Results 1  10
of
15
Homeomorphic Embedding for Online Termination
 STATIC ANALYSIS. PROCEEDINGS OF SAS’98, LNCS 1503
, 1998
"... Recently wellquasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of program analysis, specialisation and transformation techniques. In this paper, ..."
Abstract

Cited by 61 (8 self)
 Add to MetaCart
Recently wellquasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of program analysis, specialisation and transformation techniques. In this paper,
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 ..."
Abstract

Cited by 43 (3 self)
 Add to MetaCart
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.
A proof of Higman's lemma by structural induction
, 1993
"... This paper gives an example of such an inductive proof for a combinatorial problem. While there exist other constructive proofs of Higman's lemma (see for instance [10, 14]), the present argument has been recorded for its extreme formal simplicity. This simplicity allows us to give a complete descri ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
This paper gives an example of such an inductive proof for a combinatorial problem. While there exist other constructive proofs of Higman's lemma (see for instance [10, 14]), the present argument has been recorded for its extreme formal simplicity. This simplicity allows us to give a complete description of the computational content of the proof, first in term of a functional program, which follows closely the structure of the proof, and then in term of a program with state. The second program has an intuitive algorithmic meaning. In order to show that these two programs are equivalent, we introduce an intermediary program, which is a firstorder operational interpretation of the functional program. The relation between this program and the program with state is simple to establish. We can thus claim that we understand completely the computational behaviour of the proof. It is possible to give still another description of this algorithm, in term of process computing in parallel. In this form, the connection with NashWilliams non constructive argument is quite clear (though this algorithm was found first only as an alternative description of the computational content of the inductive proof). This inductive proof was actually found from the usual non constructive argument by using the technique described in [3]. These two facts give strong indication that this algorithm can be considered as the computational content of the NashWilliams argument.
Higman's Lemma in Type Theory
 PROCEEDINGS OF THE 1996 WORKSHOP ON TYPES FOR PROOFS AND PROGRAMS
, 1997
"... This thesis is about exploring the possibilities of a limited version of MartinLöf's type theory. This exploration consists both of metatheoretical considerations and of the actual use of that version of type theory to prove Higman's lemma. The thesis is organized in two papers, one in which type t ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
This thesis is about exploring the possibilities of a limited version of MartinLöf's type theory. This exploration consists both of metatheoretical considerations and of the actual use of that version of type theory to prove Higman's lemma. The thesis is organized in two papers, one in which type theory itself is studied and one in which it is used to prove Higman's lemma. In the first paper, A Lambda Calculus Model of MartinLöf's Theory of Types with Explicit Substitution, we present the formal calculus in complete detail. It consists of MartinLof's logical framework with explicit substitution extended with some inductively defined sets, also given in complete detail. These inductively defined sets are precisely those we need in the second paper of this thesis for the formal proof of Higman's lemma. The limitations of the formalism come from the fact that we do not introduce universes. It is known that for other versions of type theory, the absence of universes implies the impossib...
Ramsey's Theorem in Type Theory
, 1993
"... We present formalizations of constructive proofs of the Intuitionistic Ramsey Theorem and Higman's Lemma in MartinLof's Type Theory. We analyze the computational content of these proofs and we compare it with programs extracted out from some classical proofs. Contents 1 Introduction 2 2 The proofs ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We present formalizations of constructive proofs of the Intuitionistic Ramsey Theorem and Higman's Lemma in MartinLof's Type Theory. We analyze the computational content of these proofs and we compare it with programs extracted out from some classical proofs. Contents 1 Introduction 2 2 The proofs 4 2.1 An inductive formulation of almostfullness (AF ID ) : : : : : : : : : : 5 2.1.1 Intuitionistic Ramsey Theorem (IRT ID ) : : : : : : : : : : : : 7 2.1.2 Higman's Lemma (HL ID ) : : : : : : : : : : : : : : : : : : : : 12 2.2 A negationless inductive formulation of almostfullness (AF I ) : : : : : 17 2.2.1 Intuitionistic Ramsey Theorem (IRT I ) : : : : : : : : : : : : : 17 2.3 Equivalence between the various formulations of almostfullness : : : 20 3 The programs 22 3.1 A higher order program : : : : : : : : : : : : : : : : : : : : : : : : : 24 3.2 A first order program : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 4 Computational content of classical proofs 28 4.1 A cl...
An Inductive Version of NashWilliams’ MinimalBadSequence Argument for Higman’s Lemma
 In P. Callaghan, e.al., Types for Proofs and Programs, Lecture Notes in Computer Science 2277
, 2001
"... Abstract. Higman’s lemma has a very elegant, nonconstructive proof due to NashWilliams [NW63] using the socalled minimalbadsequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was done ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. Higman’s lemma has a very elegant, nonconstructive proof due to NashWilliams [NW63] using the socalled minimalbadsequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was done by Coquand and Fridlender [CF94]. Here we present a proof in a theory of inductive definitions that works for arbitrary decidable well quasiorders. 1
Extending Homeomorphic Embedding in the Context of Logic Programming
 Departement Computerwetenschappen
, 1997
"... Recently wellquasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of program analysis, specialisation and transformation techniques. However, as we illustrate in the paper, the homeomorphic embedding relation as it is usually defined su ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Recently wellquasi orders in general, and homeomorphic embedding in particular, have gained popularity to ensure the termination of program analysis, specialisation and transformation techniques. However, as we illustrate in the paper, the homeomorphic embedding relation as it is usually defined suffers from several inadequacies which make it less suitable in a logic programming context. We present several increasingly refined ways to remedy this problem by providing more sophisticated treatments of variables and present a new, extended homeomorphic embedding relation.
Infinite strings generated by insertions
"... —The process of generation of strings over a finite alphabet by inserting characters at an arbitrary place in the string is considered. A topology on the set of strings is introduced, in which closed sets are defined to be sets of strings that are invariant with respect to the insertion of character ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
—The process of generation of strings over a finite alphabet by inserting characters at an arbitrary place in the string is considered. A topology on the set of strings is introduced, in which closed sets are defined to be sets of strings that are invariant with respect to the insertion of characters. An infinite insertion string is defined to be a set of infinite sequences of insertions ending in the same open sets. The number of infinite insertion strings is proved to be countable. It is proved also that there is a relationship between the countability of the completion of partially wellordered sets by infinite elements and the fulfillment of an analogue of Dickson’s and Higman’s lemmas for them. 1.
A constructive proof of Higman’s lemma
 SME Conference Proceedings Bethlelem
, 1984
"... Abstract. Higman’s lemma, a specific instance of Kruskal’s theorem, is an interesting result from the area of combinatorics, which has often been used as a test case for theorem provers. We present a constructive proof of Higman’s lemma in the theorem prover Isabelle, based on a paper proof by Coqua ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. Higman’s lemma, a specific instance of Kruskal’s theorem, is an interesting result from the area of combinatorics, which has often been used as a test case for theorem provers. We present a constructive proof of Higman’s lemma in the theorem prover Isabelle, based on a paper proof by Coquand and Fridlender. Making use of Isabelle’s newlyintroduced infrastructure for program extraction, we show how a program can automatically be extracted from this proof, and analyze its computational behaviour. 1
Applications of inductive definitions and choice principles to program synthesis
"... Abstract. We describe two methods of extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and nonconstructive choice principles. The first method removes any reference to infinite sequences and transforms the theorem into a system of inductive defi ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. We describe two methods of extracting constructive content from classical proofs, focusing on theorems involving infinite sequences and nonconstructive choice principles. The first method removes any reference to infinite sequences and transforms the theorem into a system of inductive definitions, the other applies a combination of Gödel’s negativeand Friedman’s Atranslation. Both approaches are explained by means of a case study on Higman’s Lemma and its wellknown classical proof due to NashWilliams. We also discuss some prooftheoretic optimizations that were crucial for the formalization and implementation of this work in the interactive proof system Minlog. 1