Recirsive Functions

8/2/2019 Recirsive Functions

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The Witness Function Method andProvably Recursive Functions of Peano Arithmetic

Samuel R. BussDepartment of Mathematics

University of California, San Diego

La Jolla, CA 92093-0112, USA

[email protected]

Abstract

This paper presents a new proof of the characterization of the provably recursivefunctions of the fragments I n of Peano arithmetic. The proof method alsocharacterizes the k -denable functions of I n and of theories axiomatized bytransnite induction on ordinals. The proofs are completely proof-theoretic anduse the method of witness functions and witness oracles.

Similar methods also yield a new proof of Parsons theorem on the conservativityof the n +1 -induction rule over the n -induction axioms. A new proof of theconservativity of B n +1 over I n is given.

The proof methods provide new analogies between Peano arithmetic andbounded arithmetic.

1 Introduction

The witness function method has been used with great success to characterize some classesof the provably total functions of various fragments of bounded arithmetic [2, 4, 18, 23,16, 17, 5, 6, 1, 7, 8]. In this paper, it is shown that the witness function method canbe applied to the fragments I n of Peano arithmetic to characterize the functions which

are provably recursive in these fragments. This characterization of provably recursivefunctions has already been performed by a variety of methods; including: via Gentzensassignment of ordinals to proofs [9, 27], with the G odel Dialectica interpretation [12, 13],and by model-theoretic methods (see [20, 15, 26]). The advantage of the methods in this

Supported in part by NSF grants DMS-8902480 and INT-8914569.


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paper is, rstly, that they provide a simple, elegant and purely proof-theoretic methodof characterizing the provably total functions of I n and, secondly, that they unify theproof methods used for fragments of Peano arithmetic and for bounded arithmetic.

The witness function method is related to the classical proof-theoretic methods of Kleenes recursive realizability, Godels Dialectica interpretation and the Kreisel no-counterexample interpretation; however, the witness function method does not requirethe use of functionals of higher type. We feel that the witness function method providesan advantage over the other methods in that it leads to a more direct and intuitiveunderstanding of many formal systems. The classical methods are somewhat more generalbut are also more cumbersome and more difficult to understand (consider the difficultyof comprehending the Dialectica interpretation or no-counterexample interpretation of aformula with more than three alternations of quantiers, for instance). On the other hand,the more direct and intuitive witness function method has been extremely valuable for theunderstanding of why the provably total functions of a theory are what they are and alsofor the formulation of new theories for desired classes of computational complexity and,conversely, for the formulation of conjectures about the provably total functions of extanttheories. The main support for our favorable opinion of the witness function methodis, rstly, its successes for bounded arithmetic and, secondly, the results of this papershowing its applicability to Peano arithmetic.

While checking references for this paper, the author read Mints [19] for the rsttime it turns out that Mintss proof that the provably recursive functions of I 1 areprecisely the primitive recursive functions is based on what is essentially the witnessfunction method. This theorem of Mints is, in essence, Theorem 9 below. Mintss use

of the witness function method predates its independent development by this author forapplications to bounded arithmetic. The present paper expands the applicability of thewitness function method to all of Peano arithmetic.

The outline of this paper is as follows: section 2 develops the necessary backgroundmaterial on Peano arithmetic, the subtheories I n , transnite induction axioms, leastordinal principle axioms, the sequent calculus and the correct notion of free-cut free proof for transnite induction/least number principle axioms. In section 3, the central notionsof the witness function method and witness oracles are developed and the n -denablefunctions of I n and I 0 + T I (m , n ) are characterized. This includes the denitionof -primitive recursive (in

k) functions and normal forms for such functions. Then

the provably recursive (i.e., 1 -dened) functions of I n are characterized by proving aconservation theorem for T I (m , n ) over T I (m +1 , n 1). Section 4 outlines a proof of Parsons theorem on the conservativity of the n +1 -induction rule over the n -inductionaxiom. Section 5 contains a proof of the n +1 -conservativity of B n +1 over I n .Section 6 concludes with a discussion of the analogies between the methods of this paperand the methods used for bounded arithmetic.


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2 Preliminaries

2.1 Arithmetic and Ordinals

Peano arithmetic (PA) is formulated2

in the language 0, S , +, and . It has inductionaxiomsA(0) (x)(A(x) A(S (x))) (x)A(x)

for all formulas A, plus it has a nite base set of axioms, namely, Robinsons theory Q of seven axioms dening 0, S , + and and, in addition, the axiom

(x)(y)(x y (z)(x + z = y))

which denes . A bounded quantier is of the form (x t) or (x t) where t isany term not involving x . The usual quantiers, ( x) and (x), are called unbounded

quantiers . The 0 -formulas, or bounded formulas, are the formulas in which everyquantier is bounded. The classes n and n of formulas are dened by induction on n ,so that 0 = 0 = 0 and so that n +1 is the set of formulas of the form (x)B whereB n and so that n +1 is dened dually. The theory I n is dened to be the theoryin the language of Peano arithmetic with the same eight non-induction axioms as PA andwith induction axioms for all formulas A n .

The collection axioms provide an alternative way to dene fragments of Peanoarithmetic. A collection axiom is of the form

(x t)(y)A(x, y) (z)(x t)(y z)A(x, y).

We let B n denote the set of collection axioms for all A n ; Bn is dened similarly.It is well-known that I 0 + B n +1 I n and I n B n . It is also well-knownthat I 0 + B n +1 is n +1 -conservative over I n and we shall reprove this in section 5below. An important feature of the collection axioms is that it provides a quantierexchange principle that allows moving bounded quantiers inside the scope of unboundedquantiers. The classes n and n can be generalized to classes Gn and Gn by allowingbounded quantiers to appear anywhere in the formula (instead of only in the 0 matrix)but counting only the alternations of unbounded quantiers. For example, the hypothesisand conclusion of the collection axiom above are Gn -formulas if A n . The theory

I 0 + B n , and hence I n , can prove that every Gn -formula is equivalent to a n -formula.

Remark: Some authors include function symbols for all primitive recursive functions inthe language of PA. We do not adopt this convention; however, as is well-known, everyprimitive recursive function is provably recursive ( 1 -denable, see below) in I 1 andhence the theories I n , for n 1 are not signicantly affected by the addition of symbols

2 Our formulation of PA is similar to the usual one in [21] except that it has different non-inductionaxioms and has instead of < . It is easily seen that our denition of I n and P A is equivalent tothe usual one apart from the inessential replacement of < by .


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for primitive recursive functions. Thus the theorems and proofs of this paper also applyto theories with symbols for primitive recursive functions.

Denition Let T be a subtheory of PA and f : Nk N . The function f is i -denablein T iff there is a formula A(x1, . . . , x k , y) i such that

(1) T (x)(!y)A( x, y), and

(2) {( n, m): N A( n, m)} is the graph of f , i.e., A( n, m) holds iff f ( n) = m for allintegers n, m .

The function f is provably recursive in T iff f is 1 -denable in T .

The intuitive idea of provably recursive is that the theory T should prove thatsome Turing machine M , which computes f , halts on all appropriate inputs. SinceA( x, y) can be taken to be a 1 -formula expressing there is a w which codes a haltingM -computation with input x and output y, it is clear that any function which is provablyrecursive in this intuitive sense is also 1 -denable. Conversely, if f is 1 -denable in T ,then there is Turing machine M which computes f , provably in T . Namely, M performsa brute-force search for values of y and the unboundedly existentially quantied variablesof A. Thus 1 -denable coincides with the intuitive notion of provably recursive.

One reason that the provably recursive functions of T are of particular signicance isthat if f is provably recursive in T , then T may conservatively extended by adding f asa new function symbol with f ( x) = y A( x, y) as a new axiom. If T is a fragment I nthen f may be used freely in induction formulas (without affecting quantier complexity).Similarly, if T can prove that a 1 -formula and a 1 -formula are equivalent then T canconservatively extended by adding a new predicate symbol with arguments including thefree variables of the two formulas and adding a new axiom dening the predicate symbolto be equivalent to the formulas. The new predicate may also be used freely in inductionformulas. Such new predicates are called 1 -dened predicates of T .

Recall that I 1 (and even I 0 ) can formalize many metamathematical notions; of particular importance are the sequence coding functions x0, . . . , x k , ( x0, . . . , x k )i = xi ,and Len ( x0, . . . , x k ) = k + 1.

The ordinals are set-theoretically dened to be those sets which are transitive andwell-founded by . We write for the ordering of ordinals, so means .It is well-known how to dene ordinal addition, multiplication and exponentiation. TheCantor normal form for an ordinal is the unique expression

= 1 n1 + 2 n2 + r n r

where 1 2 r are ordinals and n1, . . . , n r are positive integers (i.e., nonzero,nite ordinals). Here is the rst innite ordinal; we let 0 = 1, 1 = and, generally,n +1 = n . Thus n is a stack of n s. The limit of n as n is called 0 ; hence

0 is the least ordinal such that 0 = 0 . For 0 , the Cantor normal form can


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be extended so that the exponents i are also written in Cantor normal form, and withexponents in the latter Cantor normal forms also in Cantor normal form, etc. (eventuallythe process must stop). For example,

0

3 + 0

2 4 + 0

is a Cantor normal form; usually this is expressed more succinctly as 3 + 2 4 + 1. In

this paper, we shall always use ordinals 0 and by Cantor normal form always meansthe extended version with exponents also in Cantor normal form. 0 is its own Cantornormal form.

By using Godel numbering, integers can encode Cantor normal forms and this canbe intensionally formalized 3 in I 1 ; with care, these can even be formalized in I 0 . Inparticular, I 0 can dene the relation IsOrdinal (x) expressing that x is the Godel numberof an ordinal, the relation x y, and the functions for ordinal addition, multiplication

and exponentiation. To avoid excessive notation, we use the same notation for actualand for metamathematical operations; for example, + 1 also denotes its own G odelnumber. However, there will occasionally be situations where context is not sufficient todistinguish between ordinals and their G odel numbers: this occurs when n may be eitheran integer or a nite ordinal; to resolve ambiguity, we write n for the Godel number of the ordinal n and we write n for the integer n . To improve readability, we use , , , . . .and ,, , . . . as variables that range over G odel numbers of ordinals. For example, theformula ( )( ) abbreviates the rst-order formula

IsOrdinal ( ) (x)(IsOrdinal (x) x ).

Note that corresponds to an unbounded quantier unless is known to code anite ordinal.

Transnite induction on ordinals may be used to provide alternate axiomatizations forfragments of Peano arithmetic:

Denition Let be a set of formulas and let 0 . Then TI (, ) is the set of axioms

( )[( )A( ) A( )] A() (1)

where A is a formula in , possibly with other free variables as parameters.The least ordinal principle axioms LOP (, ) are

A() ( )[A( ) ( )(A( ))] (2)

where A and A may have parameter variables. For a xed formula A, theaxioms (1) and (2) are called TI (, A) and LOP (, A), respectively.

TI ( , ) is the theory TI (, ).LOP ( , ) is the theory LOP (, ).

3 Intensionally formalized means that I 1 can prove simple syntactic facts about ordinal encodingsand about operations on encoded ordinals.


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A slight variation on the least ordinal principle and transnite induction axioms is

TI (, ) : ( )[( )A( ) A( )] ( )A( )

LOP (, ) : ( )A( ) ( )[A( ) ( )(A( ))].For one of the classes n or n , TI (, ) is equivalent to TI (, ) since the formerobviously implies the latter and since TI (, A) may be inferred from TI (, B ) whereB() is A() ( A( )), where is a new variable acting as a parameter.Similarly, LOP and LOP are equivalent for one of the classes n or n .

This paper is concerned primarily with the axioms TI ( m , n ) and LOP ( m , n )where m 2 and n 0. The next two propositions give equivalences among such axioms(see [26] for generalizations of these propositions).

Proposition 1 Let m 2 and n 0.

(a) I 0 + TI ( m , n ) I 0 + LOP ( m , n )

(b) I 0 + TI ( m , n ) I 0 + LOP ( m , n )

(c) I 0 + LOP ( m , n ) I 0 + LOP ( m , n +1 )

(d) I 0 + TI ( m , n ) I 0 + TI ( m , n +1 )

Proof (a) and (b) are trivial since TI (, A) and LOP (, A) are logically equivalent(essentially, contrapositives). For (c), if A n +1 then A() must be (y)B(, y) whereB n . Now, LOP (, A) follows from LOP ( + , C ) where C () is the n -formula

expressing encodes an ordinal + y , with y integers, such that B(, y) holds.

Also, if m , then + m ; so (c) is proved. Finally, (d) follows immediatelyfrom (a), (b) and (c). 2

It is important to note that Proposition 1 holds for n = 0; it is easy to see thatthe proof of (c) is valid for n = 0 since C is is a 0 -formula if B is. This has asconsequence that I 0 + TI ( m , 0) is equivalent to I 0 + TI ( m , 1) andsince I 0can express every primitive recursive predicate as a 1 formula, it follows that transnite( m ) induction on 0 -formulas implies the same amount of transnite induction on

primitive recursive predicates. In addition, relative to I 0 , TI ( m , 0) is equivalentto LOP ( m , 0), which in turn is equivalent to LOP ( m , 1). Since every primitiverecursive predicate can be expressed as a 1 -formula, it follows that transnite ( m )induction on 0 -formulas implies the m least ordinal principle for primitive recursivepredicates. We shall, in section 3, frequently informally argue that various complicatedmetamathematical constructions can be formalized in theories I 0 + TI ( m , n 1);since m 2 always holds, these theories can prove the usual induction and least numberprinciples for primitive recursive predicates, which is sufficient for formalizing all themetamathematical constructions in section 3.


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Proposition 2 Let n 1.

I n I n I 0 + TI (, n ) I 0 + TI (, n ) I 0 + LOP ( 2, n )

I 0 + TI ( 2, n 1)

Proof It is clear that I n I 0 + TI (, n ) and by standard techniques these areequivalent to I n and I 0 + TI (, n ). In light of Proposition 1, it suffices to show thatLOP ( 2, n ) follows from I 0 + TI (, n ). To accomplish this, we show, by inductionon k , that LOP ( k , n ) follows from the latter theory. For k = 1 this is proved bythe kind of reasoning used to prove Proposition 1(a),(b). To show LOP ( k+1 , n ); letA() n , let 0 k+1 and reason informally with the assumptions TI (, n ) andLOP ( k , n ): further set C () to be the formula ( i)A( + i), so C () n . Nowassume A(0) holds; since 0 = 1 + i1 for some 1 k and some nite i1 , C (1)

holds also. By LOP ( k

, n ), there is a least 2 such that C (2) holds and now byTI (, n ), there is a least i2 such that A( 2 + i2). Clearly = 2 + i2 is the leastordinal such that A() holds. 2

2.2 Arithmetic and the Sequent Calculus

This section describes how the sequent calculus and free cut elimination are applied tothe fragments of arithmetic dened above. The reader is presumed to be familiar with thesequent calculus (refer to [27] or Chapter 4 of [2] for the necessary background material).We shall assume the language of rst-order logic contains symbols , , , , and

; this leads to a large number of rules of inference but we shall omit most cases from ourproofs in any event. It will be assumed that bounded quantiers are part of the syntax of rst-order logic with the sequent calculus containing the four appropriate rules of inferencefor bounded quantiers. 4 See [2] for the full denition of the sequent calculus LKB withbounded quantier rules of inference.

To formalize the proof theory of arithmetic with the sequent calculus, it is customaryto use special induction inferences in place of induction axioms. An induction inference isof the form

, A(a)A(Sa ), , A(0)A(t),

where t may be any term, a is a free variable called the eigenvariable and a must notappear in the lower sequent. The induction inference for A is equivalent to the inductionaxiom for A, because the side formulas and are allowed. Thus I k is formalizedin the sequent calculus with a nite set of axiom schemes plus the induction inferencesfor k formulas. The nite set of axiom schemes for I k consists of the following initial

4 This assumption is not absolutely necessary and the reader may prefer to think of the boundedquantiers as abbreviations in this case the proofs by induction on the number of inferences in afree-cut free proof must be slightly modied.


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sequents:Sr = St r = t r 0 = 0St = 0 r (St ) = r t + rr + 0 = r r = 0 , (x r )(Sx = r )r + St = S (r + t) r t(x t)(r + x = t)r + s = tr t

where r , s and t are allowed to be any terms. Of course the usual logical initial sequentsAA with A atomic and the initial sequents for equality are also allowed. It is importantfor us that every initial sequent consists of only 0 formulas.

The theory I 0 + TI ( m , n ) is formalized in the sequent calculus with the sameinitial sequents, with induction inferences for 0 -formulas and for transnite induction,with the LOP ( m , n ) inferences dened below.

Let be a closed term with value the Godel number of an ordinal and let B() be aformula; the LOP (, B ) inference is

LOP (, B ) : , B (), , ( )B( )B( ), where is an eigenvariable and may occur only as indicated. It is not hard to see thatthe inference rule LOP (, B ) is equivalent to the axiom form of LOP (, B ): to derive theinference rule from the axiom, recall that the axiom LOP (, B ) is

B( )( )[B() ( )(B( ))], (3)and use the derivation

(3) , B (), , ( )B( )( )(B() ( )(B( ))) ,

B( ), where the double horizontal line indicates omitted inferences. Conversely, to see that theLOP (, B ) follows from the inference rule, use

, B ()( )[B( ) ( )(B( ))], ( )B( )B( )( )[B( ) ( )(B( ))]

where the upper sequent is, of course, provable in I 0 .The LOP ( m , ) inferences are the set of inferences LOP (, B ) for m and

B . The principal formula of an LOP inference is the formula B( ) in the lowersequent; the auxiliary formulas are the three formulas in the upper sequent other than and . An important property of the LOP ( m , n 1) inferences is that the principalformula and the auxiliary formulas are all in n .5

5 This is the reason we use LOP inferences instead of TI inferences. The TI (, n 1 ) inferenceswould be

, ( )B ( ), , B ( ) , B ( )

where B n 1 and is an eigenvariable. These inferences contain a n auxiliary formula.


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Below we shall extensively study the theory I 0 + TI ( m , n 1), which is equiva-lent to I 0 + LOP ( m , n 1) and is henceforth is to be formalized in the sequent cal-culus with initial sequents given above, the I 0 -induction rule and the LOP ( m , n 1)inference rule. This theory enjoys the important property of free-cut elimination . Wesay that a cut in a sequent calculus proof is free unless one of its cut formulas is a directdescendent of a formula in an axiom (initial sequent) or of a principal formula of anI 0 inference or of a principal formula of an LOP ( m , n 1) inference. The free-cutelimination theorem implies that if I 0 + LOP ( m , n 1) proves a sequent then thereis a proof (in the same theory and of the same sequent) which contains no free cuts. Sucha proof is called free-cut free . This free-cut elimination theorem is proved by a elementarytriple induction argument (equivalently, induction to 3 ) by the same argument used forthe cut elimination theorem for rst-order logic. In particular, the free-cut eliminationtheorem can be proved in I 1 .

A formula A is a subformula of B in the wide sense if A can be obtained from somesubformula C of B by substituting freely terms for variables in C . In a free-cut freeproof, each formula A is either (1) a direct descendent of a formula in an axiom or of aprincipal formula of an I 0 or LOP inference, or (2) a subformula in the wide sense of such a formula, or (3) a subformula in the wide sense of an auxiliary formula of an I 0inference or an LOP inference, or (4) a subformula in the wide sense of a formula in theendsequent of the proof. This is because each formula in the proof has a (not necessarilydirect) descendent which is a cut formula (so (1) or (2) applies), or which is an auxiliaryformula of an induction or LOP inference (so (3) applies), or which is in the endsequent(so (4) applies).

The above gives the following important proposition:Proposition 3 ( n 1) Let T be a theory I n or I 0 + TI ( m , n 1) . Suppose is a consequence of T and every formula in and is in n . Then there is a T -proof of in which every formula is in n .

3 Denable Functions of I n3.1 Witness Functions and Ordinal Primitive Recursion

A witness oracle for an existential property ( x)A(x, z) is an oracle which when queriedwith values for z responds either with a value for x such that A(x, z) or with the statementthat there is no such value for x . If A is a decidable predicate then a witness oracle for Ais clearly equivalent to an oracle for the function

U xA ( z) =1 + ( x)A(x, z) if (x)A(x, z)0 otherwise

where (x)A(x, z) is the least value for x such that A(x, z) holds. The advantage of viewing a witness oracle as a function is that it allows the denition of being primitiverecursive relative to a witness oracle:


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Denition Let n 1. The set of functions which are primitive recursive in n is denedinductively by:

(1) The constant function 0, the successor function S (x) = x + 1, and the projection

functions ni (x1, . . . , x n ) = xi are primitive recursive in n .

(2) The set of functions primitive recursive in n is closed under composition.

(3) If g : Nk N and h : Nk+2 N are primitive recursive in n then so is thefunction f dened by

f (0, z) = g( z)f (m + 1 , z) = h(m, z, f (m, z)) .

(4) If A( z) is a formula (x)B(x, z) where B n 1 then U A is primitive recursivein n .

The set of functions primitive recursive in 0 is just the set of primitive recursive functions,and is dened, as usual, by (1), (2) and (3).

It is important for the denition of primitive recursive in n that the functions U A areincluded instead of just the characteristic functions of A. For example, if n = 1, thesetwo functions are Turing equivalent; however, for primitive recursive processes these arenot equivalent since even if ( x)B is guaranteed to be true and if B is primitive recursive,a primitive recursive process can not nd a value for x making B true without knowing

(at least implicitly) an upper bound on the least value for x .A primitive recursive in n function may ask any (usual) query to a n or a n

oracle. This is because, for example, if A( z) n , then A is equivalent to a formula(x)B where B n 1 and a witness oracle U (x )B can be used to determine if A( z) istrue.

Denition Let be (the G odel number of) an ordinal. The set of -primitive recursive functions is dened inductively by the closure properties of (1), (2) and (3) above and by

(5) If g : Nk N , h : Nk+1 N and : Nk N are -primitive recursive then so isthe function f dened by

f (, z) =h(, z, f ((, z), z)) if (, z)

g(, z) otherwise

where (, z) means that and (, z) are the G odel numbers of ordinalsobeying the inequalities.

A function is said to be -primitive recursive iff it is -primitive recursive for some .


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Combining the notions of witness oracles and ordinal primitive recursion gives:

Denition Let n 0 and be (the G odel number of) an ordinal. The set of functionswhich are -primitive recursive in n is dened inductively by the closure properties of

(1)-(5) above (omitting (4) if n = 0).A function is said to be -primitive recursive in n iff it is -primitive recursive

in n for some .

It is well-known, and not too hard to show, that a function is primitive recursive in n iff it is -primitive recursive and iff it is -primitive recursive in n .

3.2 Normal Forms for Ordinal Primitive Recursive Functions

This section presents three normal forms for the denitions of m -primitive recursivefunctions. These are called the zeroth, rst and second normal forms and will be helpfulfor the proofs of the characterization of provably total functions of various fragments of Peano arithmetic.

Recall that that the set of functions m -primitive recursive in n is, by denition,the smallest set of functions satisfying the closure properties (1)-(5): the Zeroth NormalForm Theorem states that the closure (3) under primitive recursion may be dropped atthe expense of adding more base functions.

Theorem 4 (Zeroth Normal Form). Let m 2 and n 0. The functions m -primitive recursive in n can be inductively dened by

(0.1) Every primitive recursive function is m -primitive recursive in n .

(0.2) The set of functions m -primitive recursive in n is closed under composition.

(0.3) If n 1 and A( z) is (x)B(x, z) where B n 1 , then U A is m -primitiverecursive in n .

(0.4) If 0 m and if g : Nk N , h : Nk+1 N and : Nk N are m -primitiverecursive in n then so is the function f dened by

f (, z) = h(, z, f ((, z), z)) if (, z) 0

g(, z) otherwise .

Proof The fact that m -primitive recursive in n functions satisfy conditions (0,1)-(0.4) is obvious. The idea for the other direction is quite simple; namely, that -primitiverecursion may be used to simulate ordinary primitive recursion. For example, if f isdened by primitive recursion from g and h by

f (0, z) = g( z)f (m + 1 , z) = h(m, z, f (m, z))


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then f can also be dened via -primitive recursion as follows. For n N , let n denotethe G odel number of the nite ordinal n . Dene

F (, z) = g( z) if = 0

H (, z, F (Pred (), z)) otherwisewhere

Pred () = 1 if is (the G odel number of)

a successor ordinal

otherwise

and

H (, z, w) = h(m, z, w) if = m + 1 with m Narbitrary otherwise.

Now Pred is primitive recursive and H is denable by composition from h and primitiverecursive functions; furthermore, f (m, z) = F ( m , z). Thus f is dened from g and hand some primitive recursive functions using composition and -primitive recursion. 2

Note that the proof of Theorem 4 shows that (0.1) could be weakened to include only theusual base functions (1) and a few specic primitive recursive functions for manipulatingGodel numbers of nite ordinals.

Theorem 5 (First Normal Form). Let m 2 and n 0. The set of functions

m -primitive recursive in n is the smallest set of functions satisfying the four conditions(1.1)-(1.4):

(1.1)-(1.3): same as (0.1)-(0.3).

(1.4) If 0 m and if g and are unary functions which are m -primitive recursivein n then so is the function f dened by

f () = f (()) if () 0g() otherwise.

In (1.4), we say that f is dened by parameter-free 0 -primitive recursion from g and .Proof For this proof only, let F denote the smallest set of functions which satisesthe closure conditions of (1.1)-(1.4). Obviously, the Zeroth Normal Form implies thatevery function in F is m -primitive recursive in n . To show that F contains everyfunction m -primitive recursive in n , it will suffice to show that F is closed underthe m -primitive recursion of (0.4). For this, suppose f is dened by

f (, z) = h(, z, f ((, z), z)) if (, z) 0g(, z) otherwise.


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To give a denition of f using parameter-free m -primitive recursion, we shall useordinals that code the parameters z and which code a history of the computation of f ( )with 0 . In order to code the history of the computation of f , we need ordinals 0, 1, . . . , s so that 0 = and i+1 = ( i , z) i and so that ( s , z) s ; alsowe need values as , . . . , a 0 so that as = g( s , z) and a i = h( i , z, ai+1 ) for all i < s ; itwill follow that f (, z) is equal to a0 . We shall code and index this computation by thefollowing scheme. We use ordinals of the form 2 i + z, 0, . . . , i 1 to code the rstphase of the computation of f , where z, 0, . . . , i 1 denotes the nite ordinal equalto the G odel number of the sequence containing the entries z and the Godel numbers 0, . . . , i 1 . To code the second phase of the computation we use ordinals of the form i + z, 0, . . . , i 1, a i . Since 0 m there is an ordinal 0 m 1 such that0 0 . Dene

F () =F (K ()) if K () 2+ 0

G() otherwise

where K and G are dened so that

K (2 i + z, 0, . . . , i 1 ) = 2 ( i , z) + z, 0, . . . , iif i 0 and ( i , z) i

K (2 i + z, 0, . . . , i 1 ) = i + z, 0, . . . , i 1, g( i , z)where 0 i N and ( i , z ) i

K ( (i + 1) + z, 0, . . . , i , a ) = i + z, 0, . . . , i 1, h( i , z, a)for i N

K ( z, a ) = z, a

G( z, a ) = a

where, in the last two equations, z, a denotes the G odel number of the niteordinal z, a . K and G may be arbitrarily dened for other inputs. Clearly F isdened by 2+ 0 -primitive recursion from G and K . And f is denable in terms of F and g using composition:

f (, z) = F (2 + z ) if 0

g(, z) otherwise

We have used only m -primitive recursion (since 2+ 0 m ) and composition todene f from g, h , and primitive recursive functions. Hence f F .Q.E.D. Theorem 5

The nal and best normal form for m -primitive recursive in n functions is not aninductive denition, but is a true normal form.

Theorem 6 (Second Normal Form)


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(a) Let m 2 and n 1. A function F ( z) is m -primitive recursive in n iff thereare a 0 m , a A( y) of the form (x)B(x, y) with B n 1 , and primitiverecursive functions , g and so that F ( z) = f ( ( z)) where f ( ) is dened by

f ( ) = f ((, U A ( ))) if (, U A ( )) 0g( ) otherwise

(b) Let m 2. A function F ( z) is m -primitive recursive iff there are a 0 m and primitive recursive functions , g and so that F ( z) = f ( ( z)) where

f ( ) = f (( )) if ( ) 0g( ) otherwise

An important feature of the second normal form theorem is that is now required to beprimitive recursive, instead of merely m -primitive recursive in n .

Proof We shall prove (a); the proof of (b) is essentially identical. First, every primitiverecursive function can be expressed in the form (a): to prove this, if F is primitiverecursive, let 0 = 0, let ( z) = z , let (, a ) = 0 and g( z ) = F ( z) . Thefunctions and are clearly primitive recursive and g is primitive recursive since F is.Second, if A(y) is (x)B(x, y) where B n 1 , then the function U A can be expressedin the form (a) by letting 0 = 2, letting (y) = + y, letting ( + y, i ) = i and( i , a) = i and letting g( i ) = i .

Next we show that the set of functions denable in the form (a) is closed undercomposition. Suppose F 1 and F 2 are dened by F 1(v, z) = f 1( 1(v, z)) and F 2( z) =

f 2( 2( z)) where

f i ( ) =f i (i (, U A i ( ))) if i (, U A i ( )) 0,igi ( ) otherwise

for i = 1 , 2. By assumption, i , i and gi are primitive recursive functions. We mustshow F ( z) = F 1(F 2( z), z) is also denable in this way. Pick m 1 to be an ordinalsuch that 0,1, 0,2 . We set F ( z) = f (1+ 2 + z ) and dene f ( ) as in (a) with0 = 1+ 3 and with dened so that, if ,

(1+ 2 + z ) =

1+ + 2( z) + z if 2( z) 0,2

1(g

2(

2( z))) if

2( z)

0,2and 1(g2( 2( z))) 0,11+ 3 otherwise

(1+ + + z ) =

1+ + 2(, U A 2 ( )) + zif 2(, U A 2 ( )) 0,2

1(g2( ), z) if not 2(, U A 2 ( )) 0,2and 1(g2( ), z) 0,1

1+ 3 otherwise


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and, if 0,1 , ( ) = 1(, U A 1 ( )). Also dene g so that, for all ,

g(1+ 2 + z ) = g1( 1(g2( 2( z))))g(1+ + + z ) = g1( 1(g2( )))

g( ) = g1( )

This almost denes f ( z) in the desired form (a); however, there is a problem since ()is dened using both U A 1 and U A 2 (and not using them in correct manner either). To xthis, we dene a new A(y) = (x)B(x, y) so that () is a primitive recursive function of only and U A (). For this, suppose Ai = (x)B i (x, y) where B i n 1 . Dene B by

B(x, ) B2(x, ) if = 1+ + + mB1(x, ) if 1+ arbitrary otherwise .

Since B1, B2 n 1 , so is B . That completes the proof that the set of functions denablein the form (a) are closed under composition.

Finally, we must show that the functions denable in the form (a) are closed underparameter-free m -primitive recursion. For this, suppose f is dened from functions gand , which are dened in form (a), and from an ordinal 0 m as in (1.4) and furthersuppose that is dened in the normal form (a) by

() = f 1( 1())

f 1( ) =f 1(1(, U A 1 ( ))) if 1(, U A 1 ( )) 0,1g

1( ) otherwise

where 0,1 m and 1 , 1 and g1 are primitive recursive functions. Pick 0 , 1 to bethe least ordinals such that 0 0 and 0,1 1 ; hence 0, 1 m 1 . We nowdene F () = f (1+ 1 + 0 + ) where f will be dened in the second normal form (a)with primitive recursive functions , g and ordinal 0 where is dened by

(1+ 1 + 0 + ) =1+ 1 + 1 if 01+ 1 + 0 + otherwise

(1+ 1 + 1 ) =

1+ 1 + 1( ) if 1( ) 0,11+ 1 g1( 1( )) if 1( ) 0,1 and g1( 1( )) 1+ 1 + 0 otherwise

(1+ 1 + ) =

1+ 1 + 1(, U A 1 ( ))if 1(, U A 1 ( )) 0,1

1+ 1 g1( ) + 1if 1(, U A 1 ( )) and g1( )

1+ 1 + 0 if 1(, U A 1 ( )) and g1( )


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(provided 0,1 ), and g is dened by

g (1+ 1 + 0 + ) = g (1+ 1 + ) = if 0 and 1

and 0 = 1+ 1 + 0 + . Any values of and g left unspecied may be arbitrary. Now,inspection shows that

F () = F (()) if () 0 otherwise

and, by construction, F is denable in form (a). Now the function f is denable byf () = g(F ()) and since g and F are expressible in form (a) it follows by the earlierpart of this proof that their composition f is too.Q.E.D. Theorem 6

One further renement can be made to the second normal form theorem: instead of allowing arbitrary U A s with A n , it is possible to allow only a single, xed, suitablychosen U A . Of course, such an A is many-one complete for n . It is necessary to modifythe ordinal coding methods in the above proof to establish this renement the detailsare left to the reader.

3.3 Some Denability Theorems

The next theorems characterize the n denable functions of I n ; their proof will be a

straightforward use of the witness function method.

Theorem 7 Let m 2 and n 1. The n -denable functions of the theory I 0 + TI ( m , n 1) are precisely the functions which are m -primitive recursivein n 1 .

Theorem 8 Let n 1. The n -denable functions of the theory I n are precisely the functions which are primitive recursive in n 1 .

Theorem 9 The 1 -denable (provably recursive) functions of I 1 are precisely theprimitive recursive functions.

There are (at least) three prior prooftheoretic proofs of Theorem 9. Parsons [22] gave aproof based on the G odel Dialectica interpretation, Mints [19] gave a proof which usesa method very close to the witness function method except presented with a functionallanguage, and Takeuti [27] gives a proof based on Gentzen-style assignment of ordinals toproofs.


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Proof Theorems 8 and 9 are corollaries of Theorem 7 since I n andI 0 + TI ( m , n 1) are the same theory. Although, only the proof of Theorem 7is given below, it should be remarked that the other two theorems can be proved directlyby a similar and easier argument.

The easier half of the proof is to show that every m -primitive recursive in n 1function is n -denable in I 0 + TI ( m , n 1). Recall that every primitive recursivefunction is 1 -denable in I 1 so this half of the m = 2 and n = 1 case of Theorem 7follows. For other values of m and n , suppose F is m -primitive recursive in n 1 andthat F is dened by F ( z) = f ( ( z)) where

f ( ) = f ((, U A ( ))) if (, U A ( )) 0g( ) otherwise

in accordance with the Second Normal Form, so g, and are primitive recursive

functions, 0 m and A(y) is (x)B(x, y) where B n 2 (in the simpler case wheren = 1, (, U A ( )) is replaced by ( ) and U A is not used at all). Obviously it willsuffice to show that f is n -denable by I 0 + TI ( m , n 1).

A sequence of ordinals 0, . . . , k is an f -computation series if i+1 0 , i+1 =( i , U A ( i )) and i+1 i , for all 0 i < k . To metamathematically dene anf -computation series, we use (if n > 1),

w codes an f -computation series w is a sequence of Godel numbers of ordinals of length k + 1

and (i < k ) (y)[B((w)i , y) (y < y )(B((w)i , y))

(w)i+1 = ((w)i , y + 1)]

(y)(B((w)i , y)) (w)i+1 = ((w)i , 0)

and (i < k )(( w)i+1 (w)i (w)i+1 0).

(Recall that if w = 0, . . . , k , then ( w)i = i .) Since I 0 + TI ( m , n 1) containsI n , it also contains the collection axiom B n . Thus the subformula ( y < y )( ) aboveis equivalent to a n 2 formula, and by applying prenex operations, the formula w codesan f -computation series is equivalent to a n formula. By applying prenex operations

in a different order, and using B n , this formula is also equivalent to a n -formula. If n = 1, then instead dene

w codes an f -computation series w is a sequence of Godel numbers of ordinals of length k + 1and (i < k )( i+1 = ( i) i i+1 0),

so, in this case, it is a primitive recursive property. 66 It is possible to strengthen the second normal form theorem to make this a 0 -formula.


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The graph of the function f ( ) can now be dened by using the fact that y = f ( )iff y = g( ) where is the least ordinal such that there is an f -computation series , . . . , . More formally, letting fCS (w) be the formula w is an f -computation

series,

y = f ( ) ( , . . . , ) y = g( ) fCS ( , . . . , )

(( , U A ( )) 0) .

Since fCS ( ) is equivalent to a n -formula and since z = U A ( ) can be expressed as an 1 -formula, the relation y = f ( ) is a n -property, provably in I 0 + TI ( m , n 1).The theory also proves

a least s.t. , . . . , (fCS ( , . . . , ))

since fCS ( ) and by LOP ( m , n ) since 0 m .7 Thus I 0 + TI ( m , n 1)can n -dene the function f as it proves ( )(!y)(y = f ( )) where y = f ( ) denotesthe n -formula dening the graph of f . Likewise,

(z)(!y)( )( = ( z) y = f ( ))

is also provable and n -denes the function F . That completes the rst half of the proof of Theorem 7.

To prove the rest of Theorem 7, assume that I 0 + TI ( m , n 1) proves(x)(!y)A(x, y), with A n we must show that x y is a m -primitiverecursive in n 1 function. Since I 0 + TI ( m , n 1) proves (x)(y)A, there mustbe a free-cut free proof in the theory I 0 + TI ( m , n 1) of the sequent

(y)A(c, y)where c is a new free variable. Only n formulas can appear in this free-cut freeproof. The general idea of the proof is to show that this free-cut free proof embodiesan algorithm for computing y from c. Indeed, the free-cut free proof can be interpretedas explicitly containing a m -primitive recursive in n 1 algorithm. Since the proofsof the normal form theorems were constructive, the free-cut free proof also contains animplicit description of a m -primitive recursive in n 1 algorithm in the second normalform. Our proof below that an algorithm can be extracted from the free-cut free proof is quite constructive and can be formalized in I 0 + TI ( m , n 1) the upshot isthat there is a m -primitive recursive in n 1 function f which is n -dened byI 0 + TI ( m , n 1) in the form given by the Second Normal Form Theorem suchthat I 0 + TI ( m , n 1) (x)A(x, f (x)). As a corollary to the proof method, if I 0 + TI ( m , n 1) proves (x)(y)B(x, y) with B n then there is a B(x, y) nsuch that ( x)(!y)B(x, y) and B(x, y) B(x, y) are provable. 8

7 LOP ( m , n ) is a consequence of I 0 + TI ( m , n 1 ) by Proposition 1.8 This fact is readily proved directly anyway. If B n 1 then let B be the formula B (x, y )

(y < y )(B (x, y )), which is equivalent to a n formula by B n . For general B n , incorporateoutermost existential quantiers of B into the the ( y) and proceed similarly.


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We shall see later that the proof is formalizable, not only in I 0 + TI ( m , n 1),but also in I 0 + TI ( m +1 , n 2), provided n > 1.

Rather than just considering the free-cut free proof of (y)A, we more generallyconsider proofs of sequents of n -formulas. Since every principal and auxiliaryformula of a LOP ( m , n 1) inference is in n and every formula in the endsequent isin n , it follows that every formula in the free-cut free proof is in n . For convenience,assume also that the proof is in free variable normal form (so free variables are not reused).

Denition Let i 1 and A( x) i . If A i 1 then W it iA is dened to be theformula A. Otherwise, A is uniquely expressible in the form (y0) (yk )B( x, y) whereB i 1 . Then Wit iA (w, x) is the formula

B( x, (w)0, . . . , (w)k ).

Note that W it iA i 1 . If W it iA (w, x) holds, we say w witnesses the truth of A( x).

Main Lemma 10 ( n 1, m 2) Suppose I 0 + TI ( m , n 1) proves the sequent A1, . . . , Ak B1, . . . , B and that each Ai and B j is in n and that c are all the variables free in the sequent. Then there are functions f 1, . . . , f which are m -primitive recursivein n 1 and are n -denable in I 0 + TI ( m , n 1) such that I 0 + TI ( m , n 1)proves

W it nA 1 (w1, c), . . . , Wi tnA k (wk , c)Wit nB 1 (f 1( w, c), c), . . . , Wi t nB (f ( w, c), c).

Informally, the f 1, . . . , f will, given witnesses for all of A1, . . . , Ak , produce a witness forat least one of B1, . . . , B .

The proof of the Main Lemma is by induction on the number of inferences in a free-cutfree proof of the sequent. In the base case, there are zero inferences, so the sequent isan axiom and consists of 0 -formulas for these axioms, the lemma is trivial. Forthe induction step, the proof splits into cases depending in the nal inference of theproof. Most of the cases are straightforward; for example, if the last inference is an :left inference then the proof ends with

A1, . . . , AkB0( c, s), B2, . . . , BA1, . . . , Ak(z0)B0( c, z0), B2, . . . , B

where s = s( c) is a term with free variables from c only and where B1 is (z0)B0 andis of the form (z0) (zr )B ( z, c) with B n 1 (possibly r = 0). The inductionhypothesis is that

W it nA 1 (w1, c), . . . , Wi tnA k (wk , c)

Wit nB 0 ( c,s )(f 0( w, c), c), . . . , Wi t nB (f ( w, c), c) (4)is provable in I 0 + TI ( m , n 1) for appropriate functions f 0, f 2, . . . , f . If B1 n 2then Wit nB 0 is just B0 and Wit

nB 1 is just B1 ; and a single :right inference applied to (4)

gives

W it nA 1 (w1, c), . . . , Wi tnA k (wk , c)Wit nB 1 (f 1( w, c), c), . . . , Wi t nB (f ( w, c), c) (5)


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where f 1 is arbitrary. Otherwise, let f 1( w, c) be dened so that

f 1( w, c) = s( c), a1, . . . , a r where f 0( w, c) = a1, . . . , a r

if r > 0, and f ( w, c) = s( c) if r = 0. Clearly f 1 is m -primitive recursive in n 1since f 0 is and, also clearly, I 0 + TI ( m , n 1) proves (5) for this f 1 .We leave the rest of the simpler cases to the reader and consider only the two substantial

cases of :right and LOP ( m , n 1) as last inference. (Part of the :left case is alsosubstantial, but is very similar to :right .)

(:right ) Suppose the last inference is

A1, . . . , AkB0(b, c), B2, . . . , BA1, . . . , Ak(z0)B0(z0, c), B2, . . . , B

where the free variable b does not occur except as indicated and B1 is (z0)B0( z, c). SinceB1 is in n and has outermost quantier universal, it must therefore actually be in n 1and be of the form (z0) (zr )B ( z, c) where B n 2 . Also Wit nB 0 and Wit

nB 1 are

just B0 and B1 , respectively. The induction hypothesis is that I 0 + TI ( m , n 1)proves

Wit nA 1 (w1, c), . . . , Wi tnA k (wk , c)B0(b, c),Wit nB 2 (g2( w, b, c), c), . . . , Wi t nB (g ( w, b, c), c)

for functions g2, . . . , g which are m -primitive recursive in n 1 . The difficulty isthat these functions take b as an argument, but b is not free in the endsequent so we cannot just set f i = gi . The solution to this difficulty is to let C (v, c) be the n 2 -formulaB ((v)0, . . . , (v)r , c) and use the function U vC to nd a value, if any, for b such thatB0(b, c) holds: dene

f i ( w, c) = gi ( w, (U vC ( c) 1)0, c).

When B1( c) is false, U vC ( c) 1 codes a sequence b0, . . . , br such that B0(b0, . . . , br )and (U vC ( c) 1)0 equals b0 . Thus I 0 + TI ( m , n 1) proves

Wit nA 1 (w1, c), . . . , Wi tnA k (wk , c)B1( c),Wit nB 2 (f 2( w, c), c), . . . , Wi t nB (f ( w, c), c)

and f 2, . . . , f k are m -primitive recursive in n 1 since g2, . . . , gk are and since (v)C is in n 1 .

LOP ( m , n 1): Suppose the last inference is

0, A1(, c), A2, . . . , AkB1, . . . , B , ( )A1(, c)A1(0, c), A2, . . . , AkB1, . . . , B

where A1 n 1 , where 0 is a closed term with value a G odel number of anordinal m , where is a free variable, which appears only as indicated, and where


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( )A1( ) is an abbreviation for the formula ( )( A1( )). The inductionhypothesis states that I 0 + TI ( m , n 1) proves

0, A1(, c),Wit nA 2 (w2, c), . . . , Wi tnA k (wk , c)

Wit nB 1 (g1( w, , c), c), . . . , Wi t nB (g ( w, , c), c),g +1 ( w, , c) A1(g +1 ( w, , c), c)

for appropriate functions g1, . . . , g +1 . Dene

H (, c) = if A1(, c)0 otherwise

H is m -primitive recursive in n 1 since A1 n 1 . Now dene

F ( w, , c) = F ( w, H (g +1 ( w, , c), c), c) if H (g +1 ( w, , c), c) 0

otherwise .Clearly F is also m -primitive recursive in n 1 . Finally set

f i ( w, c) = gi ( w, F ( w, 0, c), c);

it is easy to check that I 0 + TI ( m , n 1) proves

A1(0, c),Wit nA 2 (w2, c), . . . , Wi tnA k (wk , c)

B1( c),Wit nB 2 (f 2( w, c), c), . . . , Wi t nB (f ( w, c), c)since F ( w, 0, c) gives the ordinal at which g +1 fails to give a smaller ordinal satisfying A1and with this ordinal, one of g1, . . . , g must produce a witness for the correspondingB1, . . . , B .

Q.E.D. Lemma 10 and Theorems 7, 8 and 9

The above proof did not consider the case where the last inference of the proof is aninduction inference: since induction is restricted to 0 -formulas and the witness formulafor a 0 -formula is just the formula itself, that case is completely trivial. However,I n is, by Proposition 2 a consequence of I 0 + TI ( m , n 1) and it must, a priori, bepossible to handle I n induction inferences by the witness function method as above. Infact, it is quite simple an I n -induction inference is handled by primitive recursion in n 1 . This leads to a direct proof of Theorems 8 and 9; we leave the details of this directproof to the reader.

We have now nished the characterization of the n -denable functions of I 0 + TI ( m , n 1) and of I n . It remains to characterize the k -denable functionsof these theories when k < n . (In section 6, we discuss the case k > n too). Thecentral result needed for this characterization is that the theory I 0 + TI ( m , n 1) isn +1 -conservative over I 0 + TI ( m +1 , n 2):


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Theorem 11 Let m 2 and n 1.

(a) I 0 + TI ( m , n 1) TI ( m +1 , n 2) .

(b) If I 0 + TI ( m , n 1) A where A n +1 ,then I 0 + TI ( m +1 , n 2) A.

Part (a) of this theorem is due to Gentzen [10]; the proof can be found in Lemma 3.4of [26] or Theorem 12.3 of [27] and is also repeated below. Part (b) extends theprior result of Schmerl [24] that I 0 + TI ( m , n 1) i s n 1 -conservative overI 0 + TI ( m +1 , n 2); Schmerls proof was based on reection principles. A weakerversion of (b) with 2 -conservativity in place of n +1 -conservativity can be found in [26].

Proof (a) By Proposition 1, it will suffice to show that the theory I 0 + TI ( m , n )can prove TI ( m +1 , n 1). Let A() n 1 and let HY P A be the formula( )[( )A( ) A( )] and let m +1 . We reason inside I 0 + TI ( m , n )to prove A() assuming HY P A . Let A() be the formula ( )A( ); by HY P A ,A() A( + 1). Let J ( ) be the formula

() A() A( + ) .

Clearly, J n . We shall use transnite induction on J to prove J (0) for some xed0 m such that 0 . Since A(0) holds trivially, J (0) implies A( 0 ) which, inturn implies A(). Thus it suffices to prove HY P J :

( )[( )J ( ) J ( )]

since, using TI ( m , n ), this implies J (0) holds for this particular 0 . First notethat J (0) holds by our observation that A() A( + 1). Now let be an arbitrarynon-zero ordinal and suppose ( )J ( ): we must prove J ( ) . I f is a successorordinal, = + 1, it suffices to show J ( ) J ( + 1), i.e.,

() A() A( + ) ( ) A( ) A( + +1 ) .

Assume J ( ) holds and let be arbitrary such that A( ) and let + +1 ; wemust show A( ). By consideration of Cantor normal forms, + n for somenite n . From J ( ), it follows that

() A() A( + k) () A() A( + (k + 1))

holds for all (nite) k . By ordinary n -induction, this implies that

() A() A( + k)

holds for all nite k . Thus A( ) holds. Finally, suppose is a limit ordinal and assume( )J () and assume A(). If + then + for some so A( )holds by J (). Since was arbitrary, J ( ) follows. That completes the proof of (a).

The proof of (b) consists of a partial formalization of the Main Lemma 10 in the theoryI 0 + TI ( m +1 , n 2). First an important lemma is necessary:


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Lemma 12 Let m 2 and n 2. I 0 + TI ( m +1 , n 2) can n -dene precisely them -primitive recursive in n 1 functions.

Proof By the just established part (a) of Theorem 11, every n -denable function

of I 0 + TI ( m +1 , n 2) is also n -dened by I 0 + TI ( m , n 1) and hence, byTheorem 7, is m -primitive recursive in n 1 . To show the converse, suppose F ( z)is dened from primitive recursive functions g, , and , from A(y) = ( x)B(x) withB n 2 , and from an ordinal 0 m as in the Second Normal From Theorem; soF ( z) = f ( ( z)) where

f ( ) = f ((, U A ( ))) if (, U A ( )) 0g( ) otherwise .

Recall the denition of an f -computation series 0, . . . , k used in the proof of Theorem 7to code a partial computation of f . In the proof of Theorem 7, the existence of a

maximal length f -computation series beginning with 0 = ( z) was proved by ndingthe least k such that there exists an f -computation series from 0 to k . The existenceof k was proved via LOP ( m , n ): this was the key step in n -dening F inI 0 + TI ( m , n 1).

To n -dene f and F in I 0 + TI ( m +1 , n 2) requires a more subtle argument.The basic motivation for this argument is that one could try to minimize the ordinals of the form

0 + 1 + + k 1 + k 2

with 0, . . . , k an f -computation series but this is too simplistic because of thepresence of the U A function. Instead, we encode partial computations of f by a sequenceof ordinals

0, 0, 1, 1, . . . , k , kwhere 0, . . . , k is an f -computation series and where each i and encodes the valueof U A ( i ):

Denition Let be the G odel number of an ordinal . Then D() is the integerdened by

D() = 0 if = n + 1 if = n

Denition An f -computation ordinal (fCO) is (the G odel number of) an ordinal of theform

2 0 + 0 +

2 1 + 1 + 2 k 1 + k 1 +

2 k + k + 2 k + k

(only the nal summand is repeated), where

(i ) i+1 i 0 , for 0 i < k ,

(ii ) i , for 0 i < k ,

(iii ) i+1 = ( i , D ( i )), for 0 i < k ,


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(iv ) For 0 i k ,

if i = n , then B( i , n) and for all m < n , B( i , m)

if i = , then (m)B( i , m),

(v ) It is not the case that ( k , D (k )) k 0 .

A psuedo- f -computation ordinal (PfCO) is dened exactly like an f -computation ordinalexcept that ( v ) is omitted and ( iv ) is replaced by

( iv ) For 0 i k , if i = n then B( i , n).

We write fCO (, z) and PfCO (, z) for formulas expressing the condition that is anfCO or PfCO, respectively, with 0 = ( z).

The quantier complexity of PfCO is easily analyzed since (i )-(iii ) are primitive recursive

and ( iv ) is n 2 since B n 2 and by Bn 2 -collection (which is a consequenceof I 0 + TI ( m +1 , n 2) since this theory contains I n 1 ). Thus PfCO is a n 2formula. Letting 1 =

2 0 + +1 we have that 1 m +1 and, therefore, if ( z) 0 andPfCO (, z), then 1 . We henceforth assume w.l.o.g. that ( z) 0 . Now, thereexists such that PfCO (, z); namely,

2 (x)+ 2. Hence, by LOP ( m +1 , n 2),there is a minimum ordinal denoted min such that PfCO (min , z). We claim thatfCO ( min , z) also holds. To prove this, suppose

min = 2 0 + 0 + +

2 k + k + 2 k + k ;

the only way fCO ( min ) can fail is if condition (iv ) or (v ) is violated. First suppose ( iv )fails for some value of i . Then, if i = but B( i , m) holds, then

2 0 + 0 + +

2 i 1 + i 1 + 2 i + m +

2 i + m (6)

is a psuedo f -computation ordinal min violating the choice of min . Likewise, if i = n but B( i , m) holds with m < n , then the same ordinal (6) is a psuedof -computation ordinal min . Hence (iv ) must hold. Now suppose ( v ) fails. Then,

2 0 + 0 + +

2 k 1 + k 1 + 2 k + k +

2 k +1 + + 2 k +1 +

where k+1 = ( k , D (k )) is a psuedo f -computation ordinal min , which is again a

contradiction. Hence ( v ) must also hold and min is an fCO.Thus I 0 + TI ( m +1 , n 2) can dene F ( z) by proving

(z)(!y) () PfCO (, z) ( )( PfCO ( , z))

= 2 0 + 0 + +

2 k + k 2y = g( k ) . (7)

PfCO is a n 2 -formula so the subformula ( )( ) is in n 1 and the subformula()( ) is a n -formula; thus this is a n -denition of F ( z) in I 0 + TI ( m +1 , n 2).Q.E.D. Lemma 12


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Lemma 12 stated that the n -denable functions of I 0 + TI ( m +1 , n 2) arepreciselythe m -primitive recursive in n 1 functions; the lemma was proved using the secondnormal form for such functions. However, this use of the second normal form was notessential for the proof: I 0 + TI ( m +1 , n 2) can also prove that the m -primitiverecursive in n 1 functions are closed under composition and under m -primitiverecursion. These closure properties are proved in I 0 + TI ( m +1 , n 2) by formalizingthe proofs of the three normal form theorems. Since the proofs of the normal form theoremwere completely constructive, this formalization is straightforward (and left to the reader).

We are now ready to return to the proof of part (b) of Theorem 11, for which itsuffices to prove that if B( c) i s a n -formula and I 0 + TI ( m , n 1) proves thesequent B( c), then so does I 0 + TI ( m +1 , n 2). In fact, more than this is true:a sequent of n -formulas is a consequence of I 0 + TI ( m , n 1) if and onlyif it is a consequence of I 0 + TI ( m +1 , n 2) this is a corollary of the next lemma.

Main Lemma 13 ( n 2, m 2) Suppose I 0 + TI ( m , n 1) proves the sequent A1, . . . , Ak B1, . . . , B and that each Ai and B j is in n and that c are all thevariables free in the sequent. Then there are functions f 1, . . . , f which are m -primitive recursive in n 1 and are n -denable in I 0 + TI ( m +1 , n 2) such that I 0 + TI ( m +1 , n 2) proves

W it nA 1 (w1, c), . . . , Wi tnA k (wk , c)Wit nB 1 (f 1( w, c), c), . . . , Wi t nB (f ( w, c), c).

The proof of Lemma 13 is exactly like the proof of Lemma 10 except that now thedenitions of the functions f 1, . . . , f k and the proofs that they produce the correct

witnesses are now carried out in I 0 + TI ( m +1 , n 2) the reader should referback to the earlier proof to verify that it works out as claimed. 2

Now suppose A1, . . . , A k B1, . . . , B is a sequent of n -formulas which is provablein I 0 + TI ( m , n 1). By the just stated lemma and from the denition of Wit ,I 0 + TI ( m , n 1) proves

Wit nA 1 (w1, c), . . . , Wi tnA k (wk , c)B1( c), . . . , B ( c)

which, via :left inferences gives

A1( c), . . . , Ak ( c)B1( c), . . . , B ( c).Q.E.D. Theorem 11

Theorem 14 Let m 2 and n 1 and 1 k n 1. Then I 0 + TI ( m , n 1) TI ( m + k , n 1 k ) and I 0 + TI ( m , n 1) is conservativeover the theory I 0 + TI ( m + k , n 1 k ) with respect to n +2 k -consequences.

Proof Apply Theorem 11 k times. 2


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Corollary 15 Let n 1. The theory I n contains and is 3 -conservative over thetheory I 0 + TI ( n +1 , 0) .

Proof Take m = 2; since I n is equal to I 0 + TI ( 2, n 1) the previous theorem

with k = n 1 yields the corollary. 2

Now we are ready to prove the theorem characterizing the j -denable functions of I 0 + TI ( m , n 1) and of I n for all 1 j n .

Theorem 16 Let m 2 and 1 j n .

(a) If j > 1 then the j -denable functions of I 0 + TI ( m , n 1) are precisely the functions which are m + n j -primitive recursive in j 1 .

(b) (For j = 1 .) The 1 -denable functions (i.e., the provably recursive functions) of I 0 + TI ( m , n 1) are precisely the functions which are m + n 1 -primitiverecursive.

Theorem 17 Suppose 1 j n . The functions which are j -denable in I n areprecisely the functions which are n j +2 -primitive recursive in j 1 .

Theorem 18 Let n 1. The provably total functions of I n are precisely the n +1 -primitive recursive functions.

Proof The proof of Theorem 16 is phrased for j > 1, but applies equally well to the j = 1case. Suppose F ( z) is j -dened by I 0 + TI ( m , n 1) proving (z)(!y)A(y, z)where A j . By Theorem 14 with k = n j , I 0 + TI ( m + n j , j 1) also provesthe j +1 -sentence (z)(!y)A; that is, it also j -denes f . Hence, by Theorem 7,F ( z) is m + n j -primitive recursive in j 1 . Conversely, every m + n j -primitiverecursive in j 1 function is j -denable in I 0 + TI ( m + n j , j 1), and hencein I 0 + TI ( m , n 1), by Theorems 7 and 14. That proves Theorem 16. The-orems 17 and 18 are corollaries of Theorem 16, since I n is the same theory asI 0 + TI ( 2, n 1). 2

Theorem 18 immediately implies the well-known fact that the provably total functionsof Peano arithmetic are precisely the 0 -primitive recursive functions.

4 n+1 -induction rule versus n induction axiom

This section presents a sketch for a proof of Parsons theorem on the conservativity of arestricted n +1 -induction rule over the usual n -induction axiom this proof is basedon the witness function method. For reasons of length we omit the details of the proof.

The n +1 -strict induction rule allows inferences of the form

A(0) A(b)A(b + 1)A(t)


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where b is the eigenvariable and occurs only as indicated, t is any term and A is inn +1 . Note that no side formulas are allowed (otherwise it would be equivalent to then +1 -induction axiom). The strict induction rule is equivalent to what Parsons calls theinduction rule modied only slightly to t in the framework of the sequent calculus.By free-cut elimination any sequent of n +1 -formulas which is provable in I 0 plus then +1 -strict induction rule has a proof in which every formula is in n +1 .

Notation n +1 -IR denotes the theory of arithmetic I 0 plus the n +1 -strict inductionrule. This system is always presumed to be formalized in the sequent calculus.

It is not too difficult to see that n +2 -IR proves the n induction axioms, for all n 0.To prove this, if A(b) n , use the strict induction rule on the formula

[A(0) (x)(A(x) A(x + 1))] A(b)

with respect to the variable b.

Theorem 19 (Parsons [22]) Let n 1. A n +1 -sentence is a theorem of I n iff it is a consequence of n +1 -IR.

Parsonss proof of Theorem 19 was based on the Godel Dialectica interpretation; otherproof-theoretic proofs of Theorem 19 have been given in [19, 25]. The main novelty of ourproof outlined below is that it uses the witness function method directly.

Proof (Outline): The easy direction is that if I n A where A n +1 , then n +1 -IRalso proves A. Since A n +1 , A is expressible as (x)B( x) where B n ; it suffices toshow that n +1 -IR B( c). By free-cut elimination, there is a I n -proof P of B( c) suchthat every formula occuring in P is a n -formula. We now can prove by induction on thenumber of inferences in this proof that every sequent in P is a consequence of n +1 -IR .The only difficult case is the induction inferences, which are of the form

, A(b)A(b + 1) , , A(0)A(t),

Letting D(b) be the formula ( A(b))( ), the upper sequent is logically equivalentto D(b) D(b + 1) and the lower sequent is logically equivalent to D(0) D(t). And

if n +1 -IR proves the upper sequent, then it also proves the lower sequent by use of thestrict induction rule on the formula D(0) D(b), which, as a Boolean combination of n -formulas is logically equivalent to a n +1 -formula.

For the hard direction of Theorem 19, we need the next lemma. We let P RA n be a setof function symbols for the functions which are primitive recursive in n . By Theorem 8,each function symbol in PRA n 1 represents a function which is n -denable in I n we may augment the language of I n with these function symbols, provided we are carefulnot to use them in induction formulas. In the next lemma, the notation xi denotes avector of variables and || xi || denotes the number (possibly zero) of variables in the vector.


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Lemma 20 Suppose Ai ( xi , c) and B j ( y j , c) are n -formulas, for 1 i k and 1 j , and that n +1 -IR proves the sequent

(x1)A1( x1, c), . . . , (xk )Ak ( xk , c)(y1)B1( y1, c), . . . , (y )B ( y , c). (8)Let f 1, . . . , f k be new function symbols so that f i has arity || xi || + || c|| . Then there areterms t i ( y1, . . . , y , c) in the language P RA n 1 {f 1, . . . , f k} , for 1 i , such that I n proves

(x1)Wit nA 1 (f 1( x1, c), x, c), . . . , (xk )WitnA k (f k ( xk , c), x, c)

Wit nB 1 (t1, y1, c), . . . , Wi t nB (t , y , c). (9)Theorem 19 follows immediately from Lemma 20 with k = 0 and = 1 and from thefact that every P RA n 1 -function is denable in I n . For reasons of length, we omit the

proof of Lemma 20: the general idea of the proof is a relatively straightforward use of thewitness function method; however, it requires the development of some deep facts aboutprimitive recursive (in n ) functions. An important feature of the lemma is that eachterm t i may involve all of y1, . . . , y .

A second theorem of Parsons is that Theorem 19 also holds with the addition of theB n -collection axiom:

Theorem 21 (Parsons [22]) Let n 1. The n +1 -consequences of n +1 -IR + B n arethe same as the n +1 -consequences of I n .

Proof (Outline) Recall that Bn 1 is equivalent to B n , relative to the base theory I 0 .The Bn 1 axioms contain unbounded quantiers in the scope of bounded quantiers, soit is not possible to use free-cut elimination to force a proof in n +1 -IR + B n to containonly n +1 -formulas. We let +n denote the set of formulas which have n blocks of likeunbounded quantiers, starting with a block of universal quantiers, allowing arbitrarybounded quantiers to be included in the rst block of unbounded quantiers (see the nextsection for a careful denition of the analogous class +n ). Now, temporarily dene the setof n formulas to be the formulas which are of one of the following forms: (1) (y)B( x)where B +n 1 or (2) (z t)(y1)B(y1, z, c) where B n 1 . We also dene the

n +1 formulas to be the formulas which are either n +1 or

n . Since the Bn 1 axiomscan be formulated in the form AA with A and A both in n , the free-cut eliminationtheorem implies that if is a sequent of n +1 -formulas provable in n +1 -IR + Bn ,then this sequent has a proof in which every formula is a n +1 -formula. The notion of witness can be generalized as follows: if A( c) is a n -formula in one of the above forms;then, if A is of form (1), W itnA (w, c) is dened just like Wit nA (w, c) was and, if A is of form (2) then WitnA (w, c) is dened to be the formula

(z t)Wit n(y1 )B ((w)z , z, c).


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Lemma 22 Suppose Ai ( xi , c) and B j ( y j , c) are n -formulas, for 1 i k and 1 j , and that n +1 -IR + B n proves the sequent

(x1)A1( x1, c), . . . , (xk )Ak ( xk , c)(y1)B1( y1, c), . . . , (y )B ( y , c).Let f 1, . . . , f k be new function symbols so that f i has arity || xi || + || c|| . Then there areterms t i ( y1, . . . , y , c) in the language P RA n 1 {f 1, . . . , f k} , for 1 i , such that I n proves

(x1)W itnA 1 (f 1( x1, c), x, c), . . . , (xk )WitnA k (f k ( xk , c), x, c)

WitnB 1 (t1, y1, c), . . . , Wi t nB (t , y , c). (10)We omit the proof of the lemma and the rest of Theorem 21.

Finally, it should be remarked that n +1 -IR + B n does not contain I n . Thiscan be proved by noting that n +1 -IR + I n is not n +2 -conservative over I n . Forexample, with n = 1, let A(k, m) be the Ackermann function so that the functionsf k (m) = A(k, m) are all primitive recursive and so that each primitive recursive functionis eventually dominated by f k for sufficiently large k . Let A(k,m,y ) be the graph of theAckermann function; it is well-known that A(k,m,y) is 0 (for us it is sufficient that itis 1 ). Now, it is easy to see that I 1 proves (x)(y)A(0,x ,y) and

(x)(y)A(b,x,y)(x)(y)A(b + 1 ,x ,y).

Thus 2-IR+ I 1 (k)(x)(y)A

(k,x,y). But the Ackermann function is not primitiverecursive, hence not 1 -denable in I 1 . Thus 2-IR + I 1 is not 2 -conservative overI 1 and thus not equal to 2-IR and not a subtheory of 2-IR + B 1 .

To show n +1 -IR + B n I n for n > 1, use essentially the same argument, but useprimitive recursive in n 1 in place of primitive recursive and use a suitable replacementof the Ackermann function that dominates the functions primitive recursive in n 1 .

5 Conservativity of Collection over Induction

In this section we prove the well-known theorem that the B n +1 -collection axioms aren +2 -conservative over I n . The proof method does not use the witness function methodper se, but it involves an induction on the length of free-cut free proofs similar to themethods of earlier sections. Earlier proofs of this theorem include Parsons [22] andParis-Kirby [21]; see in addition, [3, 25]. The advantage of our proof below is that it givesa direct and elementary proof-theoretic proof.

Recall that the B n +1 -collection axioms are equivalent to the Bn -collection axioms.In the sequent calculus, the Bn -collection axioms are of the form

(x a)(y)A(x, y)(z)(x a)(y z)A(x, y)


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where A n and may contain free variables besides x, y . In the above sequent thereare bounded quantiers outside of unbounded quantiers so the formulas are not, strictlyspeaking, n +1 -formulas. Accordingly, we dene a generalized form of n +1 -formulasthat will be allowed to appear in free-cut free proofs.

Denition The class +n +1 of formulas is dened inductively by

(1) n +n +1 ,

(2) If A +n +1 , then (x)A, (x t)A and (x t)A are in +n +1 , where t is any term

not involving x .

If s is a term and A is a +n +1 -formula, then A s is the formula obtained by boundingunbounded existential quantiers in the outermost block of quantiers of A by the term s ;

namely,Denition Fix n and suppose A +n +1 .

(1) If A n , then A s is A.

(2) If A is (x)B and A /n , then A s is (x s)B .

(3) If A is (Qx t)B then A s is (Qx t)(B s ).

Let be a sequent A1, . . . , Ak B1, . . . , B of +n +1 -formulas. Then s is theformula

k

i=1

A si and s is the formula j =1

B s j . This notation should cause no confusion

since antecedents and succedents are always clearly distinguished.

If c = c1, . . . cs is a vector of free variables, then c u abbreviates the formulac1 s cs u . (c u) and (c u) abbreviate the corresponding vectors of bounded quantiers.

Theorem 23 ( n 1) Suppose is a sequent of +n +1 -formulas that is provable in I 0 + B n +1 . Let c include all the free variables occurring in . Then

I n (u)(v)(c u) u v .

Intuitively, the theorem is saying that given a bound u on the sizes of the free variablesand on the sizes of the witness for the formulas in , there is a bound v for the values of a witness for a formula in .

Theorem 23 immediately implies the main theorem of this section:

Theorem 24 I 0 + B n +1 is n +2 -conservative over I n .

Recall that I 0 + B n +1 I n . Before proving Theorem 23, we establish the followinglemma (due to Clote and Hajek).


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Lemma 25 ( n 1) Let B( c, d) n . Then

I n (u)(v)(c u)[(x)B( c, x) (x v)B( c, x)].

The formula of Lemma 25 is called the n -strong replacement principle.Proof Let s be the length of the vector c. We reason inside I n . Let C ( c, d) be the n -formula B( c, d). Let Num (u, ) be the formula expressing

c1, d1, . . . , c , d s.t. c1, . . . , c are distinct s -tuples u and C ( ci , di ) holdsfor all 1 i .

Of course, this asserts that there are distinct values of c u for which (x)C ( c, x)holds. Now Num is a n -formula and Num ( c,(u + 1) s + 1) is clearly false; so by I n ,there is a value 0 such that Num ( c, 0) but not Num ( c, 0 + 1). Given c1, d1, . . . , c 0 , d 0witnessing Num ( c, 0), let v = max {d1, . . . , d 0 } . It follows that

(c u) (x)C ( c, x) (x v)C ( c, x)

which is what we needed to prove. 2

Proof of Theorem 23: By free-cut elimination, has a sequent calculus proof P in which every formula is a +n +1 -formula. (Since we allow bounded quantiers in

+n +1 -

formulas, it is convenient to work in the sequent calculus LKB with inference rules forbounded quantiers [2].) We prove the theorem by induction on the number of inferencesin P . The proof splits into cases depending on the last inference of P . The hardest case,

:right is saved for last.Case (1): If P has no inferences and is an initial sequent, then either isa logical, equality or arithmetic axiom, containing only 0 -formulas, and the theorem istrivial, or is a B n +1 axiom. In the latter case, taking v = u , it is immediatethat I n proves

(x a)(y u)A(x, y) (z u)(x a)(y z)A(x, y)

and the theorem holds.

Case(2): Suppose the last inference of P is a structural inference, a propositional inference

or a :left or :left inference. The inference may have either one or two premisses: or

1 1 2 2

It is easily checked that, in the rst case we have that I n proves u u and v v and, in the second case we have that I n proves u u1

u2 and

v1 v

2 v . In the rst case, the induction hypothesis states that I n proves

(v)(c u) u v


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from which (v)(c u)( u v) follows. In the second case, by the inductionhypothesis, I n proves

(vi )(c u) ui vi

i

for i = 1 , 2. Taking v = max {v1, v2} and noting that I n proves vi v vii vi ,we get that I n proves (v)(c u)( u v ).

Case (3): Suppose the nal inference of P is an :right inference:

B( c, t( c)), (x)B( c, x),

We reason inside I n as follows: given arbitrary u , there is (by the induction hypothesis)a v such that

(c u) u B v ( c, t( c)) v .

Letting v = max {v , t(u , . . . , u )} we have that t( c) v for all c u (since the languagehas 0, S , + and as the only function symbols). This v makes the theorem true. Thecase where the last inference of P is a :right is similar.

Case (4): Suppose the last inference of P is an :left :

A( c, d), (x)A( c, x),

where d is the eigenvariable occuring only where indicated. The induction hypothesis isthat I n proves

(u)(v)(c, d u) A u ( c, d) u v .

This is equivalent to

(u)(v)(c u) (d u)A u ( c, d) u v

which is what we needed to prove.

Case (5): The :left inference is a little more subtle. If the nal inference of P is

d t( c), A( c, d), (x t( c))A( c, x),

we reason inside I n as follows. Let u be arbitrary, there is a v such that

(c, d u) d t( c) A u ( c, d) u v . (11)

Let u = max {u, t ( u)} ; by the induction hypothesis, there is a v such that (11) holds withu , v in place of u, v . Now let c u and suppose (x t)A u ( c, x) u . Clearly, thisimplies (x u )(x t A u u ). Taking d to be this x , we have v holds.


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Case (6): Suppose the last inference of P is a Cut :

1 1, A A, 2 21, 2 1, 2

We reason inside I n . Suppose u is arbitrary and u1 u

2 . Pick v1 , depending onlyon u by the induction hypothesis, so that v1 A v1 . Let u2 = max {v1, u} . By theinduction hypothesis, there is a v v1 depending only on u2 so that if A v1 holds, then v2 holds. Now clearly either

v1 or

v2 holds. Since v depends only on u , this proves

this case.

Case (7): Suppose the nal inference of P is a :right :

B( c, d), (x)B( c, x),

Note B n since (x)B must be a +n +1 -formula. We reason inside I n . Let u bearbitrary. By n -strong replacement (Lemma 25) there is a u u such that

(c u) (x)B( c, x) (x u )B( c, u ) .

Let v u be given by the induction hypothesis so that

(c, d u ) u B( c, d) v . (12)

Now let c u be arbitrary such that u . We need to show (x)B( c, x) v . Supposenot, then there is a d u such that B( c, d), and by (12), v holds, which is acontradiction.

The case where the nal inference of P is a :left inference is similar, althoughLemma 25 is not needed.Q.E.D. Theorem 23

It would be interesting to give a similar proof that n +1 -IR + B n is n +1 -conservativeover n +1 -IR , in place of the more complicated and omitted proof of Theorem 21 above.

6 Analogies between Bounded and Peano ArithmeticThe witness function method has been extensively used characterizing denable functionsof fragments of bounded arithmetic the work in section 3 above gives an approachto Peano arithmetic which is very similar to some of the proofs used earlier in boundedarithmetic.

First, Theorem 8, which characterized the n -denable functions of I n is analogousto the main theorem of Buss [2] which characterized the bn -denable functions of S n2(which is axiomatized with bn -PIND axioms). In I n , the n -denable functions


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are precisely the functions primitive recursive in n 1 ; whereas, in S n2 , the bn -denablefunctions are precisely the functions polynomial time computable with respect to a (usual) pn 1 -oracle. It should be noted that a usual

pn 1 -oracle is equivalent to a witness oracle

for pn 1 with respect to polynomial time computation, since there is an a-priori boundon the size of a witness and a witness value may be queried bit-by-bit. The proofs of thesetwo theorems are analogous as well.

Second, Theorem 11, which stated that I 0 + TI ( m , n 1) is n +1 -conservative over I 0 + TI ( m +1 , n 2) is analogous to the result of [4] thatS n2 is bn -conservative over T

n 12 . To see the analogy more sharply, note

on one hand I 0 + TI ( m , n 1) and I 0 + TI ( m +1 , n 2) are equivalent toI 0 + TI ( m , n ) and I 0 + TI ( m +1 , n 1) (respectively), which are axiomatizedwith transnite induction on n -formulas up to ordinals m and on n 1 formulasup to ordinals m ; and on the other hand, S n2 may be axiomatized by induction(PIND) on b

n-formula up to lengths |x| and T n 1

2may be axiomatized by induction

on bn -formulas up to 2 |x | . So both conservation theorems give situations where thecomplexity of induction formulas may be reduced by one block of quantier alternationin exchange for exponentiating the length of induction. Another theorem of this type isthe result of [6] that Rn3 is bn -conservative over S

n 13 .

Witness oracles have been applied to bounded arithmetic in [18] and in [6]. Anotherarea of contact between bounded arithmetic and Peano arithmetic may be found inKaye [14] who gives a proof that I n = B n +1 based on methods used earlier by [18] toshow that if T n +12 = S

n +12 then the polynomial time hierarchy collapses.

We conclude with a partial characterization of the j -denable functions of I n when

j > n :Denition Let A be a formula; w.l.o.g. all negations in A are on atomic formulas. Thecounterexample oracles of A are the witness oracles U (x ) B for (x)B a subformula of A.

Theorem 26 Let j > n 1. Suppose I n (x)(!y)A(x, y) where A j . Then the function f : x y, such that (x)A(x, f (x)) , is primitive recursive in n 1 and in thecounterexample oracles for A.

The same holds for I 0 + TI ( m , n 1) with primitive recursive replaced by m -primitive recursive.

The proof of this theorem is analogous to the proof of Theorems 7 and 8 except thatthe :right cases of the proof now have to accommodate the fact that a :right quantiermay be an ancestor of a quantier in ( y)A(c, y). Of course a counterexample oracle forA is exactly what is needed for this case.

Theorem 26 can be extended to partially characterize the b j -denable functions of T n 12 or S n2 when j > n ; namely,

Theorem 27 (See [18, 23, 16]) Let j > n 1.


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(a) Suppose A b j and S n2 (x)(!y)A(x, y) . Then the function f such that (x)A(x, f (x)) can be computed by a polynomial time Turing machine with an oracle for pn 1 and with the counterexample oracles of A.

(b) Suppose A b j and T n 12 (x)(!y)A(x, y) . Then the function f such that (x)A(x, f (x)) can be computed by a polynomial time Turing machine which makesa constant number of queries to an oracle for pn 1 and to the counterexample oraclesof A.

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