The Witness Function Method and Provably Recursive Functions of Peano Arithmetic Samuel R. Buss * Department 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 recursive functions of the fragments I Σ n of Peano arithmetic. The proof method also characterizes the Σ k -definable functions of I Σ n and of theories axiomatized by transfinite induction on ordinals. The proofs are completely proof-theoretic and use the method of witness functions and witness oracles. Similar methods also yield a new proof of Parson’s theorem on the conservativity of the Σ n+1 -induction rule over the Σ n -induction axioms. A new proof of the conservativity of BΣ n+1 over I Σ n is given. The proof methods provide new analogies between Peano arithmetic and bounded arithmetic. 1 Introduction The witness function method has been used with great success to characterize some classes of 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 can be applied to the fragments I Σ n of Peano arithmetic to characterize the functions which are provably recursive in these fragments. This characterization of provably recursive functions has already been performed by a variety of methods; including: via Gentzen’s assignment 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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The Witness Function Method andProvably Recursive Functions of Peano Arithmetic
This paper presents a new proof of the characterization of the provably recursivefunctions of the fragments IΣn of Peano arithmetic. The proof method alsocharacterizes the Σk -definable functions of IΣn and of theories axiomatized bytransfinite 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 Parson’s theorem on the conservativityof the Σn+1 -induction rule over the Σn -induction axioms. A new proof of theconservativity of BΣn+1 over IΣn 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 classes
of 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 can
be applied to the fragments IΣn of Peano arithmetic to characterize the functions which
are provably recursive in these fragments. This characterization of provably recursive
functions has already been performed by a variety of methods; including: via Gentzen’s
assignment of ordinals to proofs [9, 27], with the Godel 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.
paper is, firstly, that they provide a simple, elegant and purely proof-theoretic method
of characterizing the provably total functions of IΣn and, secondly, that they unify the
proof methods used for fragments of Peano arithmetic and for bounded arithmetic.
The witness function method is related to the classical proof-theoretic methods of
Kleene’s recursive realizability, Godel’s Dialectica interpretation and the Kreisel no-
counterexample interpretation; however, the witness function method does not require
the use of functionals of higher type. We feel that the witness function method provides
an advantage over the other methods in that it leads to a more direct and intuitive
understanding of many formal systems. The classical methods are somewhat more general
but are also more cumbersome and more difficult to understand (consider the difficulty
of comprehending the Dialectica interpretation or no-counterexample interpretation of a
formula with more than three alternations of quantifiers, for instance). On the other hand,
the more direct and intuitive witness function method has been extremely valuable for the
understanding of why the provably total functions of a theory are what they are and also
for the formulation of new theories for desired classes of computational complexity and,
conversely, for the formulation of conjectures about the provably total functions of extant
theories. The main support for our favorable opinion of the witness function method
is, firstly, its successes for bounded arithmetic and, secondly, the results of this paper
showing its applicability to Peano arithmetic.
While checking references for this paper, the author read Mints [19] for the first
time — it turns out that Mints’s proof that the provably recursive functions of IΣ1 are
precisely the primitive recursive functions is based on what is essentially the witness
function method. This theorem of Mints is, in essence, Theorem 9 below. Mints’s use
of the witness function method predates its independent development by this author for
applications to bounded arithmetic. The present paper expands the applicability of the
witness function method to all of Peano arithmetic.
The outline of this paper is as follows: section 2 develops the necessary background
material on Peano arithmetic, the subtheories IΣn , transfinite induction axioms, least
ordinal principle axioms, the sequent calculus and the correct notion of free-cut free proof
for transfinite induction/least number principle axioms. In section 3, the central notions
of the witness function method and witness oracles are developed and the Σn -definable
functions of IΣn and I∆0 + TI(ωm, Πn) are characterized. This includes the definition
of α-primitive recursive (in Σk ) functions and normal forms for such functions. Then
the provably recursive (i.e., Σ1 -defined) functions of IΣn are characterized by proving a
conservation theorem for TI(ωm, Πn) over TI(ωm+1, Πn−1). Section 4 outlines a proof of
Parson’s theorem on the conservativity of the Πn+1 -induction rule over the Σn -induction
axiom. Section 5 contains a proof of the Πn+1 -conservativity of BΣn+1 over IΣn .
Section 6 concludes with a discussion of the analogies between the methods of this paper
and the methods used for bounded arithmetic.
2 Preliminaries
2.1 Arithmetic and Ordinals
Peano arithmetic (PA) is formulated2 in the language 0, S , +, · and ≤ . It has induction
axioms
A(0) ∧ (∀x)(A(x) → A(S(x))) → (∀x)A(x)
for all formulas A , plus it has a finite base set of axioms, namely, Robinson’s theory Q of
seven axioms defining 0, S , + and · and, in addition, the axiom
(∀x)(∀y)(x ≤ y ↔ (∃z)(x + z = y))
which defines ≤ . A bounded quantifier is of the form (∃x ≤ t) or (∀x ≤ t) where t is
any term not involving x . The usual quantifiers, (∀x) and (∃x), are called unbounded
quantifiers. The ∆0 -formulas, or bounded formulas, are the formulas in which every
quantifier is bounded. The classes Σn and Πn of formulas are defined by induction on n ,
so that Σ0 = Π0 = ∆0 and so that Σn+1 is the set of formulas of the form (∃~x)B where
B ∈ Πn and so that Πn+1 is defined dually. The theory IΣn is defined to be the theory
in the language of Peano arithmetic with the same eight non-induction axioms as PA and
with induction axioms for all formulas A ∈ Σn .
The collection axioms provide an alternative way to define fragments of Peano
We let BΣn denote the set of collection axioms for all A ∈ Σn ; BΠn is defined similarly.
It is well-known that I∆0 + BΣn+1 ² IΣn and IΣn ² BΣn . It is also well-known
that I∆0 + BΣn+1 is Πn+1 -conservative over IΣn and we shall reprove this in section 5
below. An important feature of the collection axioms is that it provides a ‘quantifier
exchange’ principle that allows moving bounded quantifiers inside the scope of unbounded
quantifiers. The classes Σn and Πn can be generalized to classes ΣGn and ΠG
n by allowing
bounded quantifiers to appear anywhere in the formula (instead of only in the ∆0 matrix)
but counting only the alternations of unbounded quantifiers. For example, the hypothesis
and conclusion of the collection axiom above are ΣGn -formulas if A ∈ Σn . The theory
I∆0 + BΣn , and hence IΣn , can prove that every ΣGn -formula is equivalent to a Σn -
formula.
Remark: Some authors include function symbols for all primitive recursive functions in
the language of PA. We do not adopt this convention; however, as is well-known, every
primitive recursive function is provably recursive (Σ1 -definable, see below) in IΣ1 and
hence the theories IΣn , for n ≥ 1 are not significantly affected by the addition of symbols
2Our 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 definition of IΣn and PA is equivalent tothe usual one apart from the inessential replacement of < by ≤ .
for primitive recursive functions. Thus the theorems and proofs of this paper also apply
to theories with symbols for primitive recursive functions.
Definition Let T be a subtheory of PA and f : Nk → N . The function f is Σi -definable
in T iff there is a formula A(x1, . . . , xk, y) ∈ Σi such that
(1) T ` (∀~x)(∃!y)A(~x, y), and
(2) {(~n,m): N ² A(~n,m)} is the graph of f , i.e., A(~n,m) holds iff f(~n) = m for all
integers ~n,m .
The function f is provably recursive in T iff f is Σ1 -definable in T .
The intuitive idea of ‘provably recursive’ is that the theory T should prove that
some Turing machine M , which computes f , halts on all appropriate inputs. Since
A(~x, y) can be taken to be a Σ1 -formula expressing “there is a w which codes a halting
M -computation with input ~x and output y”, it is clear that any function which is provably
recursive in this intuitive sense is also Σ1 -definable. Conversely, if f is Σ1 -definable in T ,
then there is Turing machine M which computes f , provably in T . Namely, M performs
a brute-force search for values of y and the unboundedly existentially quantified variables
of A . Thus ‘Σ1 -definable’ coincides with the intuitive notion of ‘provably recursive’.
One reason that the provably recursive functions of T are of particular significance is
that if f is provably recursive in T , then T may conservatively extended by adding f as
a new function symbol with f(~x) = y ↔ A(~x, y) as a new axiom. If T is a fragment IΣn
then f may be used freely in induction formulas (without affecting quantifier complexity).
Similarly, if T can prove that a Π1 -formula and a Σ1 -formula are equivalent then T can
conservatively extended by adding a new predicate symbol with arguments including the
free variables of the two formulas and adding a new axiom defining the predicate symbol
to be equivalent to the formulas. The new predicate may also be used freely in induction
formulas. Such new predicates are called ∆1 -defined predicates of T .
Recall that IΣ1 (and even I∆0 ) can formalize many metamathematical notions; of
particular importance are the sequence coding functions 〈x0, . . . , xk〉 , (〈x0, . . . , xk〉)i = xi ,
and Len(〈x0, . . . , xk〉) = k + 1.
The ordinals are set-theoretically defined to be those sets which are transitive and
well-founded by α . We write ≺ for the ordering of ordinals, so α ≺ β means α ∈ β .
It is well-known how to define ordinal addition, multiplication and exponentiation. The
Cantor normal form for an ordinal α is the unique expression
α = ωγ1 · n1 + ωγ2 · n2 + · · ·ωγr · nr
where γ1 Â γ2 Â · · · Â γr are ordinals and n1, . . . , nr are positive integers (i.e., nonzero,
finite ordinals). Here ω is the first infinite ordinal; we let ω0 = 1, ω1 = ω and, generally,
ωn+1 = ωωn . Thus ωn is a stack of n ω ’s. The limit of ωn as n → ω is called ε0 ; hence
ε0 is the least ordinal such that ε0 = ωε0 . For α ≺ ε0 , the Cantor normal form can
be extended so that the exponents γi are also written in Cantor normal form, and with
exponents in the latter Cantor normal forms also in Cantor normal form, etc. (eventually
the process must stop). For example,
ωωω0·3+ωω0·2 · 4 + ω0
is a Cantor normal form; usually this is expressed more succinctly as ωω3+ω2 · 4 + 1. In
this paper, we shall always use ordinals 4 ε0 and by Cantor normal form always means
the extended version with exponents also in Cantor normal form. ε0 is its own Cantor
normal form.
By using Godel numbering, integers can encode Cantor normal forms and this can
be intensionally formalized3 in IΣ1 ; with care, these can even be formalized in I∆0 . In
particular, I∆0 can define the relation IsOrdinal(x) expressing that x is the Godel number
of 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 actual
and for metamathematical operations; for example, ω + 1 also denotes its own Godel
number. However, there will occasionally be situations where context is not sufficient to
distinguish between ordinals and their Godel numbers: this occurs when n may be either
an integer or a finite ordinal; to resolve ambiguity, we write pnq for the Godel number of
the ordinal n and we write n for the integer n . To improve readability, we use α, β, γ, . . .
and ρ, σ, τ, . . . as variables that range over Godel numbers of ordinals. For example, the
formula (∀σ ≺ β)(· · ·) abbreviates the first-order formula
IsOrdinal(β) ∧ (∀x)(IsOrdinal(x) ∧ x ≺ β → · · ·).Note that ∀σ ≺ β corresponds to an unbounded quantifier unless β is known to code a
finite ordinal.
Transfinite induction on ordinals may be used to provide alternate axiomatizations for
fragments of Peano arithmetic:
Definition Let Ψ be a set of formulas and let κ 4 ε0 . Then TI(κ, Ψ) is the set of axioms
(∀γ 4 κ)[(∀β ≺ γ)A(β) → A(γ)] → A(κ) (1)
where A is a formula in Ψ, possibly with other free variables as parameters.
The least ordinal principle axioms LOP(κ, Ψ) are
A(κ) → (∃γ 4 κ)[A(γ) ∧ (∀β ≺ γ)(¬A(β))] (2)
where A ∈ Ψ and A may have parameter variables. For a fixed formula A , the
axioms (1) and (2) are called TI(κ,A) and LOP(κ,A), respectively.
TI(≺ κ, Ψ) is the theory ∪µ≺κTI(µ, Ψ).
LOP(≺ κ, Ψ) is the theory ∪µ≺κLOP(µ, Ψ).
3‘Intensionally formalized’ means that IΣ1 can prove simple syntactic facts about ordinal encodingsand about operations on encoded ordinals.
A slight variation on the least ordinal principle and transfinite induction axioms is
Proof It is clear that IΣn ≡ I∆0 + TI(ω, Σn) and by standard techniques these are
equivalent to IΠn and I∆0 + TI(ω, Πn). In light of Proposition 1, it suffices to show that
LOP(≺ ω2, Σn) follows from I∆0 +TI(ω, Πn). To accomplish this, we show, by induction
on k , that LOP(≺ ωk, Σn) follows from the latter theory. For k = 1 this is proved by
the kind of reasoning used to prove Proposition 1(a),(b). To show LOP(≺ ωk+1, Σn); let
A(α) ∈ Σn , let α0 ≺ ωk+1 and reason informally with the assumptions TI(ω, Πn) and
LOP(≺ ωk, Σn): further set C(α) to be the formula (∃i)A(ω ·α + i), so C(α) ∈ Σn . Now
assume A(α0) holds; since α0 = ω · α1 + i1 for some α1 ≺ ωk and some finite i1 , C(α1)
holds also. By LOP(≺ ωk, Σn), there is a least α2 such that C(α2) holds and now by
TI(ω, Πn), there is a least i2 such that A(ω · α2 + i2). Clearly α = ω · α2 + i2 is the least
ordinal such that A(α) holds. 2
2.2 Arithmetic and the Sequent Calculus
This section describes how the sequent calculus and free cut elimination are applied to
the fragments of arithmetic defined above. The reader is presumed to be familiar with the
sequent calculus (refer to [27] or Chapter 4 of [2] for the necessary background material).
We shall assume the language of first-order logic contains symbols ¬ , ∧ , ∨ , → , ∃ and
∀ ; this leads to a large number of rules of inference but we shall omit most cases from our
proofs in any event. It will be assumed that bounded quantifiers are part of the syntax of
first-order logic with the sequent calculus containing the four appropriate rules of inference
for bounded quantifiers.4 See [2] for the full definition of the sequent calculus LKB with
bounded quantifier rules of inference.
To formalize the proof theory of arithmetic with the sequent calculus, it is customary
to use special induction inferences in place of induction axioms. An induction inference is
of the formΓ, A(a)→A(Sa), ∆Γ, A(0)→A(t), ∆
where t may be any term, a is a free variable called the eigenvariable and a must not
appear in the lower sequent. The induction inference for A is equivalent to the induction
axiom for A , because the side formulas Γ and ∆ are allowed. Thus IΣk is formalized
in the sequent calculus with a finite set of axiom schemes plus the induction inferences
for Σk formulas. The finite set of axiom schemes for IΣk consists of the following initial
4This assumption is not absolutely necessary and the reader may prefer to think of the boundedquantifiers as abbreviations — in this case the proofs by induction on the number of inferences in afree-cut free proof must be slightly modified.
sequents:Sr = St→r = t →r · 0 = 0St = 0→ →r · (St) = r · t + r→r + 0 = r →r = 0, (∃x ≤ r)(Sx = r)→r + St = S(r + t) r ≤ t→(∃x ≤ t)(r + x = t)
r + s = t→r ≤ t
where r , s and t are allowed to be any terms. Of course the usual logical initial sequents
A→A with A atomic and the initial sequents for equality are also allowed. It is important
for us that every initial sequent consists of only ∆0 formulas.
The theory I∆0 + TI(≺ ωm, Σn) is formalized in the sequent calculus with the same
initial sequents, with induction inferences for ∆0 -formulas and for transfinite induction,
with the LOP(≺ ωm, Πn) inferences defined below.
Let τ be a closed term with value the Godel number of an ordinal and let B(α) be a
formula; the LOP(τ, B) inference is
LOP(τ, B) :α 4 τ, B(α), Γ→∆, (∃β ≺ α)B(β)
B(τ), Γ→∆
where α is an eigenvariable and may occur only as indicated. It is not hard to see that
the inference rule LOP(τ, B) is equivalent to the axiom form of LOP(τ, B): to derive the
inference rule from the axiom, recall that the axiom LOP(τ, B) is
B(τ)→(∃α 4 τ)[B(α) ∧ (∀β ≺ α)(¬B(β))], (3)
and use the derivation
(3)
α 4 τ, B(α), Γ→∆, (∃β ≺ α)B(β)
(∃α 4 τ)(B(α) ∧ (∀β ≺ α)(¬B(β))), Γ→∆
B(τ), Γ→∆
where the double horizontal line indicates omitted inferences. Conversely, to see that the
Since fCS(· · ·) is equivalent to a Σn -formula and since z = UA(β′) can be expressed as a
Πn−1 -formula, the relation y = f(β) is a Σn -property, provably in I∆0 + TI(≺ ωm, Σn−1).
The theory also proves
∀β ∃ a least β′ s.t. ∃〈β, . . . , β′〉(fCS(〈β, . . . , β′〉))since fCS(〈β〉) and by LOP(≺ ωm, Σn) since κ0 ≺ ωm .7 Thus I∆0 + TI(≺ ωm, Σn−1)
can Σn -define the function f as it proves (∀β)(∃!y)(y = f(β)) where y = f(β) denotes
the Σn -formula defining the graph of f . Likewise,
(∀~z)(∃!y)(∃β)(β = τ(~z) ∧ y = f(β))
is also provable and Σn -defines the function F . That completes the first half of the proof
of Theorem 7.
To prove the rest of Theorem 7, assume that I∆0 + TI(≺ ωm, Σn−1) proves
(∀x)(∃!y)A(x, y), with A ∈ Σn — we must show that x 7→ y is a ≺ ωm -primitive
recursive in Σn−1 function. Since I∆0 + TI(≺ ωm, Σn−1) proves (∀x)(∃y)A , there must
be a free-cut free proof in the theory I∆0 + TI(≺ ωm, Σn−1) of the sequent
→(∃y)A(c, y)
where c is a new free variable. Only Σn formulas can appear in this free-cut free
proof. The general idea of the proof is to show that this free-cut free proof embodies
an algorithm for computing y from c . Indeed, the free-cut free proof can be interpreted
as explicitly containing a ≺ ωm -primitive recursive in Σn−1 algorithm. Since the proofs
of the normal form theorems were constructive, the free-cut free proof also contains an
implicit description of a ≺ ωm -primitive recursive in Σn−1 algorithm in the second normal
form. 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(≺ ωm, Σn−1) — the upshot is
that there is a ≺ ωm -primitive recursive in Σn−1 function f which is Σn -defined by
I∆0 + TI(≺ ωm, Σn−1) in the form given by the Second Normal Form Theorem such
that I∆0 + TI(≺ ωm, Σn−1) ` (∀x)A(x, f(x)). As a corollary to the proof method, if
I∆0 + TI(≺ ωm, Σn−1) proves (∀x)(∃y)B(x, y) with B ∈ Σn then there is a B∗(x, y) ∈ Σn
such that (∀x)(∃!y)B∗(x, y) and B∗(x, y) → B(x, y) are provable.8
7LOP(≺ ωm,Σn) is a consequence of I∆0 + TI(≺ ωm,Σn−1) by Proposition 1.8This fact is readily proved directly anyway. If B ∈ Πn−1 then let B∗ be the formula B(x, y) ∧
(∀y′ < y)(¬B(x, y′)), which is equivalent to a Σn formula by BΣn . For general B ∈ Σn , incorporateoutermost existential quantifiers of B into the the (∃y) and proceed similarly.
We shall see later that the proof is formalizable, not only in I∆0 + TI(≺ ωm, Σn−1),
but also in I∆0 + TI(≺ ωm+1, Σn−2), provided n > 1.
Rather than just considering the free-cut free proof of →(∃y)A , we more generally
consider proofs of sequents Γ→∆ of Σn -formulas. Since every principal and auxiliary
formula of a LOP(≺ ωm, Πn−1) inference is in Σn and every formula in the endsequent is
in Σn , it follows that every formula in the free-cut free proof is in Σn . For convenience,
assume also that the proof is in free variable normal form (so free variables are not reused).
Definition Let i ≥ 1 and A(~x) ∈ Σi . If A ∈ Πi−1 then WitiA is defined to be the
formula A . Otherwise, A is uniquely expressible in the form (∃y0) · · · (∃yk)B(~x, ~y) where
B ∈ Πi−1 . Then WitiA(w, ~x) is the formula
B(~x, (w)0, . . . , (w)k).
Note that WitiA ∈ Πi−1 . If WitiA(w, ~x) holds, we say w witnesses the truth of A(~x).
Main Lemma 10 (n ≥ 1, m ≥ 2) Suppose I∆0 + TI(≺ ωm, Σn−1) proves the sequent
A1, . . . , Ak→B1, . . . , B` and that each Ai and Bj is in Σn and that ~c are all the variables
free in the sequent. Then there are functions f1, . . . , f` which are ≺ ωm -primitive recursive
in Σn−1 and are Σn -definable in I∆0 + TI(≺ ωm, Σn−1) such that I∆0 + TI(≺ ωm, Σn−1)
proves
WitnA1(w1,~c), . . . ,WitnAk
(wk,~c)→WitnB1(f1(~w,~c),~c), . . . ,WitnB`
(f`(~w,~c),~c).
Informally, the f1, . . . , f` will, given witnesses for all of A1, . . . , Ak , produce a witness for
at least one of B1, . . . , B` .
The proof of the Main Lemma is by induction on the number of inferences in a free-cut
free proof of the sequent. In the base case, there are zero inferences, so the sequent is
an axiom and consists of ∆0 -formulas — for these axioms, the lemma is trivial. For
the induction step, the proof splits into cases depending in the final inference of the
proof. Most of the cases are straightforward; for example, if the last inference is an ∃:left
inference then the proof ends with
A1, . . . , Ak→B0(~c, s), B2, . . . , B`
A1, . . . , 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 and
is of the form (∃z0) · · · (∃zr)B′(~z,~c) with B′ ∈ Πn−1 (possibly r = 0). The induction
hypothesis is that
WitnA1(w1,~c), . . . ,WitnAk
(wk,~c)
→WitnB0(~c,s)(f0(~w,~c),~c), . . . ,WitnB`
(f`(~w,~c),~c) (4)
is provable in I∆0 + TI(≺ ωm, Σn−1) for appropriate functions f0, f2, . . . , f` . If B1 ∈ Σn−2
then WitnB0is just B0 and WitnB1
is just B1 ; and a single ∃:right inference applied to (4)
gives
WitnA1(w1,~c), . . . ,WitnAk
(wk,~c)→WitnB1(f1(~w,~c),~c), . . . ,WitnB`
(f`(~w,~c),~c) (5)
where f1 is arbitrary. Otherwise, let f1(~w,~c) be defined so that