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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 54 (2 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.
Solving Symbolic Ordering Constraints
, 1990
"... We show how to solve boolean combinations of inequations s ? t in the Herbrand Universe, assuming that is interpreted as a lexicographic path ordering extending a total precedence. In other words, we prove that the existential fragment of the theory of a lexicographic path ordering which extends a ..."
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Cited by 49 (10 self)
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We show how to solve boolean combinations of inequations s ? t in the Herbrand Universe, assuming that is interpreted as a lexicographic path ordering extending a total precedence. In other words, we prove that the existential fragment of the theory of a lexicographic path ordering which extends a total precedence is decidable. Keywords: simplification orderings, ordered strategies, term algebras, constraint solving. 1. Introduction The first order theory of term algebras over a language (or alphabet) with no relational symbol (other than equality) has been shown to be decidable 1;2 . See also Refs 3 and 4. Introducing into the language a binary relational symbol interpreted as the subterm ordering makes the theory undecidable 5 . Venkataraman also shows in the latter paper that the purely existential fragment of the theory, i.e. the subset of sentences whose prenex form does not contain 8, is decidable. Venkataraman was concerned with some applications in functional programm...
Prooftheoretic investigations on Kruskal's theorem
 Ann. Pure Appl. Logic
, 1993
"... In this paper we calibrate the exact prooftheoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction. Introduction S.G. Simpson in his ..."
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Cited by 36 (4 self)
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In this paper we calibrate the exact prooftheoretic strength of Kruskal's theorem, thereby giving, in some sense, the most elementary proof of Kruskal's theorem. Furthermore, these investigations give rise to ordinal analyses of restricted bar induction. Introduction S.G. Simpson in his article [10], "Nonprovability of certain combinatorial properties of finite trees", presents prooftheoretic results, due to H. Friedman, about embeddability properties of finite trees. It is shown there that Kruskal's theorem is not provable in ATR 0 . An exact description of the prooftheoretic strength of Kruskal's theorem is not given. On the assumption that there is a bad infinite sequence of trees, the usual proof of Kruskal's theorem utilizes the existence of a minimal bad sequence of trees, thereby employing some form of \Pi 1 1 comprehension. So the question arises whether a more constructive proof can be given. The need for a more elementary proof of Kruskal's theorem is especially felt ...
Canonical Conditional Rewrite Systems
"... Conditional equations have been studied for their use in the specification of abstract data types and as a computational paradigm that combines logic and function programming in a clean way. In this paper we examine different formulations of conditional equations as rewrite systems, compare their ex ..."
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Cited by 12 (1 self)
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Conditional equations have been studied for their use in the specification of abstract data types and as a computational paradigm that combines logic and function programming in a clean way. In this paper we examine different formulations of conditional equations as rewrite systems, compare their expressive power and give sufficient conditions for rewrite systems to have the "confluence " property. We then examine a restriction of these systems using a "decreasing" ordering. With this restriction, most of the basic notions (like rewriting and computing normal forms) are decidable, the "critical pair" lemma holds, and some formulations preserve canonicity.
Algorithms for ordinal arithmetic
 In 19th International Conference on Automated Deduction (CADE
, 2003
"... Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is A ..."
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Cited by 11 (5 self)
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Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is ACL2. In ACL2, every function defined must be shown to terminate using the ordinals up to ɛ0. We use a compact notation for the ordinals up to ɛ0 (exponentially more succinct than the one used by ACL2) and define efficient algorithms for ordinal addition, subtraction, multiplication, and exponentiation. In this paper we describe our notation and algorithms, prove their correctness, and analyze their complexity. 1
Ordinal arithmetic in ACL2
 In ACL2 Workshop 2003
, 2003
"... Abstract. Ordinals form the basis for termination proofs in ACL2. Currently, ACL2 uses a rather inefficient representation for the ordinals up to ɛ0 and provides limited support for reasoning about them. We present algorithms for ordinal arithmetic on an exponentially more compact representation tha ..."
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Cited by 9 (6 self)
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Abstract. Ordinals form the basis for termination proofs in ACL2. Currently, ACL2 uses a rather inefficient representation for the ordinals up to ɛ0 and provides limited support for reasoning about them. We present algorithms for ordinal arithmetic on an exponentially more compact representation than the one used by ACL2. The algorithms have been implemented and numerous properties of the arithmetic operators have been mechanically verified, thereby greatly extending ACL2’s ability to reason about the ordinals. We describe how to use the libraries containing these results, which are currently distributed with ACL2 version 2.7. 1
Ordinal arithmetic: Algorithms and mechanization
 Journal of Automated Reasoning
, 2006
"... Abstract. Termination proofs are of critical importance for establishing the correct behavior of both transformational and reactive computing systems. A general setting for establishing termination proofs involves the use of the ordinal numbers, an extension of the natural numbers into the transfini ..."
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Cited by 5 (3 self)
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Abstract. Termination proofs are of critical importance for establishing the correct behavior of both transformational and reactive computing systems. A general setting for establishing termination proofs involves the use of the ordinal numbers, an extension of the natural numbers into the transfinite which were introduced by Cantor in the nineteenth century and are at the core of modern set theory. We present the first comprehensive treatment of ordinal arithmetic on compact ordinal notations and give efficient algorithms for various operations, including addition, subtraction, multiplication, and exponentiation. Using the ACL2 theorem proving system, we implemented our ordinal arithmetic algorithms, mechanically verified their correctness, and developed a library of theorems that can be used to significantly automate reasoning involving the ordinals. To enable users of the ACL2 system to fully utilize our work required that we modify ACL2, e.g., we replaced the underlying representation of the ordinals and added a large library of definitions and theorems. Our modifications are available starting with ACL2 version 2.8. 1.
Chew's Theorem Revisited  Uniquely Normalizing Property of Nonlinear Term Rewriting Systems 
, 1992
"... . This paper gives a purely syntactical proof, based on proof normalization techniques, of an extension of Chew's theorem. The main theorem is that a weakly compatible TRS is uniquely normalizing. Roughly speaking, the weakly compatible condition allows possibly nonlinear TRSs to have nonroot o ..."
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Cited by 4 (2 self)
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. This paper gives a purely syntactical proof, based on proof normalization techniques, of an extension of Chew's theorem. The main theorem is that a weakly compatible TRS is uniquely normalizing. Roughly speaking, the weakly compatible condition allows possibly nonlinear TRSs to have nonroot overlapping rules that return the same results. This result implies the consistency of CLpc which is an extension of the combinatory logic CL with parallelif rules. 1 Introduction The ChurchRosser (CR) property is one of the most important properties for term rewriting systems (TRSs). When a TRS is nonterminating, a wellknown condition for CR is Rosen's theorem, which states that a leftlinear weakly nonoverlapping TRS is CR  or, simply, that a leftlinear nonoverlapping TRS is CR[11, 13]. A pair of reduction rules is said to be overlapping if their applications interfere with each other (i.e., they are unified at some nonvariable position), and a TRS is said to be nonoverlapping if none of ...
A Complex Example of a Simplifying Rewrite System
"... . For a string rewriting system, it is known that termination by a simplification ordering implies multiple recursive complexity. This theoretical upper bound is, however, far from having been reached by known examples of rewrite systems. All known methods used to establish termination by simplific ..."
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Cited by 2 (1 self)
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. For a string rewriting system, it is known that termination by a simplification ordering implies multiple recursive complexity. This theoretical upper bound is, however, far from having been reached by known examples of rewrite systems. All known methods used to establish termination by simplification yield a primitive recursive bound. Furthermore, the study of the order types of simplification orderings suggests that the recursive path ordering is, in a broad sense, a maximal simplification ordering. This would imply that simplifying string rewrite systems cannot go beyond primitive recursion. Contradicting this intuition, we construct here a simplifying string rewriting system whose complexity is not primitive recursive. This leads to a new lower bound for the complexity of simplifying string rewriting systems. Introduction Rewriting systems serve as a model of computation in many fields of theoretical computer science, for instance automated theorem proving, algebraic specificat...
The Hydra battle revisited
 In: Rewriting, Computation and Proof; Essays Dedicated to JeanPierre Jouannaud on the Occasion of his 60th Birthday. Lecture Notes in Computer Science
, 2007
"... To JeanPierre on this momentous occasion. Abstract. Showing termination of the Battle of Hercules and Hydra is a challenge. We present the battle both as a rewrite system and as an arithmetic while program, provide proofs of their termination, and recall why their termination cannot be proved withi ..."
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To JeanPierre on this momentous occasion. Abstract. Showing termination of the Battle of Hercules and Hydra is a challenge. We present the battle both as a rewrite system and as an arithmetic while program, provide proofs of their termination, and recall why their termination cannot be proved within Peano arithmetic. As a second labour he ordered him to kill the Lernaean hydra. That creature, bred in the swamp of Lerna, used to go forth into the plain and ravage both the cattle and the country. Now the hydra had a huge body, with nine heads, eight mortal, but the middle one immortal.... By pelting it with ery shafts he forced it to come out, and in the act of doing so he seized and held it fast. But the hydra wound itself about one of his feet and clung to him. Nor could he eect anything by smashing its heads with his club, for as fast as one head was smashed there grew up two. { Pausanias, Description of Greece, 2.37.4 1