J. Gallier, What's so Special about Kruskal's Theorem and the Ordinal \Gamma 0 ?, A Survey of Some Results in Proof Theory, Annals of Pure and Applied Logic, Vol. 53, 199--260 (1991).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the strong normalization property) of R LKsp can be machine checked. 3.3. Interpretation of LK and LK sp proofs There are in the literature several proofterm calculi which represent sequent calculi such that each proof can be recovered from the term which represents it, see for instance in [Gal91], Pfe94] Pin93] or [Tah92] These proof term calculi are useful in implementing sequent calculi in logical frameworks and in logic programming (cf. Pfe94] Pin93] but actually, when we want to deal with termination properties it is sufficient to deal with proof name transformations as is ....

....well partial ordered signature; the Kruskal order (T (F) K ) induced by (F ; on ground terms over F is a well partial order. Finite versions of this theorem turn out not to be provable in arithmetic theory and even in more powerful mathematical theories, for a presentation on this matter see [Gal91]; Kruskal s original proof is not constructive; for a constructive proof see [Wei94] As a consequence of Kruskal s tree theorem the Kruskal order on ground terms over F admits a total well founded extension defined by Kamin and Levy : Definition 4.3 (Lexicographic path order) Let (F ; be an ....

J. Gallier, What's so Special about Kruskal's Theorem and the Ordinal \Gamma 0 ?, A Survey of Some Results in Proof Theory, Annals of Pure and Applied Logic, Vol. 53, 199--260 (1991).

.... of see the extensive work of Robertson and Seymour, e.g. 38, 39, 40] On the other hand, the claim that for downward closed subclasses the theory of polynomial time complexity is trivial on wqo classes should be taken with a grain of salt since the argument just given is highly non e#ective [17, 18]. In practice this proposition is used in conjunction with well quasi ordering theorems of Kruskal s type, and from such results one gets no way to extract e#ective bounds on the size of the set # occurring in the proof, or the degree of the corresponding polynomial, and thus one has very little ....

J. Gallier. What's so special about Kruskal's Theorem and the ordinal # 0 ? A survey of some results in proof theory. Annals of Pure and Applied Logic 53 (1991), 199-260.

....As we saw in examples 3.16, 3.17, in general po will not be well founded, but we can impose conditions both on the quasi order and on the status in order to obtain well founded quasi orders. A generalized way of proving well foundedness of terms orders is through Kruskal s theorem ([17, 19, 11]) Roughly this theorem implies that any simplification ordering (an order closed under substitutions and contexts and satisfying the subterm property) is wellfounded; clearly it cannot be applied to non simplification orderings, so Kruskal s theorem cannot help us prove well foundedness of our ....

Gallier, J. H. What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory. Annals of Pure and Applied Logic 53 (1991), 199--260.

....is compatible with some monotonic well founded order. An order on T (F) is said to have the subterm property if F ( t; t for all F 2 F and t 2 T (F) A monotonic order satisfying the subterm property is called a simplification order. A direct consequence of Kruskal s theorem ([10, 7]) is that any simplification order over a finite signature is well founded. A TRS over a finite signature is called simply terminating if it is compatible with a simplification order. In [13] it is described how simplification orders extend to infinite signatures. A TRS is called ....

Gallier, J. What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory. Annals of Pure and Applied Logic 53 (1991), 199--260.

....Y ) Theta p(X; X) Proposition 1. The relation Theta is a wqo on the set of expressions over a finite For a complete proof, reusing Higman s and Kruskal s results [20, 28] in a very straightforward manner, see, e.g. 33] For constructive proofs of Higman s Lemma [20] see [60, 53] See also [13] and [57] Another, non constructive one can be found in [54] To ensure, e.g. local termination of partial deduction, we have to ensure that the constructed SLDNF trees are such that the selected atoms do not embed any of their ancestors (when using a well founded order as in Example 1, we had ....

J. H. Gallier. What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory. Technical report, University of Pennsylvania, October 1993.

....regard the class of algebras as a system of ordinal notations. This system has some new features in comparison with the traditional systems. Notably every notation has a meaning in our system. For example, consider the ordinal Gamma 0 , which has a meaning as the least strongly critical ordinal [16]. But what has a meaning is the ordinal itself, not the notation assigned to it. So the notations assigned to Gamma 0 differ in one system to others. In Bachmann hierarchy it is denoted by ( Omega ; 0) in Buchholz notation [3] 0 Omega Omega , etc. In our system, it is denoted by X: 2X ....

.... Vazsonyi s conjecture was well partial orderedness for non ordered trees, but Kruskal indeed proved the theorem for ordered trees and derived Vazsonyi s conjecture from that [22] For a simpler proof by Nash Williams using the socalled minimal bad sequence argument, we refer the reader to [27, 16, 35]. The proof is worth comment in two respects; it uses a non constructive argument in an essential way, and also an impredicative argument. In fact, to formalize the proof one needs a fragment of second order arithmetic having Pi 1 1 comprehension axiom or its substitute, e.g. bar induction on ....

J. H. Gallier, What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory, Ann. Pure Appl Logic 53 (1991) 199--260.

....subterm property is called a simplification order. The transitive closure of the rewrite relation of the rewrite rules F (x 1 ; x 2 ; xn ) x i for all F 2 F and all i = 1; 2; n, where n is the arity of F , is called homeomorphic embedding. Kruskal s theorem (Kruskal (1960) Gallier (1991)) states that any order extending homeomorphic embedding is well founded. A direct consequence is that any simplification order over a finite signature is well founded. A TRS over a finite signature is called simply terminating if it is compatible with a simplification order. In Middeldorp and ....

J. Gallier. What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory. Annals of Pure and Applied Logic, 53:199--260, 1991.

....paper we remove the monotonicity condition and replace it by some decomposability condition. For orderings satisfying the subterm property and this decomposability condition we prove well foundedness in a way that is inspired by Nash Williams proof of Kruskal s theorem ( 10] as it appears in [6]) but which is much simpler. A similar technique, for a particular order, has already been used by Kamin and L evy ( 9] Standard orderings like recursive path order ( 1, 12] and semantic path order (spo) 9, 2] trivially satisfy our conditions, yielding a simple proof of well foundedness for ....

Gallier, J. H. What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory. Annals of Pure and Applied Logic 53 (1991), 199--260.

....mentioned and proved in Section 10. Some years before we had exchanged views on possible intuitionistic versions of Ramsey s Theorem, see Veldman and Bezem 1993, and Coquand 1994. Kruskal s Theorem is of course a Ramseyan Theorem, so it was natural that we should study its constructive content. J.H. Gallier, in his survey paper Gallier 1991 mentions the nding of a constructive 51 proof of Kruskal s Theorem as a major problem. Various people were searching constructive proofs of Ramseyan theorems, see for instance Murthy and Russell, and Richman and Stolzenberg 1993. In the latter paper the Finite ....

J.H. Gallier (1991), What's so special about Kruskal's theorem and the ordinal 0 ? A survey of some results in proof theory. Annals of Pure and Applied Logic 53, pp. 199-260.

....a fact that is an obvious consequence of Lemma 5.6. 2 The proof we have provided for the fact that is well founded is a direct one and has the virtue of giving us specific insight into the nature of this relation. However, an alternative proof can be provided by invoking Kruskal s tree theorem [Gal91, Kru60] thereby exhibiting relationships between and the notions of simplification orderings [Der82] and Kamin and L evy s extended recursive path orderings (described, for example, in [Hue86] Towards this end, we note that expressions of the form [ t; ol; nl; e] ffe 1 ; nl; ol; e 2 gg and ....

Jean H. Gallier. What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory. Annals of Pure and Applied Logic, 53:199--260, 1991.

....in [Ros82] originates from Nash Williams (see [NW63] Lemma 3.6 (Theorem 10.23 in [Ros82] If (T; is a wqo, then so is (T ; Next, we will prove Kruskal s Theorem (see [Kru60] in the setting of term rewriting. We will follow the proof given by Nash Williams in [NW63] see also [DJ90, Der79, Gal91]) To this end, we first define the following TRSs. Definition 3.7 Let F be a signature, V be a countable set of variables, and let (F [ V; be a quasi ordering. We define three (possibly infinite) TRSs (F ; F emb ) F ; F arg ) F ; F op ) by F arg = ff(x 1 ; x m ) x i j f ....

J. Gallier. What's so Special about Kruskal's Theorem and the Ordinal \Gamma 0 ? A Survey of Some Results in Proof Theory. Annals of Pure and Applied Logic 53, pages 199--260, 1991.

....that criterion. Proposition 3.6 Let be a binary relation on T (F ; V) which is simplifying. If for every reduction sequence the set of all function symbols occurring in the (terms of that) reduction sequence is finite, then is terminating. Proof: This follows from Kruskal s Tree Theorem, see [Gal91, Ohl92a]. 2 Definition 3.7 Let (F ; R) be a TRS. Set F 0 = l r2R (Fun(r) n Fun(l) i.e. F 0 consists of all those function symbols which occur at the right hand side r but not at the left hand side l of some rule l r 2 R. We say that (F ; R) introduces only finitely many function symbols if ....

J. Gallier. What's so Special about Kruskal's Theorem and the Ordinal \Gamma 0 ? A Survey of Some Results in Proof Theory. Annals of Pure and Applied Logic 53, pages 199--260, 1991.

....p(X; X) Proposition 2.5 The relation Theta is a wqo on the set of expressions over a finite alphabet. For a complete proof, reusing Higman s and Kruskal s results [27, 36] in a very straightforward manner, see, e.g. 41] For constructive proofs of Higman s Lemma [27] see [70, 61] See also [18] and [67] Another, non constructive one can be found in [62] In the presence of an infinite alphabet Theta is not a wqo, as the following sequence shows: f(0) f(1) f(2) we have f(i) 6 Thetaf (j) for i 6= j) To ensure, e.g. local termination of partial deduction, we have to ensure ....

J. H. Gallier. What's so special about Kruskal's theorem and the ordinal \Gamma 0? A survey of some results in proof theory. Technical report, University of Pennsylvania, October 1993.

....ordered trees correspond to much larger ordinals in that hierarchy. In particular, some orderings based on Kruskal s Tree Theorem [ Kruskal, 1960 ] correspond to the first impredicative ordinal, Gamma 0 , and even to larger ones [ Friedman, 19 ; Simpson, 1985; Smory nski, 1986; Dershowitz, 1987; Gallier, 1991 ] The significance of Gamma 0 for computer science is discussed in [ Gallier, 1991 ] 5 Conclusions It has been argued [ Gries, 1979 ] that the natural numbers suffice for termination proofs, since the (maximum) number of iterations of any terminating deterministic (or bounded ....

.... orderings based on Kruskal s Tree Theorem [ Kruskal, 1960 ] correspond to the first impredicative ordinal, Gamma 0 , and even to larger ones [ Friedman, 19 ; Simpson, 1985; Smory nski, 1986; Dershowitz, 1987; Gallier, 1991 ] The significance of Gamma 0 for computer science is discussed in [ Gallier, 1991 ] 5 Conclusions It has been argued [ Gries, 1979 ] that the natural numbers suffice for termination proofs, since the (maximum) number of iterations of any terminating deterministic (or bounded nondeterministic) program loop is fixed, depending only on the values of the variables and inputs ....

J. Gallier. What's so special about Kruskal's Theorem and the ordinal \Gamma 0 . A survey of some results in proof theory. Annals of Pure and Applied Logic, 53(3), September 1991.

....of function symbols with lexicographic status, there is a natural number bounding the arities of the function symbols in the class. That is 8f 2 F : f) lex ) 9n 0 : 8g 2 hfi : arity(g) n) 3) Before proving well foundedness of rpo , we need some additional definitions and results from [7]. Definition 3.4 A quasi order over a set S is a well quasi order, abbreviated to wqo, iff every quasi order extending it (including itself ) is well founded. There are several equivalent characterizations of wqo s. We also use the following (see [7] Every infinite sequence (s i ) i0 of ....

....some additional definitions and results from [7] Definition 3.4 A quasi order over a set S is a well quasi order, abbreviated to wqo, iff every quasi order extending it (including itself ) is well founded. There are several equivalent characterizations of wqo s. We also use the following (see [7]) Every infinite sequence (s i ) i0 of elements of S contains some infinite subsequence (s OE(i) i0 such that s OE(i 1) s OE(i) for all i 0 . A traditional way of proving well foundedness of rpo is via Kruskal s theorem. Given our extended definition of rpo , we cannot apply Kruskal s ....

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Gallier, J. H. What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory. Annals of Pure and Applied Logic 53 (1991), 199--260.

.... starting from fairly basic systems, and building more and more powerful systems by induction on ordinals [Fef75, Fef68b, Fef64] or building the progression inside the system itself [Fef68a, Fef78] Applications have led to the identification of Gamma 0 as the least impredicative ordinal [Gal91], or to the discovery of weak subsystems of second order Peano arithmetic which are still strong enough to formalize most of mathematics [Sim85] Returning to what is indended nowadays as ramified higher order logic, Church [Chu76] presents a system of ramified second order logic that enables him ....

....notations. It must allow us to compute the sum of an ordinal and an integer, to compare by = or any two ordinals in polynomial time. If we use only integers as levels, say in binary, this is trivial (the size of is then j j = max(1; dlog 2 e) This is also certainly possible up to Gamma 0 [Gal91], by using Schutte s function and natural ordinal sums: this yields ordinal notations where = is just structural comparison and is the lexicographic path ordering. The size of an ordinal notation, there, is the number of signs needed to write it on paper. The essence of our algorithms will be ....

Jean Gallier. What's so special about Kruskal's Theorem and the ordinal \Gamma 0 . A survey of some results in proof theory. Annals of Pure and Applied Logic, 53(3):199-- 260, September 1991.

....n = 1 and fi 1 = OE fl (ffi) ff fl) 1 This gives all (the predicative) ordinals up to Gamma 0 , the first ordinal whose definition requires things infinite. One can also consider ordered trees with ordinary nodes, treating its leftmost subtree as the root in the supertree ordering. See Gallier [ 1991 ] for an exposition on properties of these ordinals. 1 This patches the order preserving mapping given in [Dershowitz, 1987] An embedding of trees into Gamma 0 is given by [Gallier, 1991] and others. It avoids supernodes, but ignores all subtrees but the two largest. Ordinal 0 1 2 n 1 ....

....ordered trees with ordinary nodes, treating its leftmost subtree as the root in the supertree ordering. See Gallier [ 1991 ] for an exposition on properties of these ordinals. 1 This patches the order preserving mapping given in [Dershowitz, 1987] An embedding of trees into Gamma 0 is given by [Gallier, 1991] and others. It avoids supernodes, but ignores all subtrees but the two largest. Ordinal 0 1 2 n 1 2 Ordered tree ffl ffl ffl . ffl ffl . ....

Jean Gallier. What's so special about Kruskal's Theorem and the ordinal \Gamma 0 . A survey of some results in proof theory. Annals of Pure and Applied Logic, 53(3):199--260, September 1991.

....well founded. As we saw in examples 3.16, 3.17, in general po will not be well founded, but we can impose conditions both on the quasi order and on the status in order to obtain well founded quasi orders. A generalized way of proving well foundedness of terms orders is through Kruskal s theorem ([17, 19, 11]) Roughly this theorem implies that any simplification ordering (an order closed under substitutions and contexts and satisfying the subterm property) is wellfounded; clearly it cannot be applied to non simplification orderings, so Kruskal s theorem cannot help us prove well foundedness of our ....

Gallier, J. H. What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory. Annals of Pure and Applied Logic 53 (1991), 199--260.

.... A TRS (F ; R) is simply terminating if it is compatible with a simplification order on T (F ; V) Since we are only interested in signatures consisting of function symbols with fixed arity, we have no need for the deletion property (cf. 4] It should also be noted that many authors (e.g. [3, 4, 5, 14, 19, 39]) do not require that simplification orders are closed under substitutions. Since we don t really want to check whether a simplification order orients all instances of rewrite rules from left to right in order to conclude termination, and concrete simplification orders like the recursive path ....

....the strict part of mul by j mul in order to avoid confusion with the multiset extension mul of the strict part of , which is a smaller relation. The above definition of multiset extension of a preorder can be shown to be equivalent to the more operational ones in Dershowitz [5] and Gallier [14], but since we define the multiset extension of a preorder in terms of the well known multiset extension of a partial order, we get al..l desired properties basically for free. In particular, using the fact that multiset extension preserves wellfounded partial orders, it is very easy to show that ....

J. Gallier, What's so Special about Kruskal's Theorem and the Ordinal \Gamma 0 ? A Survey of Some Results in Proof Theory, Annals of Pure and Applied Logic 53 (1991) 199--260.

....operations (abstraction and application) Second, the application of AC can be avoided by explicitly constructing a choice function for L . Some results about the mathematical means required to prove the general form of Kruskal s Theorem for finite trees are presented by Simpson (1985) and Gallier (1991). 1.8 Applications Definition 14 Let R be a notion of reduction on L . M R N is an R cycle if M j N , and an R self embedding if M vL N . The following shows that the second observation in the introduction is generally correct. Corollary 12 If M 0 R M 1 R . then M 0 R M i ....

Gallier, J. 1991. What's so Special about Kruskal's Theorem and the Ordinal \Gamma 0? Annals of Pure and Applied Logic 31:199--260.

....than the primitive recursive functionals of T, but there is no reason to stop at ffl 0 . Indeed, since we may define the Veblen hierarchy of functions ff : ord ord for all ff: ord , we have a system of notation for all the ordinals less than Gamma 0 , the first strongly critical ordinal (see [Gal91] for a very readable discussion of the significance of Gamma 0 ) The particular Veblen hierarchy to which we refer is that starting from 0 (fi) fi ; then the function ff for ff 0 enumerates the common fixed points of all the functions fl for fl ff. For example, 1 enumerates ....

J.H. Gallier. What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory. Annals of Pure and Applied Logic, 53(3):199--260, 1991.

....nM B is the multiset difference of A and B with respect to , i.e. A nM B = A if 6 9x 2 A; y 2 B; x y and A nM B = A n fxg) nM (B n fyg) if x 2 A; y 2 B and x y. 2.1.2. The Homeomorphic Embedding Relation In this section we define the homeomorphic embedding relation for variable arity terms (Gallier, 1991), and #(u; v) the number of times the term u embeds in the term v. Definition 2.1. We define emb , the varyadic homeomorphic embedding relation, on V (F [ X ) as follows: t = g(t 1 ; t n ) emb f(s 1 ; s m ) s if and only if either: 1 there is an i, 1 i n such that t i emb s, ....

Gallier, J. H. (1991). What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory. Annals of Pure and Applied Logic, 53:199--260.

....3 Theta 5 by striking out the 3. Proposition 2.8 The relation Theta is a wqo on the set of expressions over a finite alphabet. For a complete proof, reusing Higman s Lemma [15, 19] in a straightforward manner, see e.g. 24] For constructive proofs of Higman s Lemma [15] see [42, 36] See also [12] and [40] Another, non constructive one can be found in [37] To ensure e.g. local termination of partial deduction, we have to ensure that the constructed SLDNF trees are such that the selected atoms do not embed any of their ancestors (when using a well founded order as in Example 1.3, we had ....

J. H. Gallier. What's so special about Kruskal's theorem and the ordinal \Gamma 0 ? A survey of some results in proof theory. Technical report, University of Pennsylvania, October 1993.

....called FFF, of Kruskal s classical theorem, dealing in its original form with infinite sequences of finite trees) Friedman s construction is elementary and predicative. However, the proof of FFF demonstrably requires mathematical induction up to the first impredicative denumerable ordinal G 0 [Gallier 1991]. Those and similar phenomena [Simpson 1987] confirm (or, if one prefers, make axiomatic sense of) our Theses 2 and 3, opening the way for their formalization, as follows : Thesis 4.In many interesting cases, the termination can be proved only in the axiomatic domain which vastly, and ....

Jean H. Gallier [1991]: What's so Special about Kruskal's Theorem and the Ordinal G0 ? A Survey of Some Results in Proof Theory, Ann. Pure Appl. Logic 53, 199-260.

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Gal91 J. Gallier. What's so Special about Kruskal's Theorem and the Ordinal \Gamma 0 ?, A Survey of Some Results in Proof Theory. Annals of Pure and Applied Logic, Volume 53, 199--260 (1991).

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