Learnability and Numberings of Families of Computably Enumerable Sets

Transcription

1 Learnability and Numberings of Families of Computably Enumerable Sets Klaus Ambos-Spies These notes contain most of the material covered in the course Learnability and Numberings of Families of Computably Enumerable Sets held at the University of Heidelberg in the summer term

2 Contents 1 General numberings Basic definitions and facts Comparing numberings: the u.s.l. (L(S), ) Minimal numberings More on minimal numberings: decidable numberings and positive numberings Comparing the upper semilattices (L(S), ) Exercises Appendix: solutions to selected exercises Computable numberings of families of (partial) computable functions Basic definitions and facts The Rogers semilattices of computable families of total computable functions The Rogers semilattices of computable families of partial computable functions Principal numberings Computable numberings of F rec Exercises Appendix: solutions to selected exercises Computable numberings of families of computably enumerable sets Basic definitions and facts Families of partial computable functions interpreted as families of c.e. sets Computable numberings of the family E of all c.e. sets Notation Exercises Appendix: solutions to selected exercises Inductive inference: Gold s model of learnability in the limit 51 5 Learning from text Explanatory learning from text Other learning criteria from text Locking sequences Modifications of the learning model Appendix: solutions to selected exercises Learning from informant Basic definitions and facts Learnability from informant w.o. mind changes Explanatory-learning from informant Behaviourly correct learning from informant Comparing the learnability criteria Appendix: solutions to selected exercises

3 7 Appendix: Proof of Lemma

4 1 General numberings 1.1 Basic definitions and facts Definition 1.1 Let S be any set. A (total!) function α : ω S is a numbering of S if α is onto. A numbering α is single valued if α is one-to-one. If α is a numbering of S then (S, α) is a numbered set. The set of the numberings of S is denoted by Nu(S). Intuitively, x is (the code of) a name or description of the element α(x) of S. Since α is onto, any element y of S gets a name. Since we do not require that α is one-to-one, it may happen, however, that y has more than one name or description. Formally, we call x an (α-)index of y if α(x) = y. The α-index set of an element y of S is the set Ind α (y) = {x : α(x) = y}. Example (Canonical indices of finite subsets of ω) The canonical numbering γ of the set FIN of finite subsets of ω is defined by γ 1 ( ) = 0 & γ 1 ({a 0,..., a k } = k 2 ai (a 0 < a 1 <... a k ) Note that γ is single valued. The γ-index e of a finite set A is called the canonical index of A and we write A = D e. 2. (Pairing functions) The (computable) function f : ω 2 ω defined by i=0 f(x, y) = (x + 2)2 + 3x + y 2 is a bijection. So the inverse function f 1 is a single valued numbering of ω 2. Similarly, one can show that there are computable bijections f n : ω n ω and f : ω ω (where ω is the set of finite sequence of nonnegative integers) which yield single valued numberings of ω n (n 1 and ω, respectively. (In the following we write x 0,..., x n in place of f n (x 0,..., x n ) and f (x 0,..., x n ); it will be clear from the context which one we mean). 3. (Polynomials) We obtain a single-valued numbering α of the set P n of polynomials of degree n with coefficients in ω by letting α( a 0,..., a n ) be the polynomial p(x) = a n x n + a n 1 x n a 0. Obviously, a set S possesses a numbering if and only if S is nonempty and countable: Proposition 1.3 For any set S, there is a numbering of S (i.e., Nu(S) ) if and only if S is countable and nonempty. In order to avoid trivialities, in the remainder of this section we tacitly assume that S is nonempty and countable. How many numberings does such a set possess? Proposition 1.4 (a) For any countable set S with S 2 there are continuum many (i.e. 2 ω ) numberings of S. (b) For any one-element set S, there is a unique numbering. 4

5 Proof. (b) is obvious. For a proof of (a), fix S such that S 2. Since ω and S are countable, there are 2 ω functions from ω to S. So, in particular, there are at most 2 ω numberings of S. It remains to show that there are at least 2 ω numberings of S. Since there are continuum many subsets of ω it suffices to assign a numbering α A of S to any set A ω such that α A αâ for A Â. For the definition of α A fix s 0, s 1 S such that s 0 s 1 and let α be any numbering of S. Then α A is defined by α A (2n) = α(n) and { s 0 if n A α A (2n + 1) = otherwise. 1.2 Comparing numberings: the u.s.l. (L(S), ) Numberings are ordered by effective transformations as follows. Definition 1.5 Let α i : ω S i be a numbering of S i (i = 0, 1). Then α 0 is reducible to α 1 (α 0 α 1 ) via f if f is a (total!) computable function f : ω ω and α 0 = α 1 f (or α 0 = α 1 f for short; i.e., α 0 (x) = α 1 (f(x)) for all x 0); α 0 is reducible to α 1 (α 0 α 1 ) if there is a computable function f such that α 0 is reducible to α 1 via f; and α 0 and α 1 are equivalent (α 0 α 1 ) if α 0 α 1 and α 1 α 0. If α 0 α 1 via f and x is an α 0 -name (i.e., an α 0 -index) of y then f effectively translates x into an α 1 -name (i.e., an α 1 -index) f(x) of y. Hence f is also called a translation function from α 0 to α 1. Proposition 1.6 is a preordering (i.e., reflexive and transitive) of the class of numberings. Hence is an equivalence relation. Proof. It suffices to note that α α via f(x) = x and that, for α, β and γ such that α β via f and β γ via g, α γ via g f. Note that β α implies that β is a numbering of a subset of the set numbered by α (but not necessarily a numbering of the same set). So if α β then α and β are numberings of the same set. Proposition 1.7 Let α i : ω S i be a numbering of S i (i = 0, 1). If α 0 α 1 then S 0 S 1 (hence if α 0 α 1 then S 0 = S 1 ). In the following we often consider the restriction of the reducibility relation to the numberings of a fixed set S. For ease of notation, we denote this restriction by too, and we let Nu(S) be the class of numberings of S. Note that, by Proposition 1.6, (Nu(S), ) is a preordering. Lemma 1.8 For any (nonempty and countable) set S, (Nu(S), ) is a preordering. By Lemma 1.8 and Proposition 1.7, is an equivalence relation and induces a partial ordering on the equivalence classes. We call the equivalence classes degrees (note that this is not a standard notion in the literature on numberings) and denote the partial ordering on the degrees induced by by too. s 1 5

6 Definition 1.9 For any numbering α, the degree of α is given by deg(α) = [α] = {β : α β}; and, for numberings α and β, deg(α) deg(β) if and only if α β. L(S) = Nu(S)/ is the set of the degrees of numberings of S. Lemma 1.10 (L(S), ) is a partial ordering. Moreover, for any a in L(S), any numbering in a is a numbering of S. Proof. This is immediate by Lemma 1.8 and Proposition 1.7. In fact, for any S, (L(S), ) is an upper semilattice (u.s.l.), i.e., any two degrees have a least upper bound. Lemma 1.11 (L(S), ) is an upper semilattice (u.s.l.). Moreover, for any numberings α and β of S, deg(α) deg(β) = deg(α β) where α β is defined by α β(2n) = α(n) and α β(2n + 1) = β(n). Proof. Let α and β be numberings of S. Then, obviously, α β is a numbering of S too and α α β (β α β) via f(x) = 2x (f(x) = 2x + 1). Moreover, for any numbering γ such that α γ and β γ - say via g and h, respectively - α β γ via f where f(2x) = g(x) and f(2x + 1) = h(x). In the following we look at the structure of the u.s.l. (L(S), ). If S is a oneelement set then, by Proposition 1.4, there is a unique numbering of S. So the u.s.l. (L(S), ) consists of a single degree which in turn has a single element. Proposition 1.12 Assume that S = 1. The upper semilattice L(S) consists of a single degree a and a = 1. For a set with at least two elements, the upper semilattice L(S) is uncountable and any degree consists of countably infinitely many numberings. This easily follows from Proposition 1.4 and the following observation. Proposition 1.13 Let α be a numbering and let f be a computable function. There is a unique numbering β such that β α via f (namely β = α f). Hence, in particular, there are only countably many numberings which are reducible to α. Note that the second part of Proposition 1.13 follows from the first part since there are only countably many computable functions. Proposition 1.14 Let α be a numbering of a set S with at least two elements. Then [α] is countable and infinite. Proof. The first claim is immediate by Proposition For a proof of the second claim note that, given two different elements s 0, s 1 of S, the numberings α n defined by s 0 if m < n α n (m) = s 1 if m = n α(m (n + 1)) if m > n are equivalent to α and α n α n for n n. 6

7 Proposition 1.15 Let S be a countable set with at least two elements. L(S) = 2 ω. Moreover, any principal ideal in (L(S), ) is countable. Then Proof. By Propositions 1.4 and Note that Proposition 1.15 immediately implies that, for countable S with S 2, the upper semilattice (L(S), ) has no greatest element. In fact we may argue, that for any numbering α of S there are continuum many numberings of S above α, i.e., that any principal filter in (L(S), ) has cardinality 2 ω. Lemma 1.16 Let S be a countable set with at least two elements. Then, for any numbering α of S, S( α) = {[β] : β is a numbering of S and α β} has cardinality 2 ω. So, in particular, (L(S), ) has no maximal elements (hence, in particular, no greatest element). Proof. Fix a numbering α of S and, for a contradiction, assume that S( α) < 2 ω. Consider the downward closure C of S( α). By Proposition 1.15 and by assumption, C ω S( α) < 2 ω. On the other hand, since (L(S), ) is a u.s.l., C = L(S). So L(S) < 2 ω contrary to Proposition (For an alternative (direct) proof, it suffices to observe that, for any numbering α of S, α α A for the numberings α A defined in the proof of Proposition 1.4.) 1.3 Minimal numberings Having settled the question of maximal elements in the upper semilattices (L(S), ) (by Lemma 1.16), we now turn to the question of minimal elements. We treat the case of infinite and finite S separately. Minimal numberings of infinite sets For infinite S, single valued numberings provide examples of minimal numberings. Proposition 1.17 There is a single valued numbering of S if and only if S is infinite. Moreover, for infinite S, there are continuum many pairwise different single valued numberings of S. Proof. If α is a single valued numbering of S then, obviously, S is infinite. Conversely, for infinite S, let {s n } n 0 be a sequence of the elements of S without repetitions. Then, for any permutation (i.e., bijection) π : ω ω, α π defined by α π (n) = s π(n) is a single valued numbering of S, and α π α π for π π. Lemma 1.18 Let α and β be numberings of S such that α is single valued and β α. Then α β. (So, for any single valued numbering α of S, deg(α) is a minimal element of (L(S), ).) 7

8 Proof. Fix a computable function f such that β α via f. Since α and β are numberings of the same set and since α is single valued, f is onto. So the computable function g defined by g(x) = µ y [f(y) = x] is total and α β via g. Theorem 1.19 Let S be an infinite countable set S. Then (L(S), ) has continuum many minimal elements. Hence, in particular, (L(S), ) does not have a least element. Proof. This is immediate by Proposition 1.17 and Lemma Minimal numberings of finite sets Next we turn to the questions of minimal elements of (L(S), ) for finite S. Theorem 1.20 Let S be finite. There is a numbering α of S such that, for any numbering β of S, α β. Hence (L(S), ) has a least element. Proof. Fix k 1 such that S = k and let y 0,..., y k 1 be the elements of S. Define the numbering α of S by { y n if n < k α(n) = otherwise. Then, given any numbering β of S, α β via f for { x n if n < k f(n) = otherwise. y 0 x 0 where the numbers x n are chosen so that β(x n ) = y n (n < k). 1.4 More on minimal numberings: decidable numberings and positive numberings Here we look at some more examples of minimal numberings. We start with introducing the relevant concepts. Definition 1.21 Let α be a numbering of the set S. The numbering equivalence relation α of α is defined by x α y α(x) = α(y). The numbering α is decidable if α is computable; α is positive if the relation α is computably enumerable (c.e.); and α is negative if the relation α is co-c.e. (In [6] the numbering equivalence relation α is denoted by η α.) Note that the equivalence classes of α are the previously defined (α-)index sets. Namely, x α y iff y Ind α (α(x)). So the index sets of a decidable (positive) numbering are computable (c.e.). For later use (see Exercise 6.30) we record the following obvious fact. 8

9 Proposition 1.22 Let α be a numbering of S. Then α is an equivalence relation on ω with index S. (Here we call an equivalence relation ε an equivalence relation on a set A if {(x, x) : x A} ε A A, and the index of an equivalence relation is the number of its equivalence classes.) The just introduced properties of numberings are preserved downwards and they are related to each other as follows. Proposition 1.23 Let α be a decidable (positive, negative) numbering of S and let β α. Then β is decidable (positive, negative) too. Proof. Assume that α is decidable (positive, negative) and β α via f. Then x β x if and only if f(x) α f(x ). Since f is computable and α is computable (c.e., co-c.e.), this implies that β is computable (c.e., co-c.e.) too. So β is decidable (positive, negative). Proposition 1.24 Let α be a numbering of S. (i) If α is single valued then α is decidable. (ii) If α is decidable then α is positive. (iii) α is decidable if and only if α is positive and negative. Proof. For a proof of (i) it suffices to note that, for a single valued numbering α, α = {(x, x) : x 0}. (ii) and (iii) are immediate since a set (or relation) is computable if and only if it is c.e. and co-c.e. Next, we generalize Lemma 1.18 by showing that positive numberings have minimal degree. Lemma 1.25 Let α and β be numberings of S such that α is positive and β α. Then α β. (So, for any positive numbering α of S, deg(α) is a minimal element of (L(S), ).) Proof. Given a computable function f such that β α via f, a computable function g such that α β via g is obtained as follows. Since f is computable and α is c.e., the relation R = {(x, y) : x α f(y)} is c.e. too. Moreover, α(x) = β(y) for (x, y) R and, since β = α f is a numbering of S, for any x there is a y such that (x, y) R. So it suffices to let g(x) = y for the first pair (x, y) enumerated into R by a fixed computable enumeration of R. By Lemma 1.25 we can establish the existence of minimal elements in (L(S), ) by proving the existence of decidable (or positive) numberings of S. Proposition 1.26 For any nonempty countable set S there is a decidable numbering of S. Proof. If S is infinite then the claim is immediate by Proposition 1.17 since single valued numberings are decidable (Proposition 1.24 (i)). For finite S, the numbering defined in the proof of Theorem 1.20 is decidable. In the remainder of this subsection we deal with finite and infinite sets S separately. 9

10 More on minimal numberings of finite sets We have shown already that, for nonempty finite sets S, the u.s.l. (L(S), ) has a least element (Theorem 1.20). Here we observe that, for finite S, decidable, positive and negative numberings coincide, and that the numberings in the least degree of (L(S), ) are just the decidable numberings. Lemma 1.27 Let S be a nonempty finite set and let α be a numbering of S. Then α is decidable iff α is positive iff α is negative. Proof. By Proposition 1.24 it suffices to show that α is decidable if α is positive or negative. Fix k 1 such that S = k, let y 0,..., y k 1 be the elements of S, and fix n i such that α(n i ) = y i (i < k). Then, for any n 0, α(n) = y i iff n α n i. So, in particular, for any n there is a unique i < k such that n α n i. Now, assume that α is positive. Then in order to decide whether n α n it suffices to enumerate α till the unique numbers i, i < k with n α n i and n α n i are found. Then, obviously, n α n iff i = i. If α is negative, then the argument is similar. In order to decide whether n α n it suffices to enumerate the complement of α till the unique numbers i, i < k are found which satisfy n α n j for all j {0,..., k 1} \ {i} and n α n j for all j {0,..., k 1} \ {i }. Then, again, n α n iff i = i. Lemma 1.28 Let S be a finite nonempty set and let α be a numbering of S. The following are equivalent. (i) α is decidable. (ii) For any numbering β of S, α β. I.e. α is in the least degree of (L(S), ). Proof. (i) (ii). By Lemma 1.25 (and Proposition 1.24) [α] is minimal whence the claim follows by Theorem (ii) (i). By Proposition 1.26 there is a decidable numbering ˆα of S, and, by the implication (i) (ii), ˆα is in the least degree of (L(S), ). So the claim follows by Proposition By combining the two preceding lemmas, we obtain Theorem 1.29 Let S be a finite nonempty set and let α be a numbering of S. The following are equivalent. (i) α is decidable. (ii) α is positive. (iii) α is negative. (iv) For any numbering β of S, α β. I.e. α is in the least degree of (L(S), ). Note that, by Theorem 1.29, for a finite set S all decidable numberings are equivalent, whereas, by Proposition 1.17, there are nonequivalent single valued numberings hence nonequivalent decidable numberings for infinite sets S. As we show next, Lemma 1.27 fails for infinite S too. 10

11 More on minimal numberings of infinite sets We now turn to the relation between single valued numberings, decidable numberings, positive numberings and minimal numberings for infinite countable S. We first observe that there are decidable numberings which aren t single valued, while, on the other hand, the degrees of the decidable numberings and the single valued numberings coincide. Lemma 1.30 For any nonempty countable set S there is a decidable numbering of S which is not single valued. Proof. If S is finite then this immediate by Propositions 1.26 and If S is infinite then, by Proposition 1.17 fix a single valued numbering β of S and let α(2n) = α(2n + 1) = β(n). Then α is a decidable numbering of S but not single valued. Despite Lemma 1.30, however, for any infinite S and for any decidable numbering α of S there is a single valued numbering which is equivalent to α. Lemma 1.31 Let S be infinite and let α be a decidable numbering of S. There is a single valued numbering β (of S) which is equivalent to α. Proof. Inductively define f(n) by f(0) = 0 and f(n + 1) = µ m ( n n [m α f(n )]), and let β(n) = α(f(n)). Note that f picks from each α-index set the least element. Hence β is a single valued numbering of S. Moreover, since, by decidability of α, f is computable and β = α f, it follows that β α. Finally, α β via g where g(x) = y for the least y such that x α f(y). For decidable and positive numberings we get the following strong separation. Lemma 1.32 Let S be infinite. There is a positive numbering α of S which is not equivalent to any decidable numbering. Proof. By Proposition 1.23 it suffices to give a positive numbering α of S which is not decidable. Let S = {s n : n 0} where s n s m for n m. Define α by { s 0 if n K α(n) = if n = k m s m+1 where K is the halting problem and k 0 < k 1 < k 2 <... are the elements of K in order. Then α is a numbering of S and α is positive since x α y [x = y or x, y K]. On the other hand, given any element n 0 of K, n α n 0 if and only if n K (n 0). Since K is not computable it follows that α is not computable too. So α is not decidable. 11

12 Given an infinite set S, by the above there are minimal degrees in (L(S), ) which are not represented by a decidable numbering (hence, a fortiori, are not represented by a single valued numbering). It is natural to ask whether there are minimal degrees which are not even represented by any positive numbering. The next theorem (together with Proposition 1.23) shows that this is the case. In fact, for any set A, there is a minimal numbering α of S such that α T A. So there are minimal numberings with arbitrarily complex numbering equivalence relation. Theorem 1.33 Let A be any subset of ω and let S be any infinite countable set. There is a minimal numbering α of S such that α T A. Proof. Let S = {y n : n 0} where y n y m for n m. By a finite extension argument, we define a numbering α of S with the required properties. I.e., given L(s 1) ω and the initial segment α L(s 1) of α, at stage s 0 we define L(s) > L(s 1) and the finite extension α L(s) of α L(s 1) (where L( 1) = 1 and α L( 1) = α(0) = y 0 ). In order to guarantee that α is a numbering of S we ensure range(α L(s)) = {y 0,..., y s+1 }. (1) In order to guarantee that the numbering α is minimal we meet the requirements R 2e : α f e is not a numbering of S (e 0) where {f e : e 0} is a (noneffective) list of the computable functions f with coinfinite range (i.e., with range(f) ω). This implies minimality of α as follows. Given a numbering γ of S such that γ α it suffices to show that α γ. Fix f such that γ α via f, i.e., γ = α f. Then the requirements R 2e, e 0, ensure that range(f) = ω. So we may fix a finite set D = {x 0,..., x k 1 } of numbers such that range(f) D = ω. It follows that α = γ g for the computable function g defined as follows. For x D, g(x) is the least number ˆx such that f(ˆx) = x, and, for i < k, g(x i ) = ˆx i where ˆx i is the least number such that α(x i ) = γ(ˆx i ) (note that such a number ˆx i must exist since range(α) = range(γ) = S). Finally, in order to ensure that α T A, it suffices to meet the requirements R 2e+1 : α Φ A e (e 0) where Φ e is the eth 2-ary 0-1-valued Turing functional (and where on the left hand side we identify α with its characteristic function). Note that requirement R 2e can be met by ensuring that there is a number y S such that y has a unique α-index x and x range(f e ). Requirement R 2e+1 can be met as follows. Given any two elements y and y of S and any two numbers x and x, we can define α on {x, x } in such a way that {y } α({x, x }) {y, y } and R 2e+1 is met. Namely, if Φ A e (x, x ) = 1 then it suffices to let α(x) = y α(x ) = y (whence α (x, x ) = 0); otherwise, it suffices to let α(x) = α(x ) = y. By the above, the following definition of L(s) and α L(s) ensures that (1) holds and that requirement R s is met at stage s. 12

13 Stage s = 2e. Fix x e L(s 1) minimal such that x s range(f e ); let L(s) = x e + 1 and define α L(s) by letting α(x e ) = y s+1 and α(x) = y 0 for any other x with L(s 1) x < L(s). Stage s = 2e + 1. Let L(s) = L(s 1) + 2. For the definition of α L(s) distinguish the following cases. If Φ A e (L(s 1), L(s 1) + 1) = 1 then let α(l(s 1)) = s 0 and α(l(s 1) + 1) = s s+1. Otherwise, let α(l(s 1)) = α(l(s 1) + 1) = s s+1. This completes the construction of α. Note that x e is the unique α-index of y 2e+1 and x e range(f e ). So action at stage s = 2e ensures that requirement R 2e is met. Similarly, by the remarks preceding the construction, the action at stage s = 2e + 1 ensures that requriement R 2e+1 is met. Finally, the construction obviously guarantees (1). By the above we get the following relations for any infinite countable set S: {[α] : α is a single valued numbering of S} = {[α] : α is a decidable numbering of S} {[α] : α is a positive numbering of S} {[α] : α is a minimal numbering of S} 1.5 Comparing the upper semilattices (L(S), ) Above we have shown that there are at least three different nonisomorphic upper semilattices (L(S), ): 1. For any singleton S, (L(S), ) consists of a single degree. 2. For any finite set S with at least two elements, (L(S), ) has the cardinality of the continuum and possesses a least element. 3. For any countable infinite set S, (L(S), ) has the cardinality of the continuum and possesses continuum many minimal elements (hence not a least element). It is natural to ask whether there are more examples of nonisomorphic upper semilattices (L(S), ). We prove a limiting result by showing that the structure of the u.s.l. only depends on the cardinality of S. So, in particular, the u.s.l.s (L(S), ) are isomorphic for all (countable) infinite sets S. Moreover, we show that, in contrast to the trivial u.s.l. (L(S 1 ), ) of the singletons, for two-elements sets S 2, the u.s.l. (L(S), ) is very complex. Finally, the answer to our question is given by a deep result of Ershov which we state without proof. Lemma 1.34 (Isomorphism Lemma For Numberings) Let S 0 and S 1 be countable nonempty sets such that S 0 = S 1. Then (Nu(S 0 ), ) = (Nu(S 1 ), ) (hence (L(S 0 ), ) = (L(S 1 ), )). Proof. We give the proof for the case that the sets are infinite. The case of finite sets is similar. Let S i = {s i n : n 0} where s i m s i n for m n. Then an isomorphism f : Nu(S 0 ) Nu(S 1 ) is given by f(α) = ˆα where ˆα(n) = s 1 p for the unique p such that α(n) = s 0 p. For finite sets S, the Isomorphism Lemma can be extended as follows. 13

14 Lemma 1.35 (Embedding Lemma For Numberings) Let S 0 and S 1 be nonempty finite sets such that S 0 S 1. Then (Nu(S 0 ), ) is isomorphic to some ideal of (Nu(S 1 ), ). Proof. By Lemma 1.34 w.l.o.g. we may assume that S 0 < S 1. So we may fix k 0 < k 1 ω and s i n for n < k i such that S i = {s i n : n < k i } and s i n s i m for n, m < k i with n m. Then the required embedding is induced by the function f which maps a numbering α of S 0 to a numbering ˆα of S 1 defined as follows { s 1 k ˆα(x) = 0+x if x < d s 1 y if x d and α(x d) = s 0 y where d = k 0 k 1. For the correctness of the definition, first note that, for any numberings α, β of S 0, α β if and only if ˆα ˆβ. Hence f induces an embedding of (L(S 0 ), ) into (L(S 1 ), ). Moreover, for any α, β Nu(S 0 ), α β = ˆα ˆβ. So, in order to show that {deg(ˆα) : α Nu(S i )} is an ideal of (L(S 1 ), ) it suffices to show that, for any given numberings α of S 0 and β of S 1 such that β ˆα there is a numbering γ of S 0 such that ˆγ β. Fix a computable function g such that β = ˆα g, and let D 0 = {x : g(x) < d} and D 1 = {x : g(x) d}, and let y 0 < y 1 <... be the elements of D 0 in order. Note that D 0 and D 1 are computable, and D 0 d and D 1 k 0 (D 0 may be finite or infinite, and similarly for D 1.) For the definition of γ distinguish the following two cases. If D 1 is infinite then let γ(n) = β(y n ); if D 1 is finite, say D 1 = {y 0,..., y p } then let γ(n) = β(y n ) for n p and let γ(n) = β(y 0 ) for n > p. We leave the proof of ˆγ β to the reader. For 2-elements sets S, the u.s.l. (L(S), ) can be characterized in terms of the many-one degrees which shows that this upper semilattice is rather complicated (in particular has a highly undecidable theory). Lemma 1.36 (Characterization Lemma for Cardinality 2) Let S = 2. Then (L(S), ) = (D m, ). Proof. The desired isomorphism F : D m L(S) is induced by the bijection from the power set of ω in the set of numberings of S which maps a set A to the numbering α A of S = {s 0, s 1 } where α A (x) = s 0 iff x A. Then A m B via f if and only if α A α B via f. Hence deg m (A) deg m B iff [α A ] [α B ]. Ershov (1975) has extended Lemmas 1.34 (for finite sets), 1.35 and 1.36 by showing that, for any two finite sets S 0 and S 1 with at least two elements, (L(S 0 ), ) and (L(S 1 ), ) are isomorphic. (So, the three different types of u.s.l.s (L(S), ) presented at the begin of this subsection are the only ones which exist.) Theorem 1.37 (Ershov [5]) Let S be a finite set with at least two elements. Then (L(S), ) = (D m, ). For the quite involved proof we refer to the literature (e.g. Odifreddi [11]). 14

15 1.6 Exercises Exercise 1.38 Give a numbering of the family of circles where the coordinates of the center and the radius are rational. Exercise 1.39 For any nonempty countable set S let (Nu (S), ) be the preordering of the numberings of the nonempty subsets S of S (i.e., Nu (S) = S S Nu(S )) and let (L (S), ) be the partial ordering of the corresponding equivalence classes. (a) Show that (L (S), ) is the downward closure of (L(S), ). (b) Show that (L (S), ) is a u.s.l. where the join is given by [α] [β] = [α β]. Exercise 1.40 (a) Let S 0 and S 1 be nonempty countable sets such that S 0 S 1 and S 0 is finite. Then, for any numbering α of S 1 there is a numbering β of S 0 such that β α. (b) Let S be an infinite countable set. There is a numbering α of S such that for any numbering β α either β is a numbering of a finite subset of S or β is a numbering of S. Exercise 1.41 For any nonempty finite set S, (L(S), ) is a distributive u.s.l. I.e., for any numberings α, β, γ of S, [γ] [α] [β] γ α α γ β β ([γ] = [γ α ] [γ β ]) holds. Show this without using Theorem (Hint: Use the characterization of the join given in Lemma 1.11.) Exercise 1.42 For any countable set S with at least two elements, (L(S), ) is not a lattice. Hint: Fix a (noneffective) list {y n } n 0 of the elements of S (repetitions allowed) such that y 0 y 1 and a (noneffective) list {f e } e 0 of all total computable functions. Then, by a finite extension argument, construct a pair of numberings α and β of S which does not have a greatest lower bound in (L(S), ). I.e., for any number e = e 0, e 1, ensure that either αf e0 βf e1 or there is a numbering γ e of S such that γ e α, β and γ e αf e0 (where the latter requires to ensure that γ e αf e0 f n for all n 0). Call the first alternative the finitary outcome, the second alternative the infinitary outcome. Let l(s) be the number such that α l(s) and β l(s) are defined by the end of stage s of the construction (where l( 1) = 0 and l(s) < l(s + 1)). A general constraint of the construction is that α and β may differ on ω [e] only for numbers < l(e) (which will be used in order to achieve the infinitary outcome if the finitary outcome cannot be achieved). At stage e attempt to define l(e) > l(e 1) and the finite extensions α l(e) and β l(e) of α l(e 1) and β l(e 1), respectively, in such a way that the finitary outcome is achieved. Note that this is possible if, for some number x and some i 1, f ei (x) l(e 1) and f ei (x) ω [e]. (Namely in this case make sure that l(e) is greater than f e0 (x) and f e1 (x) and ensure that α(f e0 (x)) β(f e1 (x)). Since S 2 this can be done. Moreover, this is compatible with the general constraint since this forces us to make α(y) β(y) for some y only if y = f e0 (x) = f e1 (x) whence y ω [e] and y < l(e).) In addition, at stage e we have to work on all e e for which the finitary outcome could not be achieved at stage e. For such e, by the general constraint, we obtain a numbering γ e by 15

16 letting γ e (x) = α( e, x ) = β( e, x ) for x l(e ). In order to ensure that γ e is a numbering of S we have to make sure that, for any e e, there is a number x such that l(e 1) e, x < l(e) and such that α( e, x ) = β( e, x ) = y e e. (This can be done by assigning these values for some e, x not involved in the diagonalization part (if any) for getting the finitary outcome for e.) Moreover, there has to be a witness for γ e αf e 0 f e e in the interval [l(e 1), l(e)]. Since, by failure of the finitary outcome, the range of f e 0 f e e does not contain any number > l(e ) in ω [e ] this diagonalization can be easily obtained without violating the global constraints. Finally, in order to make sure that α and β are numberings of S, we have to ensure that there is an α- and an β-index for y e in the interval [l(e 1), l(e)], but this task is like the preceding ones and straightforward too. Exercise 1.43 Let S be a nonempty countable set, and let ε be any equivalence relation on ω. There is a numbering α of S with α = ε if and only if ε has index S. Acknowledgements. Most parts of this chapter follow the survey article by Ershov [6]. 16

17 1.7 Appendix: solutions to selected exercises Solution to Exercise 1.40 (b). Note that, for any computable f with finite range, α f is a numbering of a finite set. So, given a (noneffective) list {f n : n 0} of all total computable functions with infinite range, it suffices to give a numbering α of S such that, for n 0, α f n is a numbering of S too. For the definition of such a numbering, first define infinite subsets A n of range(f n ) such that A n A n = for n n, and let a n m be the m-th element of A n in order of magnitude. (The m-th element a n m of such A n can be defined by induction on n, m by letting a 0 0 be the least element in the range of f 0 and, for n, m > 0, 0 let a n m be the least element in the range of f n which is greater than a n m for all n, m < n, m.) Then given a list y n, n 0, of the elements of S, the desired numbering is obtained by letting α(a n m) = y m for all n, m 0 and by letting α(x) = y 0 for x n 0 A n. Then α(a n ) = S whence, by A n range(f n ), α f n = S (for n 0). Solution to Exercise Let S = {y 0,..., y k 1 } (k 1), let α, β, γ be numberings of S and let f be a computable function such that γ α β via f. Then, by Lemma 1.11, it suffices to show that there are numberings γ α and γ β of S such that γ α α, γ β β and γ γ α γ β. Such numberings can be defined as follows. γ(x) = α(f(x)/2) if f(x) is even and x k γ α (x) = y x if x < k otherwise and y 0 γ(x) = β((f(x) 1)/2) γ β (x) = y x y 0 if f(x) is odd and x k if x < k otherwise. Since γ is a numberings of S, the second clauses in the definitions of γ α and γ β ensure that γ α and γ β are numberings of S. The required reductions are easily established using the fact that α, β and γ are numberings of S. Solution to Exercise The only if part is immediate by Proposition So, for the converse direction, let S = k, let y 0,..., y k 1 be the elements of S, fix an equivalence relation ε on ω with index n, and fix representatives x i, i < k, of the equivalence classes of ε. Then α defined by α(x) = y i if (x, x i ) ε (i < k) is a numbering of S and α = ε. 17

18 2 Computable numberings of families of (partial) computable functions Assume that S is a countable nonempty family of (partial) functions of type ω ω. Then any numbering α of S corresponds to a representation function u of S, where such a function u is a (partial) function of type ω ω ω such that S is the family of the branches of u, i.e., S = {u e : e 0} (where the eth branch u e of u is defined by u e (x) = u(e, x) for all x 0 (e 0)). Note that the representation function u α corresponding to α is defined by u α (e, x) = (α(e))(x). Conversely, for any representation function u of S there is a unique numbering α of S such that u is the representation function corresponding to α. Namely, u = u α for the numbering α defined by α(e) = u e (e 0). So numberings of S may be identified with representation functions of S, and it is natural to say that a numbering α is computable if the corresponding representation function is (partial) computable. Below we call (partial) computable representation functions of S universal functions, and we call S computable if it possesses a universal function. Note that any computable family of (partial) functions consists of (partial) computable functions (but - as we will show below - there are families of (partial) computable functions which are not computable). Throughout this section we assume that F is a nonempty family of (partial) computable functions (of type ω ω). And we let F rec and F tot be the families of all partial computable functions respectively all total computable functions (of this type). 2.1 Basic definitions and facts Definition 2.1 Let F be a nonempty family of (partial) computable functions. (a) A universal function u of F is a (partial) computable function u : ω ω ω such that F = {u e : e 0}. (b) F is computable if F possesses a universal function. (c) A numbering α of F corresponds to the universal function u of F if α(e) = u e for e 0. In this case we write u = u α and α = α u. (d) A numbering α of F is computable if it corresponds to a universal function of F. Computable families of (partial) computable functions are also called uniformly computable. Note that a family F is computable iff it possesses a computable numbering. (Also note that the definition of universal function here does not coincide with the definition of universal functions in the course Theoretical Computer Science.) Theorem 2.2 The family F rec of the partial computable functions is computable. For a proof of Theorem 2.2 we refer to the course Theoretical Computer Science. 18

19 Theorem 2.3 The family F tot of the total computable functions is not computable. Proof. For a contradiction assume that F tot is computable and fix a universal function u of F tot. Then the function f defined by f(x) = 1 x (x 0) is computable but, by definition, f does not agree with any of the branches u x of u. So u is not universal for F tot contrary to assumption. For showing that a family F is computable, the following straightforward closure properties may be useful. Proposition 2.4 (a) Any finite family F of partial computable functions is computable. (b) Let F 0 and F 1 be computable families of partial computable functions. Then the family F 0 F 1 is computable to. Proof. Exercise We now turn to the computable numberings of a computable family F. We first observe that any numbering which is reducible to a computable numbering is computable again and that the join of computable numberings is computable. Lemma 2.5 (a) Let α and β be numberings such that β is a computable numbering of F and α β. Then α is a computable numbering of a computable subfamily of F. (b) Let α and β be numberings such that β is a computable numbering of F and α β. Then α is a computable numbering of F. (c) Let α and β be computable numberings of F. Then α β is a computable numbering of F too. Proof. Straightforward. The preceding lemma shows that the equivalence classes of the computable numberings of a computable family F of (partial) computable functions contain only computable numberings of F and that the partial ordering of these equivalence classes is an ideal in the upper semilattice (L(F) ) of the equivalence classes of all numberings of F. Definition 2.6 Let F be a computable family of (partial) computable functions. The partial ordering (R(F), ) of the equivalence classes of the computable numberings of F is called the Rogers semi-lattice of F. Lemma 2.7 Let F be a computable family of (partial) computable functions. The Rogers semilattice (R(F), ) of F is a countable ideal in (L(F), ). Proof. Obviously, (R(F), ) is a subset of (L(F), ). By Lemma 2.5 (a), (R(F), ) is closed downwards in (L(F), ), and, by Lemmas 2.5 (c) and 1.11, (R(F), ) is closed under suprema. So (R(F), ) of F is an ideal in (L(F), ). Finally, R(F) is countable since there are only countable many computable functions hence only countably many computable numberings of F. In the next subsection we have a closer look at the Rogers semilattices for computable families of total computable functions. 19

20 2.2 The Rogers semilattices of computable families of total computable functions In this subsection we prove some basic facts on the Rogers semilattices of computable families of total computable functions. So, throughout this subsection, we assume that F is a computable family of total computable functions. We first observe that any computable numbering of F bounds a decidable computable numbering. So, in particular, the Rogers semilattice (R(F), ) of F has a minimal element. Then, in the second part of this subsection, we characterize the families F for which the Rogers semilattice is trivial (i.e., consists of a single equivalence class). Finally we give a sufficient criterion for computable families F of total computable functions guaranteeing that the Rogers semilattice (R(F), ) of F does not have maximal elements (hence is infinite) and has incomparable elements. Minimal elements of (R(F), ) Theorem 2.8 Let F be a computable family of total computable functions, and let α be a computable numbering of F. There is a decidable numbering β of F such that β α. Proof. Define g : ω ω ω by g(n, l) = µ x ( y l [α(x)(y) = α(n)(y)]) (n, l 0). Then, as one can easily check, (i) g(n, l) n (n, l 0), (ii) g is computable, and (iii) there is a number l n such that α(g(n, l)) = α(n) for l l n (n 0) hold. We claim that the numbering β defined by β( n, l ) = α(g(n, l)) (n, l 0) has the required properties. By Lemma 2.5 (a), it suffices to show that β α, β is a numbering of F, and β is decidable. Now, β α by definition of β since g is computable. For a proof of β Nu(F) note that, by β α, β is a numbering of a subfamily of F. So, since α is a numbering of F, it suffices to show that, for any number n, there is a number m such that β(m) = α(n). But, by definition, of β this is immediate by property (iii) of g. Finally, in order to show that β is decidable, it suffices to show that β( n, l ) = β( ˆn, ˆl ) g(n, l) = g(ˆn, ˆl) holds (l, n 0). The implication is immediate by definition of β. The proof of the implication is by contraposition: By symmetry, w.l.o.g. assume that g(n, l) < g(ˆn, ˆl). Then, by definition of g, α(g(n, l)) ˆl + 1 α(g(ˆn, ˆl)) ˆl + 1 hence β( n, l ) β( ˆn, ˆl ) by definition of β. 20

21 Corollary 2.9 For any computable family F of total computable functions, the Rogers semilattice (R(F), ) has a minimal element; in fact, any ideal of (R(F), ) has a minimal element. Proof. This is immediate by Theorem 2.8 since, by Proposition 1.24 and Lemma 1.25, any decidable numbering is minimal. Another consequence of Theorem 2.8 is that any minimal computable numbering of a family F as above is decidable. (Recall that, for noncomputable numberings, the corresponding fact fails for infinite F by Lemma 1.32.) Corollary 2.10 Let F be a computable family of total computable functions, and let α be a computable numbering of F. The following are equivalent. (i) deg(α) is a minimal element of R(F). (ii) α is decidable. Proof. The implication (ii) (i) is immediate by Proposition 1.24 and Lemma For a proof of the implication (i) (ii), assume that deg(α) is a minimal element of R(F). By Theorem 2.8 there is a decidable numbering β of F such that β α. So, by minimality of α, α is equivalent to the decidable numbering β. By Proposition 1.23 this implies that α is decidable. For infinite F the above corollaries can be further extended where we use the following definition. Definition 2.11 Let F be a computable family of (partial) computable functions. A Friedberg numbering of F is a single valued computable numbering of F. Corollary 2.12 Let F be an infinite computable family of total computable functions. There is a Friedberg numbering of F; in fact, for any computable numbering α of F there is a Friedberg numbering β of F such that β α. Proof. This is immediate by Theorem 2.8 and Lemma Corollary 2.13 Let F be an infinite computable family of total computable functions, and let α be a computable numbering of F. The following are equivalent. (i) deg(α) is a minimal element of R(F). (ii) α is decidable. (iii) α is equivalent to a Friedberg numbering of F. Proof. By Corollary 2.10, it suffices to show the equivalence (ii) (iii). But the nontrivial implication in this equivalence holds by Lemma

22 Families F with trivial Rogers semilattices We now give a characterization of the computable families F of total computable functions for which the Rogers semilattices are trivial, i.e., consist of a single equivalence class. Definition 2.14 A computable family F is strongly computable if any computable numbering of F is decidable. Lemma 2.15 Let F be computable. The following are equivalent. (i) F is strongly computable. (ii) For any computable numbering α of F there is a computable function such that, for n, m 0, (iii) R(F) = 1. d : ω ω ω α(n) = α(m) α(n)(d(n, m)) = α(m)(d(n, m)). (2) Proof. (i) (ii). Given a computable numbering α of F define d by { 0 if n α m d(n, m) = µ x (α(n)(x) α(m)(x)) otherwise. Obviously, d satisfies (2). Moreover, d is computable since, by choice of α and by assumption (i), α is computable and decidable. (ii) (iii). The proof is by contraposition. Assume that R(F) 2. Then there are computable numberings α and β of F such that β α hence α < α β. So the computable numbering α β of F is not minimal hence not decidable. (iii) (i). The proof is by contraposition. Assume that α is a computable numbering of F which is not decidable. By Theorem 2.8 there is a numbering β of F which is computable and decidable. Since decidability is invariant under equivalence, this implies that R(F) 2. Next we show that, any finite family F of total computable functions is strongly computable (Theorem 2.16) and that the family F c of the constant functions is strongly computable too (Lemma 2.18). Theorem 2.16 Let F be a finite computable family. hence R(F) = 1. F is strongly computable Proof. Let F = {f n : n < k} where f n f n for n, n < k with n n. By computability of F, k 1 and f 0,..., f k 1 are computable. Moreover, by f n f n, we may fix x n,n such that f n (x n,n ) f n (x n,n ) (n n, n, n < k). So, for any computable numbering α of F, given m we can effectively compute the unique n such that α(m) = f n by computing the values of α(m)(x n,n ) for all n, n < k with n n. Obviously this implies that α is decidable. 22

23 Corollary 2.17 Let F and ˆF be finite computable families of functions on ω. Then (R(F), ) = (R( ˆF), ). Lemma 2.18 Let F be a computable family of constant functions. Then F is strongly computable (hence R(F) = 1). In particular, for the family F c of all constant functions, F c is strongly computable (hence R(F c ) = 1). Proof. It suffices to note that, for any numbering α of F, α(n) = α(m) if and only if α(n)(0) = α(m)(0). So, obviously, any computable numbering of F is decidable. By the preceding lemmata, the Rogers semilattices of finite (computable) families are trivial and there are even infinite computable families with trivial Rogers semilattices. We conclude this subsection by showing that there are computable families with nontrivial Rogers semilattices. Lemma 2.19 Let F = {f n : n 0} where { 0 if n K x f n (x) = 1 otherwise for a fixed computable enumeration {K s } s 0 of the halting problem K. Then F is computable but not strongly computable (hence R(F) 2). Proof. Let α be the numbering of F defined by α(n) = f n. Then, obviously, α is computable. But α is not decidable. For a proof of the latter it suffices to note that n α n 0 n K where n 0 is any fixed number such that n 0 K. It follows that - in contrast to Corollary 2.17 and Lemma there are infinite computable families with nonisomorphic Rogers semilattices. Theorem 2.20 There are infinite computable families F and ˆF such that (R(F), ) = (R( ˆF), ). Proof. By Lemmas 2.18 and Lemma 2.19 can be extended to show that the family F defined there has no maximal computable numbering (hence has infinitely many computable numberings). We show this by extracting a property of the particular F in Lemma 2.19 which is sufficient for showing this. (Note that the family F in Lemma 2.19 depends on the computable enumeration {K s } s 0 which is chosen. For instance, if we choose {K s } s 0 so, that K s is a proper subset of K s+1 then F consists of the constant function g 0 0 with value 0 and the functions g n (n 0) where { 0 if x n g n (x) = 1 otherwise. So, for the definition of F we do not have to refer to the halting problem K.) 23

24 Families F with Rogers semilattices without maximal elements Definition 2.21 Let F be a family of total functions (of type ω ω). A function f is a limit point of the family F if f F and holds. x 0 g F (g f & y x [f(y) = g(y)]) (3) Note that the constant function f(x) = 0 is a limit point of the family F defined in Lemma Also note that any family which has a limit point is infinite but that there are infinite families without limit points (for instance the family of constant functions). Theorem 2.22 Let F be a computable family which possesses a limit point. Then the Rogers semilattice (R(F), ) does not have any maximal elements. Proof. Given a computable numbering α of F it suffices to show that there is a computable numbering β of some subfamily of F such that β α. Then α β is a computable numbering of F and α < α β. Fix a limit point f of F. Then, since α is a computable numbering of F, the function g defined by g(s) = e s, x s, where e s, x s is the least number e, x such that x > s, α(e) x = f x and α(e)(x) f(x), is total and computable. The computable numbering β is defined as follows. Given n let β(n)(s) = f(s) for all s such that ϕ n,s (n). If ϕ n,s (n), fix t s minimal such that ϕ n,t (n), and distinguish the following cases (where g(t) = e t, x t ). If α(ϕ n (n))(x t ) = f(x t ) then let β(n)(s) = α(e t )(s). Otherwise, let β(n)(s) = f(s). Obviously, β(n) is computable uniformly in n. Moreover, either β(n) = f or β(n) = α(e t ) for some number e t. In either case, β(n) F whence β is a computable numbering of a subfamily of F. Finally, for a proof of β α, for a contradiction assume that β α via the computable function h and fix n such that h = ϕ n. Then, by totality of h, ϕ n (n). But, by definition of β(n), this implies that, for t minimal such that ϕ n,t (n) and for the unique e t and x t such that g(t) = e t, x t, β(n)(x t ) α(ϕ n (n))(x t ) whence β(n) α(h(n)) contrary to assumption. By a variation of the proof of the preceding theorem we can show that any computable family which possesses a limit point has incomparable computable numberings (see [4]). We state this in a more general form which allows to deduce this fact from the existence of m-incomparable noncomputable c.e. sets. Theorem 2.23 Let F be a computable family which possesses a limit point. There are computable numberings β e of F such that holds. e, e (W e m W e & W e, ω β e β e ) (4) 24