Zeitschr. J math. Logik und Crundlogen d Moih. Bd37. S.97-111 (1991)

@ 1991 DEch Ug. d. Wiss.

EFFECTIVIZING INSEPARABILITY

by JOHN CASE in Newark, Delaware (U.S.A.)’)

SMULLYAN’S notion of effectively inseparable pairs of sets is not the best effective/construc- tive analog of KLEENE’S notion of pairs of sets inseparable by a recursive set. We present a corrected notion of effectively inseparable pairs of sets, prove a characterization of our no- tion, and show that the pairs of index sets effectively inseparable in SMULLYAN’S sense are the same as those effectively inseparable in ours. In fact we characterize the pairs of index sets effectively inseparable in either sense thereby generalizing RICE’S Theorem. For subrecursive index sets we have sufficient conditions for various inseparabilities to hold. For inseparabil- ity by sets in the same subrecursive class we have a characterization. The latter essentially generalizes KOZEN’S (and ROYER’S later) Subrecursive Rice Theorem, and the proof of each result about subrecursive index sets is presented “Rogers style” with care to observe subrecur- sive restrictions.

There are pairs of sets effectively inseparable in SMULLYAN’S sense, but not effectively inse- parable in ours. The proof of this involves a non-effective construction by finite extensions with the unusual and interesting feature that alternate stages in the construction apply an in- stance of SMULLYAN’S Double Recursion Theorem effective in the previous stage. Our con- struction yields as a corollary that the pairs of sets effectively inseparable in SMULLYAN’S sense, but not in ours, are plentiful in the sense of Baire Category. By way of contrast with the previous result we show that, for pairs of r.e. sets, our notion and SMULLYAN’S are coexten- sive. We call our notion effective A:-inseparability and generalize it, SMULLYAN’S notion, and all our results (except those about subrecursive index sets) to the A: level, for each n > 1 . (ROYER and CASE apply effective A!-inseparability to obtain results in structural complexity theory.) For subrecursive index sets we have a suficient condition for effective d’$inseparability. The proof of this latter result is made compact by an application of ROVER and CASE’S Hybrid Recursion Theorem which facilitates interaction between a subrecursive programming system and a programming system for the partial limiting-recursive functions.

N denotes the set of natural numbers, (0, 1 , 2 , 3 , . . .) . Lower case letters, with or without de- corations, near the front and rear of the alphabet range over N, and f, g, and h range over (to- tal) functions with arguments and values from N. A, B , and C range over subsets of N, and denotes the complement of A . card(A) denotes the cardinality of A, and XA denotes the char- acteristic function of A, the function which is 1 on A and 0 on x. Let Q, denote a fixed ac- ceptable programming system (numbering) for the partial recursive functions: N --* N (cf. e.g.

I ) This paper is dedicated to the memory of JOHN MYHILL. It grew out of a joint monograph with J. ROYER, and owes an especial intellectual debt to that work and the methods in MYHILL’S paper on crea- tive sets and SMULLYAN’S monograph. A suggestion of ROYER’S provided a cleaner version of Theorem 3. He also made some helpful suggestion regarding presentation. The research was supported in part by NSF grant # CCR-8713846.

7 Ztschr. f. math. Log&

98 1. CASE

[4], [35], [45], [46], [48], [Sl]), and let W , denote the domain of qP. W , is. then, the r.e. set ( s N) accepted by pprogram p . Following MEYER we let denote convergence ofa ComPuta- tion, and denote divergence ([49]). Let h , y . ( x , y ) denote a f ied Pairing function (a recur- sive, bijective mapping: N x N -+ N ([49])) with respective inverses x1 and n2. sineartime de- notes the class of functions computable on multi-tape Turing machines ((231) within a linear time bound of the length of the input (in binary). We suppose that k, y . ( x , y ) , nl, and RZ are each in gineartime. An example based on bit interlacing with standard binary representa- tions ([3]) of numbers is provided in [39], [40]. Let Axl, ..., x n . ( x l , ..., x , ) be a fixed recursive bijection: Nn-, N based on A x , y . ( x , y ) . For example, we could take ( X I , x2, x3) = (x,, ( x 2 , x 3 ) ) . We let @ denote a fmed Blum complexity measure associated with Q, (cf. e.g. [4], [171, (351). GP(z) is, intuitively, the run time of p-program p on input z. We say that a number w appears in W, in exactly z (respectively, s z) steps iff @,(w) = z (respectively, S z ) . A number w appears in W, before W, iff @,(w) 4 < @,(w) 6 Q) . A number w appears in W, at the same time as it appears in W, iff @,(w) = @,(w) 4 < Q) . f I A denotes the graph off restricted to the set A . f 1 < denotes f l iWc < ,), and f 1 , denotes f Furthermore, A aB denotes (A - B ) u ( B - A), the symmetric difference of A and B. Any other unexplained notation or terminology is from [49]. For example, A : , ,TO,, and lI: represent the usual levels in the arith- metical hierarchy.

We provide a brief, partial history of the notions that preceded the subject of this paper. GODEL’S First Incompleteness Theorem ([19], [33]) indirectly motivated DEKKER’S notion of productive set ([15], [49]). This notion was based on POST’S earlier notion of creative set ([38], [36], [49]). GODEL’S Theorem directly motivated KLEENE’S notion of recursively inseparable sets ([26], [49]). We explain these notions just below.

A is productive iff there is an effective procedure which, given any x such that W, E A , re- turns a value in ( A - W,). Here is an example from [49] extracting a recursion-theoretic es- sence of GODEL’S Theorem. Godel number the set of sentences of arithmetic onto N and identify sentences with their Godel numbers. The set of true sentences of arithmetic is pro- ductive. POST [38] was interested in productive sets with r.e. complements, for example, the set of sentences not provable in Peano Arithmetic.

A is recursively inseparable from B iff A is disjoint from B and there is no recursive set C such that A E C S B. (The reader may find a simple Euler-Venn diagram helpful to see the relation between the sets in this definition and others of this paper.) KLEENE [26] noted that the set of sentences P provable in Peano Arithmetic is recursively inseparable from the set of sentences R refutable in Peano Arithmetic. If a complete, recursive axiomatization of arith- metic existed, its deductive closure C would be a recursive set separating P from R .

DEKKER 1151 essentially defined a set A to be completely productive (abbreviated: c-produc- tive) iff (3 recursive f) (Vx) [ f ( x ) E Wxa A]. MYHILL ([15], [49]) showed that the c-productive and the productive sets coincide. Intuitively, A is c-productive iff there is an effective proce- dure to find, given any x, a counter-example to W, = A. The c-productive sets are, then, those sets that fail to be r.e. in a certain effective sense.

Now, to get to a central concept of this paper, just as there is an effective sense in which sets can fail to be r.e., there are effective senses of one set failing to be recursively separable from another. For example, we define just below effective A!-inseparabilify. Intuitively, A is ef- fectively A!-inseparable from B iff [A is disjoint from B and there is an effective procedure to find, given x and y , a counter-example to A S W, =

( 3 r e c u ~ i v e ~ ( V x , y ) ~ ( x , y ) € ( ~ ~ E ) u ( ~ ~ A ) u ( ~ n n ) u ( ~ n E ) u ( W , a ~ ) ) .

S g ] . Definit ion I . A 13 eflectiuely A:-inseparable from B iff A n B = 0 and

99 EFFECI‘IWZING INSEPARABILm

Here is a characterization of effective A :-inseparability.

Theore m 1. A is effectively A:-inseparable from B isf A n B = 0 and (3 recursiuen (Vx, y ) ( f ( x , y ) E ( W, n B ) u ( W, n A ) u (F n q)) .

Proof. (e) is immediate. (*): Suppose A is effectively d:-inseparable from B as wit- nessed by the recursive function f . By SMULLYAN’S Double Parametric Recursion Theorem ([58], [49]), there are recursive functions g and h such that, for all x and y ,

{ f ( g ( x , Y ) , ~ ( x J ) ) ) i f f ( g ( x , y ) , h ( x , y ) ) appears in W, before q,

{ f ( g ( x , y ) , h ( x , y ) ) } i f f ( g ( x , y ) , h ( x , y ) ) appears in W, before (or at the same time as it appears in) W,,

%%Y, = { 0 , otherwise, rz. otherwise. %,, =

We show that ( V x , y ) ( f ( g ( x , y ) , h ( x , y ) ) ~ ( W , n B ) u (W,n A ) u (Fn F)). Case 1. f ( g ( x , y ) , h ( x , y ) ) appears in W, before W,. Then

WP(%,) = { f ( g ( x , y > . h(x ,y ) )I and W,,,,) = 0 .

Hence, f ( g ( w ) , h ( x , y ) ) 6 (G n A ) u ( A n KcLyJ u (Wd.rsv)~W,,,,,). Therefore, since f witnesses the effective A:-inseparability of A from E, f ( g ( x , y ) , h ( x , y ) ) E B . Hence, f ( g ( x , y ) , W,Y)) E 4 n B.

Case 2 . f ( g ( x , y ) , h ( x , y ) ) appears in W, before (or at the same time as it appears in) W,. Then, by an argument symmetric to that of Case 1, f ( g ( x , y ) , h ( x , y ) ) E W, n A .

Case 3 . Neither Cases 1 nor 2. Then, f ( g ( x , y ) , h ( x , y ) ) E K n q. 0

SMULLYAN [58] effectivized KLEENE’S notion of recursive inseparability in an intensionally, and, as we shall see (Theorem 6), extensionally, different way. Definition 2 is essentially SMULLYAN’S definition.

(3 recursiven ( v x , ~ ) ( A E W, E q E B * f ( x , y ) E K n q). If C is a class of partial recursive functions, then Pc denotes lp 1 qg E C), the index set

([49]) determined by C. The next theorem characterizes effectively A :-inseparable pairs of in- dex sets and implies that for pairs of index sets our notion and SMULLYAN’S agree. (Results re- lated to special cases may be found in [16] and [211.)

Def in i t ion 2 (SMLJLLYAN). A is effectively inseparable from B iff A n B = 0 and

Theorem 2 . Suppose C and D are disjoint classes of partial recursive functions. Then (a)

(a) Pc is effectively A:-inseparable from PD. (b) Pc is effectively inseparable from PD. (c) Pc is recursively inseparable from PD. (d) Both 9 c and !PD are non-empw.

Proof. Suppose the hypothesis. If either of Pc or !PD is empty, clearly they are separated by a recursive set. Suppose, then, qc E C and Q]d E D. By Definitions 1 and 2, it suffices to show

through (d) are equivalent.

(2 recursiven ( v x , ~ ) ( f ( x , y ) E ( w , n pD) u ( w , n JJC) u CF n %)I. By KLEENE’S Parametric Recursion Theorem, there is a recursive function f such that, for all

100 J. CASE

x, Y, and 2, P ) d ( Z )

p c ( z )

t otherwise.

iff(x,y) appears in W, before W,, iff(x,y) appears in W y before (or at the same time as it appears in) W,,

Case 1. f ( x , y ) appears in W, before W,. Then, qf(&,,) = pd; hence,f(x,y) E PD. Therefore,

Case 2 . f(x,y) appears in W, before (or at the same time as it appears in) W,. Then, by

Case 3 . Neither Cases 1 nor 2. Then, f(x,y) E (E n q). 0

Theorem 2 obviously generalizes RICE’S Theorem ([44], [49]). Theorem 3 below provides a sufficient condition for disjoint pairs of subrecursive index

sets (defined two paragraphs below) to be effectively A:-inseparable. We provide next the preliminaries for that theorem and other results about subrecursive index sets.

A subrecursive class is an r.e. class of (total) recursive functions: N ---* N. Suppose S is a subrecursive class. tp is a programming system (or effective numbering) for S iff tp is a recursive function such that S = {Az. tp(r, z ) I r E N} . If p is a programming system for a subrecursive class, we write vr for Rz. p(r , z ) and speak of r as a p-program for vr. Generally subrecursive classes studied in computer science are explicitly based on bounding the time or space com- plexity of the functions allowed in the class; those studied in mathematics are implicitly so based (cf. e.g. [12], [14], [50], [39], [40]). For example, Polytime, the class of functions each computable on a multi-tape Turing machine within time given by a polynomial in the length of the input (in binary), is a subrecursive class. If one Godel numbers (onto N) the multi-tape Turing machines each explicitly clocked to halt within time bounded by some polynomial in the length of the input and sets tpr = the function computed by the machine with Godel num- ber r, then p is a subrecursive programming system for Polytime ([2], [22], [29], [39], [40]).

Suppose S and v are fixed such that S is a subrecursive class and p is a subrecursive pro- gramming system for S. If C s S, then Yc denotes { r I p? E C } , the subrecursive index set deter- mined by C ([29]). We make two basic assumptions about S and p.

Assumption 1. S contains gineartime and it is closed under (inner and outer) composition with functions in Sineartime.

We present some consequences of Assumption 1. Recall from above that we chose Ax,y.(x,y), nl, and nz in gineartime. Thanks to

Assumption 1, these are clearly useful to simulate the effect of having functions of multiple arguments in S. As another example, let

PfW, ( z ) =

f k y ) E W , n PD.

an argument symmetric to that of Case 1, f(x,y) E W, n Pc.

y i f x > O , z i f x = O .

cond = A (x,y, I ) .

Then cond is clearly in gineartime. Hence, S is closed under definition by cases. This is ex- ploited below in the proof of Theorem 3 (and also the proofs of Theorems 4 ,5 , and 8 and Re- marks 1 and 3). Let

rnax = A ( xl , . . . , xn ) .[the maximum of x, , . . . , x,] . Clearly rnax is in Xineartime. Hence, S is also closed under taking the maximum of a con- stant number of functions. We exploit this in the proof of Theorem 4 below. We al- s~ note

EFFEClTVIZING INSEPARABILITY 10 1

(and exploit below) that, in effect, the Boolean functions A (and), v (or), and i (not) are in zineartime.

Let Quadratictime be the class of functions: N + N computable within time bounded by some degree-two polynomial in the length of the input. Then S = Ouadratictime satisfies As- sumption 1, but is not itself closed under composition. The same is true for S = Vubictime, defined in the obvious way, etc.

Assumpt ion 2 . y safkfies the Kleene s-m-n Theorem wifh an s-1-1 function in %ineartime (cf. [291, 1381, WI, [521).

Essentially s-1-1 provides substitution of data into programs, of course filtered through the Godel numbering. This is an extremely elementary operation if an efficient Godel number- ing is chosen. Therefore, it is quite plausible to be able to perform this operation in linear time for a “natural”, efficiently numbered, subrecursive programming system for a class S sa- tisfying our assumptions. MARCOUX [32] shows that, as a control structure (cf. [45], [46], [47], [51]), s-1-1 is more fundamental than composition with respect to instance complexity. [38] and (391 (generalizing 1291) provide extremely simple sufficient conditions for y to have an instance of an s-1-1 function in gineartime. Now, the Kleene form of his Strong Recursion Theorem ([49, p. 21411, but not the Rogers Pseudo-Fixed Point form ([49, p. 1801, [45], [46], see also [47], [51]), holds in most naturally occurring subrecursive programming systems com- puting at least the functions in gineartime ([29]). In fact, it easily follows from our assump- tions on S and y that y has a gineartime-effective Parametric Recursion Theorem ([39], [40]) (‘Lt’ineartime-effective’ merely means that the witnessing function is in gineartime). Our proof below of Theorem 3 contains an application of this recursion theorem. Rorm and CASE [39], [40] show how to efficiently Godel number multi-tape Turing machines and for- mulate clocking mechanisms so that explicitly clocked systems for Lt’ineartime, Quadratic- time, ..., Polytime, etc., based on this numbering of Turing machines, satisfy the assump- tions above. They also present an axiomatic treatment of clocked systems and show how these inherit, from the underlying general purpose machines, relatively efficient control structures ([S I]) involving s-1-1, composition, recursion theorems, etc.

T h e o r e m 3 . Suppose C and D are disjoint subsets of S. Suppose there are y-programs b, c and d such that, for a e r y t, yb lc , c Lz.yl,(( t, z)) E C and ylbl < , c Lz.yd(( t, z)) E D. Then YC i s effec- tively A :-inseparable from YD.

We noted above that S = Ceubictime satisfies our Assumption 1. If we suppose ?,IJ is a subrecursive programming system for this class satisfying Assumption 2, then, by Theorem 3, { r I yr E (Quadratictime - gineartime)} is effectively A:-inseparable from [ r Iyr E gineartime}; [ r I ( 3 t ) (y , = Az.Ol< u Az.11 t)} is effectively A:-inseparable from [ r 1 y, = Lz.O}; and (rI(3t) (yr = Az.O(< u Az.llL !)} is effectively A:-inseparable from ( ~ I ( ~ ~ ) ( ~ ~ , = . Z I . O I , , W ~ ~ . ~ I , , ) } .

Proof of Theorem 3 . Suppose the hypotheses. For this proof we take CP to be a special, “delayed” Blum complexity measure such that one can compute in gineartime both

@Jz) if @,,(z) 6 t, t + 1 otherwise.

I, t.[CPP(z) s t l and Ap, z, t .

ROYER and CASE [39], [40] show that, for any p, such an associated @ always exists. In the rhetoric below in this proof (and others which employ such a special @), locutions of the form “so-and-so appears in W, with such-and-such restrictions on the number of steps” impli-

102 ].CASE

r Vdk(( t, z)) if each ep,j,.v appears in either W , or W, within a

number of steps least upper bounded by t , where t s z, and eij. k itself appears in W, before W,, if each ec,j,.k, appears in either W, or W, within a number of steps least upper bounded by t, where t 5 z, and e i , itself appears in W , before (or at the same time as it appears in) W,,

p,(( t, z))

citly refer, then, to this special 0. By the Seineartime-effective Kleene Parametric Recursion Theorem for ([39], [40]), there is a Seineartime function f such that, for all x , y , and z ,

vd(( t, z))

vC(( t, z ) )

if f ( x , y ) appears in W , in exactly t steps, where t s z, and f ( x , y ) appears in W, before W,, if f ( x , y ) appears in W, in exactly t steps, where t 6 z, and f ( x , y ) appears in W, before (or at the same time as it appears in) W,,

Vb(z ) otherwise.

The rest of the proof is a straightforward modification of that of Theorem 2. 0

In Theorem 4 just below we make the sufficient condition of Theorem 3 somewhat less constructive to obtain a sufficient condition for recursive inseparability of subrecursive index sets.

Theorem 4 . Suppose C and D are disjoint subsets of S. Suppose, for each i, j and k with i 5 I , j 5 m and k s n, there are corresponding p-programs bi, cj and dk such that, for every t, there are corresponding i, j and k, with is I, j 5 m and k 5 n, for which vb,l <, c Az.y,(( t, 2)) E C and wb,/ <, c Az.pdk(( t, z ) ) E D. Then Yc is recursively inseparable from Y,,.

Remark 1. For both Theorems 3 and 4, the sufficient conditions are not necessary.

Re mark 2 . Thesufficient condition of Theorem 4 does not imply thatYc is effectively A7-inseparable from YD.

Proof of Theorem 4. Suppose the hypotheses. For this proofonly wemake the convention that i,jand k(withorwithout decorations) are restricted inrange thus: i 5 1,j s mand k 6 n.Also for this proof we again take CP to be a special, “delayed” Blum complexity measure such that one can compute in sineartime both

@,,(z) if @,,(z) 6 t , t + 1 otherwise.

Ap, z, t.[@,,(z) 6 t ] and Ap, z, t .

As we noted in the proof of Theorem 3, ROYER and CASE [39], [40] show that, for any q, such an associated CP always exists.

Let x and y be futed. By the ( ( I + 1) 3 ( m + 1). ( n + 1))-ary Recursion Theorem for q ~ , there are ( ( 1 + 1) * ( m + 1). ( n + 1)) self-other referential y-programs eij, k such that, for all z,

EFFECTMZING INSEPARABILITY 103

Case 1. Each e,,.],, k' appears in either W, or K. Let t be the least upperbound ofthe number of to so appear. By the hypotheses of the theorem, there is, associated steps required for each

with this f a triple (i,j, k) such that

(2) ' v b , l < t 12. V, ( ( t, z >) E c and ' v b , I < I C 12. ' v d k ( ( t, z >> E D.

Consider, then, e,, Ir. Subcase 1.1. e,,,appears in W, before W y . Then by (l),

'vt,,, t = 'vb, 1 < r u 'vdt (( 1, z ))I p r ;

hence, by (2),

'vt&,,,,, = l z . ' v d , ( ( f * z)) yD.

Therefore, ell,, ~ E W, n YD . Subcase 1.2. e,,,,appears in W, before (or at the same time as it appears in) W,. Then by an

argument symmetric to that of Subcase 1.1, e,,,k E W, n .Yc. Case 2. Not Case 1. Then, for some triple ( i j , k), e,,,k E

Proof of Remark 1. We take @ to be a special, "delayed" Blum complexity measure as in the proofs of Theorems 3 and 4. Let C = {y1,l(3f)('v~~,,) = A r O ] , , u Az.lIzf)} and D = (tp, I(3t) (vvA0, = Az.Oi< u 12.21 J} . C and D are non-trivial since S contains Zineartime, and, clearly, C and D are disjoint subsets of S that do not satisfy the hypotheses of Theorems 3 and 4. We show that, nonetheless, Yc is effectively A:-inseparable from YD. The techniques of [39], [40] easily establish a Xieartime-effective Delayed Recursion Theorem ([lo]) for y. Hence, there is a Zineartime function f such that, for all w , x, y and z,

n q. c]

- 2 iff(x,y) appears in W, in zsteps

andf(x,y) appears in W, before W,, 1 iff(x,y) appears in W, in z steps

andf(x,y) appears in W, before (or at the same time as it appears in) W,, I 0 otherwise.

' vvf (&y) (Y) (Z) =

It is easy to argue, much as in the proofs of previous theorems, that, for all x and y,

f ( x , y ) E ( w , n yD) u CW, n ye) u (K n TI. 0

Proof of Re mark 2 . We adapt a trickofFw~~'sfrorn[13]. Let A be aset such that both A a n d 2 are immune [49, p. 1081. Let C = (1z.O). Let

D = { A Z . O ~ < u l Z . 1 1 p r I t E A } u {AZ.O~< r u 11.21 p 1 t E 21 . Since %ineartime S S, we can choose rp-program(s) bo and co such that wb, = yQ = 12.0. Further- more, for each k E {0, l} , we can choose a y-program dk such that

0 i f z < t , k + 1 otherwise. Vdk = A(?, 2).

Clearly, then, C, D, bo, co, do and d , satisfy the hypotheses ofTheorem 4. Suppose for contradic- tion that f is a recursive function such that

(3) (VX,Y) (f(x,y) E (% n %J u (& n pe) u CE n c)).

104 J. CASE

By the s-m-n Theorem for q, there is a recursive function g such that, for all s,

%(s) = { r I ly.01 c c w r } . Let q be a pprogram such that

W, = { r 1 0 2 ) [ur,(z) > 01).

(VS) (f(g(s), 4) E %(r) n Y D ) .

- - Clearly, W, n Yc = 0 and (Vs) (WBcS) n W, = 0). Hence, by (31,

(4)

Now, by (4), for each s, s 5 card({z 1 lyf(g(s).4)(z) = 0}) < a. Therefore, {lyf(gcn,p) I s E N} is infi- nite. Let t ( r ) = min,((3v > 0) ( l y r ( t ) = u)) . T is clearly partial recursive. For each u E {1,2}, let A,, = {7(f(g(s), q))1 u E range(y,(gcaq))}. Clearly, then, either A l is an infinite r.e. subset of A or A 2 is an infinite r.e. subset of 2. This is a contradiction. Hence, by Theorem 1, YC is not effectively dpinseparable from YD. 0

We next proceed to define inseparability by subrecursive sets, where the class of subrecursive sets associated with a subrecursive class S (called the class of S-sets) is

{CI(Yg E S) (C = g- W I . A special case occurs when g E Y is xc for some C. Then g-’(l) = C, and, so, C is an S-set.

Defini t ion 3 . A is S-inseparable from B iff A n B = 0 and there is no S-set C such that A E C E B .

Theorem 5 just below characterizes S-inseparability of associated subrecursive index sets. It also holds with the even less restrictive assumptions on S and ly ROYER (51, p. 1731 employed for his Subrecursive Rice Theorem (together with the assumption that xB and xN E S ) .

Theorem 5. Suppose C and D are disjoint subsets of S. Then (a) and (b) are equivalent. (a) Yc is S-inseparable from YD. Proof. Suppose the hypotheses. We note that 0 and N are S-sets since their characteristic

functions are computable in linear time. If either of Yc or YD is empty, clearly they are sepa- rated by an S-set, namely, either 0 or N. Suppose, then, qc E C and qd E D. Suppose C is an S-set. By the Kleene Recursion Theorem for ly (cf. [39], [40]), there is a yo-program e such that, for all x, y and z,

(b) Both .YC and YD are non-empty.

- Suppose for contradiction that Yc E C S PD. Hence, e E C lye = Vd E D, * e E YD, * e d C,

ye = yc E C, * e E Yc, Theorem 5 essentially generalizes KOZEN’S [29] (and ROYER’S later [51, p. 173141) Subrecur-

sive Rice Theorem. CASE introduced the notions of r.e. inseparability and effective r.e. inseparability ([13]).

Some of the theorems of [13] are lifted to Scorn’s CPO’s (cf. [54]) in [60]. A is said to be r.e. inseparable from B iff A n B = 0 and there is no r.e. set C such that A f C E B. A is effectively r.e. inseparubte from B iff A n B = 0 and (3 recursive f) (Vx) (fix) E ( A n TJ u (W, n B ) ) . ROYER 1521 applies both these notions to characterize the presence of proof speed-up between theories. ROYER and CASE [40] apply the lift to the 2: level (from the r.e. = level) of both these notions to characterize relative program succinctness phenomena between subrecur-

e E C. Therefore, e E C- e d C , a contradiction. 0

EFFECTWIZING INSEPARABILITY 105

sive/complexity-bounded programming systems (cf. [29], [Sl]) and to obtain a tight incomple- teness theorem about subrecursive program succinctness coupled with information loss.

T he0 rem 6 . There are disjoint sets A and B such that A is r.e. inseparable from B (therefore, A is vacuously effectively inseparable from B ) , but A is not effectively A!-inseparable from B.

Proof. A and B are obtained by a non-effective construction by finite extensions in con- secutive stages s 2 0. A s and B' denote, respectively, the finitely much of A and B , respecti- vely, determined by the beginning of stage s. A' and B o are both empty. A and B are con- structed to explicitly satisfy, for each s, the requirements 3: and %: below, where %: is satisfied during stage 2s and !7t: during stage 2s + 1. The finding of x, and y , in stage 2s + 1 is effective in s and (canonical indices for) A z s + and Bzsc by SMULLYAN'S Double Recursion Theorem. The construction ensures that, if a number is put into one of A and B, it is exclu- ded from the other; hence, A n B = 0 .

5%;: ( ~ w , ) ( w , E ( y n B ) u ( ~ n A ) ) .

9:: (rP3 torar*(~x*,Ys)(P'((x, ,Y,)) E Ks u ys A (Ps((&.Ys))E w , * P s ( ( x s , Y s ) ) c f B )

A (PS((X3,YS))E Y,*PS((X3&))cf A ) ) ) .

Clearly, then, if A and B satisfy 5%: for every s, A is r.e. inseparable from B. If A and B sa- tisfy %: for every s, A is not effectively &inseparable from B .

begin stage 2s;

then i f ( g w ) ( w E W n F ?

let w, be the least such w ; put w, into B (so that w, E W, n B]

let w, be the least element of F n BZs; put ws into A {so that w, E

else (hence, y is finite; therefore, is infinite] - -

n A } endif

end {stage 2s).

begin stage 2s + 1 ; find x, and y, such that

{rPS((XS>YS))} ifPS((XS9YS)) I cf Bzs+ l

{Ps,((xs~Ys)~~ ifPs((Xs7Ys))L E BZS+'7

c 0 otherwise ; wx. =

otherwise ; %= { * if ~ ~ ( ( x , , y ~ ) ) 1 d (Azs+' u B Z S + l )

then put q.~,((x~,y~)) into A {so that P,((x,,Y~)) cannot later be put into BJ

endif end {stage 2s + 11.

It is straightforward to verify that A and B are as required.

106 J. CASE

In recursion theory the subsets of N are usually topologized by identifying them with their characteristic functions in 2N, placing the discrete topology on 2 ( = {o, 1)) I and the cone- sponding product topology on 2N (cf. [37], [49]). Similarly each pair (A, B) of disjoint subsets of N can be identified with a naturally corresponding function in 3N, namely,

2 i fzEA, 12. 1 i fzEB, 1 0 otherwise.

We place the discrete topology on 3 (= (0, 1,2)) and the corresponding product topology on 3N. Now, by a theorem of HAUSDORFF'S (cf. e.g. [63, p. 216]), both these product topologies are homeomorphic to the Cantor-set topology, a complete, metrizable space; hence, the product topologies satisfy Baire's Theorem.

Corollary 1 just below says that the pairs (A , B ) witnessing the truth of Theorem 6 just above are plentiful in the sense of Baire Category.

Corollary 1. ((A, B) I (A, B) witnesses the truth of Theorem 6} i s co-meager.

Proof. At any stage in the construction of the proof of Theorem 6 a finite number of ele- ments have been committed to each of A, B, 2 and g. Modify this construction so that, for each i E (0, l} , every reference to 2s + i is changed to 3s + i + 1 and an "opponent" in a new stage 3s is allowed to commit a finite number of elements to A, B , 3 and B, provided A is kept disjoint from B and each stage honors the commitments of the previous. The A (respec- tively, B ) resulting from this modified construction is understood to be the set of numbers explicitly committed to A (respectively, B). Clearly the A and B from the modified construc- tion still satisfy Theorem 6. By the standard connection between infinite games and Baire Category (first proved by BANACH, cf. e.g. [25]), we have Corollary 1. 0

The sets A and B constructed in the proof of Theorem 6 are clearly recursive in the halting problem (Note: In that construction A = B) . Of course A cannot be r.e. However, if all we wanted was to show that effective inseparability and effective A ;-inseparability are not coex- tensive, conceivably A and B could both be r.e. Theorem 7 just below implies they could not. We did not explore whether, in Theorem 6, B can be r.e. and A recursive in the halting prob- lem. We also did not investigate whether a measure-theoretic ([49]) analog of Corollary 1 holds.

Theorem 7. Suppose A and B are r.e. Then A is effectively inseparable from B iff A is effec- tively A:-inseparable from B.

Proof.Suppose A and B are disjoint r.e. sets. It clearly suffices to show (*). Suppose A is effectively inseparable from B as witnessed by the recursive function f. By the Double Recur- sion Theorem there are recursive functions g and h such that, for all x and y ,

A u {f(g(x,y), h(x,y)N iff(g(x,y), h ( x , ~ ) f appears in &before W,,

B u {f(g(x,y), ~ ( x J J ) ) } iff(x,y), h(x,yN appears in %before (or

%iY) = {A otherwise ;

at the same time as it appears in) W,, wh(%su)= { B otherwise.

We show that A is effectively d!-inseparable from B as witnessed by Ax,y.f(g(x,y), h(x,y)).

EFFECTMZING INSEPARABILITY 107

Case 1. f ( g ( x , y ) , h ( x , y ) ) appears in W, before W y . Then

wB(,y) = A u I f (g (x , y )+ h ( x , y ) ) } and w h ( q y ) = B . - -

Suppose for contradiction that f ( g ( x , y ) , h ( x , y ) ) d B . Then A E W,,,, s W,,,,, s B. Therefore, by Definition 2, f ( g ( x , y ) , h ( x , y ) ) E W,,,,) n Wh(,y). This contradicts that W p ( x y ) = A u I f ( g ( x , y ) , h ( x , y ) ) } . Hence, f ( g ( x , y ) , M y ) ) E K n B.

Case 2 . f ( g ( x , y ) , h ( x , y ) ) appears in W y before (or at the same time as it appears in) W,. Then, by an argument symmetric to that of Case 1, f ( g ( x , y ) , h ( x , y ) ) E W, n A .

Case 3 . Neither Cases 1 nor 2. Then, f ( g ( x , y ) , h ( x , y ) ) E

We indicate, briefly, how to generalize up into the arithmetical hierarchy all the insepara- bility notions above. We let cp" denote a fixed acceptable programming system (numbering) for the set of partial functions: N 4 N which are partial recursive in 0(') (cf. 911, (491). We let W; denote the domain of pi. W i is, then, the Z:+ set ( S N) accepted by q"-program p . We present the definition of effective A:, ,-inseparability to illustrate the pattern of the generali- zations. The other inseparability notions above are similarly generalized.

--

n q. 0

Defin i t ion 4.A is effectiveb Z:+ ,-inseparable from B iff A n B = 0 and

( ~ r e c u r s i v e ~ ( ~ x , y ) ~ ( x , y ) € ( ~ ~ n ~ ) u ( ~ n ~ ) u ( ~ ~ n A ) U (w,"n B ) ~ ( w ; A W , ) ) . Since the inseparability concepts defined above are clearly invariant under choice of ac-

ceptable programming system, we have, for example, that the n = 0 case of Definition 4 agrees with Definition 1.

It is clear that by employing the appropriate relativization of recursion theoretic tools each of our results above except Theorems 3 and 4 and Remarks 1 and 2 can be generalized to each higher level of the arithmetical hierarchy. Theorems 3 and 4 provide sufficient condi- tions for effective A:-inseparability (respectively, recursive inseparability) of subrecursive in- dex sets. Theorem 8 below provides a sufficient condition for effective A:-inseparability of subrecursive index sets. We have not explored any other levels for sufficient conditions in- volving inseparability of subrecursive index sets.

We note that ROYER and CASE [40] use effective 4;-inseparability to gain insight into the- orems of LADNER [30], SCHONING (531 and AMBOS-SPIES [l] in structural complexity theory and to obtain independence results about complexity. These latter results are in the style of independence results due to REGAN [41], [42], [43], KOWALCZYK [28], HARTMANIS [20], and KuRTZ, O'DONNELL, and ROYER 1271.

Theorem 8. Suppose C and D are disjoint subsets of S. Suppose there are y-programs c and d such that, for every finite set A, wdla u IP,IAE C and yclA u ~ ~ I J E D. Then Yc is effectively A;-inseparable from YD.

The quantifier 'V" means 'for aU but finitely many'. If we suppose, as for the examples fol- lowing the statement of Theorem 3 above, that t is a subrecursive programming system for S= Vubictime and that it satisfies Assumption 2, then, by Theorem 8, the first example after the statement of Theorem 3 also provides an example of effective A:-inseparability. It is easy to argue that each of the other two example pairs of sets after the statement of Theorem 3 has a A: separating set. However, by Theorem 8 once again,

{ r Irange(vr) E {O, 11 A (V-2) [ V r ( Z ) = 011

is effectively A!-inseparable from { r Irange(yl,) E {O, I} A (V-2) [yr (z ) = l]},

108 ].CASE

Proof of Theorem 8. Suppose the hypotheses. Since effective d$inseparability is inva- riant under choice of acceptable programming system, we are free to judiciously choose one. Now, the functions recursive in 0") are well known to coincide with the functions which are the limit of some recursive function (cf. e.g. [56], [57], [59]). This and its relativization were first noticed and used by POST ([55]) and have been employed (sometimes with rediscovery) many times. CASE [ l l ] exploited an extension to partial functions. MEYER [34] proved a lemma that easily generalizes to the fact that each function recursive in 0") is the limit of some primitive recursive function. ROVER and CASE [39], [40] construct an Seineartime func- tion L, such that, if we define

p,B* = Az. lim L,(p, z, t ) , 1 - 0 0

then p* is an acceptable programming system (numbering) for the set of partial functions partial recursive in 0"). They further define a particular associated relativized Blum complex- ity measure (1311) @* based on the modulus of convergence ([56], [57]) for L*. In particular the predicate Ap, z, t . [@,*(z) 5 t] is in rf? (generalizing SHOENFIELD'S Modulus Lemma, c.f. e.g. [59]); hence, this predicate is in A;. Let W; denote the domain of 9;. W; is;then, the Z: set (G N) accepted by p*-program p . We say that a number w appears in W: before W; iff @,Z(w) 5 < @>:(w) 5 a. A number w appears in Wz at the same time as it appears in Wy' iff @:(w) = @;(w) 5 < m . The Hybrid Recursion Theorem of ROYER and CASE [39], [40] holds between v and p* and can be used to obtain self-other reference and facilitate interaction between these systems. By the Hybrid Recursion Theorem, then, there are Seineartime func- tions f and g such that, for all x, y , and z,

0 if f(x, y ) appears in W: before W; , 1 iff(x,y) appears in W; before (or at

the same time as it appears in) W:,

otherwise. (In this application of the Hybrid Recursion Theorem there is no direct self-reference, but there is circular reference since, for each x and y , p-program f(x,y) and p*-program g(x,y) each refers to the other.) It suffices to show that

- - (VX,Y)(~(X,Y) E (C n YD) u (W; n Yc) u (W: n W ; ) ) .

Case 1. f ( x , y ) appears in W; before W;. Then

= 0 = lim L ( g ( x , y ) , O , z) . 2-00

Hence, A = {z IL*(g(x,y),O, z) > 0} is a finite set. Therefore, vy(xy, = vc)clA u p&. which, by hypothesis, is in D. Hence, f ( x , y ) E Yo. Therefore, f ( x , y ) E W: n YD.

Case 2 . f ( x , y ) appears in W; before (or at the same time as it appears in) W:. Then, by an argument symmetric to that of Case 1, f ( x , y ) E W; n Yc.

Case 3 . Neither Cases I nor 2 . Then, f ( x , y ) E W: n W;. 0 Remark 3. The sufficient condition of Theorem 8 is not necessary.

Proof. Let C= {rlrange(vvk0)) S {0,1} A (V"Z)(~,,(~)(Z) = O)} and D= {rlrange(tpvAo)) S (0,l) A ( W z ) (vvL0)(z) = 1)). C and D are non-trivial since S contains Sheartime, and, clearly,

- -

EFFECTNIZING INSEPARABILITY 109

C and D are disjoint, but do not satisfy the hypotheses of Theorem 8. Nonetheless, we show, employing terminology and results from the proof of Theorem 8, that Yc is effectively .&inseparable from YD. The techniques of [39], (401 easily establish various delayed forms [lo] of the Hybrid Recursion Theorem. In particular, there are ginear t ime functions f and g such that, for all w, x, y and z,

( 0 iff( x, y ) appears in W: before W; ,

1 iff(x,y) appears in W; before (or at the same time as it appears in) W:, P;(x+y,(z) =

kT otherwise.

The rest of the proof is a straightforward modification of the proof of Theorem 8. 0

We have not explored whether the proofs of Theorems 3 and 4 and Remarks 1 and 3 might, upon suitable generalization, lead to characterization theorems. Perhaps algebraic manipula- tions in the style of [61] and [62] (see also [S], [6 ] , [7], [8], [9], [181, [24]), but for subrecursive systems, would be useful to elegantly bring the sufficient, but not necessary conditions of this paper closer to characterizations.

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(Eingegangen am 24. April 1989)