on matroids, but will also use the main tools for a direct proof that graphs with bounded branchwidth are wellquasiordered under taking minors. This proof also provides a model for the proof of the result on matroids, with all specific matroid technicalities strippedoff. 1
A Notation for Lambda Terms I: A Generalization of Environments
 THEORETICAL COMPUTER SCIENCE
, 1994
"... A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ffconversion in comparing terms. This notation also provides for a class of terms ..."
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Cited by 33 (12 self)
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A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ffconversion in comparing terms. This notation also provides for a class of terms that can encode other terms together with substitutions to be performed on them. The notion of an environment is used to realize this `delaying' of substitutions. The precise mechanism employed here is, however, more complex than the usual environment mechanism because it has to support the ability to examine subterms embedded under abstractions. The representation presented permits a ficontraction to be realized via an atomic step that generates a substitution and associated steps that percolate this substitution over the structure of a term. The operations on terms that are described also include ones for combining substitutions so that they might be performed simultaneously. Our notatio...
The NarrowingDriven Approach to Functional Logic Program Specialization
 New Generation Computing
, 2002
"... Partial evaluation is a semanticsbased program optimization technique which has been investigated within di#erent programming paradigms and applied to a wide variety of languages. Recently, a partial evaluation framework for functional logic programs has been proposed. ..."
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Cited by 33 (19 self)
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Partial evaluation is a semanticsbased program optimization technique which has been investigated within di#erent programming paradigms and applied to a wide variety of languages. Recently, a partial evaluation framework for functional logic programs has been proposed.
Controlling Conjunctive Partial Deduction of Definite Logic Programs
, 1996
"... "Classical" partial deduction, within the framework by Lloyd and Shepherdson, computes partial deduction for separate atoms independently. As a consequence, a number of program optimisations, known from unfold/fold transformations and supercompilation, cannot be achieved. In this paper, we show that ..."
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Cited by 32 (9 self)
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"Classical" partial deduction, within the framework by Lloyd and Shepherdson, computes partial deduction for separate atoms independently. As a consequence, a number of program optimisations, known from unfold/fold transformations and supercompilation, cannot be achieved. In this paper, we show that this restriction can be lifted through (polygenetic) specialisation of entire atom conjunctions. We present a generic algorithm for such partial deduction and discuss its correctness in an extended formal framework. We concentrate on novel control challenges specific to this "conjunctive" partial deduction. We refine the generic algorithm into a fully automatic concrete one that registers partially deduced conjunctions in a global tree, and prove its termination and correctness. We discuss some further control refinements and illustrate the operation of the concrete algorithm and/or some of its possible variants on interesting transformation examples.
Complete Monotonic Semantic Path Orderings
 In Proc. 17th CADE, LNAI 1831
, 2000
"... Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS). In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination ..."
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Cited by 31 (8 self)
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Although theoretically it is very powerful, the semantic path ordering (SPO) is not so useful in practice, since its monotonicity has to be proved by hand for each concrete term rewrite system (TRS). In this paper we present a monotonic variation of SPO, called MSPO. It characterizes termination, i.e., a TRS is terminating if and only if its rules are included in some MSPO. Hence MSPO is a complete termination method. On the practical side, it can be easily automated using as ingredients standard interpretations and generalpurpose orderings like RPO. This is shown to be a sufficiently powerful way to handle several nontrivial examples and to obtain methods like dummy elimination or dependency pairs as particular cases. Finally, we obtain some positive modularity results for termination based on MSPO. 1 Introduction Rewrite systems are sets of rules (directed equations) used to compute by repeatedly replacing parts of a given formula with equal ones until the simplest po...