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CDT314
FABER
Formal Languages, Automata and Models of Computation
Lecture 14
School of Innovation, Design and Engineering Mälardalen University
2012
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Content
Recursive Functions
Primitive Recursion
Ackermann function & -recursive functions
Relations Among Function Classes
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Recursion
In computer programming, recursion is related to performing computations in a loop.
Visualisation of a recursive structure
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A Sierpinksi carpet is a famous fractal objects in mathematics. Creating one is an iterative procedure. Start with a square, divide it into nine equal squares and remove the central one. That leaves eight squares around a central square hole. In the next iteration, repeat this process with each of the eight remaining squares and so on (see above).
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Recursion used in Problem Modelling
Reducing the complexity by
Breaking up computational sequence into its simplest forms.
Synthesizing components into more complex objects by replicating simple component sequences over and over again.
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"A reduction is a way of converting one problem into another problem in such a way that a solution to the second problem can be used to solve the first problem."
Michael Sipser, Introduction to the Theory of Computation
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Self-referential definitions can be dangerous if we're not careful to avoid circularity.
This is why our definition of recursion includes the word well-defined.
(Recursion can be seen as a concept of well-defined self-reference.)
*Gertrude Stein: ''A rose is a rose is a rose''
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We can write a pseudocode to determine whether somebody is an immigrant:
FUNCTION isAnImmigrant(person): IF person immigrated herself, THEN:
return true ELSE:
return isAnImmigrant(person's parent) END IF
This is a recursive function, since it uses itself to compute its own value.
Recursive definition: an immigrant…
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Functions
From math classes, we have seen many ways of defining and combining numerical functions.
Inverse
Composition
Derivatives
Iteration
(iterated function is composed with itself)
…. etc.
1f
gf
),(),(),( xfxfxf
),(),(),( 321 xfxfxf
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Functions
Look at what happens when we use only some of these. - How can we define standard interesting functions? - How do these relate to e.g. TM computations? We have seen TMs as functions. They are awkward!
As alternative, look at a more intuitive definition of functions.
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Notation
For brevity, we will limit ourselves to functions on natural numbers
Notation will also use n-tuples of numbers
},2,1,0{ N
),,( 1 nmm
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Natural Numbers
Start with standard recursive definition of natural numbers (Peano axioms)
A natural number is either
0 or
successor(n), where n is a natural number.
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What is a recurrence?
A recurrence is a well-defined mathematical function written in terms of itself.
It is a mathematical function defined recursively.
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Fibonacci sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...
The first two numbers of the sequence are both 1, while each following number is the sum of the two numbers before it.
(We arrived at 55 as the tenth number, since it is the sum of 21 and 34, the eighth and ninth numbers.)
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F is called a recurrence, since it is defined in terms of itself evaluated at other values.
)2()1()(
)(1)1(1)0(
nFnFnF
casesbaseFF
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A recursive process is one in which objects are defined in terms of other objects of the same type.
Using some sort of recurrence relation*, the entire class of objects can then be built up from a few initial values and a small number of rules.
Recursion & Recurrence
(*Recurrence is a mathematical function defined recursively.)
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Computable Functions
Any computable function can be programmed using while-loops (i.e., "while something is true, do something else").
For-loops (which have a fixed iteration limit) are a special case of while-loops.
(Computable functions could of course also be coded using a combination of for- and while-loops. )
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Total Function
A function defined for all possible input values.
Primitive Recursive Function
A function which can be implemented using only for-loops.
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103
An example function
1)( 2 nnfDomain Range
10)3( f
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Building functions from basic ones
Based on C Busch, RPI, Models of Computation
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Zero function: 0)( xzero
Successor function: 1)( xxsucc
Projection functions: 1211 ),( xxxp
2212 ),( xxxp
Basic Primitive Recursive Functions
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Building functions
Composition
)),(),,((),( 21 yxgyxghyxf
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Composition, Generally
Given
NNh
NNf
NNg
NNg
k
m
km
k
:
:
:
...
:1
),,( 1 mggfh
)),,(,),,,((),,( 1111 kmkk nngnngfnnh
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Primitive Recursion “Template”
)),(),,(()1,( 2 yxfyxghyxf
)()0,( 1 xgxf
For primitive recursive functions recursion is in only one argument.
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Any function built from
the basic primitive recursive functions
is called Primitive Recursive Function.
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0)( xzero
)())(( xzeroxsucczero
Basic Primitive Zero function
(a constant function)
0)0()1()2()3( zerozerozerozero
Example
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Basic Primitive Identity function
...
xxidentity
xx
210)(
210
))(())((
0)0(
xidentsuccxsuccidentity
identity
Recursive definition
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Basic Primitive Successor function
...
1321)(
210
xxsucc
xx
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))(()( xzerosuccxone
Using Basic Primitive Zero function
and a Successor function we can construct Constant functions
etc..
))(()( xonesuccxtwo
))(()( xtwosuccxthree
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)2(
))1((
)))0(((
))))((((
)))(((
))(()(
succ
succsucc
succsuccsucc
xzerosuccsuccsucc
xonesuccsucc
xtwosuccxthree
Example
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A Primitive Recursive Function ),( yxadd
xxadd )0,( (projection)
)),(()1,( yxaddsuccyxadd (successor function)
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5
)4(
))3((
)))0,3(((
))1,3(()2,3(
succ
succsucc
addsuccsucc
addsuccadd
Example
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5
14
1)13(
1)1)0,3((
1))1,3(()2,3(
add
addadd
Example
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Basic Primitive Predecessor function
...
1100)(
210
xxpred
xx
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Predecessor
xxsuccpred
pred
))((
0)0(
1)( xxpred
)())((
0)0(
xGxsuccpred
pred
Predecessor is a primitive recursive function with no direct self-reference.
x) identity(G(x) templaterecursive primitive
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Subtraction
)),(())(,(
)0,(
xysubpredxsuccysub
yysub
xyxysub ),(
)1)()1(( xyxy
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1
)2(
))3((
)))0,3(((
))1,3(()2,3(
pred
predpred
subpredpred
subpredsub
Example
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0)0,( xmult
)),(,()1,( yxmultxaddyxmult
),( yxmult
Multiplication
))()1(( xxyyx
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x
xxadd
xxaddxadd
xxaddxaddxadd
xaddxaddxaddxadd
xmultxaddxaddxaddxadd
xmultxaddxaddxadd
xmultxaddxadd
xmultxaddxmult
4
)3,(
))2,(,(
))),(,(,(
))))0,(,(,(,(
)))))0,(,(,(,(,(
))))1,(,(,(,(
)))2,(,(,(
))3,(,()4,(
Example
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1),0( xexp
)),,((),1( yyxexpmultyxexp
),( yxexpA Primitive Recursive Function,exponentiation
)( 1 yyy xx
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Example
4
)),(
)),,((
)),),,(((
)),),),,1((((
)),),),),,0(((((
)),),),,1((((
)),),,2(((
)),,3((),4(
yyyyy
yyyymult
yyyymultmult
yyyymultmultmult
yyyymultmultmultmult
yyyyyexpmultmultmultmult
yyyyexpmultmultmult
yyyexpmultmult
yyexpmultyexp
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Primitive Recursion in Logic
A predicate (Boolean function) with output in the set {0,1} which is interpreted as {yes, no}, can be used to define standard functions.– Logical connectives , ,, , …– Numeric comparisons =, < ,, …– Bounded existential quantification in, f(i)– Bounded universal quantification in, f(i)– Bounded minimization min i in, f(i)
where result = 0 if f(i) never true within bounds.
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Recursive Predicates
and?zero ?_ zeronon
1110))?(?(?_
0001)),(()?(
3210
xzerozerozeronon
xxonesubxzero
x
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),( yxand ),( yxor ),( yxless )(xnon returns
1
0
00 yx 00 yx
00 yx00 yx
yx
yx
0x
0x
More Recursive Predicates
))),(?((),(
))),(?((),(
))),(?((),(
yxsubzerononyxless
yxaddzerononyxor
yxmultzerononyxand
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)),((),(_ yxequalnonyxequalnon
))),(()),,(((),( xylessnonyxlessnonandyxequal
More Recursive Predicates...
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Example
Recursive predicates can combine into powerful functions.
What does this compute?
Tests primality!
))1()1((,,)(? njinijnjinjnin
A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.
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Function
0
0),,(
xify
xifzzyxif
if
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yzyxsuccif
zzyif
),),((
),,0(
)(),),((
)(),,0(
yGzyxsuccif
zBzyif
identityG Bwith
our construction
primitive recursive template
)),(),,(()1,( 2 yxfyxghyxf
)()0,( 1 xgxf
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Division example: x/4
rdqx quotient remainderx
0
1
2
3
4
5
6
7
8
0400
1401
2402
3403
0414
1415
2416
3417
0428
0
0
0
0
1
1
1
1
2
0
1
2
3
0
1
2
3
0
quotientq remainderr 4d
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Division as Primitive Recursion
))),,((
,
),,((),(
ddxsubremain
x
dxlessifdxremain
)))),,(((
,0
),,((),(
ddxsubquotsucc
dxlessifdxquot
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Division example: x/4
))),,((
,
),,((),(
ddxsubremain
x
dxlessifdxremain
rdqx quotient remainderx
0
1
2
3
4
5
6
7
8
0400
1401
2402
3403
0414
1415
2416
3417
0428
0
0
0
0
1
1
1
1
2
0
1
2
3
0
1
2
3
0
quotientq
remainderr
4d
)))),,(((
,0
),,((),(
ddxsubquotsucc
dxlessifdxquot
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Division as Primitive Recursion
)0
)),,(((
),)),,(((()),((
0),0(
dxsubremainsucc
ddxremainsucclessifdxsuccremain
dremain
)),()),),((?(()),((
0),0(
dxquotdxsuccremainzeroadddxsuccquot
dquot
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)),(?(),( dxremainzerodxdivisible
Recursive Predicate divisible
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)),((),(_ yxequalnonyxequalnon
Recursive Predicate
)),(?(),( dxremindzerodxdivisible
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Theorem The set of primitive recursive functions
is countable.
Proof Each primitive recursive function
can be encoded as a string.
Enumerate all strings in proper order.
Check if a string is a function.
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There is a function that is not primitive recursive.
Proof (by Cantor diagonal argument)Enumerate the primitive recursive functions
,,, 321 fff
Theorem
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Define function
1)()( ifig i
g differs from every if
g is not primitive recursive
END OF PROOF
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A specific function that is not
primitive recursive:
Ackermann’s function:
)),(,1()1,(
)1,1()0,(
1),0(
yxAxAyxA
xAxA
yyA
Grows very fast,
faster than any primitive recursive function
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The Ackermann function is the simplest example of a well defined total function which is computable but not primitive recursive, providing a counterexample to the belief in the early 1900s that every computable function was also primitive recursive.
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Recursive Functions
0),(such that smallest )),(( yxgyyxgy
Ackermann’s function is a
Recursive Function
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Primitive recursive functions
Recursive Functions
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Primitive Recursion: Extended Example
Needs following building blocks:– constants– addition– multiplication– exponentiation– subtraction
A polynomial function:27 73),( yxyxyxf
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Addition
add(0,n) =add(m+1,n) =
nsucc(add(m,n))
Multiplication:
mult(0,n) =mult(m+1,n) =
0add(mult(m,n),n)
nmnmadd ),(
nmnmmult ),(
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Exponentiation:
exp(0,n) =exp(m+1,n) =
1mult(exp(m,n),n)
= one(n)
Subtraction
sub(0,n) =sub(m+1,n) =
0 = zero(n)succ(sub(m,n))
nmnmsub ),(
mnnm ),(exp
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Primitive Recursion: Extended Example
f = sub ◦ (add ◦ (f1,f2), f3)
f1(x,y) = mult(3,exp(7,x)) f1 = mult ◦ (three, exp ◦ (seven))
f2(x,y) = mult(x,y) f2 = mult
f3(x,y) = mult(7,exp(2,y)) f3 = mult ◦ (seven, exp ◦ (two))
f(x,y) = sub(add(f1(x,y),f2(x,y)),f3(x,y))
27 73),( yxyxyxf
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Primitive Recursion
All primitive recursive functions are total.i.e., they are defined for all values.
Primitive recursion lack some interesting functions:“True” subtraction – when using natural numbers.“True” division – undefined when divisor is 0.Trigonometric functions – undefined for some values.…
67
Partial Recursive vs. Recursive
A function is partial recursive it can be defined by the previous constructions.
A function is recursive it is partial recursive and total.
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Division:div(m,n) = m n
div(m,n) = min i, sub(succ(m),add(mult(i,n),n)) = 0
div(m,n) = minimum i such thati mnin m-(n-1)in+n m+1(m+1) – (in+n) 0(m+1) (in+n) = 0
Example
69
Relations Among Function Classes
Functions TMs– Define TMs in terms of the
function formers.– Straightforward, but long.
TMs Functions– Define functions where
subcomputations encode TM behavior.
– Simple encoding scheme.– Straightforward, but very
messy.
partial recursive= recognizable
recursive= decidable
primitiverecursive
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otherwise 0,
)(
even isn if,1
neven
))(,1()1(
1)0(
kevensubkeven
even
Additional Examples of Primitive Recursion
A recursive function is a function that calls itself by using its own name within its function body.
Even
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))1(),(()1(
1)0(
xxfactmultxfact
fact
Factorials
1)1(! nnn
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),(),)),(((()),((
),0(),0(
),()?(
),()(
yxisSquareyxsuccsquareequaloryxsuccisSquare
yequalyisSquare
xxisSquarexsquare
xxmultxsquare
)),(,()),(,(),),,(((),(
)0,()?(
yymultxequalysuccxhxyymultlessifyxh
xhxsquare
Is a number a square?
Forward recursion (-recursion)
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number. naturalany of square anot is 5
0))5,0(
))5,0(.........................................................................
))5,1(..........................................................................
))5,2(..........................................................................
))))5,3(),5,4(((),5,5(((
)5,4(),5,5((()5,5(
)5,5()5?(
isSquare
isSquareetc
isSquareetc
isSquareetc
isSquaresquareequalorsquareequalor
isSquaresquareequalorisSquare
isSquaresquare
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etc
multequalhmultlessifh
hsquare
...
))0,0(,5(),1,5(),5),0,0((()0,5(
)0,5()5?(
