- Email: [email protected]
Chapter 13 Enumeration
Chapter 13
Enurneration In this chapter we are concerned primarily with classes of predicates, which can be generated in a number of ways. We assume such classes satisfy relevant closure properties and look at the functionals that are defined from them. If P is a class of predicates, then F g contains the functionals whose graph is in P. We shall see that Pg may satisfy important closure properties, depending, of course, on the closure properties of the class P. The relations between the classes 3 and 3 d and also between 3 and 3 p g , g’ were discussed in Sanchis [27] in terms of numerical functions. Here we consider functionals and predicates with non-total arguments, and this creates new problems. We propose new techniques which are useful in dealing with this situation. In dealing with classes of the form 3 d (or 3 ~ it )is important to take into account that the predicates are determined by their dual characteristic functionals (or by their partial characteristic functionals in the case of 3 p ) , and these functionals are not uniquely determined by the predicates. Still, the classes 3 d satisfy a number of closure properties, and these are sufficient for the applications in this chapter. Two new constructions which are used to define new predicates from given predicates are introduced in this chapter. Both are special forms of unbounded numerical quantification, existential or universal. Logically, they can be considered to be equivalent, since one can be defined from the other. In computability theory there is a natural hierarchy where existential quantification is more constructive than universal quantification. Here we are concerned primarily with the former. 189
L.E. Sanchis
190
The operation of unbounded quantification takes two forms, well known as existential and universal quantification. They apply to (k 1,m)-ary predicates and produce (k,m)-ary predicates, k, m 2 0. The formal definitions are as follows:
+
Universal U n b o u n d e d Quantification. Let Q be a (k We introduce a (k,m)-ary predicate P such that
P ( z ;a)
=
+ 1,m)-ary predicate.
(VY)Q(Y,2; Q) Q(y, z;a)holds for all y
We say that P is obtained from Q by universal unbounded quantification. Existential Unbounded Quantification. Let Q be a (k+1, m)-ary predicate. We introduce a ( k ,m)-ary predicate P such that
P ( z ;a)
3
( ~ Y ) Q ( Y2;, a) there is y such that Q(y,z; a)holds
We say that P is obtained from Q by existential unbounded quantification. Both quantifications, universal and existential, are non-finitary, for they determine a boolean value in terms of an infinite collection of values. So, they differ from the bounded quantifications of Chapter 4, which are finitary. For this reason we cannot expect that classes of the form 3 d or Fp are closed under unbounded quantification, even if 3 satisfies strong closure properties. Still, we shall see that this is possible in special cases. The determination of this type of closure is an important characterization of a given class 3 of functionals. In dealing with predicates in terms of functionals we must use characteristic functionals, and we know these are to some extent ambiguous, for in general a predicate has many partial characteristic functionals and many dual characteristic functionals. For this reason we must be very careful in translating definition by cases given by predicates into definition by cases given by functionals. On the other hand, this translation may be necessary if we want to insure monotonicity in the specification. For example, a specification of the form
h(z;a)
N
f i ( z ; a ) if
N
f2(z;a)
if
Q(z;a) -&(%;a),
is not necessarily monotonic and the mapping h is not necessarily a functional. A
191
Chapter 13. Enumeration monotonic approximation can be obtained by the specification h(z;a)
2:
f i ( z ; a ) if
N
fZ(z;a)
if
0 $ 0,
X Q ( Z ; ~ )N
XQ(z;a)
but this specification depends on the particular dual characteristic functional XQ. These problems do not appear in dealing with numerical functions, for a numerical predicate has a unique partial characteristic function and a unique dual characteristic function, and the translation above is faithful to the original specification. To handle this situation it is convenient to introduce the following notation. If f and g are (k,m)-ary functionals we say that f is a similar extension of g if f is an extension of g in the usual sense and f is also similar to g. The latter condition means that f(z;a ) N g(z; a ) whenever the a are total functions. We write g Es f to denote that f is a similar extension of g. This relation is clearly reflexive, transitive, and anti-symmetric.. When we say that a functional f is a similar extension of a functional g, this means that f is more defined than g, but only for non-total arguments. Since functionals are monotonic, and the values for non-total arguments are determined by the values for total arguments, the changes that f may introduce by being more defined than g are predictable from the common values of f and g for total arguments. So, we may say that f contains more explicit information, but such information was already implicit in g. Let F and F' be classes of functionals. We say that F' is a similar extension of F if 7 F',and whenever f is a functional in F' there is a functional g in F such that g Cs f. We write F c s 3' to denote that F' is a similar extension of F. This relation is clearly reflexive, transitive, and anti-symmetric. If F is a class of functionals, then 3; is the class that contains all functionals that are similar extensions of functionals in F. Clearly, F G s Fs. Furthermore, if F CS F',then F' CS Fs.
c
13.1 Let f be a (k,m)-ary functional and Gf the graph predicate o f f . I f f ' is a selector functional for G,, then f is a similar extension o f f ' . Hence, iff' is an element of the class F ,then f is an element of the class Fs.
EXAMPLE
13.2 If g is a k-ary numerical function, then the only similar extension of g is g itself. If 3 is a class of numerical functions, then F = 3;. EXAMPLE
L.E. Sanchis
192
c
The subscript “s” is clearly monotonic, so if F F’,then FS E F’s. Furthermore, it can be concatenated with other subscripts. For example, if F is a class of functionals, then Fsd and FSpare classes of predicates, FSdg and Fspg are classes of functionals. Let P be a predicate and F a class of functionals. If P is d-computable in the class PR(F) we say that P is primitive recursive in F. We denote by PRd(F) the class of all predicates that are primitive recursive in F. If 7 is empty we write simply PRd = PRd(0). w e say that the predicates in PRd are primitive recursive. If the predicate P is d-computable in the class MR(F) we say that P is precursive in 3. If P is p-computable in MR(F) we say that P is partially p-recursive in 3.We denote by MRd(F) the class of all predicates p-recursive in F ,and by MRp(7) the class of all predicates partially p-recursive in F. If F is empty we use the notations MRd and MRp. The predicates in MRd are p-recursive, and the predicates in M R p are partially p-recursive. If the predicate P is d-computable in the class RC(F) we say that P is recursive in 3. If P is p-computable in RC(F) we say that P is partially recursive in F.We denote by RCd(F) the class of all predicates recursive in F ,and by RCp(F) the class of all predicates partially recursive in F. If F is empty we use the notation RCd = the class of recursive predicates, and RCp = the class of partially recursive predicates. Note that in relation to the class PR(F) we introduce only the class PRd(F) of predicates that are primitive recursive in F. Of course, the class PRp(F) is also defined, but it is not used in our discussion. On the other hand, the classes MRp(F) and RCp(F) are important extensions of the classes MRd(F) and RCd, respectively. Finally, note that PR(F) C MR(3) C RC(F), hence PRd(F) MRd(F) & ( F ) and MRp(F) RCp(F).
c
c
Theorem 13.1 Let 3 be a class of functionals. Then,
(i) The class PRd(T) (MRd(F)) (RCd(F)) is closed under conjunction, disjunction, negation, universal bounded quantification, existential bounded quantification, distribution, and numerical substitution with quasi-total functionals primitive recursive (p-recursive) (recursive) in F.
(ii) The class MRp(7) (RCp(F)) is closed under conjunction, universal bounded quantification, distribution, and numerical substitution with functionals p recursive (recursive) in F.
Chapter 13. Enumeration
193
(iii) The class RCd(3) is closed under functional substitution with quasi-total functionals recursive in 3. Parts (i) and (ii) follow from Theorem 3.1, Theorem 3.3, and Theorem 4.6. Part (iii) is clear from the definitions, noting that the requirement that the functionals be quasi-total is necessary to make sure that conditions PCF 1 and DCF 1 are satisfied. 0 PROOF.
13.3 If 3 is a class of functionals closed under p-recursive operations, then the functional U D is 3-computable. If P is a predicate which is d-computable in F there is a dual characteristic functional x p which is 3-computable. From this we get a partial characteristic functional $ p , where EXAMPLE
This proves that 3 d C 3 p . It follows that MRd(3)
MRp(F) and RCd(3) C
RCP ( 3 ) . If P is a class of predicates and Q is a predicate obtained from a predicate in P by existential unbounded quantification we say that Q is P-enumerable. We denote by Pe the class of all predicates that are P-enumerable. If Q is a predicate obtained by universal unbounded quantification from a predicate in P we say that Q is Pcoenumerable. We denote by Pu the class of all predicates that are P-coenumerable. We denote by P c the class of all predicates that are obtained by the negation of a predicate in the class P. It follows that Pe = Pcuc, PU = Pcec, PCC = P, Pe, = PCU , and PUC = Pce. Theorem 13.2 Let P be a class of predicates closed under numerical substitution with primitive recursive numerical functions. Then,
(i) P e l P,, and P c are closed under numerical substitution with primitive recursive functions.
(ii) P
Pe and P
Pu.
(iii) Pe is closed under existential unbounded quantification, and PU i s closed under universal unbounded quantification.
(iv) If P is closed under distribution, then P, and PU are closed under distribution.
L.E. Sanchis
194
(v) If P is closed under conjunction, then Pe and
P, are closed under conjunction.
(vi) I f P is closed under disjunction, then Pe and PU are closed under disjunction. (vii) If P is closed under universal bounded quantification, then Pe and P, are closed under universal bounded quantafication. (viii) If P is closed under existential bounded quantification, then Pe and PU are closed under existential bounded quantification. The proof of part (i) is straightforward, noting that in the case of P, the conclusion follows because the primitive recursive functions are total. Part (ii) is also straightforward since by numerical substitution with projection functions we can add to a predicate in P a new dummy variable that can be eliminated by existential or universal quantification. To prove (iii) note that if Q is (k 2, m)-ary predicate in P , and P is a predicate obtained by two existential quantifications in the form: PROOF.
+
P ( z ;QL)
PY)PV)Q(Y, v , 2; a),
P ( z ;QL)
(3y)Q([yIi, [YIz,
then 2; a),
P is in Pe. A similar argument applies to P". Part (iv) is straightforward from the assumption. For parts (v) to (viii) we consider first the class Pe. If P = PI A Pz, where PI and Pz are in Pe, then there are predicates Q1 and Q z in P such that
so
P ( z ;a)
(3Y)Qi(y,z;
A ( ~ Y ) Q z ( Y2; , a) ( ~ Y ) ( Q ~ ( [ Y ] ~ , ~A; QQ Lz () [ Y ] z ; ~ ; Q L ) ) ,
so P is in Pe.If P = PIV Pz we use the same technique of exporting the existential
quantifiers, but here there is no need to contract them, for they can be identified. To prove part (vii), assume P is a predicate of the form
where Q is in the class P. It follows that
195
Chapter 13. Enumeration
so P is in the class Pe. Part (viii) for Pe is trivial, for it reduces to the interchange
of two existential quantifiers. To prove (v) to (viii) for P, we use duality. For example, assume P is closed under conjunction. Then Pc is closed under disjunction, hence Pce is also closed 0 under disjunction, and Pcec = P , is closed under conjunction.
Corollary 13.2.1. Assume
F is closed under primitive recursive operations. Then
(i) Fde is closed under existential unbounded quantification, distribution, conjunction, disjunction, universal bounded quantification, existential bounded quantification, and numerical substitution with quasi-total T-computable functionals.
(ii) Tpe is closed under existential unbounded quantification, distribution, conjunction, universal bounded quantification, and numerical substitution with F-computable functionals. (iii) If Pi is Fd-enumerable and P2 is Fp-enumerable, then P = PI v P2 is Fpenumerable. PROOF. Parts (i) and (ii) follow easily from Theorem 13.2 with Theorem 3.1, Theorem 3.3, and Theorem 4.6. To prove part (iii), assume Q1 is a predicate in Fd and Qz is a predicate in 3p such that Pi(2;Q) ( ~ Y ) Q ~ ( Y2;, a) Pz(2;Q) ( ~ Y ) Q z ( Y a; , a). It follows that
P ( z ;0) I (%)(Qi(y, a;Q) V Qz(Y, hence P is in
2; a)),
Fpe by Theorem 3.3 (iii).
0
Theorem 13.3 Let T be a class of functionals closed under p-recursive operations. Then 3 d = 3 d e fI Fdu.
The inclusiond' . E Tde n Tdu follows from Theorem 13.2 (ii). To prove the converse, assume P is a predicate such that PROOF.
P ( z ;a)
=
2; Q) (VY)QZ(Y, 2; a),
(3Y)Ql(Y,
L.E. Sanchis
196
where Q1 and Q2 are predicates d-computable in F. Assume x1 is a dual characteristic functional of Q1 which is F-computable, and xz is a dual characteristic functional of Q2 which is F-computable. We introduce a functional f such that
Clearly, f is 3-computable, so the functional
is also F-computable. We claim that x is a dual characteristic functional of P . To prove this we must check that conditions DCF 1 and DCF 2 are satisfied. To check DCF 1, assume the functions a are total. It follows that x l ( y l z ;a) and ~ 2 ( y , z ; a are ) defined for all y, hence f ( y , z ; a ) is defined for all y. We must show that there is some y such that f ( y , z ; a)N 0. If there is y such that u x l ( y , z ; a)N 0, then f ( y , z ; a)N 0. Otherwise, we have p(z;a), so there is y such that x2(y,z; a)N 0, hence f ( y , z ; a)N 0. This means x(z;a)is defined whenever the a are total functions. To prove DCF 2, note that if x(z;a)N v, then either v = 0 or v = 1. If v = 0 there is y such that x1(y,z; a)N 0, hence P ( z ;a)holds. If v = 1 there is y such 0 that x l ( y , z ; a)E 1, and f(y, z;a)N 0, so P ( z ;a)does not hold.
13.4 Let F be an arbitrary class of functionals. It follows from Theorem 13.3 that M R d ( F ) = M R d e ( F ) n M R d U ( F ) and R C d ( 7 ) = RCde(F) n RCdu(F).
EXAMPLE
Corollary 13.3.1 Let MRde
F be an arbitrary class of functionals. Then M R d ( F ) =
MRdec(F).
From Theorem 13.3we know that M R d ( F ) = M R d e ( F ) f l M R d U ( F ) . Since M R d ( F ) is closed under negation, we get
PROOF.
MRd(F)
*
= MRde(F) MRdcec(7) = MRde(F) n M R d e c ( F ) . 0
The result of Corollary 13.3.1can be expressed in the following form: a predicate P is p-recursive in the class F if and only if both P and the negation of P are M R d (F)-enumerable.
Chapter 13. Enumeration
197
Let P be a class of predicates. We recall that Pg is a class of functionals where f is in P g if and only if the graph predicate G, is in F. It is convenient to introduce another subscript "#," where P# = Peg. A functional in P# is said to be ?-enumerable.
13.5 Let P be a class of predicates closed under numerical substitution with primitive recursive numerical functions. It follows that P Pe, hence Pg C Peg = P#. If P is also closed under existential unbounded quantification we have Pe P , hence P# Pg,so Pg = P#. EXAMPLE
c
13.6 Assume the class T is closed under primitive recursive operations and f is a quasi-total functional in 3. It follows that a dual characteristic functional xf of G, is given by
EXAMPLE
X,(Y, a;a)2 eq(Y,f(z;a)), hence f is in the chss Tdg C Td#.
13.7 Assume the class 7 is closed under primitive recursive operations, contains UD , and f is a functional in 3. It follows that a a partial characteristic functional t+b, of G, is given by: EXAMPLE
hence 3 E 3 p g
Tp#.
P. Assume P is closed under numerical substitution with primitive recursive functions, distribution, conjunction, disjunction, universal bounded quantification, and existential
Theorem 13.4 Let P be a class of predicates such that PRd
unbounded quantification. Then Pg is closed under p-recursive operations. PROOF. From the assumption PRd C P it follows that Pg contains the basic functionals and the function s (see Example 13.6) and, furthermore, the equality predicate = is in P . We must prove that Pg is closed under numerical substitution, distribution, definition by cases, primitive recursion, and unbounded minimalization. The general approach is to prove that whenever a functional h is obtained by one of the rules, then G h can be expressed using the closure properties of P. If h is obtained by numerical substitution in the form
L.E.Sanchis
198
where the prefix (3yl . . .y a ) stands for ( 3 ~ ~ , . () 3 . ~ ~ ) . Closure under distribution follows immediately from the closure of P under distribution. If h is obtained using definition by cases in the form h ( z ; a ) N [ s ( z ; a )----* we can express
Gh
f 1 ( z ; a )f,2 ( z ; a ) ]
as follows:
Now assume the functional h is given by primitive recursion in the form
h(0,z;a)
h(v
+ 1,z;a)
We express the predicate
Gh
N
g(2;a)
N
f ( h ( v ,2; a ) ,v , z ; a).
as follows:
( 3 ~ ) G g ( [ ~z;]a) i, A (Vz < v ) G f ( [ ~ ] z [+~~] ,z + l t2; z , a) A [w]u+1= Y .
% ( Y , v , z ; a)
Finally, we consider an application of unbounded minimalization where
We express the predicate
Gh
in the form
0
The crucial assumption in the proof of Theorem 13.4 is that the class P is closed under existential unbounded quantification. The importance of existential quantification can be seen from the treatment of primitive recursion and unbounded minimalization. In both cases the quantifiers asserts the existence of a computation where the output is the value of the functional h.
Corollary 13.4.1. If T is a class of functionals closed under primitive recursive operations, then F,j# is closed under p-recursive operations.
199
Chapter 19. Enumeration
F ,hence w e apply Theorem 13.4 with P = Tde, noting that PR PRd C ’d. C F d e . The closure properties follow from Theorem 3.1, Theorem 3.3, 0 Theorem 4.6, and Theorem 13.2. PROOF.
Corollary 13.4.2. Let 7 be a class of functionals closed under p-recursive oper-
ations. Then
(i)
T p e is closed under disjunction and existential bounded quantification..
(ii) T p # is closed under p-recursive operations. To prove (i), assume P = PI V Pz where P1,Pz are in T p e , so there are predicates Q1, Qz in Fp and PROOF.
We introduce a predicate Q such that
To prove that Q is in Tpwe show there is a partial characteristic functional of Q in F.We set
where $1 is a partial characteristic functional of Q1 in T , and $2 is a partial characteristic functional of Q2 in T . Since Q is in Tp, and we can express P in the form
P ( z ;a)
=
(~Y)Q(Y a; , 4,
and it follows that P is in 3 p e . Closure under existential bounded quantification is trivial. To prove (ii), we apply Theorem 13.4 with P = T p e . From Example 13.3 we have PRd E T p .The closure properties follow from Theorem 3.1, Theorem 3.3, and 0 Theorem 4.6, with Theorem 13.2.
Theorem 13.5 Let T be a class of functionals closed under primitive recursive operations, and P a class of predicates closed under substitution with primitive
200
L.E. Sanchis
recursive functions. Assume every predicate in P has a selector in the class 3. I f f is a (k, m ) - a r y P-enumerable functional, then there is a (k, m ) - a r y functional h in f, where the class 3, and a unary primitive recursive function g, such that f’
ss
f’(z;a)= g ( h( z ; a)). PROOF.
Since f is in
P# there is a predicate Q in P
such that
We set Q’(y,z; a)3 Q([y]l, [y]2,z; a),take h a selector functional for Q’ in 3, and 0 a)N g(h(z ; a)). set g(y) = [yI2. It follows that f’ ss f, where f’(z;
Corollary 13.5.1 Let T be a class of functionals closed under primitive recursive operations. If 3 has the p-selector property, then T CS 3 p # . From Example 13.7, noting that UD is the only selector of a false predicate, we know that 3 3 p # . Hence from Theorem 13.5 it follows that 3 ssF P # .
PROOF.
s
0
Corollary 13.5.2 If T is a class that contains only quasi-total functionals, then MR(3)
GS
PRd#(F).
PROOF. From Example 13.6 we know that 3
s PR(3) c PRd#(F).
Since PRd# ( T ) is closed under p-recursive operations we have M R ( F ) E PRd#(F). From Theorem 13.5, with the construction in Example 11.6, we get M R ( F ) PRd#(F)
cs
0
These results apply to numerical functions, where CS is equivalent to =. Note that if P is a k-ary numerical predicate, then P has exactly one dual characteristic function x p , which is a total function. Furthermore, if C is a class of numerical functions, then MR(C) = RC(C) by Theorem 12.6. The result of Corollary 13.5.2 is essentially a generalization of the well known Kleene normal form for recursive numerical functions. The original version (see Kleene [14]) reduces the recursive functions by the application of existential unbounded quantification to a primitive recursive predicate. The same reduction holds for functions that are recursive in a total recursive function. The extension
20 1
Chapter 19. Enumeration
we present here applies to functionals that are p-recursive in given quasi-total funcwhich is a kind of weak tionals. The relation is expressed by the inclusion equivalence. If we restrict the reduction to numerical functions we get the original Kleene normal form. An equivalent formulation is given in the next theorem. Note that these results apply only to p-recursive operations and not to recursive operations in general. A formulation for recursive operations is given in Chapter 17.
s8,
Theorem 13.6 Let 3 be a class that contains only quasi-total functionals, and f a k-ary function. The following conditions are equivalent:
(i) f is p-recursive in 3 (ii) There is a k + 1-ary numerical predicate Q primitive recursive in 3, and a unary primitive recursive function g such that
To prove that (i) implies (ii), note that from Corollary 13.5.2 it follows that f is in PRd#(F), and the construction of Theorem 13.5 shows that there is a k-ary function f’ such that f ’ f and f ‘ ( z )= g ( h ( z ) ) , where g is a primitive recursive function, and h is a selector function for a predicate Q in PRd(3). Now since f and f ’ are numerical functions, this means f = f ‘ . Furthermore, the selector where Q is primitive recursive in 3 function h is of the form h ( z ) N (~Y)XQ(Y,Z), (see Example 11.6). 0 The implication from (ii) to (i) is trivial. PROOF.
cs
EXERCISES
13.1 Assume 3 and 3’are classes of functionals such that 3 CS 3’.Prove that 3 d = FA and 3 p = 3;. 13.2 If 3 is a class of functionals, then Ft denotes the class of all quasi-total functionals in 3. Assume 3 is closed under p-recursive operations and prove:
L.E. Sanchis
202 Fts = F d # t
(b) F d = F d # d . 13.3 Let C and C’ be classes of numerical functions. Prove that C if C = C’.
csC’ if and only
13.4 Assume F is a class of functionals closed under primitive recursive operations, P is a class of predicates closed under substitution with primitive recursive numerical functions, and every predicate in P has a selector functional in F . Prove that every predicate in Pe has a selector functional in 3. 13.5 Assume the class F is closed under basic operations and F p is closed under existential unbounded quantification. Prove that F p is closed under disjunction. 13.6 Assume a class 7 of functionals is closed under basic operations and has the p-selector property. Prove that F p is closed under existential unbounded quantification. 13.7 Let 7 be an arbitrary class of functionals. Prove Fpgs = Fpg. 13.8 Let 3 be closed under p-recursive operations and assume F has the p-selector property. Prove:
13.9 Let P be a class of predicates closed under substitution with numerical primitive recursive functions, negation, conjunction, and universal bounded quantification. Assume PI and Pz are P-enumerable (k,rn)-ary predicates. Prove there are ?-enumerable (k,rn)-ary predicates Q1 and Qz such that the following conditions are satisfied: (a) PI is a boolean extension of
Q1 and P2 is a boolean extension of
(b) Q1 V Qz is a boolean extension of PI V P2. N
(c)
Qz is a boolean extension of Q1.
Q2.
Chapter 13. Enumeration
203
13.10 Assume F is a class of functionals closed under primitive recursive operations, and F has the p-selector property. Let PI and P2 be Fp-enumerable ( k , m)-ary predicates. Prove there are 3p-enumerable ( k , m)-predicates Q1 and Qz such that the following conditions are satisfied: (a) PI is a boolean extension of
(b)
Q1 V Q 2
Q1
and Pz is a boolean extension of
Q2.
is a boolean extension of PI V P2.
N
( c ) Q2 is a boolean extension of
Q1.
Notes Enumeration plays a relatively minor role in the theory of recursive functionals. Still, it is important in dealing with p-recursive operations. For example, Corollary 13.5.2 characterizes the functionals that are p-recursive in a class F that contains only quasi-total functionals in terms of the functionals that are primitive recursive in F. It is easy to see that this result is a generalization of the traditional Kleene normal form for numerical functions recursive in total functions. To get similar results for the class R C F we must require the class F to be normal, and we must use hyperenumeration rather than enumeration. These results are proved in Chapter 17. Note that the relation 3 5, F' between classes of functionals is intended as a form of weak equality or equivalence. For example, it implies 3 d = 3 : and 3 p = 3.;.
Enurneration In this chapter we are concerned primarily with classes of predicates, which can be generated in a number of ways. We assume such classes satisfy relevant closure properties and look at the functionals that are defined from them. If P is a class of predicates, then F g contains the functionals whose graph is in P. We shall see that Pg may satisfy important closure properties, depending, of course, on the closure properties of the class P. The relations between the classes 3 and 3 d and also between 3 and 3 p g , g’ were discussed in Sanchis [27] in terms of numerical functions. Here we consider functionals and predicates with non-total arguments, and this creates new problems. We propose new techniques which are useful in dealing with this situation. In dealing with classes of the form 3 d (or 3 ~ it )is important to take into account that the predicates are determined by their dual characteristic functionals (or by their partial characteristic functionals in the case of 3 p ) , and these functionals are not uniquely determined by the predicates. Still, the classes 3 d satisfy a number of closure properties, and these are sufficient for the applications in this chapter. Two new constructions which are used to define new predicates from given predicates are introduced in this chapter. Both are special forms of unbounded numerical quantification, existential or universal. Logically, they can be considered to be equivalent, since one can be defined from the other. In computability theory there is a natural hierarchy where existential quantification is more constructive than universal quantification. Here we are concerned primarily with the former. 189
L.E. Sanchis
190
The operation of unbounded quantification takes two forms, well known as existential and universal quantification. They apply to (k 1,m)-ary predicates and produce (k,m)-ary predicates, k, m 2 0. The formal definitions are as follows:
+
Universal U n b o u n d e d Quantification. Let Q be a (k We introduce a (k,m)-ary predicate P such that
P ( z ;a)
=
+ 1,m)-ary predicate.
(VY)Q(Y,2; Q) Q(y, z;a)holds for all y
We say that P is obtained from Q by universal unbounded quantification. Existential Unbounded Quantification. Let Q be a (k+1, m)-ary predicate. We introduce a ( k ,m)-ary predicate P such that
P ( z ;a)
3
( ~ Y ) Q ( Y2;, a) there is y such that Q(y,z; a)holds
We say that P is obtained from Q by existential unbounded quantification. Both quantifications, universal and existential, are non-finitary, for they determine a boolean value in terms of an infinite collection of values. So, they differ from the bounded quantifications of Chapter 4, which are finitary. For this reason we cannot expect that classes of the form 3 d or Fp are closed under unbounded quantification, even if 3 satisfies strong closure properties. Still, we shall see that this is possible in special cases. The determination of this type of closure is an important characterization of a given class 3 of functionals. In dealing with predicates in terms of functionals we must use characteristic functionals, and we know these are to some extent ambiguous, for in general a predicate has many partial characteristic functionals and many dual characteristic functionals. For this reason we must be very careful in translating definition by cases given by predicates into definition by cases given by functionals. On the other hand, this translation may be necessary if we want to insure monotonicity in the specification. For example, a specification of the form
h(z;a)
N
f i ( z ; a ) if
N
f2(z;a)
if
Q(z;a) -&(%;a),
is not necessarily monotonic and the mapping h is not necessarily a functional. A
191
Chapter 13. Enumeration monotonic approximation can be obtained by the specification h(z;a)
2:
f i ( z ; a ) if
N
fZ(z;a)
if
0 $ 0,
X Q ( Z ; ~ )N
XQ(z;a)
but this specification depends on the particular dual characteristic functional XQ. These problems do not appear in dealing with numerical functions, for a numerical predicate has a unique partial characteristic function and a unique dual characteristic function, and the translation above is faithful to the original specification. To handle this situation it is convenient to introduce the following notation. If f and g are (k,m)-ary functionals we say that f is a similar extension of g if f is an extension of g in the usual sense and f is also similar to g. The latter condition means that f(z;a ) N g(z; a ) whenever the a are total functions. We write g Es f to denote that f is a similar extension of g. This relation is clearly reflexive, transitive, and anti-symmetric.. When we say that a functional f is a similar extension of a functional g, this means that f is more defined than g, but only for non-total arguments. Since functionals are monotonic, and the values for non-total arguments are determined by the values for total arguments, the changes that f may introduce by being more defined than g are predictable from the common values of f and g for total arguments. So, we may say that f contains more explicit information, but such information was already implicit in g. Let F and F' be classes of functionals. We say that F' is a similar extension of F if 7 F',and whenever f is a functional in F' there is a functional g in F such that g Cs f. We write F c s 3' to denote that F' is a similar extension of F. This relation is clearly reflexive, transitive, and anti-symmetric. If F is a class of functionals, then 3; is the class that contains all functionals that are similar extensions of functionals in F. Clearly, F G s Fs. Furthermore, if F CS F',then F' CS Fs.
c
13.1 Let f be a (k,m)-ary functional and Gf the graph predicate o f f . I f f ' is a selector functional for G,, then f is a similar extension o f f ' . Hence, iff' is an element of the class F ,then f is an element of the class Fs.
EXAMPLE
13.2 If g is a k-ary numerical function, then the only similar extension of g is g itself. If 3 is a class of numerical functions, then F = 3;. EXAMPLE
L.E. Sanchis
192
c
The subscript “s” is clearly monotonic, so if F F’,then FS E F’s. Furthermore, it can be concatenated with other subscripts. For example, if F is a class of functionals, then Fsd and FSpare classes of predicates, FSdg and Fspg are classes of functionals. Let P be a predicate and F a class of functionals. If P is d-computable in the class PR(F) we say that P is primitive recursive in F. We denote by PRd(F) the class of all predicates that are primitive recursive in F. If 7 is empty we write simply PRd = PRd(0). w e say that the predicates in PRd are primitive recursive. If the predicate P is d-computable in the class MR(F) we say that P is precursive in 3. If P is p-computable in MR(F) we say that P is partially p-recursive in 3.We denote by MRd(F) the class of all predicates p-recursive in F ,and by MRp(7) the class of all predicates partially p-recursive in F. If F is empty we use the notations MRd and MRp. The predicates in MRd are p-recursive, and the predicates in M R p are partially p-recursive. If the predicate P is d-computable in the class RC(F) we say that P is recursive in 3. If P is p-computable in RC(F) we say that P is partially recursive in F.We denote by RCd(F) the class of all predicates recursive in F ,and by RCp(F) the class of all predicates partially recursive in F. If F is empty we use the notation RCd = the class of recursive predicates, and RCp = the class of partially recursive predicates. Note that in relation to the class PR(F) we introduce only the class PRd(F) of predicates that are primitive recursive in F. Of course, the class PRp(F) is also defined, but it is not used in our discussion. On the other hand, the classes MRp(F) and RCp(F) are important extensions of the classes MRd(F) and RCd, respectively. Finally, note that PR(F) C MR(3) C RC(F), hence PRd(F) MRd(F) & ( F ) and MRp(F) RCp(F).
c
c
Theorem 13.1 Let 3 be a class of functionals. Then,
(i) The class PRd(T) (MRd(F)) (RCd(F)) is closed under conjunction, disjunction, negation, universal bounded quantification, existential bounded quantification, distribution, and numerical substitution with quasi-total functionals primitive recursive (p-recursive) (recursive) in F.
(ii) The class MRp(7) (RCp(F)) is closed under conjunction, universal bounded quantification, distribution, and numerical substitution with functionals p recursive (recursive) in F.
Chapter 13. Enumeration
193
(iii) The class RCd(3) is closed under functional substitution with quasi-total functionals recursive in 3. Parts (i) and (ii) follow from Theorem 3.1, Theorem 3.3, and Theorem 4.6. Part (iii) is clear from the definitions, noting that the requirement that the functionals be quasi-total is necessary to make sure that conditions PCF 1 and DCF 1 are satisfied. 0 PROOF.
13.3 If 3 is a class of functionals closed under p-recursive operations, then the functional U D is 3-computable. If P is a predicate which is d-computable in F there is a dual characteristic functional x p which is 3-computable. From this we get a partial characteristic functional $ p , where EXAMPLE
This proves that 3 d C 3 p . It follows that MRd(3)
MRp(F) and RCd(3) C
RCP ( 3 ) . If P is a class of predicates and Q is a predicate obtained from a predicate in P by existential unbounded quantification we say that Q is P-enumerable. We denote by Pe the class of all predicates that are P-enumerable. If Q is a predicate obtained by universal unbounded quantification from a predicate in P we say that Q is Pcoenumerable. We denote by Pu the class of all predicates that are P-coenumerable. We denote by P c the class of all predicates that are obtained by the negation of a predicate in the class P. It follows that Pe = Pcuc, PU = Pcec, PCC = P, Pe, = PCU , and PUC = Pce. Theorem 13.2 Let P be a class of predicates closed under numerical substitution with primitive recursive numerical functions. Then,
(i) P e l P,, and P c are closed under numerical substitution with primitive recursive functions.
(ii) P
Pe and P
Pu.
(iii) Pe is closed under existential unbounded quantification, and PU i s closed under universal unbounded quantification.
(iv) If P is closed under distribution, then P, and PU are closed under distribution.
L.E. Sanchis
194
(v) If P is closed under conjunction, then Pe and
P, are closed under conjunction.
(vi) I f P is closed under disjunction, then Pe and PU are closed under disjunction. (vii) If P is closed under universal bounded quantification, then Pe and P, are closed under universal bounded quantafication. (viii) If P is closed under existential bounded quantification, then Pe and PU are closed under existential bounded quantification. The proof of part (i) is straightforward, noting that in the case of P, the conclusion follows because the primitive recursive functions are total. Part (ii) is also straightforward since by numerical substitution with projection functions we can add to a predicate in P a new dummy variable that can be eliminated by existential or universal quantification. To prove (iii) note that if Q is (k 2, m)-ary predicate in P , and P is a predicate obtained by two existential quantifications in the form: PROOF.
+
P ( z ;QL)
PY)PV)Q(Y, v , 2; a),
P ( z ;QL)
(3y)Q([yIi, [YIz,
then 2; a),
P is in Pe. A similar argument applies to P". Part (iv) is straightforward from the assumption. For parts (v) to (viii) we consider first the class Pe. If P = PI A Pz, where PI and Pz are in Pe, then there are predicates Q1 and Q z in P such that
so
P ( z ;a)
(3Y)Qi(y,z;
A ( ~ Y ) Q z ( Y2; , a) ( ~ Y ) ( Q ~ ( [ Y ] ~ , ~A; QQ Lz () [ Y ] z ; ~ ; Q L ) ) ,
so P is in Pe.If P = PIV Pz we use the same technique of exporting the existential
quantifiers, but here there is no need to contract them, for they can be identified. To prove part (vii), assume P is a predicate of the form
where Q is in the class P. It follows that
195
Chapter 13. Enumeration
so P is in the class Pe. Part (viii) for Pe is trivial, for it reduces to the interchange
of two existential quantifiers. To prove (v) to (viii) for P, we use duality. For example, assume P is closed under conjunction. Then Pc is closed under disjunction, hence Pce is also closed 0 under disjunction, and Pcec = P , is closed under conjunction.
Corollary 13.2.1. Assume
F is closed under primitive recursive operations. Then
(i) Fde is closed under existential unbounded quantification, distribution, conjunction, disjunction, universal bounded quantification, existential bounded quantification, and numerical substitution with quasi-total T-computable functionals.
(ii) Tpe is closed under existential unbounded quantification, distribution, conjunction, universal bounded quantification, and numerical substitution with F-computable functionals. (iii) If Pi is Fd-enumerable and P2 is Fp-enumerable, then P = PI v P2 is Fpenumerable. PROOF. Parts (i) and (ii) follow easily from Theorem 13.2 with Theorem 3.1, Theorem 3.3, and Theorem 4.6. To prove part (iii), assume Q1 is a predicate in Fd and Qz is a predicate in 3p such that Pi(2;Q) ( ~ Y ) Q ~ ( Y2;, a) Pz(2;Q) ( ~ Y ) Q z ( Y a; , a). It follows that
P ( z ;0) I (%)(Qi(y, a;Q) V Qz(Y, hence P is in
2; a)),
Fpe by Theorem 3.3 (iii).
0
Theorem 13.3 Let T be a class of functionals closed under p-recursive operations. Then 3 d = 3 d e fI Fdu.
The inclusiond' . E Tde n Tdu follows from Theorem 13.2 (ii). To prove the converse, assume P is a predicate such that PROOF.
P ( z ;a)
=
2; Q) (VY)QZ(Y, 2; a),
(3Y)Ql(Y,
L.E. Sanchis
196
where Q1 and Q2 are predicates d-computable in F. Assume x1 is a dual characteristic functional of Q1 which is F-computable, and xz is a dual characteristic functional of Q2 which is F-computable. We introduce a functional f such that
Clearly, f is 3-computable, so the functional
is also F-computable. We claim that x is a dual characteristic functional of P . To prove this we must check that conditions DCF 1 and DCF 2 are satisfied. To check DCF 1, assume the functions a are total. It follows that x l ( y l z ;a) and ~ 2 ( y , z ; a are ) defined for all y, hence f ( y , z ; a ) is defined for all y. We must show that there is some y such that f ( y , z ; a)N 0. If there is y such that u x l ( y , z ; a)N 0, then f ( y , z ; a)N 0. Otherwise, we have p(z;a), so there is y such that x2(y,z; a)N 0, hence f ( y , z ; a)N 0. This means x(z;a)is defined whenever the a are total functions. To prove DCF 2, note that if x(z;a)N v, then either v = 0 or v = 1. If v = 0 there is y such that x1(y,z; a)N 0, hence P ( z ;a)holds. If v = 1 there is y such 0 that x l ( y , z ; a)E 1, and f(y, z;a)N 0, so P ( z ;a)does not hold.
13.4 Let F be an arbitrary class of functionals. It follows from Theorem 13.3 that M R d ( F ) = M R d e ( F ) n M R d U ( F ) and R C d ( 7 ) = RCde(F) n RCdu(F).
EXAMPLE
Corollary 13.3.1 Let MRde
F be an arbitrary class of functionals. Then M R d ( F ) =
MRdec(F).
From Theorem 13.3we know that M R d ( F ) = M R d e ( F ) f l M R d U ( F ) . Since M R d ( F ) is closed under negation, we get
PROOF.
MRd(F)
*
= MRde(F) MRdcec(7) = MRde(F) n M R d e c ( F ) . 0
The result of Corollary 13.3.1can be expressed in the following form: a predicate P is p-recursive in the class F if and only if both P and the negation of P are M R d (F)-enumerable.
Chapter 13. Enumeration
197
Let P be a class of predicates. We recall that Pg is a class of functionals where f is in P g if and only if the graph predicate G, is in F. It is convenient to introduce another subscript "#," where P# = Peg. A functional in P# is said to be ?-enumerable.
13.5 Let P be a class of predicates closed under numerical substitution with primitive recursive numerical functions. It follows that P Pe, hence Pg C Peg = P#. If P is also closed under existential unbounded quantification we have Pe P , hence P# Pg,so Pg = P#. EXAMPLE
c
13.6 Assume the class T is closed under primitive recursive operations and f is a quasi-total functional in 3. It follows that a dual characteristic functional xf of G, is given by
EXAMPLE
X,(Y, a;a)2 eq(Y,f(z;a)), hence f is in the chss Tdg C Td#.
13.7 Assume the class 7 is closed under primitive recursive operations, contains UD , and f is a functional in 3. It follows that a a partial characteristic functional t+b, of G, is given by: EXAMPLE
hence 3 E 3 p g
Tp#.
P. Assume P is closed under numerical substitution with primitive recursive functions, distribution, conjunction, disjunction, universal bounded quantification, and existential
Theorem 13.4 Let P be a class of predicates such that PRd
unbounded quantification. Then Pg is closed under p-recursive operations. PROOF. From the assumption PRd C P it follows that Pg contains the basic functionals and the function s (see Example 13.6) and, furthermore, the equality predicate = is in P . We must prove that Pg is closed under numerical substitution, distribution, definition by cases, primitive recursion, and unbounded minimalization. The general approach is to prove that whenever a functional h is obtained by one of the rules, then G h can be expressed using the closure properties of P. If h is obtained by numerical substitution in the form
L.E.Sanchis
198
where the prefix (3yl . . .y a ) stands for ( 3 ~ ~ , . () 3 . ~ ~ ) . Closure under distribution follows immediately from the closure of P under distribution. If h is obtained using definition by cases in the form h ( z ; a ) N [ s ( z ; a )----* we can express
Gh
f 1 ( z ; a )f,2 ( z ; a ) ]
as follows:
Now assume the functional h is given by primitive recursion in the form
h(0,z;a)
h(v
+ 1,z;a)
We express the predicate
Gh
N
g(2;a)
N
f ( h ( v ,2; a ) ,v , z ; a).
as follows:
( 3 ~ ) G g ( [ ~z;]a) i, A (Vz < v ) G f ( [ ~ ] z [+~~] ,z + l t2; z , a) A [w]u+1= Y .
% ( Y , v , z ; a)
Finally, we consider an application of unbounded minimalization where
We express the predicate
Gh
in the form
0
The crucial assumption in the proof of Theorem 13.4 is that the class P is closed under existential unbounded quantification. The importance of existential quantification can be seen from the treatment of primitive recursion and unbounded minimalization. In both cases the quantifiers asserts the existence of a computation where the output is the value of the functional h.
Corollary 13.4.1. If T is a class of functionals closed under primitive recursive operations, then F,j# is closed under p-recursive operations.
199
Chapter 19. Enumeration
F ,hence w e apply Theorem 13.4 with P = Tde, noting that PR PRd C ’d. C F d e . The closure properties follow from Theorem 3.1, Theorem 3.3, 0 Theorem 4.6, and Theorem 13.2. PROOF.
Corollary 13.4.2. Let 7 be a class of functionals closed under p-recursive oper-
ations. Then
(i)
T p e is closed under disjunction and existential bounded quantification..
(ii) T p # is closed under p-recursive operations. To prove (i), assume P = PI V Pz where P1,Pz are in T p e , so there are predicates Q1, Qz in Fp and PROOF.
We introduce a predicate Q such that
To prove that Q is in Tpwe show there is a partial characteristic functional of Q in F.We set
where $1 is a partial characteristic functional of Q1 in T , and $2 is a partial characteristic functional of Q2 in T . Since Q is in Tp, and we can express P in the form
P ( z ;a)
=
(~Y)Q(Y a; , 4,
and it follows that P is in 3 p e . Closure under existential bounded quantification is trivial. To prove (ii), we apply Theorem 13.4 with P = T p e . From Example 13.3 we have PRd E T p .The closure properties follow from Theorem 3.1, Theorem 3.3, and 0 Theorem 4.6, with Theorem 13.2.
Theorem 13.5 Let T be a class of functionals closed under primitive recursive operations, and P a class of predicates closed under substitution with primitive
200
L.E. Sanchis
recursive functions. Assume every predicate in P has a selector in the class 3. I f f is a (k, m ) - a r y P-enumerable functional, then there is a (k, m ) - a r y functional h in f, where the class 3, and a unary primitive recursive function g, such that f’
ss
f’(z;a)= g ( h( z ; a)). PROOF.
Since f is in
P# there is a predicate Q in P
such that
We set Q’(y,z; a)3 Q([y]l, [y]2,z; a),take h a selector functional for Q’ in 3, and 0 a)N g(h(z ; a)). set g(y) = [yI2. It follows that f’ ss f, where f’(z;
Corollary 13.5.1 Let T be a class of functionals closed under primitive recursive operations. If 3 has the p-selector property, then T CS 3 p # . From Example 13.7, noting that UD is the only selector of a false predicate, we know that 3 3 p # . Hence from Theorem 13.5 it follows that 3 ssF P # .
PROOF.
s
0
Corollary 13.5.2 If T is a class that contains only quasi-total functionals, then MR(3)
GS
PRd#(F).
PROOF. From Example 13.6 we know that 3
s PR(3) c PRd#(F).
Since PRd# ( T ) is closed under p-recursive operations we have M R ( F ) E PRd#(F). From Theorem 13.5, with the construction in Example 11.6, we get M R ( F ) PRd#(F)
cs
0
These results apply to numerical functions, where CS is equivalent to =. Note that if P is a k-ary numerical predicate, then P has exactly one dual characteristic function x p , which is a total function. Furthermore, if C is a class of numerical functions, then MR(C) = RC(C) by Theorem 12.6. The result of Corollary 13.5.2 is essentially a generalization of the well known Kleene normal form for recursive numerical functions. The original version (see Kleene [14]) reduces the recursive functions by the application of existential unbounded quantification to a primitive recursive predicate. The same reduction holds for functions that are recursive in a total recursive function. The extension
20 1
Chapter 19. Enumeration
we present here applies to functionals that are p-recursive in given quasi-total funcwhich is a kind of weak tionals. The relation is expressed by the inclusion equivalence. If we restrict the reduction to numerical functions we get the original Kleene normal form. An equivalent formulation is given in the next theorem. Note that these results apply only to p-recursive operations and not to recursive operations in general. A formulation for recursive operations is given in Chapter 17.
s8,
Theorem 13.6 Let 3 be a class that contains only quasi-total functionals, and f a k-ary function. The following conditions are equivalent:
(i) f is p-recursive in 3 (ii) There is a k + 1-ary numerical predicate Q primitive recursive in 3, and a unary primitive recursive function g such that
To prove that (i) implies (ii), note that from Corollary 13.5.2 it follows that f is in PRd#(F), and the construction of Theorem 13.5 shows that there is a k-ary function f’ such that f ’ f and f ‘ ( z )= g ( h ( z ) ) , where g is a primitive recursive function, and h is a selector function for a predicate Q in PRd(3). Now since f and f ’ are numerical functions, this means f = f ‘ . Furthermore, the selector where Q is primitive recursive in 3 function h is of the form h ( z ) N (~Y)XQ(Y,Z), (see Example 11.6). 0 The implication from (ii) to (i) is trivial. PROOF.
cs
EXERCISES
13.1 Assume 3 and 3’are classes of functionals such that 3 CS 3’.Prove that 3 d = FA and 3 p = 3;. 13.2 If 3 is a class of functionals, then Ft denotes the class of all quasi-total functionals in 3. Assume 3 is closed under p-recursive operations and prove:
L.E. Sanchis
202 Fts = F d # t
(b) F d = F d # d . 13.3 Let C and C’ be classes of numerical functions. Prove that C if C = C’.
csC’ if and only
13.4 Assume F is a class of functionals closed under primitive recursive operations, P is a class of predicates closed under substitution with primitive recursive numerical functions, and every predicate in P has a selector functional in F . Prove that every predicate in Pe has a selector functional in 3. 13.5 Assume the class F is closed under basic operations and F p is closed under existential unbounded quantification. Prove that F p is closed under disjunction. 13.6 Assume a class 7 of functionals is closed under basic operations and has the p-selector property. Prove that F p is closed under existential unbounded quantification. 13.7 Let 7 be an arbitrary class of functionals. Prove Fpgs = Fpg. 13.8 Let 3 be closed under p-recursive operations and assume F has the p-selector property. Prove:
13.9 Let P be a class of predicates closed under substitution with numerical primitive recursive functions, negation, conjunction, and universal bounded quantification. Assume PI and Pz are P-enumerable (k,rn)-ary predicates. Prove there are ?-enumerable (k,rn)-ary predicates Q1 and Qz such that the following conditions are satisfied: (a) PI is a boolean extension of
Q1 and P2 is a boolean extension of
(b) Q1 V Qz is a boolean extension of PI V P2. N
(c)
Qz is a boolean extension of Q1.
Q2.
Chapter 13. Enumeration
203
13.10 Assume F is a class of functionals closed under primitive recursive operations, and F has the p-selector property. Let PI and P2 be Fp-enumerable ( k , m)-ary predicates. Prove there are 3p-enumerable ( k , m)-predicates Q1 and Qz such that the following conditions are satisfied: (a) PI is a boolean extension of
(b)
Q1 V Q 2
Q1
and Pz is a boolean extension of
Q2.
is a boolean extension of PI V P2.
N
( c ) Q2 is a boolean extension of
Q1.
Notes Enumeration plays a relatively minor role in the theory of recursive functionals. Still, it is important in dealing with p-recursive operations. For example, Corollary 13.5.2 characterizes the functionals that are p-recursive in a class F that contains only quasi-total functionals in terms of the functionals that are primitive recursive in F. It is easy to see that this result is a generalization of the traditional Kleene normal form for numerical functions recursive in total functions. To get similar results for the class R C F we must require the class F to be normal, and we must use hyperenumeration rather than enumeration. These results are proved in Chapter 17. Note that the relation 3 5, F' between classes of functionals is intended as a form of weak equality or equivalence. For example, it implies 3 d = 3 : and 3 p = 3.;.