Streamlined Subrecursive Degree Theory

Streamlined Subrecursive Degree Theory Lars Kristiansen1 1 2

1

Jan-Christoph Schlage-Puchta2

Andreas Weiermann2

Department of Mathematics, University of Oslo Department of Mathematics, Ghent University

Introduction

The study of honest elementary degrees has its roots in subrecursion theory from the nineteen seventies. Some relevant papers are Meyer & Ritchie [12] and Machtey [9–11]. These papers deal with subrecursive function classes being generated by so-called honest functions where an honest function is defined relative to a subrecursive class S and a model of computation: a function f : N → N is honest if the number of steps in a computation of f is bounded by ψ(x, f (x)) for some ψ ∈ S. Influenced by subrecursion theory from the seventies, Kristiansen introduces the honest elementary degrees about twenty years later. These degrees are the equivalence classes induced on the honest functions by the reducibility relation “being (Kalmar) elementary in”, but now a function is regarded to be honest if it is monotone, dominates 2x and has (Kalmar) elementary graph. This notion of honesty is in certain respects equivalent to the one from the seventies, but not in every respect, and the combination of the particular reducibility relation “being elementary in” and the novel definition of an honest function is the basis for the following pivotal theorem: The Growth Theorem. An honest function f is elementary in an honest function g if, and only if, there exists a fixed k ∈ N such that f (x) ≤ g k (x). In the nineteen nineties Kristiansen uses the Growth Theorem to investigate honest elementary degrees and related subjects. This research is published in a thesis [6] and four papers [3–5, 7]. The structure of honest elementary degrees is comparable to a classical computability-theoretic degree structure, e.g., the structure of Turing degrees, but the Growth Theorem makes it possible to abandon classical computability-theoretic proof methods and investigate this structure by asymptotic analysis and methods of number theoretic nature. To prove that g ≤E f , it is sufficient to provide a fixed k such that g(x) ≤ f k (x); to prove that g 6≤E f , it is sufficient to prove that such a k does not exist. Thus, there is no need for the standard computability-theoretic machinery involving enumerations, diagonalisations and constructions with requirements to be satisfied. This makes the proofs concise and transparent. The current paper is divided into two parts. Part I, being concerned with the structure of elementary honest degrees, might be viewed as a journal version of a conference paper published in 1999 [7]. If we take that view, then we are

2

Lars Kristiansen

Jan-Christoph Schlage-Puchta

Andreas Weiermann

dealing with an extended version containing several new and significant results not found in [7]. Besides, in order to achieve a coherent and readable exposition, Part I also contains results initially published in [4] and [5], and thus, perhaps it will be more adequate to view Part I as a presentation of original research which includes a survey of some relevant research from the nineteen nineties. In Part II we generalise the degree theory found in Part I by introducing the reducibility relation “being α-elementary in”: For the succeeding discussion, fix an ordinal α less or equal to ǫ0 such that the set of ordinals preceding α is closed under sums. A function f is α-elementary in a function g when f can be generated from g by elementary operations and ordinal iteration up to α. The honest α-elementary degrees are equivalence classes induced by this reducibility relation on the honest functions. The structure of honest α-elementary degrees turns out to be very similar to the degree structure studied in the first part of the paper. Exactly how similar, is yet to be seen, but we prove a generalised version of the Growth Theorem making the honest α-elementary degrees amenable to the number-theoretic proof techniques we apply so successfully in the first part of the paper. This makes us believe that all results we prove on honest elementary degrees in Part I, also hold for honest α-elementary degrees. One motivation for our generalisation of honest elementary degree theory, is expected applications with respect to formal systems for mathematics and independence results for such systems. We intend to investigate such applications in the future, and towards the end of the paper we give an example of the type of applications we have in mind. For convenience, we will state these results a bit imprecisely in this introductory section (the precise formulations are found in Section 10): Let tot(h) denote a formalisation of the true statement “the function h is total” in the first order theory P A (Peano Arithmetic). We prove that an honest function g is ǫ0 -elementary in an honest function f if, and only if, P A + tot(f ) ⊢ tot(g). Since the structure of honest ǫ0 -elementary degrees contains minimal pairs relative to the zero degree, it becomes an immediate consequence that there exists functions h1 , h2 such that P A + tot(h1 ) 6⊢ tot(h2 ) and P A + tot(h2 ) 6⊢ tot(h1 ) and moreover, for any function f such that P A + tot(h1 ) ⊢ tot(f ) and P A + tot(h2 ) ⊢ tot(f ) we also have P A ⊢ tot(f ).

2

Preliminaries

We assume the reader is familiar with the most basic concepts of classical computability theory, see e.g. Odifreddi [13] or Rogers [15]. We also assume acquaintance with subrecursion theory and, in particular, with the elementary functions. An introduction to this subject can be found in [14] or [16]. Here we just state some important basic facts and definitions, see [14] and [16] for proofs.

Streamlined Subrecursive Degree Theory

3

The initial elementary functions are the projection functions (Iin ), the constants 0 and 1, addition (+) and modified subtraction ( . ). The elementary definition schemes are composition, that is, f (~x) = h(g1 (~x), . . . ,P gm (~x)) and x, i) bounded sum and bounded product, that is, respectively f (~ x , y) = i f (x). From now on, we reserve the letters f, g, h, . . . to denote honest functions. Small Greek letter like φ, ψ, ξ, . . . will denote number-theoretic functions not necessarily being honest. Definition. A function φ is elementary in a function ψ, written φ ≤E ψ, if φ can be generated from the initial functions ψ, 2x , max, 0, S (successor), Iin (projections) by composition and bounded primitive recursion. ⊓ ⊔ Theorem 1 (Growth Theorem). Let f and g be honest functions. Then, we have g ≤E f ⇔ g ≤ f k for some fixed k . Proof. Recall that f is monotone and dominates 2x . By induction on the buildup of a function ψ form the initial functions 0, S, Iin , 2x , max, f by composition and bounded primitive recursion, it is easy to prove that there exists k ∈ N such that ψ(~x) ≤ f k (max(~x)). Hence, if g ≤E f , we have g ≤ f k for some fixed k. Now, suppose that g ≤ f k . Since g is honest, the relation g(x) = y is elementary. We have g(x) = (µy ≤ f k (x))[g(x) = y]. Hence, g ≤E f since the functions elementary in f are closed under composition and the bounded µ-operator. ⊓ ⊔

4

The Lattice of Honest Elementary Degrees

Definition. We define the relation ≡E by f ≡E g ⇔ f ≤E g ∧ g ≤E f . Now, ≡E is an equivalence relation on the honest functions, and we will use H denote the set of ≡E -equivalence classes of honest functions. The elements of H are the honest elementary degrees. Honest elementary degrees will normally just be called degrees, and following the tradition of classical computability theory, we use boldface lowercase Latin letters a, b, c, . . . to denote our degrees. We will use deg(f ) denote the degree of the honest function f , that is, deg(f ) = {g | g ≡E f }. We define the relation 0, could replace ν in this proof.) For each k ∈ N, we will define a sequence dk,0 < dk,1 < . . . < dk,ν(k)2 . Moreover, for each k, we will have dk,ν(k)2 < dk+1,0 . Let d0,0 = 0. For each j ∈ {1, . . . , ν(k)2 }, let  f (dk,j−ν(k) ) if ν(k) divides j dk,j = 2dk,j−1 otherwise and let dk+1,0 = f ′ (dk,ν(k)2 ). Furthermore, let  dk,i+1 if dk,i ≤ x < dk,i+1 for some k, i G(x) = dk,ν(k)2 if dk,ν(k)2 ≤ x < dk+1,0 for some k, i and let g(x) = max(2x , G(x)). This completes the construction of g. The reader should note the following properties of g (and f ): (P1) (P2) (P3) (P4)

g(dk,i ) = dk,i+1 for any k and any i < ν(k)2 for any k and any i < ν(k)2 , we have g(dk,i ) = f (dk,i ) if ν(k) divides i for any k and any i < ν(k)2 , we have g(dk,i ) = 2dk,i if ν(k) does not divide i 2 g ν(k) (dk,0 ) = dk,ν(k)2 = f ν(k) (dk,0 ) for any k d

2

(P5) for any m, we have g m (dk,ν(k)2 ) = 2mk,ν(k) < dk+1,0 for all but finitely many k. These five properties is more or less straightforward consequences of the construction of g, in particular, to see that (P5) holds, note that dk+1,0 = f ′ (dk,ν(k)2 ) and f (x) ≥ 2xx . 2 (Claim I) follows straightaway from (P4). For any m we have g m (dk,0 ) = f m (dk,0 ) for each of the infinitely many k’s such that ν(k) = m. We turn to the proof of (Claim II). The proof splits into the two cases: the case when x lies in an interval of the form dk,0 , . . . , dk,ν(k) − 1, and the case when x lies in an interval of the form dk,ν(k) , . . . , dk+1,0 − 1. 2 We will first prove that we have g m (x) < f 3m+1 (x) when x is sufficiently large and lies in an interval of the form dk,0 , . . . , dk,ν(k) −1. The proofs splits into the the two sub cases m ≥ ν(k) and m < ν(k). First, assume that m ≥ ν(k). We have f 3m+1 (x) = f (3m+1)−ν(k) f ν(k) (x) ≥ f (3m+1)−ν(k) f ν(k) (dk,0 )

f is monotone

= f (3m+1)−ν(k) (dk,ν(k)2 )

(P4)

> f (dk,ν(k)2 )

as m ≥ ν(k)

≥ ≥

d 2 2dk,ν(k)2 k,ν(k) d 2 2mk,ν(k) 2 2

= g m (dk,ν(k)2 ) ≥ g

m2

(x) .

as f (x) ≥ 2xx x is large (P5) and x is large g is monotone

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Lars Kristiansen

Jan-Christoph Schlage-Puchta

Andreas Weiermann

Next, assume that m < ν(k). Fix the unique i such that dk,i ≤ x < dk,i+1 . Since m < ν(k), there will be at most one number j in the interval i, . . . , min(i + m, ν(k)2 ) such that ν(k) divides j. Hence, by (P2), (P3) and (P5), there exist m0 , m1 such that f (2x m1 )

g m (x) ≤ 2m0

≤ f 3 (x) .

(†)

Furthermore, g is monotone and x ≤ dk,ν(k)2 , and then, by (P5), we have 2

2

g m (x) ≤ g m (dk,ν(k)2 ) < dk+1,0

(‡) 2

for all but finitely many x. It follows from (†) and (‡), we have g m (x) < f 3m+1 (x) for all sufficiently large x. 2 The reader is invited to verify that we also have g m (x) < f 3m+1 (x) for sufficiently large x lying in intervals of the form dk,ν(k) , . . . , dk+1,0 − 1. To verify this, note that for any x in such an interval we have g(x) = 2x whereas f (x) ≥ 2xx . This completes the proof of (Claim II). We will briefly now argue that g is honest an honest function. The function f is honest by assumption. First we argue that dk,j = x is an elementary relation in k, j, x. Let a | b denote the relation “a divides b”. This relation is elementary. We have dk,j = x ⇔
j 6= 0 ∧ ν(k) | j ∧ ∃x0 < x [ dk,j−ν(k) = x0 ∧ f (x0 ) = x ] ∨ ( j 6= 0 ∧ ¬ ν(k) | j ∧ ∃x0 < x [ dk,j−1 = x0 ∧ 2x0 = x ] ) ∨
j = 0 ∧ ∃x0 < x [ dk,ν(k)2 = x0 ∧ 2x0 = x ] ∨

(k = 0 ∧ j = 0 ∧ x = 0) .

This can be viewed as a recursive definition of dk,j = x. All the functions, relations and operations involved are elementary. Thus, we have defined the relation dk,j = x by a recursion scheme of the form R(k, j, x) ⇔ φ(R(k0 , j0 , x0 ), R(k1 , j1 , x1 ), R(k2 , j2 , x2 )) where φ is an elementary predicate and k0 , k1 , k2 ≤ k; j0 , j1 , j2 ≤ k; and x0 , x1 , x2 ≤ x. The elementary predicates are closed under such a recursion scheme, and hence, dk,j = x is an elementary relation. Thus, ∃k, j ≤ x[dk,j = x] is an elementary predicate. Once we have realised that this predicate is elementary, it becomes easy to see that g has elementary graph. Obviously, g is monotone and dominates 2x . Thereby, g is honest. We will now prove the theorem. We have g ≤E f by the Growth Theorem since g ≤ f . Let m be any number. Pick x such that x > m and x = dk,ν(k)2 for some k. By (P5), we have g m (x) = 2xm < 2xx ≤ f (x). Hence, we have f 6≤E g by the Growth Theorem. This proves g 1 ∧ ∀β, γ < α[ β#γ < α ]} = {ω β | β > 0} . ⊓ ⊔ It is easy to see that the natural sum is associative and commutative. The reader should note that N (α#β) = N α + N β, and furthermore, that γ ≤ β iff α#γ ≤ α#β. Definition. We define the α-iterate of the unary function φ, written φα , by φ0 (x) = φ(x) and φα (x) = max{φβ φβ (x) | β < α ∧ N β ≤ N α + x} for any α such that 0 < α < ǫ0 . (Note that the set {β | β < α ∧ N β ≤ N α + x} is finite.)

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Lars Kristiansen

Jan-Christoph Schlage-Puchta

Andreas Weiermann

We will view such an iteration as a definition scheme, and we will call this scheme ordinal iteration. If ψ is defined by ψ(x) = φβ (x) for some β < α, we will say that ψ is defined by α-iteration (over φ). ⊓ ⊔ The next lemma is very fundamental, and we will occasionally apply the lemma without referring to it. Lemma 10 (Basic Properties of Ordinal Iteration). Let f be an honest function. Then, (i) fα is monotone, (ii) fα dominates 2x , (iii) fα (x) ≤ fα#β (x), (iv) fβ fβ (x) ≤ fβ#1 (x), (v) for any unary function φ, we have φα (x) ≤ fα (x) whenever φ(x) ≤ f (x), and (vi) fα fβ (x) ≤ fα#β#1 (x). Proof. We prove (i) by induction on α. When α = 0, we have fα = f , and thus, fα is monotone since f is monotone. When α > 0, the induction hypothesis yields that fβ is monotone when β < α. Hence fα (x) = max{fβ fβ (x) | β < α ∧ N β ≤ N α + x} ≤ max{fβ fβ (x + 1) | β < α ∧ N β ≤ N α + x}

def. of fα ind. hyp.

≤ max{fβ fβ (x + 1) | β < α ∧ N β ≤ N α + x + 1} = fα (x + 1) .

def. of fα

This proves (i). Furthermore, (ii) holds since 2x ≤ f (x) = f0 (x) ≤ fα (x), and (iii) follows straightforwardly from the definition of ordinal iteration since α ≤ α#β and N α ≤ N (α#β). We have β < β#1 and N β ≤ N (β#1) + x for any x. Thus, by the definition of ordinal iteration we have fβ fβ (x) ≤ fβ#1 (x). This proves (iv), and (v) is proved by a straightforward induction on α. Finally, (vi) is a consequence of (i), (iii) and (iv) since fα fβ (x) ≤ fα#β fα#β (x) ≤ fα#β#1(x). ⊓ ⊔ Lemma 11. For any α, β, we have (fα )β (x) ≤ fα#β (x). Proof. We prove this lemma by induction over β. By our definitions, we have (fα )0 (x) = fα (x) = fα#0 (x). Thus, the lemma holds when β = 0. Next, we note that η < β ∧ N η ≤ N β + x ⇒ α#η < α#β ∧ N (α#η) ≤ N (α#β) + x .

(*)

It is obvious that we have α#η < α#β if η < β. Furthermore, if N η ≤ N β + x, then N (α#η) = N α + N η ≤ N α + N β + x = N (α#β) + x. Thus, (*) holds. Now, assume β > 0. By the induction hypothesis and Lemma 10 (i), we have (fα )β (x) = max{fα )η (fα )η (x) | η < β ∧ N η ≤ N β + x} ≤ max{fα#η fα#η (x) | η < β ∧ N η ≤ N β + x}

(†)

and thus (fα )β (x) ≤ max{fα#η fα#η (x) | η < β ∧ N η ≤ N β + x} ≤ max{fα#η fα#η (x) | α#η < α#β ∧ N (α#η) ≤ N (α#β) + x} ≤ max{fγ fγ (x) | γ < α#β ∧ N γ ≤ N (α#β) + x} = fα#β (x) .

(†) (*)

Streamlined Subrecursive Degree Theory

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⊓ ⊔ Definition. Let α ≤ ǫ0 . A function φ is α-elementary in a function ψ, written φ ≤αE ψ, if φ can be generated from the initial functions ψ, 2x , max, 0, S (successor), Iin (projections) by composition, bounded primitive recursion and β-iteration where β < α. A function is α-elementary if it is α-elementary in the constant function 0. ⊓ ⊔ For each α ∈ SLim, the reducibility relation “being α-elementary in” is transitive, and thus, the relation will induce a degree structure on the honest function. The next theorem shows that this structure of honest α-elementary degrees can be investigated by number-theoretic techniques similar to those used in the first part of this paper. Theorem 13 (Generalised Growth Theorem). Let f and g be honest functions, and let α ∈ SLim. Then, we have g ≤αE f ⇔ g ≤ fβ for some fixed β < α . Proof. Assume g ≤ fβ and β < α. The relation g(x) = y is elementary since g is honest. Furthermore, the function fβ is α-elementary in f . We have g(x) = (µy ≤ fβ (x))[g(x) = y]. Hence, g ≤αE f since the α-elementary functions are closed under composition and the bounded µ-operator. This proves the right-left implication of the theorem. In the proof, we have used that honest functions have elementary graphs. To prove the converse implication, we will use that honest functions are monotone and dominate 2x . Assume that ψ ≤αE f , that is, ψ is build from f , 2x , max, 0, S, Iin by composition, bounded primitive and β-iteration where β < α. We will prove, by induction over the build-up of ψ, that there exists an ordinal β strictly less than α such that ψ(x1 , . . . , xn ) ≤ fβ (max((x1 , . . . , xn )) .

(*)

First we note that f0 (x) = f (x) and f (x) ≥ 2x . Thus, when ψ is one of the initial functions, (*) holds with β = 0. Now, assume that ψ is a composition over ξ and η1 , . . . , ηm . The induction hypothesis yields ordinals γ0 , . . . , γm < α such that ξ(~z) ≤ fγ0 (max(~z)) and ηi (~x) ≤ fγi (max(~x)), for i = 1, . . . , m. Moreover, fγ0 , . . . , fγm are monotone functions. Thus, we have ψ(~x) = ξ(η1 (~x), . . . , ηm (~x))

def. of ψ

≤ fγ0 (max(η1 (~x), . . . , ηm (~x)))

ind. hyp. on ξ

≤ fγ0 (max(fγ1 (max(~x)), . . . , fγm (max(~x))))

ind. hyp. on ηi

≤ fγ0 fγ1 #...#γm (max(~x))

Lemma 10 (iii)

≤ fγ0 #γ1 #...#γm #1 (max(~x))

Lemma 10 (vi)

and γ = γ0 # . . . #γm #1 < α since α ∈ SLim. This concludes the proof of (*) for the case when ψ is a composition.

20

Lars Kristiansen

Jan-Christoph Schlage-Puchta

Andreas Weiermann

When ψ is generated by bounded primitive recursion, we have (*) straightaway from the the induction hypothesis. We are left with the case when ψ is generated by α-iteration. So, assume ψ(x) = φβ (x) where β < α. By the induction hypothesis, we have γ strictly less than α such that φ(x) ≤ fγ (x). Hence, by Lemma 10 (v) and Lemma 11, we have ψ(x) = φβ (x) ≤ (fγ )β (x) ≤ fγ#β (x) . Moreover, γ#β < α since α ∈ SLim. This completes the proof of (*). The leftright implication of the theorem follows straightforwardly from (*). ⊓ ⊔

8

The Honest α-Elementary Degrees is a Lattice

Definition. Let α ∈ SLim. We define the relation ≡αE by f ≡αE g ⇔ f ≤αE g ∧ g ≤αE f . Now, ≡αE is an equivalence relation on the honest functions, and we will use Hα denote the set of ≡αE -equivalence of honest functions. The elements of Hα are the honest α-elementary degrees, and we use boldfaced lowercase Latin letters a, b, . . . to denote such degrees. Furthermore, let ≤ denote the ordering relation induced on Hα by ≤αE . Let degα (f ) denote the ≡αE -equivalence class of the honest function f . ⊓ ⊔ In this section we will show that min[·, ·] and max[·, ·] induce meet and join operators on the honest α-elementary degrees. Recall that min[f, g] and max[f, g] are honest functions when f and g are honest functions (Lemma 1). Lemma 12. For any honest functions f and g, we have min[fα , gβ ] ≤ min[f, g]α#β . Proof. We will do induction on the ordinal α#β. Assume α#β = 0, and recall that the definition says that ψ0 (x) = ψ(x). We have min[f, g]0 (x) = min[f, g](x) = min[f0 , g0 ](x) and the lemma holds. Assume α#β > 0. We can w.l.o.g. assume min[fα , gβ ](x) = gβ (x). The case when β > 0 and the case when β = 0 have to be treated separately. First we consider the case when β > 0. Then there exists a γ < β with N γ ≤ N β + x such that gβ (x) = gγ gγ (x). This entails that α#γ < α#β ∧ N α#γ ≤ N α#β + x .

(i)

Furthermore, we have gγ (x) ≤ gγ gγ (x) = gβ (x) ≤ fα (x) ≤ fα gγ (x) .

(ii)

Streamlined Subrecursive Degree Theory

21

Now min[fα , gβ ](x) = gγ gγ (x) = min[fα , gγ ](gγ (x))

(ii)

= min[fα , gγ ] min[fα , gγ ](x)

(ii)

≤ min[f, g]α#γ min[f, g]α#γ (x)

ind. hyp.

≤ min[f, g]α#β (x) .

(i)

This concludes the proof for the case when α#β > 0. We will now consider the case when β = 0. Then we have α > 0 as α#β > 0. We will need the following implication. g0 (x) ≤ fα (x) ⇒ g0 (x) ≤ max{min[fγ , g0 ] min[fγ , g0 ](x) | γ < α ∧ N γ ≤ N α + x} . (†) Observe that max{min[fγ , g0 ] min[fγ , g0 ](x) | γ < α ∧ N γ ≤ N α + x} either – – – –

equals fγ fγ (x) for some γ such that γ < α and N γ ≤ N α + x or equals fγ g0 (x) for some γ such that γ < α and N γ ≤ N α + x or equals g0 fγ (x) for some γ such that γ < α and N γ ≤ N α + x or equals g0 g0 (x).

In the three latter cases, the consequent of (†) is true as fγ and g are monotone and dominate 2x . In the first case, (†) holds since max{fγ fγ (x) | γ < α ∧ N γ ≤ N α + x} = fα (x). Hence, we conclude that (†) holds. Furthermore, we have min[fα , g0 ](x) = (†)

g0 (x) ≤ max{ min[fγ , g0 ] min[fγ , g0 ](x) | γ < α ∧ N γ ≤ N α + x } (‡)

≤ max{ min[f, g]γ#0 min[f, g]γ#0 (x) | γ < α ∧ N γ ≤ N α + x } = min[f, g]α#0 (x) .

The relation labelled (‡) holds by the induction hypothesis since γ#0 < α. The equality holds by the definition of min[f, g]α . This completes our proof. ⊓ ⊔ Lemma 13. For any honest functions f and g, we have max[fα , gβ ] ≤ max[f, g]α#β . Proof. We can w.l.o.g. assume max[fα , gβ ](x) = gβ (x). Then, by (v) and (iii) of Lemma 10, we have max[fα , gβ ](x) = gβ (x) ≤ max[f, g]β (x) ≤ max[f, g]α#β (x) . ⊓ ⊔ Lemma 14. Let f, g, h be honest functions, and let α ∈ SLim. Then,

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Lars Kristiansen

Jan-Christoph Schlage-Puchta

Andreas Weiermann

(i) min[f, g] ≤αE f and min[f, g] ≤αE g (ii) if h ≤αE f and h ≤αE g, then h ≤αE min[f, g]. Proof. (i) follows easily from the Growth Theorem. To prove (ii), assume h ≤αE f and h ≤αE g. By the Growth Theorem we have β, γ < α such that h ≤ fβ and h ≤ gγ . By Lemma 12, we have h(x) ≤ min(fβ (x), gγ (x)) ≤ min[f, g]β#γ (x) . We have β#γ < α, and thus, we have h ≤αE min[f, g] by another application of the Growth Theorem. ⊓ ⊔ Lemma 15. Let f, g, h be honest functions and let α ∈ SLim. Then, (i) f ≤αE max[f, g] and g ≤αE max[f, g]. (ii) if f ≤αE h and g ≤αE h, then max[f, g] ≤αE h. Proof. The proof of this lemma is similar to proof of Lemma 14: apply Lemma 13 in place of Lemma 12. ⊓ ⊔ Lemma 16. For any honest functions f, f¯, g, g¯ and any α ∈ SLim such that f ≤αE f¯ and g ≤αE g¯, we have (i) min[f, g] ≤αE min[f¯, g¯] and (ii) max[f, g] ≤αE max[f¯, g¯]. Proof. (i) follows from straightforwardly from Lemma 14, and (ii) follows from straightforwardly from Lemma 15. ⊓ ⊔ The previous lemma entails that (f ≡αE f¯ ∧ g ≡αE g¯) ⇒ (max[f, g] ≡αE max[f¯, g¯] ∧ min[f, g] ≡αE min[f¯, g¯]) when α ∈ SLim and f, f¯, g, g¯ are honest functions. By Lemma 1, we know that max[f, g] and min[f, g] are honest functions whenever f and g are. Hence, the next definition makes sense. Definition. Let f and g be any honest functions such that a = degα (f ) and b = degα (g). We define the join of a and b, written a ∪α b, by a ∪α b = degα (max[f, g]). We define the meet of a and b, written a ∩α b, by a ∩α b = degα (min[f, g]). When the ordinal α is given by the context, we will write ∪ and ∩ in place of respectively ∪α and ∩α . ⊓ ⊔ Theorem 14 (Distributive Lattice). Let α ∈ SLim. The structure honest αelementary degrees hHα , ≤, ∪, ∩i is a distributive lattice, that is, for any a, b, c ∈ H, we have (i) a ∩ b is the greatest lower bound of a and b under the ordering ≤; (ii) a ∪ b is the least upper bound of a and b under the ordering ≤; (iii) a ∪ (b ∩ c) = (a ∪ b) ∩ (a ∪ c) and a ∩ (b ∪ c) = (a ∩ b) ∪ (a ∩ c). Proof. The theorem follows from the lemmas above. We leave the details to the reader. See also the proof of Theorem 2. ⊓ ⊔

Streamlined Subrecursive Degree Theory

9

23

A Jump Operator on the Honest α-Elementary Degrees

In this section we will show that the honest α-elementary degrees, for α ∈ SLim, admit a jump operator which is a generalisation of the jump operator introduced in the first part of this paper. Indeed, the honest α-elementary degrees can admit several jump operators: one for each ordinal β where α ≤ β < ǫ0 . Lemma 17. Let f be an honest function. Then, fα is an honest function. Proof. Lemma 10 states that fα is monotone and dominates 2x . In order to prove that fα has elementary graph, assume some natural G¨odel enumeration of the ordinals less than ǫ0 , and let ⌈α⌉ denote the G¨odel number for α. The reader should recall that hx1 , . . . , xn i denotes the number encoding the sequence consisting of the numbers x1 , . . . , xn , and that (s)i denotes the i’th number in the sequence encoded by s, that is, (hx1 , . . . , xn i)i = xi . We define the ternary relation u ≺x v by ⌈β⌉ ≺x ⌈α⌉ ⇔ β < α ∧ N β ≤ N α + x . Then, when α > 0, we have fα (x) = fβ fβ (x) for some β such that ⌈β⌉ ≺x ⌈α⌉. It is easy to verify that u ≺x v is an elementary relation. Now we are going to prove that fα has elementary graph. We will work with finite nonempty trees where the nodes are natural numbers. Each node may have an arbitrary number of subtrees. More formally, we define a tree to be a pair (n, T ) where n ∈ N and T is a (possibly empty) set of trees. We define the height of the tree T , written ♯T , by ♯(n, ∅) = 1 and, for T = 6 ∅, ♯(n, T ) = 1 + max ♯T . T ∈T

We define the tree Tα,x by recursion on α. Let  ∅ [ Tα,x = (h⌈α⌉, x, fα (x)i, T ) with T = 

if α = 0 {Tβ,x, Tβ,fβ (x) } if α > 0 .

⌈β⌉≺x ⌈α⌉

(Claim I) We have ♯Tα,x ≤ fα (x) − x.

We prove this claim by induction on α. Assume α = 0. Then, we have Tα,x = (h⌈0⌉, x, fα (x)i, ∅), and thus, we also have fα (x) − x ≥ 1 = ♯T since fα (x) = f (x) ≥ 2x . Assume α > 0. We have fα (x) ≥ fβ fβ (x) for any β such that ⌈β⌉ ≺x ⌈α⌉. Fix β such that ⌈β⌉ ≺x ⌈α⌉. Since fβ is monotone and dominates 2x , we have x < fβ (x) < fβ fβ (x) ≤ fα (x). It follows that fβ (x) − x < fα (x) − x and fβ fβ (x) − fβ (x) < fα (x) − x. The induction hypothesis yields ♯Tβ,x ≤ fβ (x) − x and ♯Tβ,fβ (x) ≤ fβ fβ (x) − fβ (x), and thus, we have ♯Tβ,x < fα (x) − x and ♯Tβ,fβ (x) < fα (x) − x. But then we also have ♯Tα,x ≤ fα (x) − x. This completes the proof of (Claim I). We define the rank of an ordinal α < ǫ0 , written rk(α), by rk(ω α ) = 1+rk(α); rk(α + β) = max(rk(α), rk(β)); and rk(0) = 0.

24

Lars Kristiansen

Jan-Christoph Schlage-Puchta

Andreas Weiermann

(Claim II) For any m, n ∈ N, we have | {α | rk(α) ≤ n + 1 ∧ N α < m} | ≤ m(m

n

)

.

We prove this claim by induction on n. There are exactly m ordinals of rank 1 that have norm strictly less than m. Hence, | {α | rk(α) ≤ 0 + 1 ∧ N α < m} | = m = m(m

0

)

and the claim holds when n = 0. Any ordinal γ 6= 0 of rank n + 2 with N γ < m can be written in the form γ = ω α1 + . . . + ω αk where α1 , . . . , αk are ordinals of rank ≤ n + 1; N αi < m ( for i = 1, . . . , k); and 1 ≤ k < m. Hence, by the induction hypothesis, we have | {α | rk(α) ≤ (n + 1) + 1 ∧ N α < m} | ≤ m−1 X

| {α | rk(α) ≤ n + 1 ∧ N α < m} |k ≤

k=0

m−1 X

n

n

(m(m ) )k < (m(m ) )m = m(m

n+1

)

.

k=0

This completes the proof of (Claim II). Next we will define a set of of binary relations on trees. For any n ∈ N, we define – T ′ ⊆0 T iff T ′ = T – T ′ ⊆n+1 T iff T = (m, T ) and T ′ ⊆n T0 for some T0 ∈ T . Intuitively, T ′ ⊆n T holds if, and only if, T ′ is subtree of T and the path from the root of T ′ to the root of T is of length n. We define a tree T to be a computation tree iff T = Tα,x for some α < ǫ0 and some x ∈ N. (Claim III) Let T0 = (h⌈β⌉, x0 , y0 i, T0 ) and T = (h⌈α⌉, x, yi, T ) be computation trees such that T0 ⊆n T . Then, N β+x0 +1 ≤ N α+(n+1)y. (Claim III) is proved by a straightforward induction on n. We omit the details and state our next claim. (Claim IV) Let β > 0, and let T0 = (h⌈β⌉, x0 , y0 i, T0 ) and T = (h⌈α⌉, x, yi, T ) be computation trees such that T0 ⊆n T . Then, | {γ | γ < β ∧ N γ ≤ N β + x0 } | ≤ 2((ny+2y)

y

)

.

Streamlined Subrecursive Degree Theory

25

In order to prove this claim, we observe that δ0 < δ1 ⇒ rk(δ0 ) ≤ rk(δ1 ) (†) and rk(δ) ≤ N δ (‡) hold for any δ0 , δ1 , δ < ǫ0 . By (Claim II) and (†), we have | {γ | γ < β ∧ N γ ≤ N β + x0 + 1} | ≤ (N β + x0 + 1)((N β+x0 +1)

rk(β)−1

)

.

Furthermore, we have (N β + x0 + 1)((N β+x0 +1)

rk(β)−1

)

≤ 2((N β+x0 +1)

rk(β)

≤ 2((N α+(n+1)y) ≤ 2

)

rk(β)

)

((N α+(n+1)y)N α )

≤ 2((ny+2y)

y

(Claim III) (†), (‡) and β < α

)

as y ≥ N α

and thereby we have proved (Claim IV). We define a to be node in the tree T iff there exist n ∈ N and a set of trees T such that (a, T ) ⊆n T . Moreover, we will say that the trees in T are the immediate subtrees of a. (Claim V) Let T = (h⌈α⌉, x, yi, T ′ ) be a computation tree. Then, the 3y+2 ) number of nodes in T is bounded by 2(y . Let T0 = (h⌈β⌉, x0 , y0 i, T0 ) be any subtree of T , that is, we have T0 ⊆n T for some n. By the definition of a computation tree, we know that |T0 | = 2 · | {γ | γ < β ∧ N γ ≤ N β + x0 } | y

and by (Claim IV), we have |T0 | ≤ 2((ny+2y) Hence, (we can w.l.o.g. assume that y ≥ 2) |T0 | ≤ 2((ny+2y)

y

)+1

≤ 2(((y−x)y+2y)

)+1

y

. By (Claim I), we have n ≤ y−x.

)+1

≤ 2((y

3 y

) )+1

≤ 2(y

3y

)+1

.

3y

This shows that no node in the tree T has more than 2(y )+1 immediate subtrees. (Claim I) states that the height of T is bounded by y − x. Hence, the number of 3y+2 ) nodes in T is bounded by 2(y . This completes the proof of (Claim V). We will now define a predicate F (⌈α⌉, x, y, t) and prove that the next claim holds. (Claim VI) There exists a fixed k ∈ N such that we have fα (x) = y ⇔ ∃t ≤ 2yk F (⌈α⌉, x, y, t) for any α < ǫ0 . Moreover, F is an elementary predicate. The elementary predicates are closed under bounded qualification, and thus, it follows immediately from this claim that fα has elementary graph. Intuitively, the predicate F (⌈α⌉, x, y, t) will state that t encodes a sequence t0 , t1 , . . . , tℓ such that any node in the tree Tα,x occurs somewhere in this sequence. Let us turn to the precise definition of F . We define the predicate P0 by P0 (t) ⇔ ∀i, v, w < t [ (t)i = h⌈0⌉, v, wi → f (v) = w ]

26

Lars Kristiansen

Jan-Christoph Schlage-Puchta

Andreas Weiermann

and we define the predicates P1 and P2 by Q1 (⌈β⌉, v, w, t) ⇔ ∃j, k < t ∃z ≤ w [ (t)j = h⌈β⌉, v, zi ∧ (t)k = h⌈β⌉, z, wi ] ] ] P1 (t) ⇔ ∀i, ⌈γ⌉, v, w < t [ (t)i = h⌈γ⌉, v, wi ∧ ⌈γ⌉ 6= ⌈0⌉ → ∃⌈β⌉ < t [ ⌈β⌉ ≺v ⌈γ⌉ ∧ Q1 (⌈β⌉, v, w, t) ] ] Q2 (⌈β⌉, v, w, t) ⇔ ∃j, k < t ∃z, u ≤ w [ (t)j = h⌈β⌉, v, zi ∧ (t)k = h⌈β⌉, z, ui ] ] ] P2 (t) ⇔ ∀i, ⌈γ⌉, v, w < t [ (t)i = h⌈γ⌉, v, wi ∧ ⌈γ⌉ 6= ⌈0⌉ → ∀⌈β⌉ < t [ ⌈β⌉ ≺v ⌈γ⌉ → Q2 (⌈β⌉, v, w, t) ] ] . Finally, we define the predicate P by P (t) ⇔ P0 (t) ∧ P1 (t) ∧ P2 (t) and our main predicate F by F (⌈α⌉, x, y, t) ⇔ (t)0 = h⌈α⌉, x, yi ∧ P (t) . All encoding operations and all relations involved in the definition of F are elementary. In particular, we know that the relation f (v) = w is elementary as f is an honest function. The elementary relations and predicates are closed under bounded quantification and the operations of the propositional calculus, and thus, F is an elementary predicate. We will now prove the left-right implication of the equivalence in (Claim VI). Assume fα (x) = y. Let T be the tree Tα,x . Then, T is in form T = (h⌈α⌉, x, yi, T ). By (Claim V), we know that that the number of nodes in T is bounded by 3y+2 ) 2(y . Given (Claim III) and a reasonable G¨odel numbering of the ordinals, it should be obvious that there exists an elementary function ψ (not depending on T ) such that the G¨ odel number for any node in T is bounded by ψ(y). Now, 3y+2 ) there exists k ∈ N such that ht1 , . . . tℓ i ≤ 2yk if ℓ ≤ 2(y and ti ≤ ψ(y) (for i = 1, . . . , ℓ). This entails that there exists a number t ≤ 2yk such that F (⌈α⌉, x, y, t) holds. Just let t = ht1 , . . . , tℓ i where t1 , . . . , tℓ are the nodes of T . The right-left implication of the equivalence in (Claim VI) follows easily from the following assertion: P (t)

→

∀i, v, w [ (t)i = h⌈γ⌉, v, wi → fγ (v) = w ] .

(*)

We prove (*) by induction on γ. Let γ = 0. Assume P (t) and (t)i = h⌈γ⌉, v, wi. Now, as P0 (t) holds, we have fγ (v) = f (v) = w.

Streamlined Subrecursive Degree Theory

27

Let γ > 0. Assume P (t) and (t)i = h⌈γ⌉, v, wi. Since P1 (t) holds, we have j, k, z, β such that ⌈β⌉ ≺v ⌈γ⌉ and (t)j = h⌈β⌉, v, zi and (t)k = h⌈β⌉, z, wi. Now, we have β < α since ⌈β⌉ ≺v ⌈γ⌉, and then, our induction hypothesis yields fβ (v) = z and fβ (z) = w. This shows that w = fβ fβ (v) for some β such that ⌈β⌉ ≺v ⌈γ⌉. Now, as P2 (t) also holds, our induction hypothesis also yields that w ≥ fβ fβ (v) for all β such ⌈β⌉ ≺v ⌈γ⌉. Hence, w = max{ fβ fβ (v) | ⌈β⌉ ≺v ⌈γ⌉ } = max{ fβ fβ (v) | β < γ ∧ N β ≤ N γ + v } = fγ (v) . ⊓ ⊔

This completes the proof of (*).

Lemma 18. Let α < β, and let f be an honest function. Then, we have fα (x) ≤ fβ (N α + x). Proof. fβ (N α + x) = max{fδ fδ (N α + x) | δ < β ∧ N δ ≤ N β + N α + x} ≥ max{fδ fδ (x) | δ < α ∧ N δ ≤ N α + x} = fα (x) . ⊓ ⊔ Lemma 19 (≤αE -monotonicity of β-iteration). Let α, β ∈ SLim, and let f and g be honest functions. Then, f ≤αE g ⇒ fβ ≤αE gβ . Proof. We can w.l.o.g. assume that α ≤ β as the lemma holds trivially when β < α. Assume f ≤αE g. By the Growth Theorem, we have (i) f ≤ gγ for some γ < α. Furthermore, we have (ii) (gγ )β (x) = (gγ )δ (gγ )δ (x) for some δ such that δ < β and N δ ≤ N β + x. Thus, we have fβ (x) ≤ (gγ )β (x)

(i)

= (gγ )δ (gγ )δ (x)

(ii)

≤ gγ#δ gγ#δ (x)

Lemma 11

≤ gγ#δ#1 (x)

Lemma 10 (iv)

≤ gβ (N (γ#δ#1) + x)

γ, δ < β ∈ SLim and Lemma 18

≤ gβ (N (γ#β#1) + 2x)

as N δ ≤ N β + x

and this proves that fβ (x) ≤ gβ (N (γ#β#1) + 2x). Hence, since gβ (x) ≥ 2x , there exists m ∈ N such that fβ (x) ≤ (gβ )m (x). We have m < α, and thus, Lemma 17 and the Growth Theorem yield fβ ≤αE gβ . ⊓ ⊔ The previous lemma entails that we have f ≡αE g ⇒ fβ ≡αE gβ whenever α, β ∈ SLim and f, g are honest functions. Moreover, Lemma 17 states that fβ is honest whenever f is. Hence, the next definition makes sense.

28

Lars Kristiansen

Jan-Christoph Schlage-Puchta

Andreas Weiermann

Definition. Fix α, β ∈ SLim such that α ≤ β. For any honest α-elementary degree a we define the jump of a, written a′ , by a′ = degα (fβ ) where f is some honest function such that a = degα (f ). Furthermore, we define the zero degree, written 0, by 0 = degα (2x ). ⊓ ⊔ Lemma 20. Let α < β, and let f be an honest function. Then, we have fα (x) < fβ#1 (x) for all but finitely many x. Proof. By Lemma 18, we have fα (x) ≤ fβ (N α + x) ≤ fβ (2x) < fβ fβ (x) ≤ fβ#1 (x) for all x > N α.

⊓ ⊔

Lemma 21. Let f be an honest function. Then, fβ#1 ≤E fβ . Proof. Lemma 17 says that fβ#1 and fβ are honest functions. Furthermore, we have fβ#1 (x) = max{ fγ fγ (x) | γ ≤ β ∧ N γ ≤ N β + x + 1 } = max{ fβ fβ (x) , max{ fγ fγ (x) | γ < β ∧ N γ ≤ N β + x + 1 } } ≤ max{ fβ fβ (x) , fβ (x + 1) } ≤ fβ fβ (x + 1) ≤ fβ fβ fβ (x) . Hence, we have fβ#1 ≤E fβ by the Growth Theorem.

⊓ ⊔

Lemma 22. Let f be an honest function, and let α, β ∈ SLim be such that α ≤ β. Then, we have f