Stephen G. Simpson, Ordinal numbers and the Hilbert Basis Theorem, Journal of Symbolic Logic, 53(3) September 1988, 961-974. 2, 2  Home/Search   Document Not in Database   Summary   Related Articles   Check

....eventually solve it. It turns out that (at least) two constructive (predicative) proofs of a constructive version of Higman s lemma have been given independently by Richman and Stolzenberg [45] and Murthy and Russell [35] Steve Simpson has proven a related result for the Hilbert s basis theorem [49], and his proof technique seems related to some of the techniques of Richman and Stolzenberg. The signi cance of having a constructive proof is that one gets an algorithm which, given a constructively (and nitely presented) in nite sequence, yields the lefmost pair of embedded strings. Murthy ....

Simpson, S.G. Ordinal numbers and the Hilbert basis theorem. Journal of Symbolic Logic 53 (1988), 961-964.

....p(X; Y ) and p(X; 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 ....

S. G. Simpson. Ordinal numbers and the Hilbert basis theorem. Journal of Symbolic Logic, 53(3):961--974, 1988.

....is to find semi classical principles necessary to prove non constructive theorems. For example, the minimal number theorem MNP and 6 0 1 LEM are equivalent in HA. 2 Simpson has shown that Hilbert basis theorem is equivalent to well orderedness of in a framework of reverse mathematics [19]. In the formal systems of reverse mathematics, LEM are free for use, but the comprehension and the induction are restricted. In our setting, the former is restricted and the latter are free for use. What is the implication of this difference Is there any general framework in which we can ....

Simpson, S. G.: Ordinal numbers and the Hilbert basis theorem, Journal of Symbolic Logic, 53 (1988) 961--974

....k is of the form 8X 1 9X 2 ; 9X k and is arithmetical, i.e. contains no quanti cation over set variables. 7 Hilbert s nite basis theorem is not provable in RCA 0 , but for each n 2 IN , the indeterminate in the polynomial ring R[X 1 ; X n ] RCA 0 proves Hilbert s theorem. [Si88] Obviously, the decidability proof of R and of any other related logical system relying on the same decision procedure where termination is insured by one of the properties equivalent to Kripke s lemma is not provable in PRA. And a fortiori, as conjectured by S. Kripke, it is not provable in ....

SIMPSON, S.G., \Ordinal numbers and the Hilbert basis theorem", J. of Symbolic Logic, 53, No. 3 (1988) 961-974.

....tree Bad (A) Note that well orderedness of jT j is equivalent to well foundedness of T . Hence well orderedness of the tree jBad(A)j is equivalent to well partial orderedness of A. It is difficult, however, to calculate the ordinal jBad(A)j concretely. The main techniques are the reification [36] and the linearization, where the reification gives upper bounds and the linearization lower bounds. 2.8 definition Let A be a partial order and ff an ordinal. A reification of A by ff is a map r : Bad (A) ff 1 satisfying oe ae ) r(oe) r( Notation: oe ae denotes that the sequence oe ....

....and basic properties of Bachmann hierarchy. h (ff; i denotes the family of normal functions in the hierarchy. We use also the family h (ff; i modified so that ( becomes one to one. We use Phi for the natural sum of ordinals and Omega for the natural product. For the definition, see [36]. Therein we can find also the definition of additively indecomposable ordinals and multiplicatively indecomposable ordinals. 3 Discontinuity of Higman Embeddings In this section we give an analysis of Higman embedding from the calculation of order types. The results in this section will be ....

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S. G. Simpson, Ordinal numbers and the Hilbert basis theorem, J. Symbolic Logic 53 (1988) 961--974.

....well founded and equipped with a well founded induction scheme. 2. The WQO on 6 is decidable. Classically, the first assumption is obvious based on the WQO property of , but constructively it is not. The WQO that satisfies the assumptions above is called a constructive well quasi order (CWQO) [20]. We will briefly review the techniques used in [11] We will refer to an empty word as ffl and a set of words which contains w (i.e. fx 2 6 3 j w xg) as w ffi . As a convention, we will refer to the symbols in 6 as a; b; c; 1 1 1, the words in 6 3 as u; v; w; 1 1 1, the finite sequences ....

....3 Let b 2 6, and let A = a1 ; a2 ; 1 1 1 ; a k be a bad sequence in 6. The constant expression (b 0 A) is fx 2 6 j b x a i 6 x for each i kg; and the starred expression (6 0A) 3 is fw = c1 c2 1 1 1 cn 2 6 3 j c i 6 a j for each i n; j kg: 2 Similar idea to [16] is also found in [20]. The concatenation of A and a 2 6 is Aja. A sequential regular expression (sequential r.e. oe is a (possibly empty) concatenation of either constant or starred expressions. The size size(oe) of oe is the number of the concatenation. For a finite set 2 of sequential expressions, we define L(2) ....

S.G. Simpson. Ordinal numbers and the hilbert basis theorem. Journal of Symbolic Logic, 53(3):961--974, 1988.

....) and p(X; Y ) Theta 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 ....

S. G. Simpson. Ordinal numbers and the Hilbert basis theorem. Journal of Symbolic Logic, 53(3):961--974, 1988.

....from 2 Theta 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 ....

S. G. Simpson. Ordinal numbers and the Hilbert basis theorem. Journal of Symbolic Logic, 53(3):961--974, 1988.

....a diagonalization argument which yields a simple and very elegant proof of Higman s lemma. Jullien [Jul68] sketched an original proof and Haines [Hai69] unaware of all the previous proofs, rediscovered the result with yet a different proof. In spite of not mentioning Higman s lemma, Simpson [Sim88] presented a constructive proof of a version of the lemma in terms of ordinals. This proof is a translation to English of the proof of Schutte and Simpson [SS85] which would then be the first constructive proof of Higman s lemma. Independently of Simpson s results, Stolzenberg noticed that as a ....

....( Jul68] Hai69] Let = be the equality relation on a finite alphabet. Then = is a wqo. By the previous observation that = on finite sets is a wqo, Theorem 2.3 is an easy consequence of Theorem 2.2. 2. 3 Constructive formulations Among the first constructive proofs of Higman s lemma ( SS85] [Sim88], RS93] Mur90] and [MR90] probably the clearest formulation is the one in [RS93] THEOREM 2.4 ( RS93] For any decidable relation S, if S is a wqo so is S . With the constructive proofs, a new question arises concerning the formulation of the lemma. Theorem 2.4 requires S to be ....

S. Simpson. Ordinal Numbers and the Hilbert Basis Theorem. The Journal of Symbolic Logic, 53:961--974, 1988.

....a diagonalization argument which yields a simple and very elegant proof of Higman s lemma. Jullien [Jul68] sketched an original proof and Haines [Hai69] unaware of all the previous proofs, rediscovered the result with yet a different proof. In spite of not mentioning Higman s lemma, Simpson [Sim88] presented a constructive proof of a version of the lemma in terms of ordinals. This proof is a translation to English of the proof of Schutte and Simpson [SS85] which would then be the first constructive proof of Higman s lemma. Independently of Simpson s results, Stolzenberg noticed that as a ....

....( Jul68] Hai69] Let = be the equality relation on a finite alphabet. Then = is a wqo. By the previous observation that = on finite sets is a wqo, Theorem 3 is an easy consequence of Theorem 2. 2. 3 Constructive formulations Among the first constructive proofs of Higman s lemma ( SS85] [Sim88], RS93] Mur90] and [MR90] probably the clearest formulation is the one in [RS93] Theorem4 ( RS93] For any decidable relation S, if S is a wqo so is S . With the constructive proofs, a new question arises concerning the formulation of the lemma. Theorem 4 requires S to be decidable. Is ....

S. Simpson. Ordinal numbers and the hilbert basis theorem. The Journal of Symbolic Logic, 53:961--974, 1988.

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Stephen G. Simpson, Ordinal numbers and the Hilbert Basis Theorem, Journal of Symbolic Logic, 53(3) September 1988, 961-974. 2, 2

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