CS 5000: Lecture 38
Vladimir Kulyukin
Department of Computer Science
Utah State University
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Normal Form Theorem for Partially Computable Functions - Prof. Vladimir A. Kulyukin, Study notes of Computer Science
The normal form theorem for partially computable functions, which states that every partially computable function can be expressed in normal form using a finite number of applications of composition, recursion, and minimalization. The document also covers the concept of proper minimalization and its relationship to computable functions.
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CS 5000: Lecture 38
Vladimir Kulyukin
Department of Computer Science
Utah State University
Outline
- Normal Form for Partially Computable
Functions
- Normal Form for Computable Functions
Review: Step Counter Predicate
(^
)^ (
(^
)^ (
(^
y
t
y
x
ty x
x
x
n
n n
n
SNAP
TERM
STP
The R Predicate
(^
)
(^
)
(^ )
( )
(^
)
( )
(^ )
( )
(^
)
(^
)
(^
)^.
, ,
,...,
SNAP
&
, ,
,...,
STP
, ,
,...,
:
follows as
defined
predicate the
Consider.
,...,
function p.c.a
computes
that
program a of
number the be
Let
1
0
0 1
1
0
1
1
0
z r y x x r z l
z r y x
x
z y x
x R
R
x
x f
y
n
n n
n
n
n
=
⇔
Proof Sketch
(^
)^
(^
)
(^ )^ (
)
( )
( )
( )
(^ )^
( )
(^
)
( )
(^
)
(^ )^
( )
(^
)
(^
)
(^
)^
( )
SNAP
have we,
,...,
Since
STP,
Since
and
Then'.
, ,
Let
STP
that
such'
that know we p.c., is
Since.
Then.
some
for
Suppose.
computes that
programa
of
number the be let p.c., is
Since
1
0
1
1
0
1
0
1
1
1
1
0
zl k
zr y x x
r
x x f k zl
zr y x x
z m zr
m zr
k zl
z m m k z
z y x x
z
f
N
k
x x f x x f x x
f
y
f
n
n
n
n
n
n
n
n
n
n
Theorem 3.3 (Ch. 4):Normal Form Theorem (^
)
(^
)
(^
)^
(^
)
(^
)^
(^
). ,...,
computes that
programa of
number theis
where, , , ,...,
min
,...,
such thatLet
, , ,...,
predicate
recursive
primitivea is
Then there
function.
computable
partiallya be
,...,
Let
1
0
0
1
1
0
1
1
n
n
z
n
n
n
x xf
y z y x x R l x
xf
z y x x R x
xf
=
Proof 3.
p.c. is
tion,
minimaliza and
recursion, n,
compositio of
ns
applicatio of
number
finitea by
functions
initial the from
obtained is if that
Show
tion.
minimaliza and
recursion, n,
compositio of
ns
applicatio of
number
finitea by
functions
initial
the from
obtained is it then p.c. is if that
Show) 1^ f
f f
Proof 3.4: Part 1
(^
)^
(^
)
(^
)
tion.
minimaliza and
recursion,
n,
compositio of ns
applicatio of
number finitea by
functions
initial the from
obtained is
Thus,
then with and min with
composed is
recursion. andn
compositio of ns
applicatio
of
number finitea by
functions
initial the
from
obtained is
therefore and p.r. is
min
Ch. 3.3,
(Theorem
Theorem Form
Normal
By the
function. p.c.a be Let
0
1
1
f
l
R R
z y x x R l x x f
f
n
z
n^
Proper Minimalization
(^
)
(^
)
(^
)
tion.
minimaliza
proper
is
min
case,
the
is
this
If
such that
one
least
at
is
there
each
for
when
happens
This
happen?
this
does
When
function.
totala
be
can
min
0
1
0
(^11)
0
1
z
y
x
x
R
z y x x R z
x x z y x x R
n
z
n n
n
z
Proper Minimalization
(^
(^
(^
tion.
minimaliza
proper is , ,...,
min case, the is this If
. 1
,
,...,
such that
one least at is
there
,...,
each for
when
happens This
happen? this does
When.
program
some
for
function totala be can ,
,...,
min
1
(^11)
1
z x x R
z x x R
z
x x
z y x x R
n
z
n n
n
z
=
Proof 3.
-^
The statement follows directly from thedefinition of proper minimalization.
-^
If the minimalization is proper, thefunction is total.
-^