H. Schwichtenberg. Proof theory: Some applications of cut-elimination. In J. Barwise, editor, Handbook of Mathematical Logic, pages 867--895. North-Holland, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....applies to LK and LJ . Historically, this is the version of the cut elimination theorem proved by Gentzen [8] 1935) Gentzen s proof was later simpli ed by Tait [29] and Girard [13] especially the induction measure) A simpli ed version of Tait s proof is nicely presented by Schwichtenberg [26]. The proof given here combines ideas from Tait and Girard. The induction measure used is due to Tait [29] the cut rank) but the explicit transformations are adapted from Girard [13] 9] We need to de ne the cut rank of a formula, the depth of a proof, and the logical depth of a proof. De ....

H. Schwichtenberg. Proof theory: some applications of cut-elimination. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic, pages 867-895. North Holland, Amsterdam, 1977.

.... have the axioms and rules of many sorted classical predicate logic as well as symbols and de ning equations for all primitive recursive functionals of type level 2 in the sense of Kleene [7] i.e. ordinary primitive recursion uniformly in function parameters, for details see e.g. 6] II.1) or [21]) We also have a schema of quanti er free induction (w.r.t. to this extended language) and abstraction for number variables, i.e. y:t[y] x = t[x] x; y tuples of the same length. So PRA is the second order fragment of the (restricted) nite type system d PA from [3] It is clear ....

....how to represent real numbers and the basic arithmetical operations and relations on them in EA The results of this section a fortiori hold for PRA Here recursive is meant in the sense of [16] i.e. unnested. In contrast to this the notion of recursiveness as used e.g. in [2] [21] corresponds to nested recursion. The representation of IR presupposes a representation of Q: Let j be the Cantor pairing function. Rational numbers are represented as codes j(n; m) of pairs (n; m) of natural numbers n; m. j(n; m) represents the rational number ; if n is even, and the ....

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Schwichtenberg, H., Proof theory: some applications of cut-elimination. In: Barwise, J. (ed.), Handbook of Mathematical Logic, North{Holland, Amsterdam, pp. 867-895

....e.g. PA, then definition by cases will not be sufficient in general. In the case of PA for instance one needs all ff recursive functionals for ff 0 and these functionals are also sufficient. This was proved firstly in [16] using an substitution procedure based on [1] Later Schwichtenberg [25] gave a proof of this result using a form of cut elimination (due to [30] instead. The cut elimination procedure does not give a local interpretation of proofs, i.e. given proofs of A and A B, a realization of the n.c.i. of B is not computed out of given realizations for the n.c.i of A and A ....

....interpretation of A) The passage through higher types makes it necessary to use a normalization procedure for T in order to obtain the n.c.i. in terms of ff( 0 ) recursive functionals rather than type 2 functionals defined in terms of primitive recursion in higher types (see e. g [21] [25]) Instead of functional interpretation one could also use a combination of (negative translation plus) the Friedman Dragalin A translation and a suitable notion of realizability. If one uses here the so called minimal realizability of [3] one can avoid the use of higher types but the resulting ....

Schwichtenberg, H., Proof theory: some applications of cut-elimination. In: Barwise,

....the unobtainable ideal the meta theory should be expected to get. There seem to be two very different approaches to this problem: one is to approximate the complete meta theory to within (say) a particular proof theoretical bound (we have much of the work in formalised proof theory in mind see e.g. [14]) the other is to weaken the notion of correspondence between the meta theory and the object theory and give a meta theory which is complete with respect the new relationship. 3 The only requirement on naming is that if OE and are identical constants then OE and are identical formulae. ....

H. Schwichtenberg. Proof theory: Some applications of cut-elimination. In J. Barwise, editor, Handbook of Mathematical Logic. North Holland Publishing Company, 1977.

....for all inductive operator forms A(P; x) all L c 1 formulas A and all ordinals 0 : 1. rn(A(P A ; s) rn(P A ( s) 2. If for all 2 oc(A) then rn(A) The system T1 is formulated as a Tait style calculus for nite sets ( of L c 1 formulas (cf. e.g. [23]) If A is an L c 1 formula, then ; A is a shorthand for [ fAg, and similarly for expressions like ; A; B. T1 contains the following axioms and rules of inference. 11 I. Axioms. For all nite sets of L c 1 formulas, all closed number terms s and t with identical value, and all literals A in ....

Schwichtenberg, H. Proof theory: Some applications of cut-elimination. In Handbook of Mathematical Logic, J. Barwise, Ed. North Holland, Amsterdam, 1977, pp. 867-895. 21

....of T are then reduced to SRT by means of an asymmetric interpretation argument. 9 We let Gamma; range over finite sets of L 2 formulas; we often write (for example) Gamma; A for the union of Gamma and fAg. The Tait calculus T is an extension of the classical Tait calculus (cf. [14]) by the non logical axioms of FP 0 ( Pi 1 0 DC) It comprises the following axioms and rules of inference. I. Axioms. For all finite sets Gamma of L 2 formulas, all Sigma 1 1 formulas A and all Sigma 1 1 formulas B which are axioms of FP 0 : Gamma; A; A and Gamma; B: II. ....

Schwichtenberg, H. Proof theory: Some applications of cut-elimination. In Handbook of Mathematical Logic, J. Barwise, Ed. North Holland, Amsterdam,

.... (J) F I N ) to W d E Omega and d E Omega 0 , respectively, is provided by Theorem 18. In the case of the theory EET( J) F I N ) a straightforward formalization of infinitary derivations and cut elimination procedures is needed within d E Omega 0 , cf. Schwichtenberg [31] for similar arguments. The second step of our reductions consists in formalizing Theorem 20 in W d E Omega and d E Omega 0 , respectively. As far as this formalization is concerned, let us make the following general remarks: i) it is sufficient to consider structures SM (ff) for a ....

Schwichtenberg, H. Proof theory: Some applications of cut-elimination. In Handbook of Mathematical Logic, J. Barwise, Ed. North Holland, Amsterdam, 1977, pp. 867--895.

....operator forms A(P; x) all L c 1 formulas A and all ordinals ff 0 : 1. rn(A(P ff A ; s) rn(P ff A ( s) 2. If fi ff for all fi 2 oc(A) then rn(A) ff . The system T1 is formulated as a Tait style calculus for finite sets ( Gamma; of L c 1 formulas (cf. e.g. [60]) If A is an L c 1 formula, then Gamma; A is a shorthand for Gamma [ fAg, and similarly for expressions like Gamma; A; B. T1 contains the following axioms and rules of inference. I. Axioms. For all finite sets Gamma of L c 1 formulas, all closed number terms s and t with identical ....

Schwichtenberg, H. Proof theory: Some applications of cut-elimination. In Handbook of Mathematical Logic, J. Barwise, Ed. North Holland, Amsterdam, 1977, pp. 867--895.

....for an appropriate constant d independent of and fl. Proof. By Theorem 3. 1 there exists a proof fl[ of Delta Gamma with only atomic cuts and l(fl[ 2 Delta l( l(fl) 2kflk 1) But the elimination of atomic cuts is at most exponential in the length of proofs (see (Tait, 1968) (Schwichtenberg, 1977)) 2 The bound in Theorem 3.2 can be improved to 2 d Deltal( l(fl) For this purpose we have to replace the LK proofs in Definition 3.5 by LK proofs with the mix rule (see (Takeuti, 1987) In fact the mix rule makes the preparatory contractions of the multiple occurrences of the resolution ....

Schwichtenberg, H. (1977). Proof Theory: Some Applications of Cut-Elimination. Handbook of Mathematical Logic, ed. by J. Barwise, 867--895. North Holland: Amsterdam.

....following corollary: Corollary 2.1 (Subformula Property) In a cut free proof in LK all the formulas which occur in it are substitution instances of subformulas of the end sequent. 3 The two traditional methods for the elimination of cuts are Gentzen s method ( Gen34, Tak87] and Tait s method ([Tai68, Sch77]) Gentzen s algorithm reduces the topmost cut by shifting the cut upwards to the place in the proof where the cut formulas are infered on both sides. The cut is then reduced to cuts of minor logical size. Tait s algorithm reduces the largest cuts to minor cuts without regard of position and ....

....for an appropriate constant d independent of and fl. Proof: By Theorem 5. 1 there exists a proof fl[ of Delta Gamma with only atomic cuts and l(fl[ 2 Delta l( l(fl) 2kflk 1) But the elimination of atomic cuts is at most exponential in the length of proofs (see [Tai68] [Sch77]) 2 The bound in Theorem 5.2 can be improved to 2 d Deltal( l(fl) For this purpose we have to replace the LK proofs with cuts by LK proofs with the mix rule (see [Tak87] In fact the mix rule makes the preparatory contractions of the multiple occurrences of the resolution atom A ....

Schwichtenberg, H.: Proof Theory: Some Applications of CutElimination. Handbook of Mathematical Logic, ed. by J. Barwise, 867-- 895. North Holland: Amsterdam (1977).

....online at http:##www.idealibrary.com on 84 0890 5401#00 #35.00 Copyright # 2000 by Academic Press All rights of reproduction in any form reserved. 1 This work was supported by NSF Grant CCR 9303383. Many proofs of cut elimination have been given in the literature (see, for example, ML68, Sch77, Dra87, Her95] Yet, to our knowledge, none of them have been formalized even though this is clearly possible in principle (see, for example, Matthews [Mat94] pencil and paper analysis of cut elimination for the ( 6 , c) fragment of classical propositional logic in FS 0 ) They are difficult to ....

Schwichtenberg, H. (1977), Proof theory: Some applications of cut-elimination, in Handbook of Mathematical Logic," pp. 867#895, North-Holland, Amsterdam.

...., Xm ] We take as L n # formulas of T n # the L n # formulas without free number variables. We let #, #, range over finite sets of L n # formulas; we often write (for instance) #, # for the union of # and # . The Tait calculus T n # is an extension of the classical Tait calculus [16]. It contains the following axioms and rules of inference: 1. Ontological axioms I. For all finite sets # of L n # formulas of T n # , all closed number terms s, t with identical value, all true literals # of L 1 , all set 12 variables X and all # #, # # #: #, # and #, t # X, s # ....

Schwichtenberg, H. Proof theory: Some applications of cutelimination. In Handbook of Mathematical Logic, J. Barwise, Ed. NorthHolland, Amsterdam, 1977, 867 -- 895.

....: m:f(n; m) However the use of variables f 0(0) 0) is more convenient since it avoids the use of the operator in many cases. 3 functionals of type level 2 in the sense of Kleene [7] i.e. ordinary primitive recursion uniformly in function parameters, for details see e.g. 6] II.1) or [22]; we do not include higher type primitive recursion in the sense of [5] We also have a schema of quanti er free induction (w.r.t. to this extended language) and abstraction for number variables, i.e. y:t[y] x = t[x] x; y tuples of the same length. So PRA 2 essentially is the second order ....

....BW and A A are not conservative over PRA. Relative to PRA 2 ( AC 0;0 qf WKL) these principles are conservative over PRA but the principle Limsup is not. 6 Here recursive is meant in the sense of [17] i.e. unnested. In contrast to this the notion of recursiveness as used e.g. in [2] [22] corresponds to nested recursion. 6 2 Preliminaries We rst indicate how to represent real numbers and the basic arithmetical operations and relations on them in EA 2 . The results of this section a fortiori hold for PRA 2 instead of EA 2 . Our representation of IR relies on the following ....

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Schwichtenberg, H., Proof theory: some applications of cut-elimination. In: Barwise, J. (ed.), Handbook of Mathematical Logic, North{Holland, Amsterdam, pp. 867-895 (1977).

....are made up of subformulae of the conclusion. PROPOSITION 2. In a cut free proof of A, all the formulae which occur within the proof are contained in the set of subformulae of and A. A number of interesting facts can be derived from this property these have been studied by Schwichtenberg [36]. As we shall see in the next section, this property is not so straightforward for a natural deduction formulation. studia.tex; 17 12 1999; 11:09; p.6 On an Intuitionistic Modal Logic 7 4. A Natural Deduction Formulation of IS4 In a natural deduction system, originally due to Gentzen [18] but ....

H. Schwichtenberg. Proof theory: Some applications of cut-elimination. In J. Barwise, editor, Handbook of Mathematical Logic, chapter D.2, pages 867--896. North Holland, 1977.

....45 induction along the ordering of T (K) To see this, note that hT (K) i is primitive recursive (after some coding) and that recursive RS(K) derivations suffice for the results of Sections 6 through 10. Now, recursive RS(K) derivations can be formalized in first order arithmetic (see Schwichtenberg [1977]) But we can do even better. For a particular arithmetic theorem of KP Pi 3 Ref , say A, an n can be determined (depending on the proof of A) such that there is a cut free controlled recursive derivation of A that utilizes solely ordinals from T n (K) C(ae n ; 0) where ae 0 = 1 and ae k 1 ....

SCHWICHTENBERG H. [1977] Proof Theory: some applications of cut--elimination, in: J. Barwise, ed., Handbook of mathematical logic, North Holland, Amsterdam, 867--895.

....the finite jump lemma introduced here. For more information on the traditional ordinal analyses of the theories discussed here and in [4] as well as proofs that the bounds we give are sharp (obtained by proving instances of transfinite induction within the theories themselves) see, for example, [17, 18, 20, 28, 7, 27]. For information on theories of first order arithmetic, see [10, 11] and for more information on the relevant theories of second order arithmetic, see, for example, 26, 24, 25, 7, 2, 3] 2 2 Overview In this section we give an informal introduction to the model theoretic techniques we will ....

....is that g( a) bounds the length of the descending sequence of notations less than fi that is generated by h with parameters a. A function is OEfi recursive if it is fl recursive for some fl OE fi. For further discussion see [29] for other characterizations of the OEfi recursive functions see [23, 27]. It is not difficult to verify that each function f ff is ff recursive, so the construction described in Section 2 provides a method of showing that a theory T doesn t prove a certain ff recursive function to be total. In fact, our constructions yield an even stronger result, as described in ....

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Schwichtenberg, Helmut, "Proof theory: Some applications of cutelimination, " in [5], pages 867--895. 38

....n # formulas of T n # the L n # formulas without free number variables. We let #, #, range over finite sets of L n # formulas; we often write (for instance) #, # for the union of # and # . We first introduce the Tait calculus T n # . It is an extension of the classical Tait calculus [28] by the non logical axioms of T n # . It contains the following axioms and rules of inference: 1. Ontological axioms I. For all finite sets # of L n # formulas of T n # , all closed number terms s, t with identical value, all true literals # of L 1 , all set variables X and all # #, # ....

Schwichtenberg, H. Proof theory: Some applications of cut-elimination. In Handbook of Mathematical Logic, J. Barwise, Ed. North-Holland, Amsterdam, 1977, 867 -- 895.

....algorithm converts any proof in the calculus into a normal form known as #a cut free sequent proof. The bounding algorithm reveals how many more steps are needed in the cut free sequent proof of a given inference compared to the original. These are well known tools of proof theory #see #Schwichtenberg, 1977##, and they apply straightforwardly to Mates calculus #our reference system for de#ning the class of normal theories#. Boolos shows that every cut free sequent proof of #12# contains a tremendous number n of symbols. Moreover, given a normal system S, the normalizing and bounding algorithms can ....

Schwichtenberg, H. #1977#. Proof Theory: Some Applications of Cut-Elimination. In Barwise, J., editor, Handbook of Mathematical Logic, pages 867#895. North-Holland Publishing Company, Amsterdam.

....Principle The main tool of the proofs of inclusions (4) Theorems 11.1 and 12.1) and of the transitivity of F SnS (f ; g) Theorem 13.1) is the well known inversion principle. The rule invertibility is the fundamental principle of the cut free Gentzen type derivation systems, see, e.g. [Sch77]. The inversion principle is the key property needed to prove the minimal typing property for F . In fact, this is almost all what is needed to reconstruct F inferences into normal forms, CG92] RR nRR 2309 18 Sergei Vorobyov The inversion principle can be formulated as follows: for an ....

H. Schwichtenberg. Proof theory: some applications of cut-elimination. In J. Barwise, editor, Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, pages 867--895. North-Holland Publishing Company, 1977.

....can be generated more easily in PAc than PA. 3 PA c : Arithmetic with the Constructive Omega Rule The system PA is essentially PA enriched with the rule in place of the rule of induction. The derivations are then infinite trees of formulae; a formula is demonstrated in 1 See for example [Schwichtenberg 77] for a formalisation. 2 In other words, such that there is a primitive recursive function f for which, for every n, f(n) is a Godel number of the proof of A(n) the nth numerator of the rule. 3 See Section 3 below for description. 4 For a more formal description see [Baker 93] A Proof ....

....constructively generated are allowed, in order to capture the notion of infinite labelled trees in a finite way. The normal approach when dealing with a system with infinitary proofs such as PA is to work with numeric codes for the derivations rather than using the derivations themselves. See [Schwichtenberg 77, P886] for further details, including the case of the rule. By adding the provability relation and numeric encoding, a reflection system which necessarily extends the original one may be formed [Kreisel 65, P163] However, the necessity of using this Godel numbering approach may be avoided by ....

H. Schwichtenberg. Proof theory: Some applications of cutelimination. In Barwise, editor, Handbook of Mathematical Logic, pages 867--896. North-Holland, 1977. A Proof Environment for Arithmetic with the !-Rule 13

....the regularity in the case of conjunction inversion. The cut elimination theorem was proved originally by G. Gentzen [3] for his sequent calculus. Smullyan [8] translated this argument into the language of tableaux. Our proof is structurally similar to the sequent calculus proof of Schwichtenberg [6]. The following lemma captures the essential step: Lemma 25 (Reduction lemma) Suppose Xp : T A; As and Xn : T :A;As are regular atomically closed tableaux such that ae(Xp) oe(A) and ae(Xn) oe(A) Then there is a regular atomically closed tableau X : T As such that ae(X) ....

H. Schwichtenberg. Proof theory: Some applications of cut-elimination. In the Handbook of Mathematical Logic, J. Barwise (editor), North-Holland, 1977.

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H. Schwichtenberg. Proof theory: Some applications of cut-elimination. In J. Barwise, editor, Handbook of Mathematical Logic, pages 867--895. North-Holland, 1977.

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Schwichtenberg H. Proof-theory: some applications of cut-elimination, in: Handbook of Mathematical Logic, ed. by J. Barwise, NorthHolland (1977). 26

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Schwichtenberg H. Proof-theory: some applications of cut-elimination, in: Handbook of Mathematical Logic, ed. by J. Barwise, NorthHolland (1977).

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Schwichtenberg, Helmut, "Proof theory: some applications of cut-elimination", in Handbook of Mathematical Logic (J. Barwise, ed., 1997) 867-895.

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