Martin-Lof Random and PA-complete Sets Frank Stephan Universitat Heidelberg November 2002 Abstract A set A is Martin-Lof random i the class fag does not have 0 1-measure 0. A set A is PA-complete if one can compute relative to A a consistent and complete extension of Peano Arithmetic. It is shown that every Martin-Lof random set either permits to solve the halting problem K or is not PA-complete. This result implies a negative answer to the question of Ambos-Spies and Kucera whether there is a Martin-Lof random set not above K which is also PA-complete. 1 Introduction Gacs [4] and Kucera [7, 8] showed that every set can be computed relative to a Martin-Lof random set. In particular, for every set B there is a Martin-Lof random set A such that B T A T B K where K is the halting problem. A can even be chosen such that the reduction from B to A is a weak truth-table reduction, Merkle and Mihailovic [12] give a simplied proof for this fact. A natural question is whether it is necessary to go up to the degree of B K in order to nd the random set A. Martin-Lof random sets can be found below every set which is PA-complete, so there are Martin-Lof random sets in low and in hyperimmune-free Turing degrees. A set A is called PA-complete if one can compute relative to A a complete and consistent extension of the set of rst-order formulas provable in Peano Arithmetic. An easier Mathematisches Institut, Im Neuenheimer Feld 294, Universitat Heidelberg, 69120 Heidelberg, Germany, EU, Email: fstephan@math.uni-heidelberg.de. Frank Stephan was supported by the Deutsche Forschungsgemeinschaft (DFG), Heisenberg grant Ste 967/1{1. 1
and equivalent denition of being PA-complete is to say that given any partial-recursive and f0; 1g-valued function, one can compute relative to A a total extension of. One can of course choose such that also is f0; 1g-valued. Extending all possible f0; 1g-valued partial-recursive functions is as dicult as to compute a f0; 1g-valued DNR function. A diagonally nonrecursive (DNR) function f satises f(x) 6= ' x (x) whenever ' x (x) is dened. Kucera [7] showed that one can compute relative to any Martin-Lof random set A a DNR function f but that f is not f0; 1g-valued: Taking a suciently large c, f(x) is just the value of the string 1A(0)A(1) : : : A(x + c) interpreted as a binary number. For all x where ' x (x) is dened, the Kolmogorov complexity of f(x) is strictly larger than that of ' x (x) and it follows that f(x) 6= ' x (x). So Martin-Lof random sets and PA-complete sets have in common that one can compute relative to them DNR functions. Therefore, it is a natural question whether their degrees coincide. Kucera [7] showed that this is not the case: while the measure of the class of Martin-Lof random sets is 1, the measure of the PA-complete sets is 0. Since the notion PA-complete is invariant with respect to Turing equivalence, there are Turing degrees containing Martin-Lof random sets but no PA-complete sets. It remains to ask whether there is at least an inclusion: does every Turing degree containing a PA-complete set also contain a Martin-Lof random set? Kucera [7, 8] answered this question also negatively and constructed several examples of PA-complete sets A such that no set below A is both, Martin-Lof random and PA-complete. Ambos-Spies and Kucera [1, Open Problem 3.5] asked whether there is an A 6 T K which fails to have this property. The negative answer to this question is the main result of the present work. So the PA-complete sets constructed by Kucera share the desired property with all PA-complete sets not above K. From this result it also follows that the uniform constructions of Gacs [4] and Kucera [7, 8] to reduce a given set B to a Martin-Lof random set A are optimal with respect to the Turing degree of A: For many sets B, in particular for the PA-complete sets B, one cannot avoid that the set A is above K. 2 Only the Random Sets Above K are PA-complete Measures. The measure used is the standard measure for the innite product f0; 1g 1 of f0; 1g where the measure of f0; 1g is 1 and of each fbg f0; 1g is 1. In particular, the measure of the class of all sets B which extend a binary string is 2?jj where jj is the length of the string, that is, the cardinality of its domain. While in standard measure theory, every singleton fag has measure 0, this does no longer hold for eective versions like the 0 1-measure. Most singletons fag are not eectively measurable. The measurable ones are contained in a sequence of classes which are eectively measurable and whose measure 2
goes to 0. More formally, A is contained in a class of 0 1-measure 0 i there is a sequence U 0 ; U 1 ; : : : of subclasses of f0; 1g 1 such that every U n has measure 2?n or less; the U n are uniformly 0 1, that is, there is a recursively enumerable set R of pairs (; n) such that a set B is in U n i there is an m with (B(0)B(1) : : : B(m); n) 2 R; A 2 U n for all n. Martin-Lof random sets are those sets which are not contained in a class of 0 1-measure 0. Equivalently, one can say that A is Martin-Lof random i fag does not have 0 1-measure 0. PA-complete sets are those sets A such that every f0; 1g-valued partial-recursive function has a total f0; 1g-valued extension which is computable relative to A. The interested reader might consult the books of Li and Vitanyi [10], Odifreddi [13] and Soare [15] for a formal denition of the further concepts mentioned in this paper. Martin-Lof random sets are named after their inventor Martin-Lof [11]. The main result says that there are two types of Martin-Lof random sets: the rst type are the computationally powerful sets which permit to solve the halting problem K; the second type of random sets are computationally weak in the sense that they are not PA-complete. Every set not belonging to one of these two types is not Martin-Lof random. Theorem. Let A be Martin-Lof random. Then one can compute relative to A a complete and consistent extension of Peano-Arithmetic if and only if one can solve the halting problem relative to A. Proof. The theorem is proven by considering any A 6 T K which is PA-complete and showing that such an A cannot be Martin-Lof random. As a rst step, one constructs a partial-recursive f0; 1g-valued function. Since A is PA-complete, A permits to compute a total f0; 1g-valued extension of. In the second step, one uses some properties of and the fact that A 6 T K for the construction of a Martin-Lof test which witnesses that A is not Martin-Lof random. Construction of. The goal of the construction of is that the class of oracles B such that ' B e is a total extension of is small. More precisely, the measure of this class and also the measure of almost every approximation to it should be below 2?e?1. Furthermore, it is sucient to assign to every e an interval I e such that the same holds for the class of oracles B for which ' B e is total on I e. Now the construction is given in detail. The partial function is undened on f0; 1; 2; 3g. 3
On the intervals I e = f2 e+2 ; 2 e+2 + 1; : : : ; 2 e+3? 1g, one denes in stages as below, the construction of on I e does not interact with the construction on any other I e 0, e 0 6= e. The above mentioned approximations are obtained by considering in stage s only those numbers where has already been dened in previous stages and only those oracles, which are computed on I e with use at most s. Here the use of ' B e at input x is dened as use(e; x; B) = max(ft : the computation ' B e (x) either needs time t or queries B at tg): where use(e; x; B) = 1 if the computation ' B e (x) does not halt. Before starting the construction, let a 0 = min(i e ) and be undened everywhere. Stage s of the construction of on I e. Let P e;s;b be the 0 1-class of all oracles B such that, for all x 2 I e, ' B e (x) halts and use(e; x; B) s; if x < a s then ' B e (x) = (x); if x = a s then ' B e (x) = b. Compute the measures d e;s;0 of P e;s;0 and d e;s;1 of P e;s;1. If d e;s;0 + d e;s;1 > 2?e?1 then choose b 2 f0; 1g such that d e;s;b d e;s;1?b, let (a s ) = b and update a s+1 = a s + 1 else let unchanged and let a s+1 = a s. This completes stage s. Properties of. It is easy to see that is partial-recursive. Furthermore, whenever in stage s a new value for on I e is dened, the measure of the class of oracles B for which ' B is consistent with before but not after stage e s is at least 2?e?2. One denes in at most 2 e+2? 2 many stages s a new value (a s ) because after 2 e+2? 2 times dening a new value, the measure of the class of the oracles B for which ' B e is f0; 1g-valued and consistent with is at most 2?e?1. It follows that a s 2 I e for all stages s. So the procedure to dene on I e terminates eventually without having used up the entries on I e completely. For all stages s where no new value (a s ) is dened it holds that d e;s;0 + d e;s;1 2?e?1. A is not Martin-Lof random. Since A is PA-complete, there is an A-recursive f0; 1gvalued total function which extends. By the Padding Lemma [13, Proposition II.1.6], there is a recursive ascending one-one sequence e 0 ; e 1 ; : : : of programs which compute relative 4
to A: = ' A e k for all k. Furthermore, let b 0 ; b 1 ; : : : be a one-one enumeration of K. Now one denes for every s the following numbers e(s) and r(s) in dependence of s: e(s) = e bs and r(s) is the rst stage t > s where P e(s);t;0 [ P e(s);t;1 has at most the measure 2?e(s)?1. Note that the membership of B in P e(s);r(s);0 [ P e(s);r(s);1 depends only on the values of B at arguments up to r(s), thus this class is a 0 1-class. Since the construction is eective in s, one can dene the sequence =[ U n P e(s);r(s);0 [ P e(s);r(s);1 s where e(s)n and has that these classes U n are uniformly 0 1-classes. The measure of U n is bounded by the sum over all 2?e(s)?1 where e(s) n. Since the mappings s! b s and k! e k are one-one, so is the mapping s! e(s) = e bs. Thus the innite sum 2?n?1 + 2?n?2 + : : : = 2?n is an upper bound for the measure of U n. It follows that the sequence of the U n is a Martin-Lof test. For given k, let f(k) = maxfuse(e k ; x; A) : x 2 I ek g. The function f can be computed relative to A. Since K 6 T A, there are innitely many k 2 K such that s > f(k) for the s with k = b s. It follows for these k; s and all x 2 I ek that ' A e k (x) halts and use(e k ; x; A) r(s). So A 2 P e(s);r(s);0 [ P e(s);r(s);1. Since A is in innitely many classes P e(s);r(s);0 [ P e(s);r(s);1, A is also in all classes U n. It follows that A is not Martin-Lof random. Examples. A natural example for a Martin-Lof random set of the rst type is Chaitin's. is in the Turing-degree of K. One among several denitions for is that there is a universal prex-free Turing machine M such that is the set of positions n where the n-th binary digit of the halting probability is 1. Formally, the halting probability of M is the measure of the 0 1-class of those sets B such that there is an n for which M(B(0)B(1) : : : B(n)) halts. Martin-Lof random sets are immune and thus not recursively enumerable, but is as close to being recursively enumerable as possible: The left cut f 2 f0; 1g : is lexicographically before (0)(1) : : :g is a recursively enumerable set of nite strings. Kucera and Slaman [9] showed that all Martin-Lof random sets with recursively enumerable left cut are -like, that is, every such set can be constructed in the same way as Chaitin constructed. One can relativize the denition of to oracles, so K is the probability that a xed universal prex-free machine with access to the oracle K holds. This relativized construction satises K K T K 0, but one cannot omit the oracle K from the left hand side, that is, K 6 T K. So, K is an example for a Martin-Lof random set of the second type. 3 Applications Arslanov's Completeness Criterion [2] says that every recursively enumerable set which permits to compute a xed-point-free function is already above K. A xed-point-free function f 5
has the property that W x 6= W f (x) for all x. Jockusch, Lerman, Soare and Solovay [5] showed that Arslanov's Completeness Criterion also holds with DNR functions in place of xed-pointfree functions. They furthermore generalized the result: For every sequence A 1 ; A 2 ; : : : ; A n of sets such that A 1 < T A 2 < T : : : < T A n, A n 6 T K, A 1 is recursively enumerable and each A m+1 is recursively enumerable relative to A m for m = 1; 2; : : : ; n? 1, one cannot compute a DNR function relative to A n. The corresponding result does not hold for Martin-Lof random sets because each of them can compute a DNR function [7]. But the main result of the present work gives a natural variant of Arslanov's completeness criterion: A Martin-Lof random set A is above K i one can compute a f0; 1g-valued DNR function relative to A. Call a set B to be DNR i its characteristic function is an (automatically f0; 1g-valued) DNR function. The degrees of DNR sets are closed upward: Let fa 0 ; a 1 ; : : :g be a recursive set such that ' x (x) = 2 for all x 2 fa 0 ; a 1 ; : : :g and let B T A be a DNR set. Then one can construct a further DNR set C by C(x) = B(x) for x =2 fa 0 ; a 1 ; : : :g and C(a k ) = A(k) for all k. The set C has the same Turing degree as A. So, the Turing degrees containing a DNR set coincide with those of PA-complete sets. In particular, the Turing degrees containing both, a DNR set and a Martin-Lof random set, coincide with the upper cone of Turing degrees above the one of K. This answers negatively Open Problem 3.5 of Ambos-Spies and Kucera [1]. Furthermore, as Ambos-Spies and Kucera already noted, their Open Problem 3.6 is also solved by the negative answer to Open Problem 3.5: Whenever a Turing degree contains a DNR set and a Martin-Lof random set, then every Turing degree above this one contains such sets, too. Kucera [8] constructed a PA-complete set A < T K such that the Turing degree of A does not contain a Martin-Lof random set. Since there is a Martin-Lof random set below A, one can conclude that the Turing degrees of Martin-Lof random sets are not closed upward [1]. This result is even strengthened: Above every set A 6 T K there is a set B T A such that the Turing degree of B does not contain a Martin-Lof random set. This is shown as follows. The sets above K are exactly the sets which permit to compute a function majorizing the modulus c K of convergence of any xed given enumeration of K. Since A 6 T K, no such majorizing function is A-recursive. By the Hyperimmune-Free Basis Theorem of Jockusch and Soare [6] there is a PA-complete set B T A which is hyperimmune-free relative to A. In particular, every total B-recursive function is majorized by a total A-recursive function. Therefore no total B-recursive function majorizes c K. Thus B 6 T K. It follows that the Turing degree of B does not contain a Martin-Lof random set. Scott and Tennenbaum [14] showed that the Turing degree of a PA-complete set cannot be minimal. Since the Turing degree of K is not minimal, one only has to consider sets A 6 T K. The traditional way to prove this result is to take a Martin-Lof random set B below A and then to consider the set C = fx : 2x 2 Bg. For these sets A; B; C one has ; < T C < T B T A. On the one hand, the main result of the present work permits to conclude that already B satises ; < T B < T A; one does not need the set C. On the other 6
hand, the proof that C < T B is less involved than the proof that B < T A. So it is a matter of taste which proof one prefers. Gacs [4] and Kucera [7, 8] showed that one can nd for any set B a Martin-Lof random set A such that B T A T B K. In this result, K cannot be replaced by any set C 6 T K: Given such a set C 6 T K, there is a set B which is PA-complete, above C and not above K. Then all Martin-Lof random sets A with A T B satisfy A T K. In particular, such a set A is not in the Turing degree of B C. Acknowledgement. interesting comments. The author would like to thank Bill Gasarch and Andre Nies for References [1] Klaus Ambos-Spies and Antonn Kucera. Randomness in computability. Peter A. Cholak, Steen Lempp, Manuel Lerman and Richard A. Shore, editors, Computability theory and its applications. Current trends and open problems. Proceedings of a 1999 AMS- IMS-SIAM joint summer research conference, Boulder, CO, USA, June 13{17, 1999. Contemporary Mathematics, 257:1{14, American Mathematical Society, 2000. [2] Marat M. Arslanov. On some generalizations of a xed point theorem. Soviet Mathematics 25:1{10, 1981; translated from Izvestiya Vysshikh Uchebnykh Zavedenij, Matematika, 228:9{16, 1981. [3] Gregory J. Chaitin. An algebraic equation for the halting probability. Rolf Herken, editor, The universal Turing machine, a half-century survey, 279{283, Oxford University Press, 1988. [4] Peter Gacs. Every sequence is reducible to a random one. Information and Control, 70:186{192, 1986. [5] Carl G. Jockusch, Jr., Manuel Lerman, Robert I. Soare and Robert M. Solovay. Recursively enumerable sets modulo iterated jumps and extensions of Arslanov's completeness criterion. The Journal of Symbolic Logic, 54:1288{1323, 1989. [6] Carl G. Jockusch, Jr., and Robert Soare. 0 1 classes and degrees of theories. Transactions of the American Mathematical Society, 173:33{56, 1972. [7] Antonn Kucera. Measure, 0 1-classes and complete extensions of PA. Heinz-Dieter Ebbinghaus, Gert H. Muller and Gerald E. Sacks, editors, Recursion Theory Week, Proceedings of a Conference held in Oberwolfach, West Germany, April 15{21, 1984. Lecture Notes in Mathematics, 1141:245{259, Springer, 1985. 7
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