Computability and Logic

7.1. RECURSIVE RELATIONS 75
Given a relation R(y 1 , ..., y m ) and total functions f 1 (x 1 , ..., x n ), ..., f m (x 1 , ...,
x n ), the relation defined by substitution of the f i in R is the relation R ∗ (x 1 , ..., x n )
that holds of x 1 , ..., x n if and only if R holds of f 1 (x 1 , ..., x n ), ..., f m (x 1 , ..., x n ),
or in symbols,
R ∗ (x 1 ,...,x n ) ↔ R( f 1 (x 1 ,...,x n ),..., f m (x 1 ,...,x n )).
If the relation R ∗ is thus obtained by substituting functions f i in the relation R, then
the characteristic function c ∗ of R ∗ is obtainable by composition from the f i and the
characteristic function c of R:
c ∗ (x 1 ,...,x n ) = c( f 1 (x 1 ,...,x n ),..., f n (x 1 ,...,x n )).
Therefore, the result of substituting recursive total functions in a recursive relation is
itself a recursive relation. (Note that it is important here that the functions be total.)
An illustration may make this important notion of substitution clearer. For a given
function f , the graph relation of f is the relation defined by
G(x 1 ,...,x n , y) ↔ f (x 1 ,...,x n ) = y.
Let f ∗ (x 1 ,...,x n , y) = f (x 1 ,...,x n ). Then f ∗ is recursive if f is, since
Now f (x 1 , ..., x n ) = y if and only if
f ∗ = Cn [ f, id n+1
1
,...,id n+1 ]
n .
f ∗ (x 1 ,...,x n , y) = id n+1
n+1 (x 1,...,x n , y).
Indeed, the latter condition is essentially just a long-winded way of writing the former
condition. But this shows that if f is a recursive total function, then the graph relation
f (x 1 , ..., x n ) = y is obtainable from the identity relation u = v by substituting the
recursive total functions f ∗ and id n+1
n+1
. Thus the graph relation of a recursive total
function is a recursive relation. More compactly, if less strictly accurately, we can
summarize by saying that the graph relation f (x) = y is obtained by substituting the
recursive total function f in the identity relation. (This compact, slightly inaccurate
manner of speaking, which will be used in future, suppresses mention of the role of
the identity functions in the foregoing argument.)
Besides substitution, there are several logical operations for defining new relations
from old. To begin with the most basic of these, given a relation R, its negation or
denial is the relation S that holds if and only if R does not:
S(x 1 ,...,x n ) ↔∼R(x 1 ,...,x n ).
Given two relations R 1 and R 2 , their conjunction is the relation S that holds if and
only if R 1 holds and R 2 holds:
S(x 1 ,...,x n ) ↔ R 1 (x 1 ,...,x n )&R 2 (x 1 ,...,x n )