theory of computation (fall 2014): primitive recursively closed classes & definition by cases;...
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Theory of Computation
Primitive Recursively Closed Classes & Definition by Cases, Summations & Products, Bounded
Quantification, Bounded Quantification & Primitive Recursive Predicates
Vladimir Kulyukin
www.vkedco.blogspot.com
Outline
● Primitive Recursively Closed Classes & Definition by Cases● Summations & Products● Bounded Quantification● Bounded Quantification & Primitive Recursive Predicates
Primitive Recursively Closed Classes & Definition by Cases
Theorem 5.4 (Ch. 3): Definition by Cases
.,...,Then
otherwise. ,...,
,..., if ,...,,...,
Let . tobelong predicate theand , ,
functions Let the class. PRC a be Let
1
1
111
Cxxf
xxh
xxPxxgxxf
CPhg
C
n
n
nnn
Proof 5.4 (Ch. 3)
nn
nnn
xxPxxh
xxPxxgxxf
,...,,...,
,...,,...,,...,
11
111
Recall that * and + have been shown to be primitive recursive and a primitive recursive function belongs to every PRC class. This is Theorem 3.3 (Ch. 3).
Interpretation of Theorem 5.4 (Ch. 3)
Theorem 5.4 (Ch. 3) shows that it is possible to write if-then-else statements from the functions that we have defined previously and we know to be in the same PRC class.
if ( P(x1, …, xn) ) { return g(x1, …, xn);
}else {
return h(x1, …, xn);}
Corollary 5.5 (Ch. 3)
.Then
otherwise. ,...,
,..., if ,...,
...
,..., if ,...,
,...,
Let .,..., all and 1 allfor
,0,...,,...,let and , to
belong ,..., predicates and ,,...,
functionsary -let class, PRC a be Let
1
11
1111
1
1
11
11
Cf
xxh
xxPxxg
xxPxxg
xxf
xxmji
xxPxxPC
PPhgg
nC
n
nmnm
nn
n
n
njni
mm
Proof 5.5 (Ch. 3)
otherwise ,...,
,..., if ,...,
...
,..., if ,...,
,...,
.1Consider 3). (Ch. 5.4 Theoremby trueisstatement
the,1 If .on induction by statement thisprove We
1
1111
1111
1
n
nmnm
nn
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xxh
xxPxxg
xxPxxg
xxf
m
mm
Proof 5.5 (Ch. 3)
induction.by ,,...,Then
otherwise. ,...,
,..., if ,...,
...
,..., if ,...,
,...,
And 3). (Ch. 5.4 Theoremby ,,...,Then
otherwise. ,...,
,..., if ,...,,...,Let
:one as functions last two therewrite We
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1111
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1''
1
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''
Cxxf
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xxPxxg
xxPxxg
xxf
Cxxh
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xxPxxgxxh
n
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nmnm
nn
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nmnmn
Corollary 5.5 (Ch. 3): Practical Interpretation We can write if-then-else-if statements from previously defined functions in the same PRC
class:
if ( P1(x1, …, xn) ) {return g1(x1, …, xn);
}else if ( P2(x1, …, xn) ) {
return g2(x1, …, xn);}…else {
return h(x1, …, xn);}
Summations & Products
Theorem 6.1 (Ch. 3)
.),...,,(),...,,(
and
),...,,(),...,,(
functions thedo sothen
,),...,,( If class. PRC a be Let
011
10
1
1
y
tnn
n
y
tn
n
xxtfxxyh
xxtfxxyg
CxxtfC
Proof 6.1
We can use the definition of the PRC class. We know that C is a PRC class. We know that f is in C. If we can derive g and h from f using composition and recursion, g and h will, by definition, be in C.
Proof 6.1 (Ch. 3)
).,...,,1(),...,,(),...,,1(
);,...,,0(),...,,0(
:,...,,,...,,for srecurrence the writeusLet
111
11
011
nnn
nn
y
tnn
xxtfxxtgxxtg
xxfxxg
xxtfxxyg
Proof 6.1 (Ch. 3)
),...,,1(),...,,(),...,,1(
),...,,0(),...,,0(
:),...,,(),...,,(for srecurrence the writenow usLet
111
11
011
nnn
nn
y
tnn
xxtfxxthxxth
xxfxxh
xxtfxxyh
Starting Summation at 1
),...,,1(),...,,(),...,,1(
0),...,,0(
:srecurrence adjust thecan We
.),...,,(),...,,(
:1at summingstart want to weSuppose
111
1
111
nnn
n
y
tnn
xxtfxxtgxxtg
xxg
xxtfxxyg
Starting Product at 1
),...,,1(),...,,(),...,,1(
1),...,,0(
:follows as srecurrence adjust thecan We
.),...,,(),...,,(
:1at productsstart want to that weSuppose
111
1
111
nnn
n
y
tnn
xxtfxxthxxth
xxh
xxtfxxyh
Corollary 6.2 (Ch. 3)
.),...,,(),...,,(
);,...,,(),...,,(
:functions thesedo sothen
PRC, is and ),...,,( If
111
11
1
1
y
tnn
n
y
tn
n
xxtfxxyh
xxtfxxyg
CCxxtf
Bounded Quantification
Bounded Universal Quantifier: Definition
.0for ,1,...,,
ifonly and if TRUE 1 is ,...,,
1
1
yixxiP
xxtPt
n
ny
Bounded Existential Quantifier: Definition
.,0 oneleast at for ,1,...,,
ifonly and if (TRUE) 1 is ,...,,
1
1
yixxiP
xxtPt
n
ny
Theorem 6.3 (Ch. 3)
).,...,,()( and ),...,,()( predicates thedo so
thenC, class PRC some tobelongs),...,,( predicate some If
11
1
nyny
n
xxtPtxxtPt
xxtP
Proof 6.3 (Ch. 3)
1,...,,),...,,()(
:follows asproduct theof terms
intion quantifica universal bounded definecan We
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y
tnny xxtPxxtPt
Proof 6.3 (Ch. 3)
0,...,, ),...,,()(
:follows assummation theof terms
intion quantifica lexistentia bounded definecan We
011
y
tnny xxtPxxtPt
Corollary of Theorem 6.3 (Ch. 3)
.),...,,(&)()(),...,,(
;),...,,()(),...,,()(
:follows as
tionquantifica boundedstrict -non of in terms expressed becan it
because tion,quantifica boundedstrict for validis 6.3 Theorem
11
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nyny
nyny
xxtPyttxxtPt
xxtPyttxxtPt
Bounded Quantification &
Primitive Recursiveness
Bounded Quantification & Primitive Recursiveness
● We can now use the results on bounded quantification to show even more functions to be primitive recursive
● Bounded quantification furnishes us iterative tools that we can use to check if a predicate is true for every number in a range or for some number in a range
● We can also use the negation of a bounded quantified statement to show that there is no number for which some predicate is true
Y | X is Primitive Recursive
recursive primitive
is ) ofdivisor a is or divides ( | that Show xyxyxy
Y | X is Primitive Recursive
xtytxy x |
prime(x) is Primitive Recursive
.|1&1
)prime(
as expressed becan thisFormally, itself.
and 1 other than divisors no hasit and
1an greater th isit if prime isnumber A
xtxtttx
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x
Longer List of Primitive Recursive Functions
.
8.
,lcm 15. 7.
,gcd 14. 6.
prime 13. 5.
| 12. 4.
11. ! 3.
10. 2.
9. .1
x
yxyx
yxyx
xxp
yxx
yxx
yxyx
yxyx
y
Reading Suggestions
● Ch. 3, Computability, Complexity, and Languages, 2nd Edition, by Davis, Weyuker, Sigal
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