Theory of Computation (Fall 2014): Minimalization
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Transcript of Theory of Computation (Fall 2014): Minimalization
Theory of Computation
Minimalization
Vladimir Kulyukin
www.vkedco.blogspot.comwww.vkedco.blogspot.com
Outline
● Minimalization● Bounded Minimalization● Minimalization & PRC Classes● Unbounded Minimalization● Minimalization in Proofs of Primitive Recursiveness
Minimalization
Minimalization: Definition
• Suppose P is a predicate, P(t, x1, … , xn) for n ≥ 0• Minimalization is a technique for finding the minimal
value of t for which P(t, x1, … , xn) = 1• If there is such a t, then minimalization returns it• If there is no such t, then minimalization is undefined
Example
P(1) = 0 P(2) = 0 P(3) = 1 P(4) = ?
1 2 3 4
P(0) = 0
0
3,0...0
1111210
11110
10
nnPP
PPP
PP
P
Bounded Minimalization
Bounded Minimalization
0
0
01
1010
101
if 0
if 1,...,,
Then
true.is ),...,,(for which , of aluesmallest v theis ;1),...,,( .2
false; is ),...,,(, valuesallfor ;0),...,,( 1.
:Suppose
0
tu
tuxxtP
xxtPttxxtP
xxtPttxxtPt
u
tn
nn
nnt
Bounded Minimalization
y
u
u
tnn
n
n
xxtPxxyg
xxtPt
CxxtP
0 011
10
1
)).,...,,((),...,,(
:function following theDefine
.1),...,,(for which aluesmallest v thebe Let
. class PRC somein predicate a be ),...,,(Let
Example
.300111)),...,,((
)),...,,(()),...,,((
)),...,,(()),...,,((
)),...,,((),...,,4(
.4,3 Suppose
4
01
2
0
3
011
1
01
0
01
4
0 011
0
tn
t tnn
tn
tn
u
u
tnn
xxtP
xxtPxxtP
xxtPxxtP
xxtPxxyg
yt
Example
.211
)),...,,(()),...,,((
)),...,,((),...,,1(
.1,3 Suppose
1
01
0
01
1
0 011
0
tn
tn
u
u
tnn
xxtPxxtP
xxtPxxg
yt
Lemma
.1,...,, 10 yxxygy nt
Proof
y
u
u
tnn
u
tn
yxxtPxxyg
yy
xxtPy
uty
0 011
01
0
.1)),...,,((,...,,
elements, 1 has ,0 range
theSince .1)),...,,(( ,,0
range in the every for Then .Let
Corollary
.1),...,,(which
for of eleast valu theis ),...,,(
.1),...,,( then , If
1
1
010
0
n
n
tun
xxtP
txxyg
txxygyt
Example
.2011)),...,,((
)),...,,(()),...,,((
)),...,,((),...,,2(
.2,2 Suppose
2
01
1
01
0
01
2
0 011
0
tn
tn
tn
u
u
tnn
xxtP
xxtPxxtP
xxtPxxyg
yt
Bounded Minimalization: Definition
otherwise. 0 and ,0 if 1,...,,for which
of eleast valu theis ,...,,mins,other wordIn
otherwise 0
,...,, if ,...,,,...,,min
1
1
11
1
ytxxtP
t xxtP
xxtPtxxtgxxtP
n
nyt
nyn
nyt
Minimalization & PRC Classes
Theorem 7.1 (Ch. 03)
. tobelongs also ),...,,(
then ),,...,,(min),...,,( and
class PRC some tobelongs ),...,,( If
1
11
1
Cxxyf
xxtPxxyf
CxxtP
n
nyt
n
n
In other words, if some predicate belongs to a PRC class, its bounded minimalization will stay in that class.
Proof 7.1 (Ch. 03)
.in is ),...,,(min cases),by n (definitio 5.4 TheoremBy
.in also is ),...,,( 6.3, TheoremBy
n.compositioby themfrom obtained is ),...,,( and
C,in are and ,, because ,in is tion,minimaliza bounded
of definition in the usedfunction the),,...,,(
1
1
1
1
CxxtP
CxxtPt
xxyg
C
xxyg
nyt
ny
n
n
Unbounded Minimalization
Theorem 7.2 (Ch. 3)
computablepartially is ,...,,min,...,
thenpredicate, computable a is ,...,, If
11
1
ny
n
n
xxyPxxg
xxtP
Proof 7.2 (Ch. 03)
[A1] IF P(Y, X1, …, Xn) GOTO E Y ← Y + 1 GOTO A1
Here is a program that computes g(x1 , …, x
n) is partially computable.
Why is g not computable?
Minimalization & Proofs of Primitive Recursiveness
Floor Function
recursive. primitive
is )quotient theofpart (integer that Show x/yy
x
Floor Function
xyty
xxt
)1(min
Keep incrementing t until (t+1)y > x. Then return t.
Example
.78213 because
,372)1(min2
77
t
t
More Examples
Why?00
05
4
42
9
42
8
x
Remainder Function
recursive. primitive is ,by ofdivision
theofremainder thei.e. , that Show
yx
R(x,y)
Remainder Examples
4
3
4
7
4
31
4
7
3
1
3
7
3
12
3
7
3
1
3
4
3
11
3
4
Remainder Function
.0)0,( that Note
),( .4
),( .3
),( .2
),( .1
xR
yy
xxyxR
yxRyy
xx
y
yxR
y
x
y
x
y
yxR
y
x
y
x
Reading Suggestions
● Ch. 3, Computability, Complexity, and Languages, 2nd Edition, by Davis, Weyuker, Sigal