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Natural termination
 Theoretical Computer Science
"... Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1 ..."
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Cited by 86 (11 self)
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Abstract. We generalize the various path orderings and the conditions under which they work, and describe an implementation of this general ordering. We look at methods for proving termination of orthogonal systems and give a new solution to a problem of Zantema's. 1
Open Problems in Rewriting
 Proceeding of the Fifth International Conference on Rewriting Techniques and Application (Montreal, Canada), LNCS 690
, 1991
"... Introduction Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings #including this one# #15, 26, 41,66#; three workshop proceedings #33, 47, 77#; #ve special journal issues #5,88, 24, 40, 67#; more than ten surveys #2,7,27 ..."
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Cited by 19 (2 self)
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Introduction Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings #including this one# #15, 26, 41,66#; three workshop proceedings #33, 47, 77#; #ve special journal issues #5,88, 24, 40, 67#; more than ten surveys #2,7,27, 28, 44, 56,57,76, 82, 81#; one edited collection of papers #1#; four monographs #3, 12,55,65#; and seven books #four of them still in progress# #8,9, 35, 54, 60,75, 84#. To encourage and stimulate continued progress in this area, wehave collected #with the help of colleagues# a number of problems that appear to us to be of interest and regarding whichwe do not know the answer. Questions on rewriting and other equational paradigms have been included; manyhave not aged su#ciently to be accorded the appellation #open problem". Wehave limited ourselves to theoretical questions, though there are certainly many additional interesting questions relating to applications and implementation
Termination by abstraction
, 2004
"... Abstract. Abstraction can be used very effectively to decompose and simplify termination arguments. If a symbolic computation is nonterminating, then there is an infinite computation with a top redex, such that all redexes are immortal, but all children of redexes are mortal. This suggests applying ..."
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Cited by 9 (1 self)
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Abstract. Abstraction can be used very effectively to decompose and simplify termination arguments. If a symbolic computation is nonterminating, then there is an infinite computation with a top redex, such that all redexes are immortal, but all children of redexes are mortal. This suggests applying weaklymonotonic wellfounded relations in abstractionbased termination methods, expressed here within an abstract framework for termbased proofs. Lexicographic combinations of orderings may be used to match up with multiple levels of abstraction. A small number of firms have decided to terminate their independent abstraction schemes.
LOGICAL DREAMS
"... Abstract. We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic. Reading instructions: The real instructions, i.e., my hopes, are that you start at the beginning and read until the end. But if you are looking for advice o ..."
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Cited by 6 (0 self)
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Abstract. We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic. Reading instructions: The real instructions, i.e., my hopes, are that you start at the beginning and read until the end. But if you are looking for advice on how much not to read and still get what you like, note that the sections are essentially independent. If you wonder what the relevance of mathematical logic is to the rest of mathematics and whether you implicitly accept the usual axioms of set theory (that is, the axioms of E. Zermelo and A. Fraenkel with the Axiom of Choice, from nowonreferredtoasZFC),youshouldreadmainly§1. If you are excited about the independence phenomenon, you should read §2. If you are interested in considerations on new additional axioms and/or in looking at definable sets of reals (as in descriptive set theory) as compared to general sets of reals, and in looking more deeply into independence, you should go to §3. For more on independence in set theory, see §4. If you really are interested in cardinal arithmetic, i.e., the arithmetic of infinite numbers of G. Cantor, go to §5. I had intended to also write sections on “Pure model theory”, “Applied model theory”, and “On Speculations and Nonsense”, but only §6 materialized.
Ordinal arithmetic with list structures
 In Logical Foundations of Computer Science
, 1992
"... We provide a set of \natural " requirements for wellorderings of (binary) list structures. We showthat the resultant ordertype is the successor of the rst critical epsilon number. The checker has to verify that the process comes to an end. Here again he should be assistedbytheprogrammer givin ..."
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Cited by 5 (0 self)
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We provide a set of \natural " requirements for wellorderings of (binary) list structures. We showthat the resultant ordertype is the successor of the rst critical epsilon number. The checker has to verify that the process comes to an end. Here again he should be assistedbytheprogrammer giving a further de nite assertion to be veri ed. This may take the form of a quantity which is asserted todecrease continually and vanish when the machine stops. To the pure mathematician it is natural to give an ordinal number. In this problem the ordinal might be (n, r)! 2 +(r, s)! + k. A less highbrow form of the same thing would be to give the integer 2 80 (n, r)+2 40 (r, s)+k. Alan M. Turing (1949) 1
An Ordinal Calculus for Proving Termination in Term Rewriting
 CAAPColl.on Trees in Algebra and Programming, volume 1059 of Lecture Notes in Computer Science
, 1996
"... In this article, we are concerned with the proofs of termination of rewrite systems by the method of interpretations. We propose a modified approach to subrecursive hierarchies of functions by means of a syntactical approach to ordinal recursion. Our method appears to be appropriate for finding inte ..."
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Cited by 5 (1 self)
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In this article, we are concerned with the proofs of termination of rewrite systems by the method of interpretations. We propose a modified approach to subrecursive hierarchies of functions by means of a syntactical approach to ordinal recursion. Our method appears to be appropriate for finding interpretations in a systematic way. We provide several examples of applications. It is shown that three usual recursion schemas over the natural numbers, recursion with parameter substitution, simple nested recursion and unnested multiple recursion can be encoded directly in our system. As the corresponding ordinal terms are primitive recursively closed, we get a concise and intuitive proof for the closure of the class of primitive recursive functions under these schemes.