aa r X i v : . [ m a t h . L O ] J un NATURAL FACTORS OF THE MUCHNIK LATTICECAPTURING IPC
RUTGER KUYPER
Abstract.
We give natural examples of factors of the Muchnik lattice whichcapture intuitionistic propositional logic (IPC), arising from the concepts oflowness, 1-genericity, hyperimmune-freeness and computable traceability. Thisprovides a purely computational semantics for IPC. Introduction
Ever since the introduction of intuitionistic logic by Heyting, there have beeninvestigations into the computational content of proofs in intuitionistic logic. Thebest known of these is the investigation into realisability, which was initiated byKleene in his 1945 paper [8]. Unfortunately, Kleene’s original concept of realis-ability turns out to capture a proper extension of intuitionistic propositional logic(IPC). Nowadays, this field investigates not only Kleene realisability, but also manyvariations thereof; see e.g. the recent reference Van Oosten [23].Another approach to capture IPC in a computational way was provided by Med-vedev [11] and Muchnik [13] in respectively 1955 and 1963. Their approaches, inthe form of the
Medvedev lattice and the
Muchnik lattice , again turn out to fallshort: they realise the weak law of the excluded middle ¬ p ∨ ¬¬ p . However, thestudy of these lattices did not end here, for multiple reasons.First, the Medvedev and Muchnik lattices can be seen as generalisations of theTuring degrees (in fact, the Turing degrees can be embedded into both of theselattices). Therefore these lattices are of independent interest to computability the-orists, regardless of any logical content they might carry. Research in this directionhas increased in recent years; many details can be found in the surveys of Sorbi [19]and Hinman [5].Furthermore, even on the logical side not all is lost: it turns out that we canrepair the logical deficiency of the Medvedev and Muchnik lattices (i.e. the factthat they realise more than IPC). In [17], Skvortsova shows that there is a factorof the Medvedev lattice which exactly captures IPC, and in Sorbi and Terwijn [20]the analogous result for the Muchnik lattice is shown.These factors are obtained by taking the Medvedev or Muchnik lattice moduloa principal filter generated by some set A . If we want to capture IPC in a trulycomputational way, we would want such a set A to have some computational inter-pretation. Unfortunately, this is not the case for the sets A appearing in the resultof Skvortsova and in the result of Sorbi and Terwijn: instead of starting with somecomputationally motivated set A and proving that the factor induced by this set A captures IPC, they construct a set A which exactly has the properties they requirefor their proof, but which does not seem to have any computational interpretation. Date : 14th September 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Muchnik degrees, Intuitionistic logic, Kripke models.Research supported by NWO/DIAMANT grant 613.009.011 and by John Templeton Founda-tion grant 15619: ‘Mind, Mechanism and Mathematics: Turing Centenary Research Project’.
In [21], Terwijn asked if there are any natural sets A for which the factor capturesIPC.In the present paper, we will show that for the Muchnik lattice it is indeedpossible to choose the set A in a natural way (in the sense that it is definable usingcommonly used concepts from computability theory) and still obtain IPC as thetheory of its factor. This way, we obtain a purely computational semantics for IPC.Aside from this, our results also put the computability-theoretic concepts used todefine A into a new light. Among these concepts are lowness, 1-genericity below ∅ ′ ,hyperimmune-freeness and computable traceability. Since our framework is general,our results could be adapted to suit other concepts.In the next section we will briefly recall the structure of the Muchnik latticeand its factors. In section 3 we will describe our framework of splitting classes .In section 4 we show that our framework is non-trivial by proving that the lowfunctions and the functions of 1-generic degree below ∅ ′ fit in our framework. Next,in section 5 we prove that splitting classes naturally induce a factor of the Muchniklattice which captures IPC. Finally, in section 6 we consider whether two otherconcepts from computability theory give us splitting classes: hyperimmune-freenessand computable traceability.Our notation is mostly standard. We let ω denote the natural numbers and ω ω the Baire space of functions from ω to ω . For finite strings σ, τ we denote by σ ⊆ τ that σ is a substring of τ , by σ ⊂ τ that σ is a proper substring of τ andby σ | τ that σ and τ are incomparable. The concatenation of σ and τ is denotedby σ ⋆ τ ; for n ∈ ω we denote by σ ⋆ n the concatenation of σ with the string h n i . We assume a fixed, computable enumeration of the set of all finite binarystrings. We let ∅ ′ denote the halting problem. By { e } A ( n )[ m ] ↓ we mean thatthe e th Turing machine with oracle A and input n terminates in at most m steps.For functions f, g ∈ ω ω we denote by f ⊕ g the join of the functions f and g , i.e.( f ⊕ g )(2 n ) = f ( n ) and ( f ⊕ g )(2 n + 1) = g ( n ). For a poset ( X, ≤ ) and elements x, y ∈ X , we denote by [ x, y ] X the set of elements u ∈ X satisfying x ≤ u ≤ y .For any set A ⊆ ω ω we denote by A its complement in ω ω . When we say thata set is countable, we include the possibility that it is finite. For unexplainednotions from computability theory, we refer to Odifreddi [14], for the Muchnik andMedvedev lattices, we refer to the surveys of Sorbi [19] and Hinman [5] (but we usethe notation from Sorbi and Terwijn [20]), for lattice theory, we refer to Balbes andDwinger [1], and finally for unexplained notions about Kripke semantics we referto Chagrov and Zakharyaschev [2].2. Muchnik lattice and Brouwer algebras
We begin by briefly recalling the definition of and some elementary facts aboutthe Muchnik lattice.
Definition 2.1. (Muchnik [13]) Let A , B ⊆ ω ω (we will call such subsets of ω ω mass problems ). We say that A Muchnik reduces to B (notation: A ≤ w B ) if forevery g ∈ B there exists an f ∈ A such that f ≤ T g . If A ≤ w B and B ≤ w A wesay that A and B are Muchnik equivalent (notation: A ≡ w B ). The equivalenceclasses of Muchnik equivalence are called Muchnik degrees and the set of Muchnikdegrees is denoted by M w .To avoid confusion, we do not use ∨ for the join (least upper bound) or ∧ forthe meet (greatest lower bound) in lattices, because later on we will see that thejoin corresponds to the logical conjunction ∧ and that the meet corresponds to thelogical disjunction ∨ . Instead, we use ⊕ for join and ⊗ for meet. ATURAL FACTORS OF THE MUCHNIK LATTICE CAPTURING IPC 3
Definition 2.2. (McKinsey and Tarski [9]) A
Brouwer algebra is a bounded dis-tributive lattice together with a binary implication operator → satisfying: a ⊕ c ≥ b if and only if c ≥ a → b i.e. a → b is the least element c satisfying a ⊕ c ≥ b .First, we give a simple example of a Brouwer algebra. Definition 2.3.
Let ( X, ≤ ) be a poset. We say that a subset Y ⊆ X is upwardsclosed or is an upset if for all y ∈ Y and all x ∈ X with x ≥ y we have x ∈ Y .Similarly, we say that Y ⊆ X is downwards closed or a downset if for all y ∈ Y andall x ∈ X with x ≤ y we have x ∈ Y .We denote by O ( X ) the collection of all upwards closed subsets of X , orderedunder reverse inclusion ⊇ . Proposition 2.4. O ( X ) is a Brouwer algebra under the operations U ⊕ V = U ∩ V , U ⊗ V = U ∪ V and U → V = { x ∈ X | ∀ y ≥ x ( y ∈ U ⇒ y ∈ V ) } . Proof.
The upwards closed sets of a poset form a topology (because they are closedunder arbitrary unions and intersections). The result now follows from Balbes andDwinger [1, IX.3, Example 4]. (cid:3) It turns out that the Muchnik lattice is also a Brouwer algebra.
Proposition 2.5. (Muchnik [13])
The Muchnik lattice is a Brouwer algebra underthe operations induced by: A ⊕ B = { f ⊕ g | f ∈ A and g ∈ B } A ⊗ B = A ∪ BA → B = { g ∈ ω ω | ∀ f ∈ A ∃ h ∈ B ( f ⊕ g ≥ T h ) } Proposition 2.6.
The Muchnik lattice is isomorphic to the lattice of upsets of theTuring degrees.Proof.
We use a proof inspired by Muchnik’s proof that the Muchnik degrees canbe embedded in the Medvedev degrees (preserving 0, 1 and minimal upper bounds)from [13]. For every A ⊆ ω ω , we have that A ≡ w C ( A ) := { f ∈ ω ω | ∃ g ∈ A ( g ≤ T f ) } . Now it is directly verified that the mapping sending A to C ( A ) induces anorder isomorphism between M w and O ( D ) (as defined in Definition 2.3). Finally,every order isomorphism between Brouwer algebras is automatically a Brouweralgebra isomorphism, see Balbes and Dwinger [1, IX.4, Exercise 3]. (cid:3) The main motivation behind Brouwer algebras is that they allow us to specifysemantics containing IPC.
Definition 2.7. (McKinsey and Tarski [10]) Let ϕ ( x , . . . , x n ) be a propositionalformula with free variables among x , . . . , x n , let B be a Brouwer algebra and let b , . . . , b n ∈ B . Let ψ be the formula in the language of Brouwer algebras obtainedfrom ϕ by replacing logical disjunction ∨ by ⊗ , logical conjunction ∧ by ⊕ , logicalimplication → by Brouwer implication → and the false formula ⊥ by 1 (we viewnegation ¬ α as α → ⊥ ). We say that ϕ ( b , . . . , b n ) holds in B if ψ ( b , . . . , b n ) = 0.Furthermore, we define the theory of B (notation: Th( B )) to be the set of thoseformulas which hold for every valuation, i.e.Th( B ) = { ϕ ( x , . . . , x m ) | ∀ b , . . . , b m ∈ B ( ϕ ( b , . . . , b m ) holds in B ) } . In most literature, including Balbes and Dwinger, results are proved for Heyting algebras, theorder-dual of Brouwer algebras. However, all results we cite directly follow for Brouwer algebrasin the same way.
R. KUYPER
The following soundness result is well-known and directly follows from the ob-servation that all rules in some fixed deduction system for IPC preserve truth.
Proposition 2.8. (McKinsey and Tarski [10, Theorem 4.1])
For every Brouweralgebra B : IPC ⊆ Th( B ) .Proof. See e.g. Chagrov and Zakharyaschev [2, Theorem 7.10]. (cid:3)
As discussed in the introduction, one might hope that the computationally mo-tivated Muchnik lattice has IPC as its theory. However, it is easily verified that theweak law of the excluded middle ¬ p ∨ ¬¬ p holds in the Muchnik lattice, while itdoes not hold in IPC. Fortunately, it turns out we can still capture IPC by lookingat certain factors of the Muchnik lattice. Proposition 2.9.
Let B be a Brouwer algebra. For every principal filter F gen-erated by some element x ∈ B , B/ F is a Brouwer algebra (also denoted by B/x )under the implication defined on the equivalence classes by [ y ] → B/ F [ z ] = [( y ⊗ x ) → B ( z ⊗ x )] . Proof.
On one hand we have (because [ y ⊗ x ] = [ y ] by definition of B/ F ):[( y ⊗ x ) → B ( z ⊗ x )] ⊕ [ y ] = [(( y ⊗ x ) → B ( z ⊗ x )) ⊕ ( y ⊗ x )] ≥ [ z ⊗ x ] . On the other hand, for any element u such that [ y ] ⊕ [ u ] ≥ [ z ] we have that [ y ⊕ u ⊕ z ] = [ y ⊕ u ] so ( y ⊕ u ⊕ z ) ⊗ x = ( y ⊕ u ) ⊗ x by definition of B/x . Then distributivityshows that ( y ⊗ x ) ⊕ ( u ⊗ x ) ⊕ ( z ⊗ x ) = ( y ⊗ x ) ⊕ ( u ⊗ x )i.e. ( y ⊗ x ) ⊕ ( u ⊗ x ) ≥ ( z ⊗ x ). So, since B is a Brouwer algebra we see that u ⊗ x ≥ ( y ⊗ x ) → B ( z ⊗ x ), and therefore [ u ] = [ u ⊗ x ] ≥ [( y ⊗ x ) → B ( z ⊗ x )]. (cid:3) Taking such a factor essentially amounts to moving from the entire algebra tojust the interval [0 , x ] M w of elements below x (indeed, the factor is isomorphic tothis interval). Because the top element of [0 , x ] M w is smaller than the top elementof M w if x = 1, the interpretation of negation ¬ b , which is defined as b →
1, alsodiffers between these two algebras. Thus, taking a factor roughly corresponds tochanging the negation.The following result, an analogue of the same result for the Medvedev lattice bySkvortsova [17], shows that there exists a factor of the Muchnik lattice with IPCas its theory.
Theorem 2.10. (Sorbi and Terwijn [20])
There exists a mass problem A ⊆ ω ω such that Th( M w / A ) = IPC . The particular mass problem A from the previous theorem does not have anintuitive interpretation and is constructed in quite an ad-hoc manner. However, inthis paper we will show that natural mass problems A such that the factor M w / A captures IPC do exist. 3. Splitting classes
As announced above, we will present our results in a general framework so thatadditional examples can easily be obtained. Our framework of splitting classes abstracts exactly what we need for our proof in section 5 to work. It roughly saysthat A is a splitting class if, given some function f ∈ A , we can construct functions h , h ∈ A above it whose join is not in A while ‘avoiding’ a given finite set of otherfunctions in A . This is made precise below. ATURAL FACTORS OF THE MUCHNIK LATTICE CAPTURING IPC 5
Definition 3.1.
Let A ⊆ ω ω be a non-empty countable class which is downwardsclosed under Turing reducibility. We say that A is a splitting class if for every f ∈ A and every finite subset B ⊆ { g ∈ A | g T f } there exist h , h ∈ A suchthat h , h ≥ T f , h ⊕ h A and for all g ∈ B : g ⊕ h , g ⊕ h A .Note that, because every splitting class A is downwards closed under Turingreducibility, we in particular have that A is closed under Turing equivalence, i.e. if f ∈ A and g ≡ T f then also g ∈ A .We emphasise that we required a splitting class to be countable. There are alsointeresting examples which satisfy the requirements except for the countability: forexample, in section 6 we will see that this is the case for the set of hyperimmune-free functions. In that section we will also discuss how to suitably generalise theconcept to classes of higher cardinality.It turns our that in order to show that something is a splitting class it willbe easier to prove that one of the two alternative formulations given by the nextproposition holds. Proposition 3.2.
Let A ⊆ ω ω be a non-empty countable class which is downwardsclosed under Turing reducibility. Then the following are equivalent: (i) A is a splitting class. (ii) For every f ∈ A and every finite subset B ⊆ { g ∈ A | g T f } there exists h ∈ A such that h > T f and for all g ∈ B : g ⊕ h A . (iii) For every f ∈ A there exists h ∈ A such that h T f , and for every f ∈ A ,every finite subset B ⊆ { g ∈ A | g T f } and every h ∈ { g ∈ A | g T f } there exists h ∈ A such that h > T f , h ⊕ h A and for all g ∈ B : h T g .Proof. (i) → (ii): Let h , h ∈ A be such that h , h ≥ T f , h ⊕ h A and forall g ∈ B : g ⊕ h , g ⊕ h A . Let h = h . Because h ≡ T f would imply that h ⊕ h ≡ T h ∈ A we see that h > T f and therefore we are done.(ii) → (iii): First, for every f ∈ A we can find h ∈ A such that h T f byapplying (ii) with B = ∅ . Next, using (ii) determine h ∈ A such that h > T f and for all g ∈ B ∪ { h } : g ⊕ h A . Then the only thing we still need to showis that h T g for all g ∈ B . However, h ≥ T g would imply h ⊕ g ≡ T h ∈ A , acontradiction.(iii) → (ii): Fix g ∈ A such that g T f . Let B ⊆ { g ∈ A | g T f } befinite. Without loss of generality, we may assume that g ∈ B ; in particular, wemay assume that B is non-empty. So, let B = { g , . . . , g n } . We inductively definea sequence h , < T h , < T · · · < T h ,n of functions in A . First, we let h , = f .Next, to obtain h ,i +1 from h ,i , apply (iii) to find a function h ,i +1 > T h ,i suchthat h ,i +1 ⊕ g i +1 A and for all i + 2 ≤ j ≤ n we have g j T h ,i +1 . Then h := h ,n is as desired.(ii) → (i): Using (ii), we can find h ∈ A such that h > T f and g ⊕ h A for all g ∈ B . By applying (ii) a second time, we can now find h ∈ A such that h > T f and for all g ∈ B ∪ { h } : g ⊕ h A . Then h and h are as desired. (cid:3) Low and 1-generic below ∅ ′ are splitting classes Before we show that splitting classes allow us to capture IPC as a factor of theMuchnik lattice, we want to demonstrate that our framework of splitting classesis non-trivial. To this end, we will show that the class of low functions, and thatthe class of functions of 1-generic degree below ∅ ′ together with the computablefunctions, are splitting classes. We will denote the first class by A low and thesecond class by A gen ≤∅ ′ . We remark that the second class naturally occurs as theclass of functions that are low for EX (as proved in Slaman and Solovay [18]). R. KUYPER
Because these kinds of arguments are usually given as constructions on sets (orelements of Cantor space) rather than the functions (or elements of Baire space)which occur in the Muchnik lattice, we will work with sets instead of functions inthis section. However, we do not use the compactness of Cantor space anywhereand therefore it is only a notational matter.First, we recall some basic facts about 1-genericity over a set.
Definition 4.1. (Jockusch [6, p. 125]) Let
A, B ⊆ ω . We say that B is if for every e ∈ ω there exists σ ⊆ B such that either { e } σ ( e ) ↓ or for all τ ⊇ σ wehave { e } τ ( e ) ↑ .More generally, we say that B is A if for every e ∈ ω there exists σ ⊆ B such that either { e } A ⊕ σ ( e ) ↓ or for all τ ⊇ σ we have { e } A ⊕ τ ( e ) ↑ . Lemma 4.2. (Folklore)
Let B be 1-generic over A . Then: (i) If A is 1-generic, then A ⊕ B is 1-generic. (ii) If A is low and B ≤ T ∅ ′ , then A ⊕ B is low.Proof. (i): Assume A is 1-generic. Let e ∈ ω . We need to find a σ ⊆ A ⊕ B suchthat either { e } σ ( e ) ↓ or such that for all τ ⊇ σ we have { e } τ ( e ) ↑ .If { e } A ⊕ B ( e ) ↓ , we can choose σ ⊆ A ⊕ B such that { e } σ ( e ) ↓ . Otherwise, since B is 1-generic over A , we can determine σ B ⊆ B such that for all τ B ⊇ σ B we have { e } A ⊕ τ B ( e ) ↑ . Fix an index ˜ e such that for all C ⊆ ω and all x ∈ ω : { ˜ e } C ( x ) ↓⇔ ∃ τ B ⊇ σ B { e } C ⊕ τ B ( e ) ↓ . We first note that { ˜ e } A ( x ) ↑ by our choice of σ B . Therefore, using the 1-genericity of A , determine σ A ⊆ A such that for all τ A ⊇ σ A we have { ˜ e } τ A (˜ e ) ↑ .By choice of ˜ e we then have for for all τ A ⊇ σ A that ∀ τ B ⊇ σ B { e } τ A ⊕ τ B ( e ) ↑ , whichis the same as saying that for all τ ⊇ σ A ⊕ σ B we have { e } τ ( e ) ↑ . This is exactlywhat we needed to show.(ii) We show that both ( A ⊕ B ) ′ and its complement ( A ⊕ B ) ′ are c.e. in A ′ ⊕ B ≡ T ∅ ′ . To this end, we note that e ∈ ( A ⊕ B ) ′ if and only if ∃ σ A ⊆ A ∃ σ B ⊆ B (cid:0) { e } σ A ⊕ σ B ( e ) ↓ (cid:1) which is c.e. in A ⊕ B ≤ T A ′ ⊕ B . Next, using the fact that B is 1-generic over A ,we see that e ( A ⊕ B ) ′ if and only if ∃ σ B ⊆ B ∀ τ B ⊇ σ B (cid:0) { e } A ⊕ τ B ( e ) ↑ (cid:1) which is c.e. in A ′ ⊕ B . The result now follows by the relativised Post’s theorem. (cid:3) Theorem 4.3. A low and A gen ≤∅ ′ are splitting classes.Proof. The first class is clearly downwards closed; for the second class this is provedin Haught [4] (but also follows from the fact mentioned above that A gen ≤∅ ′ consistsof exactly those functions which are low for EX).First, we consider the class of low functions. By Proposition 3.2, we can showthat the low functions form a splitting class by proving that for every low A andevery finite B ⊆ { B ∈ ω ω | B low and B T A } there exists a set C T A suchthat A ⊕ C is low and such that for all B ∈ B we have that B ⊕ ( A ⊕ C ) ≡ T ∅ ′ .(Note that C T A ensures that A ⊕ C > T A , while B ⊕ ( A ⊕ C ) ≡ T ∅ ′ ensuresthat B ⊕ ( A ⊕ C ) is neither 1-generic nor low.) Lemma 4.2 tells us that we canmake A ⊕ C low by ensuring that C ≤ ∅ ′ and that C is 1-generic over A . Thus, itis enough if we can show:(1) If A is low and B ⊆ { B ∈ ω ω | B ≤ T ∅ ′ and B T A } is finite, then thereexists a set C ≤ T ∅ ′ which is 1-generic over A such that C T A and for all B ∈ B : B ⊕ ( A ⊕ C ) ≡ T ∅ ′ . ATURAL FACTORS OF THE MUCHNIK LATTICE CAPTURING IPC 7
In fact, we then also immediately get the result for the class of functions of1-generic degree below ∅ ′ . Namely, let A ≤ T ∅ ′ be of 1-generic degree and let B ⊆ { B ∈ ω ω | B ≤ T ∅ ′ and B T A } be finite. Just as above, it would be enoughto have a set C ≤ T ∅ ′ such that C T A , A ⊕ C is of 1-generic degree and for all B ∈ B : B ⊕ ( A ⊕ C ) ≡ T ∅ ′ . Note that this expression is invariant under replacing A with a Turing equivalent set, so because A is of 1-generic degree we may withoutloss of generality assume A to be 1-generic. Then, because A ≤ ∅ ′ is 1-generic, it isalso low. So we can find a set C as in (1). By Lemma 4.2 we then have that A ⊕ C is 1-generic, and therefore C is exactly as desired.To prove (1) we modify the proof of the Posner and Robinson Cupping Theorem[16]. Let B = { B , . . . , B k } . For every B i ∈ B , since B i ≤ ∅ ′ we can approximate B i by a computable sequence B i , B i , . . . of finite sets. We now let α i be the computation function defined by letting α i ( n ) be the least m ≥ n such that B mi ↾ ( n + 1) = B i ↾ ( n + 1). Then B i ≡ T α i . Now let α = min( α , . . . , α k ). Then, byLemma 6 of [16], any function g which dominates α computes some B i . Thus, wesee that no function computable in A can dominate α .We will now construct a set C as in (1) by a finite extension argument, i.e. as C = S n ∈ ω σ n . Fix any computable sequence τ , τ , . . . of mutually incomparablefinite strings (for example, τ n = h n i , the string consisting of n times a 0 followedby a 1). We start with σ = ∅ . To define σ e +1 given σ e , let n be the least m ∈ ω such that either (where the quantifiers are over finite strings):(2) ∀ σ ⊇ σ e ⋆ τ m (cid:0) { e } A ⊕ σ ( e ) ↑ (cid:1) or(3) ∃ σ ⊇ σ e ⋆ τ m (cid:0) | σ | ≤ α ( m ) ∧ { e } A ⊕ σ ( e )[ | σ | ] ↓ (cid:1) . Such an m exists: otherwise, for every l ∈ ω we could let β ( l ) be the least s ∈ ω such that ∃ σ ⊇ σ e ⋆ τ l (cid:0) { e } A ⊕ σ ( e )[ | σ | ] ↓ ∧| σ | = s (cid:1) . For every l such an s exists because (2) does not hold for l , while such an s hasto be strictly bigger than α ( l ) because (3) also does not hold. So, β would be afunction computable in A which dominates α , of which we have shown above thatit cannot exist.Now, if case (2) holds for n , then we let σ e +1 = σ e ⋆ τ n ⋆ ∅ ′ ( e ). Otherwise, we let σ e +1 = σ ⋆ ∅ ′ ( e ), where σ is the least σ such that (3) is satisfied.The construction is computable in A ′ ⊕ B ⊕ · · · ⊕ B k ≤ T ∅ ′ : the set of m ∈ ω forwhich (2) holds is co-c.e. in A , while for (3) this is computable in α ≤ T B ⊕· · ·⊕ B k and A . Therefore, C ≤ T ∅ ′ holds.Furthermore, per construction of σ e +1 we have either { e } A ⊕ σ e +1 ( e ) ↓ , or for all τ ⊇ σ e +1 we have { e } A ⊕ τ ( e ) ↑ . So, C is 1-generic over A .Next, for every 1 ≤ i ≤ k the construction is computable in ( A ⊕ C ) ⊕ B i : todetermine σ e +1 given σ e , use C to find the unique n ∈ ω such that C ⊇ σ e ⋆ τ n .We can now compute in A and B i if there exists some string σ ⊇ σ e ⋆ τ n of lengthat most α i ( n ) such that { e } A ⊕ σ ( e )[ | σ | ] ↓ : if so, let σ be the least such string andthen σ e +1 = B ↾ | σ | + 1. Otherwise, σ e +1 = B ↾ | σ e | + 1. Then we also see that ∅ ′ is computable in ( A ⊕ C ) ⊕ B i , because ∅ ′ ( e ) is the last element of σ e +1 . Since also A, B i , C ≤ T ∅ ′ we see that ( A ⊕ C ) ⊕ B i ≡ T ∅ ′ .Finally, because for every low A there exists some low B > T A (see e.g. Odi-freddi [14, Proposition V.2.21]), we may without loss of generality assume that sucha B is in B . Then we have B ⊕ ( A ⊕ C ) ≡ T ∅ ′ , as shown above. Now, if it werethe case that C ≤ T A , then ∅ ′ ≡ T B ⊕ ( A ⊕ C ) ≡ T B , which contradicts B beinglow. So C T A , which is the last thing we needed to show. (cid:3) R. KUYPER The theory of a splitting class
We will now show that the theory of a splitting class equals IPC. We start bymoving away from our algebraic viewpoint to Kripke semantics. The crucial stepwe need for this is the following:
Theorem 5.1.
For any poset ( X, ≤ ) , the theory of ( X, ≤ ) as a Kripke frame isthe same as theory of the lattice of upsets of X as a Brouwer algebra.Proof. See e.g. Chagrov and Zakharyaschev [2, Theorem 7.20] for the order-dualresult for Heyting algebras. (cid:3)
Proposition 5.2.
Let A ⊆ ω ω be downwards closed under Turing reducibility.Then M w / A (i.e. M w modulo the principal filter generated by A ) is isomorphic tothe lattice of upsets O ( A ) of A . In particular, Th( M w / A ) = Th( A ) (the first asBrouwer algebra, the second as Kripke frame).Proof. By Proposition 2.6, M w is isomorphic to the lattice of upsets O ( D ) of theTuring degrees D , by sending each set B ⊆ ω ω to C ( B ). Since A is upwardsclosed, we see that the isomorphism sends A to itself. Therefore, M w / A , whichis isomorphic to the initial segment [ ω ω , A ] M w of M w , is isomorphic to the initialsegment [ ω ω , A ] O ( D ) . Finally, [ ω ω , A ] O ( D ) is easily seen to be isomorphic to O ( A ),by sending each set B ∈ O ( A ) to B ∪ A . The result now follows from the previoustheorem. (cid:3) Thus, if we take the factor of M w given by the principal filter generated by A ,we get exactly the theory of the Kripke frame ( A , ≤ T ). The rest of this section willbe used to show that for splitting classes this theory is exactly IPC. To this end,we need the right kind of morphisms for Kripke frames, called p-morphisms . Definition 5.3. (De Jongh and Troelstra [3]) Let ( X , ≤ ), ( X , ≤ ) be Kripkeframes. A surjective function f : ( X , ≤ ) → ( X , ≤ ) is called a p-morphism if(1) f is an order homomorphism: x ≤ y → f ( x ) ≤ f ( y ),(2) ∀ x ∈ X ∀ y ∈ X ( f ( x ) ≤ y → ∃ z ∈ X ( x ≤ z ∧ f ( z ) = y )). Proposition 5.4.
If there exists a p -morphism from ( X , ≤ ) to ( X , ≤ ) , then Th( X , ≤ ) ⊆ Th( X , ≤ ) .Proof. See e.g. Chagrov and Zakharyaschev [2, Corollary 2.17]. (cid:3)
Theorem 5.5.
Th(2 So, if we want to show that the theory of M w / A equals IPC, it is enough to showthat there exists a p -morphism from A to 2 <ω . We next show that this is indeedpossible for splitting classes. Proposition 5.6. Let A be a splitting class. Then there exists a p -morphism α : ( A , ≤ T ) → <ω .Proof. Instead of building a p -morphism from A , we will build it from A / ≡ T (whichis equivalent to building one from A , since any order homomorphism has to send T -equivalence classes to equal strings). For ease of notation we will write A for A / ≡ T during the remainder of this proof.Fix an enumeration a , a , . . . of A . We will build a sequence α ⊆ α ⊆ . . . of finite, partial order homomorphisms from A to 2 <ω , which additionally satisfythat if a , b ∈ dom( α i ) and α i ( a ) | α i ( b ), then a ⊕ b A .We satisfy the following requirements: ATURAL FACTORS OF THE MUCHNIK LATTICE CAPTURING IPC 9 • R : α ( ) = ∅ (where is the least Turing degree) • R n +1 : a n ∈ dom( α n +1 ) • R n +2 : there are c , c ∈ dom( α n +2 ) with c , c ≥ T a n and α n +2 ( c ) = α n +1 ( a n ) ⋆ α n +2 ( c ) = α n +1 ( a n ) ⋆ α = S n ∈ ω α n is a p -morphism α : ( A , ≤ T ) → <ω . First, the odd requirements ensure that α is total.Furthermore, α is an order homomorphism because the α i are. To show that α isa p -morphism, let a ∈ A and let α ( a ) ⊆ y ; we need to find some a ≤ T b ∈ A suchthat α ( b ) = y . Because α ( a ) ⊆ y we know that y = α ( a ) ⋆ y ′ for some finite string y ′ . We may assume y ′ to be of length 1, the general result then follows by induction.Now, if we let n ∈ ω be such that a = a n then a n ∈ dom( α n +1 ), so requirement R n +2 tells us that there are functions c , c ≥ a n with α n +2 ( c ) = α ( a ) ⋆ α n +2 ( c ) = α ( a ) ⋆ 1. Now either α ( c ) = y or α ( c ) = y , which is what we neededto show. That α is surjective directly follows from the fact that ∅ is in its rangeand that it satisfies property (2) of a p -morphism.Now, we show how to actually construct the sequence. First, α is alreadydefined. Next assume α n has been constructed, we will construct α n +1 extending α n such that a n ∈ dom( α n +1 ). The set X := { α n ( b ) | b ∈ dom( α n ) and b ≤ T a n } is totally ordered under ⊆ . Since, if b , c ≤ T a n then b ⊕ c ≤ T a n . Now, if α n ( b )and α n ( c ) are incomparable then we assumed that b ⊕ c A . This contradictsthe assumption that A is downwards closed. So, we can define α n +1 ( a n ) to be thelargest element of X .We show that α n +1 is an order homomorphism; we then also automaticallyknow that it is well-defined. Thus, let b , b ∈ dom( α n +1 ) with b ≤ T b . If theyare both already in dom( α n ), then the induction hypothesis on α n already tells usthat α n +1 ( b ) ⊆ α n +1 ( b ). If b ∈ dom( α n ) and b = a n , then α n ( b ) ∈ X ,so by definition of α n +1 ( a n ) we directly see that α n +1 ( b ) ⊆ α n +1 ( a n ). Finally,we consider the case that b = a n and b ∈ dom( α n ). To show that α n +1 ( a n ) ⊆ α n +1 ( b ) = α n ( b ) it is enough to show that all elements of X are below α n ( b ),because α n +1 ( a n ) is the largest element of the set X . Therefore, let b ∈ dom( α n )be such that b ≤ T a n . Then we have that b ≤ T a n ≤ T b , and since α n is anorder homomorphism this implies that α n ( b ) ≤ T α n ( b ), as desired.Finally, we need to show that if c ∈ dom( α n ) is such that α n +1 ( c ) and α n +1 ( a n ) are incomparable, then c ⊕ a n A . If α n +1 ( c ) and α n +1 ( a n ) areincomparable, there has to be some b ≤ T a n with b ∈ dom( α n ) such that α n ( c )and α n ( b ) are incomparable (because α n +1 ( a n ) is the largest element of X ).However, then by induction hypothesis b ⊕ c A and because A is downwardsclosed this also implies that c ⊕ a n A .We now assume that α n +1 has been defined and consider requirement R n +2 .Let B = { b ∈ dom( α n +1 ) | b T a n } . Since A is a splitting class there exist c , c ∈ A such that c , c ≥ a n , c ⊕ c A and for all b ∈ B we have b ⊕ c , b ⊕ c A . Now extend α n +1 by letting α n +2 ( c ) = α n +1 ( a n ) ⋆ α n +2 ( c ) = α n +1 ( a n ) ⋆ α n +2 is an order homomorphism. Let b , b ∈ dom( α n +2 )and b ≤ T b . We again distinguish several cases: • b , b ∈ dom( α n +1 ): this directly follows from the fact that α n +1 is anorder homomorphism by induction hypothesis. • b , b ∈ { c , c } : since c ⊕ c A and therefore differs from both c and c , this can only happen if b = b , so this case is trivial. • b ∈ { c , c } , b ∈ dom( α n +1 ): note that c , c > T a n (otherwise c ⊕ c ∈ { c , c } ⊆ A ), so we see that b > T a , and then by construction of c and c we know that b ⊕ c , b ⊕ c A . This contradicts b ≤ T b , sothis case is impossible. • b ∈ dom( α n +1 ), b ∈ { c , c } : if b T a n , then again by constructionof c and c we have that b = b ⊕ b A which is a contradiction. So b ≤ T a n and therefore α n +2 ( b ) = α n +1 ( b ) ⊆ α n +1 ( a n ) ⊆ α n +2 ( b ).Finally, we show that if b ∈ dom( α n +2 ) is such that α n +2 ( b ) and α n +2 ( c ) areincomparable, then b ⊕ c A (the same then follows analogously for c ). If b = c this is clear from the definition of α n +2 . Otherwise, we have b ∈ dom( α n +1 ). If itwere the case that b ≤ T a n , then α n +2 ( b ) = α n +1 ( b ) ⊆ α n +1 ( a n ) ⊆ α n +2 ( c ),a contradiction. Thus b T a n , and therefore b ⊕ c A by construction of c . (cid:3) Theorem 5.7. For any splitting class A : Th( M w / A ) = IPC .Proof. From Proposition 5.2, Proposition 5.4, Theorem 5.5 and Proposition 5.6. (cid:3) Therefore, combining this with the results from section 4 we now see: Theorem 5.8. Th( M w / A low ) = Th( M w / A gen ≤∅ ′ ) = IPC . Further splitting classes Hyperimmune-free functions. In this section, we will look at some otherclasses and consider if they are splitting classes. First, we look at the class ofhyperimmune-free functions. Recall that a function f is hyperimmune-free if every g ≤ T f is dominated by a computable function. We can see a problem right away:the class of hyperimmune-free functions is well-known to be uncountable, while werequired splitting classes to be countable. We temporarily remedy this by onlylooking at the hyperimmune-free functions which are low (where a function f islow if f ′′ ≡ T ∅ ′′ ); after the proof, we will discuss how we might be able to look atthe entire class. As in section 4 we will present our constructions as constructions on Cantorspace rather than Baire space for the reasons discussed in that section. Theorem 6.1. The class A HIF , low of hyperimmune-free functions which are low is a splitting class. In particular, Th( M w / A HIF , low ) = IPC .Proof. We prove that (iii) of Proposition 3.2 holds. That for every hyperimmune-free low set A there exists a hyperimmune-free low set B such that B T A (orthat there even exists one such that B > T A ) is well-known, see Miller and Martin[12, Theorem 2.1]. We prove the second part of (iii) from Proposition 3.2. Ourconstruction uses the tree method of Miller and Martin [12].Let A ≤ T ∅ ′′ be hyperimmune-free and low , let B ⊆ { B ⊆ ω | B T A, B ≤ T ∅ ′′ and B HIF } be a finite subset and let C ≤ T ∅ ′′ be a hyperimmune-free (low ) set not below A .We need to construct a hyperimmune-free set A < T C ≤ T ∅ ′′ such that C ⊕ C is not of hyperimmune-free degree (i.e. of hyperimmune degree) and such that forall B ∈ B we have that C T B . There are different natural countable subsets of the hyperimmune-free degrees which formsplitting classes; for example, instead of the low hyperimmune-free functions we could also takethe hyperimmune-free functions f for which there exists an n ∈ ω such that f ≤ T ∅ ( n ) . Thisfollows from the proof of Theorem 6.1. However, since our main reason to look at these countablesubclasses is to view them as a stepping stone towards the class of all hyperimmune-free functions,we will not pursue this topic further. ATURAL FACTORS OF THE MUCHNIK LATTICE CAPTURING IPC 11 First, we remark that we may assume that not only C T A , but even that C T A ′ . Indeed, assume C ≤ T A ′ . If C ≥ T A then we see that A < C ≤ A ′ soby Miller and Martin [12, Theorem 1.2] we see that C is of hyperimmune degree,contrary to our assumption. So, C | T A . However, then A < T A ⊕ C ≤ A ′ and asbefore we then see that A ⊕ C is already of hyperimmune degree, so we may take C to be any hyperimmune-free set strictly above A which is low (such a set canbe directly constructed using the construction of Miller and Martin).Without loss of generality we may even assume that C is not c.e. in A ′ : we mayreplace C by C ⊕ C , which is of the same Turing degree as C , and is not c.e. in A ′ because otherwise C would be computable in A ′ , a contradiction.Let B = { B , . . . , B n } and fix a computable enumeration α of n × ω . We willconstruct a sequence T ⊇ T ⊇ . . . of A -computable binary trees (in the sense ofOdifreddi [14, Definition V.5.1]) such that:(i) T is the full binary tree.(ii) For all D on T e +1 : D = { e } A .(iii) For T e +2 , one of the following holds:(a) For all D on T e +2 , { e } A ⊕ D is not total.(b) For all D on T e +2 , { e } A ⊕ D is total and ∀ n ∀ σ ( | σ | = n → { e } A ⊕ T e +2 ( σ ) ( n )[ | T e +2 ( σ ) | ] ↓ ) . Furthermore, this choice is computable in ∅ ′′ .(iv) For all D on T e +3 , { α ( e ) } A ⊕ D = B α ( e ) .(v) T e +4 is the full subtree of T e +3 above T e +3 ( h∅ ′′ ( e ) i ).(vi) For every infinite branch D on all of the trees T i , the sequence T ⊇ T ⊇ . . . is computable in C ⊕ ( A ′ ⊕ D ).(vii) The sequence T ⊇ T ⊇ . . . is computable in ∅ ′′ .For now, assume we can construct such a sequence. Let D = S i ∈ ω T i ( ∅ ), then D is an infinite branch lying on all of the T i . Let C = A ⊕ D . Then the requirements(ii) guarantee that D T A and therefore C > T A . By (vii) we also have that C ≤ T ∅ ′′ . Furthermore, the requirements (iii) enforce that C is hyperimmune-free relative to A (due to Miller and Martin, see e.g. Odifreddi [15, PropositionV.5.6]), and because A is itself hyperimmune-free it is directly seen that C ishyperimmune-free. The requirements (iv) ensure that C T B i for all B i ∈ B .Next, we have that ( C ⊕ C ) ′ ≥ T C ⊕ ( A ′ ⊕ D ) ≥ T ∅ ′′ : by requirement (vi)the sequence T i is computable in C ⊕ ( A ′ ⊕ D ), while by requirement (v) we havethat T e +4 ( ∅ ) = T e +3 ( ∅ ) ⋆ ∅ ′′ ( e ) which allows us to recover ∅ ′′ ( e ). So, C ⊕ C isnot low . In fact, C ⊕ C is not even hyperimmune-free: by a theorem of Martin( C ⊕ C ) ′ ≥ T ∅ ′′ implies that C ⊕ C computes a function which dominates everytotal computable function (see e.g. Odifreddi [15, Theorem XI.1.2]), and therefore C ⊕ C is not hyperimmune-free, as desired.Finally, we show that C is low . By requirement (iv) and requirement (vii) wehave that ∅ ′′ ≥ T { e ∈ ω | { e } C is total } . Since the latter has the same Turingdegree as C ′′ , this shows that C is indeed low .We now show how to actually construct such a sequence of computable binarytrees. Let T be the full binary tree. Next, assume T e has already been defined.To fulfil requirement (ii), observe that T e (0) and T e (1) are incompatible, so atleast one of them has to differ from { e } A . If the first differs from { e } A we take T e +1 to be the full subtree above T e (0), and otherwise we take the full subtreeabove T e (1). Next, assume T e +1 has been defined, we will construct T e +2 fulfilling require-ment (iii). Let n be the smallest m ∈ ω such that either(4) m C ∧ ∃ σ ⊇ h m i∃ x ∀ τ ⊇ σ (cid:16) { e } A ⊕ T e +1 ( τ ) ( x ) ↑ (cid:17) or(5) m ∈ C ∧ ∀ σ ⊇ h m i∀ x ∃ τ ⊇ σ (cid:16) { e } A ⊕ T e +1 ( τ ) ( x ) ↓ (cid:17) , where as before h m i denotes the string consisting of m times a 0 followed by a 1.Such an m exists: indeed, if such an m did not exist, then C = n m ∈ ω | ∃ σ ⊇ h m i∃ x ∀ τ ⊇ σ (cid:16) { e } A ⊕ T e +1 ( τ ) ( x ) ↑ (cid:17)o and therefore C is c.e. in A ′ , which contradicts our assumption above.If (4) holds for n , let σ ⊇ h n i be the smallest such string and let T e +2 be thefull subtree above T e +1 ( σ ). Otherwise, we inductively define T e +2 ⊆ T e +1 . First,if we let τ be the least ˜ τ ⊇ h n i such that { e } A ⊕ T e +1 (˜ τ ) (0)[ | T e +1 (˜ τ ) | ] ↓ , then welet T e +2 (0) = T e +1 ( τ ). Inductively, given T e +2 ( σ ), let ρ be such that T e +2 ( σ ) = T e +1 ( ρ ). Now, if we let τ be the least ˜ τ ⊇ ρ such that { e } A ⊕ T e +1 (˜ τ ) ( | σ | +1)[ | T e +1 (˜ τ ) | ] ↓ , we let T e +2 ( σ ⋆ 0) = T e +1 ( τ ⋆ 0) and T e +2 ( σ ⋆ 1) = T e +1 ( τ ⋆ e = α ( e ). First, webuild a subtree S ⊆ T e +2 such that either there is no ˜ e -splitting relative to A on S (i.e. for all strings σ, τ on S and all x ∈ ω , if { ˜ e } A ⊕ σ ( x ) ↓ and { ˜ e } A ⊕ τ ( x ) ↓ , thentheir values are equal), or S (0) and S (1) are an ˜ e -splitting relative to A (in fact, S will even be an ˜ e -splitting tree relative to A ). Let n be the smallest m ∈ ω suchthat m C ∧ ∃ σ ⊇ h m i∀ τ, τ ′ ⊇ σ ∀ x (cid:16) { ˜ e } A ⊕ T e +2 ( τ ) ( x ) ↓ ∧{ ˜ e } A ⊕ T e +2 ( τ ′ ) ( x ) ↓→ { ˜ e } A ⊕ T e +2 ( τ ) ( x ) = { ˜ e } A ⊕ T e +2 ( τ ′ ) ( x ) (cid:17) (6)or m ∈ C ∧ ∀ σ ⊇ h m i∃ τ, τ ′ ⊇ σ ∃ x (cid:16) { ˜ e } A ⊕ T e +2 ( τ ) ( x ) ↓ ∧{ ˜ e } A ⊕ T e +2 ( τ ′ ) ( x ) ↓∧ { ˜ e } A ⊕ T e +2 ( τ ) ( x ) = { ˜ e } A ⊕ T e +2 ( τ ′ ) ( x ) (cid:17) (7)That such an m exists can be shown in the same way as above. If (6) holds for n , let σ be the smallest such string and let S be the full subtree above T e +2 ( σ ).Then there are no ˜ e -splittings relative to A on S . Otherwise, we can inductivelybuild S : let S ( ∅ ) = T e +2 ( h n i ) and if S ( σ ) is already defined we can take S ( σ ⋆ S ( σ ⋆ 1) to be two ˜ e -splitting extensions relative to A of S ( σ ) on T e +2 .If there are no ˜ e -splittings relative to A on S , then we can take T e +3 = S .Since, assume { ˜ e } A ⊕ D = B i for some B i ∈ B . Then, by Spector’s result (see e.g.Odifreddi [15, Proposition V.5.9]) we have that B i ≤ T A , contrary to assumption.Otherwise we can find an x ∈ ω such that { ˜ e } A ⊕ S ( h i ) ( x ) and { ˜ e } A ⊕ S ( h i ) ( x ) bothconverge, but such that their value differs. Then either { ˜ e } A ⊕ S ( h i ) ( x ) = B α ( e ) andwe take T e +3 to be the full subtree above S ( h i ), or { ˜ e } A ⊕ S ( h i ) ( x ) = B α ( e ) andwe take T e +3 to be the full subtree above S ( h i ). Then T e +3 satisfies requirement(iv).Finally, how to define T e +4 from T e +3 is already completely specified by re-quirement (v). This completes the definitions of all the T i . Note that all steps inthe construction are computable in A ′′ ≡ T ∅ ′′ .So, the last thing we need to show is that requirement (vi) is satisfied, i.e. thatfor any infinite branch D on all T i the construction is computable in C ⊕ ( A ′ ⊕ D ).This is clear for the construction of T e +1 from T e . For the construction of T e +2 from T e +1 the only real problem is that we need to choose between (4) and (5). ATURAL FACTORS OF THE MUCHNIK LATTICE CAPTURING IPC 13 However, because D is on T e +2 , we can uniquely determine n ∈ ω such that T e +1 ( h n i ) is an initial segment of D . Then (4) holds if and only if n C and(5) holds if and only if n ∈ C . So, we can decide which alternative was taken using C . Furthermore, if (4) holds then we can use A ′ to calculate the string σ used inthe computation of T e +2 .For T e +3 we can do something similar for the tree S used in the definition of T e +3 , and using D we can determine if we took T e +3 to be the subtree above S ( h i ) or S ( h i ). Finally, using D it is also easily decided which alternative wetook for T e +4 , because T e +4 is the full subtree above T e +3 ( h i i ) for the unique i ∈ { , } such that T e +3 ( h i i ) ⊆ D . Therefore we see that the construction isindeed computable in C ⊕ ( A ′ ⊕ D ), which completes our proof. (cid:3) This result is slightly unsatisfactory because we restricted ourselves to the hyper-immune-free which are low . Because the entire class of hyperimmune-free functions A HIF is also downwards closed we directly see from the proof above that the onlyreal problem is the uncountability, i.e. A HIF satisfies all properties of a splittingclass except for the countability. Our next result shows that, if we assume thecontinuum hypothesis, we can still show that the theory of the factor given by A HIF is IPC. Definition 6.2. Let A ⊆ ω ω be a non-empty class of cardinality ℵ which isdownwards closed under Turing reducibility. We say that A is an ℵ splitting class if for every f ∈ A and every countable subset B ⊆ { g ∈ A | g T f } there exist h , h ∈ A such that h , h ≥ T f , h ⊕ h A and for all g ∈ B : g ⊕ h , g ⊕ h A . Proposition 6.3. Let A ⊆ ω ω be a non-empty class of cardinality ℵ which isdownwards closed under Turing reducibility. Then the following are equivalent: (i) A is an ℵ splitting class. (ii) For every f ∈ A and every countable subset B ⊆ { g ∈ A | g T f } thereexists h ∈ A such that h > T f and for all g ∈ B : g ⊕ h A .Furthermore, if every countable chain in A has an upper bound in A , these two arealso equivalent to: (iii) For every f ∈ A there exists h ∈ A such that h T f , and for every f ∈ A ,every countable subset B ⊆ { g ∈ A | g T f } and every h ∈ { g ∈ A | g T f } there exists h ∈ A such that h > T f , h ⊕ h A and for all g ∈ B : h T g .Proof. In almost exactly the same way as Proposition 3.2. For the implication (iii) → (ii) we define an infinite sequence h , < T h , < T . . . instead of a finite one,and then let h be an upper bound in A of this chain. (cid:3) Theorem 6.4. For any ℵ splitting class A : Th( M w / A ) = IPC .Proof. We can generalise the construction in Proposition 5.6 to a transfinite con-struction over ℵ . However, instead of building a p -morphism to 2 <ω we showthat we can build a p -morphism to every finite binary tree of the form 2 Assume CH . Then A HIF is an ℵ splitting class. In particular, Th( M w / A HIF ) = IPC .Proof. First, A HIF has cardinality ℵ by CH. Next, every countable chain in A HIF has an upper bound in A HIF (Miller and Martin [12, Theorem 2.2]), so we can usethe equivalence of (i) and (iii) of Proposition 6.3. Thus, it is sufficient if we showthat the construction in Theorem 6.1 not only applies to just finite sets B , but alsoto countable sets B . However, this is readily verified. (cid:3) In particular, we see that it is consistent (relative to ZFC) to have Th( M w / A HIF ) =IPC. Unfortunately, we currently do not know if this already follows from ZFC orif it is independent of ZFC. Question 6.6. Does Th( M w / A HIF ) = IPC follow from ZFC ? Computably traceable functions. A class that is closely related to thehyperimmune-free functions is the class A trace of computably traceable functions.We first recall its definition. Definition 6.7. (Terwijn and Zambella [22]) A set T ⊆ ω × ω is called a trace ifall sections T [ k ] = { n ∈ ω | ( k, n ) ∈ T } are finite. A computable trace is a tracesuch that the function which maps k to the canonical index of T [ k ] is computable.A trace T traces a function g if g ( k ) ∈ T [ k ] for every k ∈ ω . A bound is a function h : ω → ω that is non-decreasing and has infinite range. If | T [ k ] | ≤ h ( k ) for all k ∈ ω , we say that h is a bound for T .Finally, a function f is called computably traceable if there exists a computablebound h such that all (total) functions g ≤ T f are traced by a computable tracebounded by h .Computable traceability can be seen as a uniform kind of hyperimmune-freeness.If f is computably traceable, then it is certainly hyperimmune-free: if g ≤ T f istraced by some computable trace T , then for the computable function ˜ g ( k ) =max (cid:0) T [ k ] (cid:1) we have g ≤ ˜ g . Conversely, if f is hyperimmune-free and g ≤ T f , then g has a computable trace: fix some computable ˜ g ≥ g and let T g = { ( k, m ) | m ≤ ˜ g ( k ) } . However, these traces T g need not be bounded by any uniform computablebound h . Computable traceability asserts that such a uniform bound does exist. Itcan be shown that there are hyperimmune-free functions which are not computablytraceable, see Terwijn and Zambella [22].The computably traceable functions naturally occur in algorithmic randomness.In [22] it is shown that the computably traceable functions are precisely thosefunctions which are low for Schnorr null, and in Kjos-Hanssen, Nies and Stephan ATURAL FACTORS OF THE MUCHNIK LATTICE CAPTURING IPC 15 [7] it is shown that this class also coincides with the functions which are low forSchnorr randomness.Terwijn and Zambella also showed that the usual Miller and Martin tree con-struction of hyperimmune-free degrees actually already yields a computably trace-able degree. Combining their techniques with the next lemma, we can directlysee that our constructions of hyperimmune-free degrees above can also be used toconstruct computably traceable degrees. Lemma 6.8. Let A be computably traceable, and let B be computably traceablerelative to A . Then B is computably traceable.Proof. Let h be a computable bound for the traces of functions computed by A and let h ≤ T A be a bound for the traces of functions computed by B . Because A is hyperimmune-free (as discussed above) h is bounded by a computable function˜ h .We claim: every function computed by B has a trace bounded by the computablefunction h · ˜ h .To this end, let g ≤ T B . Fix a trace T ≤ T A for g which is bounded by h (andhence is also bounded by ˜ h ). Then the function mapping k to the canonical indexof T [ k ] is computable in A , so because A is computably traceable we can determinea computable trace S for this function which is bounded by h .Finally, denote by D e,n the (at most) n smallest elements of the set D e corres-ponding to the canonical index e ; i.e. D e,n consists of the n smallest elements of D e if | D e | ≥ n , and D e,n = D e otherwise. Now let U be the computable trace suchthat U [ k ] = S e ∈ S [ k ] D e, ˜ h ( k ) . Then U is clearly bounded by h · ˜ h . It also traces g ,because g ( k ) ∈ T [ k ] and for some e ∈ S [ k ] we have T [ k ] = D e, ˜ h ( k ) . (cid:3) Theorem 6.9. The class A trace , low of computably traceable functions which arelow is a splitting class. In particular, Th( M w / A trace , low ) = IPC .Proof. As in Theorem 6.1. (cid:3) Theorem 6.10. Assume CH . Then A trace is an ℵ splitting class. In particular, Th( M w / A trace ) = IPC .Proof. As in Theorem 6.5. (cid:3) Question 6.11. Does Th( M w / A trace ) = IPC follow from ZFC ? Acknowledgements. The author thanks Sebastiaan Terwijn for helpful discus-sions on the subject and for suggesting some of the classes studied above. Further-more, the author thanks the anonymous referee for his help in correcting an errorin section 6. References [1] R. Balbes and P. Dwinger, Distributive lattices , University of Missouri Press, 1974.[2] A. Chagrov and M. Zakharyaschev, Modal logic , Clarendon Press, 1997.[3] D. H. J. de Jongh and A. S. 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