2
DEFINITION 5.1.2. TM(0,1,+,-,•,
↑
,log) is the set of all
terms built up from 0,1,+,-,•,
↑
,log, and variables v
1
,v
2
,...
.
DEFINITION 5.1.3. Each t
∈
TM(0,1,+,-,•,
↑
,log) gives rise
to infinitely many functions, one of each arity that is at
least as large as all subscripts of variables appearing in
t, as follows. Let the variables of t be among v
1
,...,v
k
, k
≥
1. Then we associate the function f:N
k
→
N given by
f(v
1
,...,v
k
) = t(v
1
,...,v
k
)
where t is interpreted according to Definition 5.1.1.
DEFINITION 5.1.4. BAF (basic functions) is the set of all
functions given by terms in 0,1,+,-,•,
↑
,log, according to
Definition 5.1.3.
It is very convenient to extend TM(0,1,+,-,•,
↑
,log) with
definition by cases, to get an alternative description of
BAF.
DEFINITION 5.1.5. ETM(0,1,+,-,•,
↑
,log) is the set of
“extended terms” of the following form:
t
1
if
ϕ
1
;
t
2
if
ϕ
2
∧
¬
ϕ
1
;
...
t
n
if
ϕ
n
∧
¬
ϕ
1
∧
...
∧
¬
ϕ
n-1
;
t
n+1
if
¬
ϕ
1
∧
...
∧
¬
ϕ
n
.
where n