- Email: [email protected]
On relative randomness
Annals of Pure and Applied Logic 63 (1993) 61-67 North-Holland
61
On relative randomness Antonin KuEera Department of Computer Science, Charles University, 118 00 Praha 1, Czechoslovakia
Malostranske’ ncime’sti 25,
Communicated by A. Nerode Received 12 April 1992
Abstract
KuEera, A., On relative randomness,
Annals of Pure and Applied Logic 63 (1993) 61-67.
It is the aim of the paper to answer a question raised by M. van Lambalgen and D. Zambella whether there can be a nonrecursive set A having the property that there is a set B such that B is l-random relative to A and simultaneously A is recursive in B. We give a positive answer to this question as well as further information about relative randomness.
1. Introduction
The concept of randomness from the computational point of view was extensively studied by several approaches for different purposes by Church, Kolmogorov, Martin-LGf, Solovay, Chaitin, Kurtz, van Lambalgen, among others. Two basic approaches use either a measure-theoretic method or Kolmogorov complexity. There are several variants in both of them but it is well known that appropriate variants are equivalent. For more details we refer e.g. to [l, 4-6, 10-131. We now summarize some basic notions and facts concerning randomness (from recursion theory point of view) which are used later. A class & c (0, l} o is of 27~~ measure zero if there is a recursive sequence (of indices) of _?$” classes {‘?&},,, such that for all it, ,u(‘Bj,) < 2-” and B G B,,. A set B is l-random relative to A (or, A-l-random) if the class {B} is not of fi” measure zero. If A is recursive, we use l-random instead of l-random relative to A. More generally, replacing @” classes by 2:” classes (for n > 0) we obtain a notion of n-randomness relative to A (or, A-n-randomness). Again, if A is recursive we use simply n-random instead of n-random relative to A. Correspondence to: A. KuEera, Department of Computer Science, Charles University, Malostranskt niimesti 25, 118 00 Praha 1, Czechoslovakia.
0168~0072/93/$06.00 @I 1993 - Elsevier Science Publishers B.V. All rights reserved
62
A. K&era
It is known that n-randomness is the same as l-randomness relative to @-i) and similarly A-n-randomness is the same as l-randomness relative to A(“-‘). Martin-Liif [14] showed that there is a universal class of ,V$ measure zero, i.e., there is a recursive sequence of EC:classes {?I,},,, such that for all ~1,‘u, z ‘$&+,, p(‘i!l,) < 2-” and Yl, contains every class of 2: measure zero. Analogously, for every set A we have a universal class of Z:‘*Ameasure zero. As a corollary we obtain that for every set A the class of l-random relative to A sets is a Z$” class. On the other hand, degrees of A-l-random sets are exactly determined by degrees of members of some @i,A class of sets (cf. [6, Lemma 31 or [9, Lemma 11). Let us note that every l-random degree is also both an FPF degree and a DNR degree. In fact, there is an interesting relation between various kinds of randomness and generalizations of FPF functions (FPF stands for fixed point free and DNR stands for diagonally nonrecursive). For more details see, e.g., [2, 6, 8,
91. Our notation and terminology are standard. We deal with sets and functions over w = (0, 1, 2, . . .}. We often identify subsets of o with their characteristic functions. aC is the eth Turing reduction functional (in some standard enumeration of all such functionals). By a degree we mean a Turing degree. For A E o let deg(A) denote the degree of A. A string is a finite sequence of O’s and 1’s. Strings may also be viewed as functions from finite initial segments of w into (0, l}. By u E A we denote that the characteristic function of A extends o. We use other usual notation for strings. We apply notions of recursion theory to strings via Godel numbering. We usually represent strings by o, t, p. Let Ext(a) denote the class of all sets A for which o E A, and for A E w let Ext(A) be the union of Ext(a) for o E A. A class Cs of subsets of w is a J$‘~” class and x is its index (or A-index) if K = Ext(W;). GSA classes are complements (in (0, 1)“) of 2:‘s” classes. We write 2: and n’l for $‘70 and flTO respectively. A I# class can be thought of as just the class of all infinite branches through a recursive tree. By p we denote Lebesgue measure. [15] or [16] is a general reference for unexplained terminology. The following definition is frequently used later. Definition. (1) A set B is l-RRA to a set A if B is l-random relative to A and A is recursive in B. (2) A set A is a basis for l-RRA if there is a set B which is l-RRA to A.
2. Limitation It is clear that 0 is a basis for l-RRA since l-random sets are just sets I-RRA to 0. On the other hand, no set B which is l-random relative to 0’ (i.e.
On relative randomness
63
0’-l-random) satisfies 0’ dT B (cf. [9]). Thus, neither 0’ nor any set in which 0’ is recursive is a basis for l-RRA. We first show that any set which is a basis for l-RRA has to satisfy a much stronger limitation, namely, any such set has to belong to the class GL,. The following lemma is easy to prove. 1. For any a, x, j it is possible to find 0’-recursively
Lemma
a rational number q
such that wherey = p({M:
I4 -Yl<2-‘,
(3
2 0))).
Together with the well-known fact (due to Sacks) that the cone above any nonzero degree has measure zero it gives the following. Corollary. For every nonrecursive such that for all x ,u({M: (3 sx)(@(M)
zA
set A there is a function g recursive in A 0 0’ 1 g(x))})
<2-Y
Using the corollary we can easily show that every set which is a basis for l-RRA has to belong to the class GL2. For, e.g. in a special case when A is recursive in 0, if A $ L, then for every function g recursive in 0’ there is a function h recursive in A such that for infinitely many x we have h(x) >g(x). This is clearly sufficient to show that the class {M: M +-A} has Z’y*” measure zero and, thus, contains no set l-RRA to A. We now prove a stronger result. Theorem
2.
If a set A
Proof. Suppose that when A is recursive each e E A’ assigns undefined otherwise. such that
is a basis for l-RRA,
a set A is a basis for l-RRA and consider a special case in 0’. Let H be a function partial recursive in A which for the least s such that @,,,(A)(e) is defined and which is We construct a recursive sequence of Zy*A classes {Cs,},,,
CS,={M:(3iSe
1 cS,=0
then A belongs to the class GL,.
)( @i(M) 2 A r H(e))}
if e E A’,
ife$A’.
By a standard method we can construct a recursive sequence of _@” classes {%),~uJ such that for all e: ‘P3,E Cs,, &!13?e)< 2~’ and ‘B, = CSe whenever p(&) < 2-’ (roughly speaking, enumerate KC as long as the measure of Ext(Q,) remains less than 23. Let g be a function recursive in 0’ satisfying the condition from the above corollary. Claim.
For all butjinitefy many e, H(e)J
*H(e)
c g(e).
A. K&era
64
To prove our claim suppose it does not hold. Then every set it4 such that belongs to the class B3, for infinitely many e. It follows that the class {M: A + M} is of @” measure zero and therefore contains no set l-RRA to A. This contradicts our assumption. We now see immediately from our claim that A’ dTO’, i.e., A E L1. A similar argument shows that A E GL1 in the general case (i.e., without the special assumption that A is recursive in 0’). q M +A
3. Existential theorem We now give a positive answer to our original question by constructing a nonrecursive set which is a basis for l-RRA. Let us note first that if B is l-RRA to A and A is nonrecursive then, roughly speaking, A cannot be coded in B uniformly. Let us remark that the class of degrees of sets which are a basis for l-RRA is downward closed. Theorem
3. There is a nonrecursive
set A which is a basis for I-RRA.
Proof. Let M be a low nonrecursive r.e. set. Let us recall that the class of sets l-random relative to M is a 2$” class. Thus, we can use the relativized Low Basis Theorem (cf. [3]) an d construct a set B which is l-random relative to M, recursive in 0’ and low. The crucial point is that M and B cannot form a minimal pair and,
moreover, there is a nonrecursive r.e. set which is recursive in both of them. This fact can be easily proved from the following: (1) every set l-random relative to M is also a l-random set, (2) every l-random set is of a DNR degree (cf. [6], [9]), (3) for every DNR degree G 0’ and nonzero r.e. degree there is a nonzero r.e. degree below both of them (cf. [7]). Let us fix some nonrecursive r.e. set A such that A is recursive both in M and B. Since B is l-random relative to M it is now easy to see that B is l-RRA to A.
Cl
The above proof shows even more. Corollary. For every nonrecursive r.e. set M there exists a nonrecursive which is both recursive in M and a basis for I-RRA.
r.e. set A
It is known that the class of l-random degrees contains the upper cone of degrees above 0’ (cf. [6], [9]). W e p rove an analogous result, namely, for every set A which is a basis for l-RRA the class of degrees of sets which are l-RRA to A contains an upper cone of degrees of the form {d: d 3 e} for some e L a’.
On relative randomness
65
Theorem 4. For every set A which is a basis for 1-RRA there LYa degree e such that the class of degrees of sets which are 1-RRA to A contains the upper cone
{d: da e}. Proof. Let B be a set which is l-RRA to a set A. Take e = deg(B’). By [9, Theorem 31 for every degree d 3 e there is a set C l-random relative to B such that deg(B 0 C) = d. Van Lambalgen proved [12, Theorem 5.101 that if X is l-random and Y is X-l-random (i.e., l-random relative to X) then both X is Y-l-random (i.e., l-random relative to Y) and X 0 Y is l-random. Relativizing this theorem we can conclude that B 0 C is l-random relative to A (the fact B >,A is substantial here). It follows that B 0 C is l-RRA to A. 0 Remark. If B satisfies B’ ==A’ in the above proof then e = a’, i.e., e = a U 0’, where a = deg(A). Such a situation can happen as we saw in the proof of Theorem 3 above. Nevertheless, a question whether for every set A which is a basis for l-RRA the class of degrees of sets which are l-RRA to A contains the upper cone of degrees {d: d 2 deg(A’)} remains open.
4. Decomposition In this section we give examples of low sets which are not a basis for l-RRA. Thus, the property ‘to be a basis for l-RRA’ nontrivially splits the class Lr (and, thus, also GL,). One way to show this is to use [12, Theorem 5.101 (mentioned already above) which easily implies that no l-random set can be a basis for l-RRA and it is well known that there are low l-random sets. We now give other examples of such sets. Theorem 5. There is a l-generic set A recursive in 0’ (and, thus, low) which is not a basis for l-RRA.
Proof. The idea is to construct a l-generic set A recursive some A-recursive function f and for infinitely many x P({M: (ai s g(x))(@@)
2 A If(x))))
in 0’ such that for
< 27,
for some nondecreasing recursive function g going to infinity. To reach it we use a technique which resembles a method used to prove the following well-known fact: for every function F recursive in 0’ there is a l-generic set G such that there is a function h recursive in G which is not dominated by F, i.e., for infinitely many x, h(x) > F(x).
66
A. KuCera
Fix some nonrecursive set G recursive in 0’. We construct 0’-recursive sequences {oX}X,,, {tx}xeo such that a, G ox+,, lh(a,) is even and x + 1
a. Find @‘-recursively) the least x such that the string ts defined by z,(2j) = G(j) for j sx, z,(2j + 1) = 1 for j
2 OS-, * Zs)}) <2-“-’
(such x exists since G is nonrecursive). Substep b. Ask whether there is a string p for which p 2 a,_, * ts and @&)(4 Case NO. Let a, = a,_, * z, and t, = max(s + 1, ts_1 + 1). Case YES. By a standard search take the first such string p of even length. Let a, = p and t, = max(s + 1, ts-1 + 1, y), where y is the least j such that @s,i(p)(s)J. To end the construction let now A be the set which extends a, for all s. It is easy to verify that A is recursive in 0’ and l-generic (substeps b are used to force the jump). It remains to prove that the class {M: M +A} is of Zy” measure zero. Let us note that during step s, substep a we (0’-recursively) put on even section of A an initial part of G and simultaneously on odd section of A a trace for recognizing the string ts such that p({M: (3
6. There is a nonrecursive
low r.e. set which is not a basis for l-RRA.
References [l] G.J. Chaitin, Algorithmic Information Theory (Cambridge Univ. Press, Cambridge, 1987). [2] C.G. Jockusch, Jr., M. Lerman, R.I. Soare and R.M. Solovay, Recursively enumerable sets module iterated jumps and extensions of Arslanov’s completeness criterion, J. Symbolic Logic 54 (1989) 1288-1323.
On relative randomness
67
(31 C.G. Jockusch, Jr. and R.I. Soare, n’i classes and degrees of theories, Trans. Amer. Math. Sot.
173 (1972) 33-56. [4] S. Kautz, Degrees of random sets, PhD thesis, Cornell University, 1991. [5] A. Kolmogorov and V. Uspensky, Algorithms and randomness, Theory Probab. Appl. 32 (1988) 389-412. (61 A. KuEera, Measure, fl-classes and complete extensions of PA, in: H.D. Ebbinghaus, G. Miiller and G. Sacks, eds., Recursion Theory Week Proceedings, Lecture Notes in Math. 1141 (Springer, Berlin, 1985) 245-259. [7] A. Kucera, An alternative, priority-free, solution to Post’s problem, in: J. Gruska, B. Rovan and J, Wiedermann, eds., Proceedings MFCS’86, Lecture Notes in Comput. Sci. 233 (Springer, Berlin, 1986) 493-500. (81 A. K&era, On the use of diagonally nonrecursive functions, in: H.-D. Eddinghaus et al., eds., Logic Colloquium ‘87 (North-Holland, Amsterdam, 1989) 219-239. [9] A. Kueera, Randomness and generalizations of fixed point free functions, in: K. Ambos-Spies, G. Miiller and G. Sacks, eds., Recursion Theory Week, Proceedings Oberwolfach 1989 (Springer, Berlin, 1990) 245-254. [lo] S.A. Kurtz, Randomness and genericity in the degrees of unsolvability, PhD thesis, University of Illinois at Urbana-Champaign, 1981. [ll] M. van Lambalgen, Random sequences, PhD thesis, Dept. of Math., University of Amsterdam, Amsterdam, 1987. [12] M. van Lambalgen, The axiomatization of randomness, J. Symbolic Logic 55 (1990) 1143-1167. [13] M. Li and P. Vitanyi, Kolmogorov complexity and its applications, in: J. van Leeuwen, ed., Handbook of Theoretical Computer Science, Vol. A (Elsevier Sci. Publ., Amsterdam, 1990) 187-254. [14] P. Martin-Ldf, The definition of random sequences, Information and Control 9 (1966) 602-619. [15] P. Odifreedi, Classical Recursion theory (North-Holland, Amsterdam, 1989). [16] R.I. Soare, Recursively Enumerable Sets and Degrees: A study of computable functions and computably generated sets (Springer, Berlin. 1987).
61
On relative randomness Antonin KuEera Department of Computer Science, Charles University, 118 00 Praha 1, Czechoslovakia
Malostranske’ ncime’sti 25,
Communicated by A. Nerode Received 12 April 1992
Abstract
KuEera, A., On relative randomness,
Annals of Pure and Applied Logic 63 (1993) 61-67.
It is the aim of the paper to answer a question raised by M. van Lambalgen and D. Zambella whether there can be a nonrecursive set A having the property that there is a set B such that B is l-random relative to A and simultaneously A is recursive in B. We give a positive answer to this question as well as further information about relative randomness.
1. Introduction
The concept of randomness from the computational point of view was extensively studied by several approaches for different purposes by Church, Kolmogorov, Martin-LGf, Solovay, Chaitin, Kurtz, van Lambalgen, among others. Two basic approaches use either a measure-theoretic method or Kolmogorov complexity. There are several variants in both of them but it is well known that appropriate variants are equivalent. For more details we refer e.g. to [l, 4-6, 10-131. We now summarize some basic notions and facts concerning randomness (from recursion theory point of view) which are used later. A class & c (0, l} o is of 27~~ measure zero if there is a recursive sequence (of indices) of _?$” classes {‘?&},,, such that for all it, ,u(‘Bj,) < 2-” and B G B,,. A set B is l-random relative to A (or, A-l-random) if the class {B} is not of fi” measure zero. If A is recursive, we use l-random instead of l-random relative to A. More generally, replacing @” classes by 2:” classes (for n > 0) we obtain a notion of n-randomness relative to A (or, A-n-randomness). Again, if A is recursive we use simply n-random instead of n-random relative to A. Correspondence to: A. KuEera, Department of Computer Science, Charles University, Malostranskt niimesti 25, 118 00 Praha 1, Czechoslovakia.
0168~0072/93/$06.00 @I 1993 - Elsevier Science Publishers B.V. All rights reserved
62
A. K&era
It is known that n-randomness is the same as l-randomness relative to @-i) and similarly A-n-randomness is the same as l-randomness relative to A(“-‘). Martin-Liif [14] showed that there is a universal class of ,V$ measure zero, i.e., there is a recursive sequence of EC:classes {?I,},,, such that for all ~1,‘u, z ‘$&+,, p(‘i!l,) < 2-” and Yl, contains every class of 2: measure zero. Analogously, for every set A we have a universal class of Z:‘*Ameasure zero. As a corollary we obtain that for every set A the class of l-random relative to A sets is a Z$” class. On the other hand, degrees of A-l-random sets are exactly determined by degrees of members of some @i,A class of sets (cf. [6, Lemma 31 or [9, Lemma 11). Let us note that every l-random degree is also both an FPF degree and a DNR degree. In fact, there is an interesting relation between various kinds of randomness and generalizations of FPF functions (FPF stands for fixed point free and DNR stands for diagonally nonrecursive). For more details see, e.g., [2, 6, 8,
91. Our notation and terminology are standard. We deal with sets and functions over w = (0, 1, 2, . . .}. We often identify subsets of o with their characteristic functions. aC is the eth Turing reduction functional (in some standard enumeration of all such functionals). By a degree we mean a Turing degree. For A E o let deg(A) denote the degree of A. A string is a finite sequence of O’s and 1’s. Strings may also be viewed as functions from finite initial segments of w into (0, l}. By u E A we denote that the characteristic function of A extends o. We use other usual notation for strings. We apply notions of recursion theory to strings via Godel numbering. We usually represent strings by o, t, p. Let Ext(a) denote the class of all sets A for which o E A, and for A E w let Ext(A) be the union of Ext(a) for o E A. A class Cs of subsets of w is a J$‘~” class and x is its index (or A-index) if K = Ext(W;). GSA classes are complements (in (0, 1)“) of 2:‘s” classes. We write 2: and n’l for $‘70 and flTO respectively. A I# class can be thought of as just the class of all infinite branches through a recursive tree. By p we denote Lebesgue measure. [15] or [16] is a general reference for unexplained terminology. The following definition is frequently used later. Definition. (1) A set B is l-RRA to a set A if B is l-random relative to A and A is recursive in B. (2) A set A is a basis for l-RRA if there is a set B which is l-RRA to A.
2. Limitation It is clear that 0 is a basis for l-RRA since l-random sets are just sets I-RRA to 0. On the other hand, no set B which is l-random relative to 0’ (i.e.
On relative randomness
63
0’-l-random) satisfies 0’ dT B (cf. [9]). Thus, neither 0’ nor any set in which 0’ is recursive is a basis for l-RRA. We first show that any set which is a basis for l-RRA has to satisfy a much stronger limitation, namely, any such set has to belong to the class GL,. The following lemma is easy to prove. 1. For any a, x, j it is possible to find 0’-recursively
Lemma
a rational number q
such that wherey = p({M:
I4 -Yl<2-‘,
(3
2 0))).
Together with the well-known fact (due to Sacks) that the cone above any nonzero degree has measure zero it gives the following. Corollary. For every nonrecursive such that for all x ,u({M: (3 sx)(@(M)
zA
set A there is a function g recursive in A 0 0’ 1 g(x))})
<2-Y
Using the corollary we can easily show that every set which is a basis for l-RRA has to belong to the class GL2. For, e.g. in a special case when A is recursive in 0, if A $ L, then for every function g recursive in 0’ there is a function h recursive in A such that for infinitely many x we have h(x) >g(x). This is clearly sufficient to show that the class {M: M +-A} has Z’y*” measure zero and, thus, contains no set l-RRA to A. We now prove a stronger result. Theorem
2.
If a set A
Proof. Suppose that when A is recursive each e E A’ assigns undefined otherwise. such that
is a basis for l-RRA,
a set A is a basis for l-RRA and consider a special case in 0’. Let H be a function partial recursive in A which for the least s such that @,,,(A)(e) is defined and which is We construct a recursive sequence of Zy*A classes {Cs,},,,
CS,={M:(3iSe
1 cS,=0
then A belongs to the class GL,.
)( @i(M) 2 A r H(e))}
if e E A’,
ife$A’.
By a standard method we can construct a recursive sequence of _@” classes {%),~uJ such that for all e: ‘P3,E Cs,, &!13?e)< 2~’ and ‘B, = CSe whenever p(&) < 2-’ (roughly speaking, enumerate KC as long as the measure of Ext(Q,) remains less than 23. Let g be a function recursive in 0’ satisfying the condition from the above corollary. Claim.
For all butjinitefy many e, H(e)J
*H(e)
c g(e).
A. K&era
64
To prove our claim suppose it does not hold. Then every set it4 such that belongs to the class B3, for infinitely many e. It follows that the class {M: A + M} is of @” measure zero and therefore contains no set l-RRA to A. This contradicts our assumption. We now see immediately from our claim that A’ dTO’, i.e., A E L1. A similar argument shows that A E GL1 in the general case (i.e., without the special assumption that A is recursive in 0’). q M +A
3. Existential theorem We now give a positive answer to our original question by constructing a nonrecursive set which is a basis for l-RRA. Let us note first that if B is l-RRA to A and A is nonrecursive then, roughly speaking, A cannot be coded in B uniformly. Let us remark that the class of degrees of sets which are a basis for l-RRA is downward closed. Theorem
3. There is a nonrecursive
set A which is a basis for I-RRA.
Proof. Let M be a low nonrecursive r.e. set. Let us recall that the class of sets l-random relative to M is a 2$” class. Thus, we can use the relativized Low Basis Theorem (cf. [3]) an d construct a set B which is l-random relative to M, recursive in 0’ and low. The crucial point is that M and B cannot form a minimal pair and,
moreover, there is a nonrecursive r.e. set which is recursive in both of them. This fact can be easily proved from the following: (1) every set l-random relative to M is also a l-random set, (2) every l-random set is of a DNR degree (cf. [6], [9]), (3) for every DNR degree G 0’ and nonzero r.e. degree there is a nonzero r.e. degree below both of them (cf. [7]). Let us fix some nonrecursive r.e. set A such that A is recursive both in M and B. Since B is l-random relative to M it is now easy to see that B is l-RRA to A.
Cl
The above proof shows even more. Corollary. For every nonrecursive r.e. set M there exists a nonrecursive which is both recursive in M and a basis for I-RRA.
r.e. set A
It is known that the class of l-random degrees contains the upper cone of degrees above 0’ (cf. [6], [9]). W e p rove an analogous result, namely, for every set A which is a basis for l-RRA the class of degrees of sets which are l-RRA to A contains an upper cone of degrees of the form {d: d 3 e} for some e L a’.
On relative randomness
65
Theorem 4. For every set A which is a basis for 1-RRA there LYa degree e such that the class of degrees of sets which are 1-RRA to A contains the upper cone
{d: da e}. Proof. Let B be a set which is l-RRA to a set A. Take e = deg(B’). By [9, Theorem 31 for every degree d 3 e there is a set C l-random relative to B such that deg(B 0 C) = d. Van Lambalgen proved [12, Theorem 5.101 that if X is l-random and Y is X-l-random (i.e., l-random relative to X) then both X is Y-l-random (i.e., l-random relative to Y) and X 0 Y is l-random. Relativizing this theorem we can conclude that B 0 C is l-random relative to A (the fact B >,A is substantial here). It follows that B 0 C is l-RRA to A. 0 Remark. If B satisfies B’ ==A’ in the above proof then e = a’, i.e., e = a U 0’, where a = deg(A). Such a situation can happen as we saw in the proof of Theorem 3 above. Nevertheless, a question whether for every set A which is a basis for l-RRA the class of degrees of sets which are l-RRA to A contains the upper cone of degrees {d: d 2 deg(A’)} remains open.
4. Decomposition In this section we give examples of low sets which are not a basis for l-RRA. Thus, the property ‘to be a basis for l-RRA’ nontrivially splits the class Lr (and, thus, also GL,). One way to show this is to use [12, Theorem 5.101 (mentioned already above) which easily implies that no l-random set can be a basis for l-RRA and it is well known that there are low l-random sets. We now give other examples of such sets. Theorem 5. There is a l-generic set A recursive in 0’ (and, thus, low) which is not a basis for l-RRA.
Proof. The idea is to construct a l-generic set A recursive some A-recursive function f and for infinitely many x P({M: (ai s g(x))(@@)
2 A If(x))))
in 0’ such that for
< 27,
for some nondecreasing recursive function g going to infinity. To reach it we use a technique which resembles a method used to prove the following well-known fact: for every function F recursive in 0’ there is a l-generic set G such that there is a function h recursive in G which is not dominated by F, i.e., for infinitely many x, h(x) > F(x).
66
A. KuCera
Fix some nonrecursive set G recursive in 0’. We construct 0’-recursive sequences {oX}X,,, {tx}xeo such that a, G ox+,, lh(a,) is even and x + 1
a. Find @‘-recursively) the least x such that the string ts defined by z,(2j) = G(j) for j sx, z,(2j + 1) = 1 for j
2 OS-, * Zs)}) <2-“-’
(such x exists since G is nonrecursive). Substep b. Ask whether there is a string p for which p 2 a,_, * ts and @&)(4 Case NO. Let a, = a,_, * z, and t, = max(s + 1, ts_1 + 1). Case YES. By a standard search take the first such string p of even length. Let a, = p and t, = max(s + 1, ts-1 + 1, y), where y is the least j such that @s,i(p)(s)J. To end the construction let now A be the set which extends a, for all s. It is easy to verify that A is recursive in 0’ and l-generic (substeps b are used to force the jump). It remains to prove that the class {M: M +A} is of Zy” measure zero. Let us note that during step s, substep a we (0’-recursively) put on even section of A an initial part of G and simultaneously on odd section of A a trace for recognizing the string ts such that p({M: (3
6. There is a nonrecursive
low r.e. set which is not a basis for l-RRA.
References [l] G.J. Chaitin, Algorithmic Information Theory (Cambridge Univ. Press, Cambridge, 1987). [2] C.G. Jockusch, Jr., M. Lerman, R.I. Soare and R.M. Solovay, Recursively enumerable sets module iterated jumps and extensions of Arslanov’s completeness criterion, J. Symbolic Logic 54 (1989) 1288-1323.
On relative randomness
67
(31 C.G. Jockusch, Jr. and R.I. Soare, n’i classes and degrees of theories, Trans. Amer. Math. Sot.
173 (1972) 33-56. [4] S. Kautz, Degrees of random sets, PhD thesis, Cornell University, 1991. [5] A. Kolmogorov and V. Uspensky, Algorithms and randomness, Theory Probab. Appl. 32 (1988) 389-412. (61 A. KuEera, Measure, fl-classes and complete extensions of PA, in: H.D. Ebbinghaus, G. Miiller and G. Sacks, eds., Recursion Theory Week Proceedings, Lecture Notes in Math. 1141 (Springer, Berlin, 1985) 245-259. [7] A. Kucera, An alternative, priority-free, solution to Post’s problem, in: J. Gruska, B. Rovan and J, Wiedermann, eds., Proceedings MFCS’86, Lecture Notes in Comput. Sci. 233 (Springer, Berlin, 1986) 493-500. (81 A. K&era, On the use of diagonally nonrecursive functions, in: H.-D. Eddinghaus et al., eds., Logic Colloquium ‘87 (North-Holland, Amsterdam, 1989) 219-239. [9] A. Kueera, Randomness and generalizations of fixed point free functions, in: K. Ambos-Spies, G. Miiller and G. Sacks, eds., Recursion Theory Week, Proceedings Oberwolfach 1989 (Springer, Berlin, 1990) 245-254. [lo] S.A. Kurtz, Randomness and genericity in the degrees of unsolvability, PhD thesis, University of Illinois at Urbana-Champaign, 1981. [ll] M. van Lambalgen, Random sequences, PhD thesis, Dept. of Math., University of Amsterdam, Amsterdam, 1987. [12] M. van Lambalgen, The axiomatization of randomness, J. Symbolic Logic 55 (1990) 1143-1167. [13] M. Li and P. Vitanyi, Kolmogorov complexity and its applications, in: J. van Leeuwen, ed., Handbook of Theoretical Computer Science, Vol. A (Elsevier Sci. Publ., Amsterdam, 1990) 187-254. [14] P. Martin-Ldf, The definition of random sequences, Information and Control 9 (1966) 602-619. [15] P. Odifreedi, Classical Recursion theory (North-Holland, Amsterdam, 1989). [16] R.I. Soare, Recursively Enumerable Sets and Degrees: A study of computable functions and computably generated sets (Springer, Berlin. 1987).