Strahm, Thomas

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REVIEWS 435(Relativizations of the P =?NP question, JSL LI 1061), which state that there are oraclesα, 	 relativizing P and NP such that Pα = NPα and P	 = NP	 , cannot be applied to hissetting.I have one general concern: If we consider a certain entity of Go¨del numbers T (anykind of Go¨delization) of closed terms in the language of BA and another entity Tx ofGo¨del numbers of terms in one ﬁxed variable x, and if we further consider an evaluationfunction val : T → N and a substitution function sub : Tx × N → T , then faithfulnessof the arithmetization implies that all polynomial growth rates can be represented in thesystem (T, Tx, val, sub). For provingGo¨del’s diagonalization lemmawe need the substitutionfunction sub to have a polynomial growth rate, and for showing the correctness of derivationsin BA by the evaluation strategy we need the evaluation function val to have a polynomialgrowth rate. But this is impossible, because then f(a, b) := val(sub(a, b)) has polynomialgrowth rate, i.e., |f(a, b)|  |max(a, b)|c for some c ∈ N. By faithfulness there is sometc+1 ∈ Tx such that |m|c+1  |val(sub(tc+1, m))|  |max(tc+1, m)|c = |m|c for m bigenough, a contradiction. I think this indicates that in order to obtain a separation via Go¨delsentences we need further ideas that diﬀer strongly from the evaluation strategy.The ﬁrst reviewed paper summarizes all necessary notions and results on strictly i-normalproofs and partial truth deﬁnitions for them. Let Prf i(w, ϕ) denote “w is the Go¨delnumber of a strictly i-normal proof of ϕ,” and let |x|k denote the k-times iterated “integer-logarithm” | · | applied to x. Takeuti shows that BA can prove weak consistency statementslike ∀w¬Prf i(|w|, →) (the empty sequent “→” representing contradiction) and weakforms of reﬂection like ∃w Prf i(|w|2, ∀x ϕ(x)) → ∀x ϕ(|x|) for simple ϕ. Of course,Go¨del’s second incompleteness theorem can only be proved for the “usual” consistencystatement Si2  ∀w¬Prf i (w, →). Takeuti deﬁnes Go¨del sentences ϕik satisfying S12 ϕik ↔ ∀x¬Prf i (|x|k, ϕik). They have similar properties to the above-described variationsof consistency. By Go¨del’s diagonalization lemma, Takeuti obtains Si2  ϕik for all k, but onthe other hand Si+12  ∀x¬Prf i(|x|k+1, ϕik) for k  2. He formulates two conjectures inthe above-described form that would imply P = NP.The second reviewed paper extends this line of research and contrasts it with Go¨delsentences based on the usual (unrestricted) derivability. That is, let PRFi (w, ϕ) denotethe formalized notion of “w is the Go¨del number of an (unrestricted) Si2-proof of ϕ,” anddeﬁne Go¨del sentences Φik satisfying S12  Φik ↔ ∀x¬PRFi(|x|k, Φik). Takeuti comparespairs (PRFi ,Φik) with (Prfi , ϕik), obtaining results like IΔ0 +exp  ∀x¬PRFi (|x|k+1, Φik)for all k, in contrast to the above-obtained results for (Prf i , ϕik). He further establishesrelationships between diﬀerent Go¨del sentences, and between Go¨del sentences and weakconsistency statements.The reviewed papers are top-level research papers in the ﬁeld of applications of Go¨delsentences to the separation problem of BA. This implies that for the study of them one needssome background in BA (at least parts of the Buss book). Like all papers of Takeuti thatI have read, I found the ones under review inspiring, showing new ideas to the separationproblem of BA.Arnold BeckmannInstitut fu¨r Algebra und Computermathematik, Technische Universita¨t Wien, WiednerHauptstraße 8–10, A-1040 Vienna, Austria. beckmann@logic.at.Andreas Weiermann. How is it that infinitary methods can be applied to finitary math-ematics? Go¨del’s T : a case study. The journal of symbolic logic, vol. 63 (1998), pp.1348–1370.The paper under review is concerned with Go¨del’s system T of primitive recursive func-tionals of ﬁnite type, which is suﬃcient for providing a functional interpretation of ﬁrst-orderPeano arithmetic PA. The main technical achievement of the paper is a novel proof of thehttps://doi.org/10.2178/bsl/1182353903Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 15:48:00, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms.436 REVIEWSstrong normalization of T by deﬁning a new assignment [ ]0 : T →  which decreases underone-step reduction 1 of terms in T , i.e., t 1 t′ entails [t′]0 < [t]0. Moreover, [ ]0 is anε0-recursive function. As a byproduct, optimal bounds for the natural fragments Tn of T areobtained.As the title of the paper suggests, this new analysis of Go¨del’s T has to be seen as aspeciﬁc case study of a more general program, which has its roots in inﬁnitary proof theory.There the general concept of miniaturization and concretion of certain (large) cardinalshas long been central in the development of ordinal notation systems. Further, as hasbeen shown by Weiermann, it is often possible to miniaturize or project down the ordinalanalysis of a (strong) system in order to obtain a perspicuous proof-theoretic analysis of acorresponding weaker system. For example, he has shown in How to characterize provablytotal functions by local predicativity (The journal of symbolic logic, vol. 61 (1996), pp. 52–69)how to pin down the standard local predicativity treatment (due to Pohlers) of the theoryof one inductive deﬁnition ID1 to a technically smooth analysis of Peano arithmetic PA. Inthis miniaturization, the collapsing function D : εΩ+1 → Ω is replaced by the miniaturizedcollapsing function  : ε0 → . The treatment of PA thus obtained indeed also producesoptimal bounds for the fragments IΣn+1 of PA.The treatment of Go¨del’s T in the paper under review can be seen as a miniaturizationof Howard’s analysis of bar recursion of type zero and is inspired by the local predicativityapproach to pure proof theory. In particular, the deﬁnition of the assignment [ ]0 mentionedabove makes crucial use of the miniaturized collapsing function . Whereas previous treat-ments of the subsystems Tn of T (in which the recursors have type level less than or equal ton + 2) used the ordinal bound n+3 (e.g. in Weiermann’s paper A proof of strongly uniformtermination for Go¨del’s T by methods from local predicativity, Archive for mathematical logic,vol. 36 (1997), pp. 445–460—aprecursor of the article under review), the present paper yieldsthe optimal strong normalization bound n+2 for Tn .This is a very important paper both from the technical as well as from the conceptualpoint of view. In a long and sparkling introduction, the author outlines his vision of howto apply a certain kind of inﬁnitary methods to questions of ﬁnitary mathematics, and hediscusses exciting connections of his results with term rewriting theory, hierarchy theory,and computational complexity. The paper is largely self-contained and, due to its extensivemotivation and conceptual discussion, it should be accessible to a wide readership.Thomas StrahmInstitut fu¨r Informatik und angewandte Mathematik, Universita¨t Bern, Neubru¨ckstrasse10, CH-3012 Bern, Switzerland. strahm@iam.unibe.ch.Ulrich Kohlenbach. Relative constructivity. The journal of symbolic logic, vol. 63(1998), pp. 1218–1238.In this article, the author continues his investigations of the rate of growth of functionsdeﬁnable in certain subsystems of analysis in ﬁnite types. In previous papers (e.g.Mathemat-ically strong subsystems of analysis with low rate of growth of provably recursive functionals,BSL VII 280) the classical systems GnA were analyzed and proof-theoretic tools such asmonotone functional interpretation (introduced by the author) were used. In the paper underreview, Kohlenbach studies the intuitionistic versions GnAi of these systems and analyzesthem via a new monotone realizability interpretation. He shows that the addition of certainstrong non-constructive and even classically refutable analytical and logical principles hasno impact on the rate of growth of deﬁnable functions.The main result of the paper is summarized in the following.Main theorem. Let E-GnAi be the system GnAi plus extensionality axioms in all ﬁnitetypes, the full axiom of choice AC in all ﬁnite types, and the independence of premissprinciple IP¬ for negated formulas. Furthermore, let A consist of the following (classicallyhttps://doi.org/10.2178/bsl/1182353903Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 15:48:00, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms.

Book Review: The journal of symbolic logic: Weiermann Andreas . How is it that infinitary methods can be applied to finitary mathematics? Gödel's T: a case study. The journal of symbolic logic, vol. 63 (1998), pp. 1348-1370

Book Review: The journal of symbolic logic: Weiermann Andreas . How is it that infinitary methods can be applied to finitary mathematics? Gödel's T: a case study. The journal of symbolic logic, vol. 63 (1998), pp. 1348-1370

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