recursive functions and their turing-computability

8/4/2019 Recursive Functions and Their Turing-Computability

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Recursive functions and theirTuring-computability

Abhijit P. Pai

29th November, 2007
http://goforward/http://find/http://goback/


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Outline

Computability notions

Primitive Recursive Functions Recursive Functions

Recursive Functions are Turing-Computable
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Computability

Church-Turing thesis states that Turing Machines form abound on what can be computed.

The computational equivalence of the following have been

shown: Turing Machines Recursive functions Post systems Type 0 Grammars

Calculus


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Terms

Function: A function from INk to IN for some k 0; (called a

k-place function). Primitive Recursive function: Class of functions defined by 3

types of initial functions and two combining rules.


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Initial Functions

The 0-place zero function from IN0 to IN such that () = 0

Let k 1 and 1 i k. The ith k-place projection functionki fom IN

ktoIN such that ki (n1,..., nk) = ni for anyn1,..., nk IN

The successor function from IN to IN such that(n) = n + 1 for each n IN

An initial function is ,, or one of the ki


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Notation

n represents the sequence n1, ..., nk where k 0

n will also be used to represent the k-tuple (n1,..., nk)


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Combining Rules - Composition

Let l > 0 and k 0

Let g be an l-place function

Let h1,...hl be k-place functions Let f be the k-place function such that for every n INk,

f(n) = g(h1(n),..., hl(n))

Then f is said to be obtained from g, h1,...hl by composition


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Combining Rules - Primitive Recursion

Let k 0, g be a k-place function, h be a (k+2)-placefunction.

Let f be the (k+1)-place function such that for every n INk,

f(n, 0) = g(n) and for every n INk and m IN,

f(n,m + 1) = h(n, m, f(n,m))

Then f is said to be obtained from g and h by primitive

recursion


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Primitive Recursive Functions

A function is primitive recursive if it is an initial function or

can be generated from the intial functions by some sequenceof composition and primitive recursion.


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Examples

Constant functions k = 0 : K0

0

is , K0j+1

() = (K0j ()) for each j 0

k > 0 : Kk+1j (n, 0) = Kkj (n)

k > 0 : Kk+1j (n,m + 1) = k+2k+2(n,m,K

k+1j (n,m))


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Examples

plus(a, m) = a + m plus(a, 0) = 11(a)

plus(a,m + 1) = h(m, n, plus(a,m)), h = 3

3 mult(a,m) = a m

mult(a, 0) = K10 mult(a,m + 1) = h(m, n,mult(a,m)), h = plus(31 ,

33)


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Why primitive recursive functions couldnt possibly define

computability

Each primitive recursive function has a finite encoding

Generate a list of all possible primitive recursive functions inlexicographic order A

1, A

2,A

3,...

Let fi be the p.r.f. defined by Ai Consider the 1-place function f such that for every

n IN, f(n) = fn(n,..., n) + 1

f is computable. But f must be fi for some i.

When evaluated with argument i, f and fi differ by 1.


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Unbounded Minimization

Let k 0 and g be a (k+1)-place function.

The unbounded minimization of g is that k-place function f

such that, for any n INk

,f(n) = { The least m such that g(n,m) = 0, if such m existsf(n) = { 0 otherwise

We write f(n) = m[g(n,m) = 0]


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Regular Functions

A (k+1)-place function g is regular iff, for every n INk,

there is an m such that g(n,m) = 0.


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Recursive Functions

A function is recursive iff it can be obtained from, ki , and by: Composition Primitive Recursion Application of Unbounded minimization to regular functions


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Computing Turing Machine Model

Input surrounded by blank on each side, written on leftmostsquares of tape

Head is initially at the blank just past the input string The TM M computes f if, for all w , if f(w) = u, then

(s, #w#) M (h,#u#)


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Turing computability of Recursive functions - SomeTMs

R# searches for the first blank square to the right of thesquare currently scanned

L# searches for the first blank square to the left of the squarecurrently scanned

SL shifts left - #w# to w# SR shifts right - #w# to ##w#

E (eraser) - #w# to #, where w can be empty

B (backup) - #u#w# to #w#, u and w can be empty

k-copier Ck(k 1) - #w1#w2#...#wk# to#w1#w2#...#wk#w1#

Clk - #w1#w2#...#wk# to#w1#w2#...#wk#w1#w2#...#wl#

f


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The initial functions are Turing-computable

is computed by a TM that moves its head one square to theright and halts.

Initial config: (s, #)

Final config: (h, ##)

Th i i i l f i T i bl


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ki is computed by EkiBi1

Eki

- #w1#w2#...#wk# to #w1#w2#...#wi# Bi1 transforms this to #wi#

Th i i i l f i T i bl


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- A TM which transforms #1

n

#(n 0) into #1

n+1

#

C iti i T i t bl


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Composition is Turing-computable

Composition l > 0, k 0, g an l-place function, h1,...hl k-place functions f, the k-place function such that for every n INk,

f(n) = g(h1(n),...,hl(n))

If g, h1...hl are Turing-computable, so is f.

Let G,H1,..., Hl be TMs that compute g, h1, ..., hl. The function f is computed by the TM

CkkH1Ckk+1H2...C

kk+l1HlGB

k

CkkH1 - #w1#w2#...#wk# to

#w1#w2#...#wk#w1#w2#...#wk# to#w1#w2#...#wk#u1# where ui = 1

hi(n)

After CkkH1...Ckk+l1Hl, #w1#w2#...#wk#u1#u2#...#ul#

C iti i T i t bl


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Composition is Turing-computable

G makes this into #w1#w2#...#wk#v# where v will be

1f(n)

Bk makes this into #v#

P i iti si is T i g t bl


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Primitive recursion is Turing-computable

Primitive Recursion k 0, g, a k-place function, h, a (k+2)-place function f, the (k+1)-place function s.t. for every n INk,

f(n, 0) = g(n)

and for every n INk

and m IN,f(n,m + 1) = h(n,m, f(n,m))

If g and h are Turing-computable, so is f.

Let G and H be TMs that compute g and h

We write w for w1#w2#...wk

Primitive recursion is Turing computable


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Add a left end marker $ to input ($ / G, H) using(SRL

2#)

k+1$Rk+1# The tape is now $#w#Im#

Change tape to$#w#Im1#w#Im2#...#w#I1#w#I0#w# using steps:

Copy Im to tape 2 Copy w to tape 3 Erase tape 1 to get $# Move right twice to get $## Repeat till tape 2 becomes zero

Decrement tape 2

Copy tape 3 to tape 1 Put # in tape 1 Copy tape 2 to tape 1 Put # in tape 1

Copy tape 3 to tape 1 Put # in tape 1

Primitive recursion is Turing computable


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Now G and H compute successive values f(n, i) fori = 0,...,m as:

$#w#Im1#w#Im2#...#w#I1#w#I0#w#

to $#w#Im1#w#Im2#...#w#I1#w#I0#v0#

to $#w#Im1#w#Im2#...#w#I1#v1#

to $#w#Im1#vm1#

to $#vm#

Now remove the $, and the output value is the valuecomputed by f.

Unbounded Minimization is Turing computable


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Unbounded Minimization is Turing-computable

g, regular (k+1)-place Turing-computable.

If f is got from g by unbounded minimization, f isTuring-computable.

Let G be the TM that computes g, we construct F thatcomputes f, where for all n INk, f(n) = the least m suchthat g(n,m) = 0.

g regular m exists for every n

F gets #w1#w2#...#wk#

Change to #w1#w2#...#wk#I0#

Create a copy of this.

Apply G to the copy. If result is nonzero, add one more 1 atthe end of the original and repeat.

At some point, for #w1#w2#...#wk#Im#, G gives 0.

Now, eliminate using B 1st k arguments and keep Im on tape.

Functions with strings as arguments


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Define a function known as the Godel number gn(w) of the

string w as the number corresponding to the base = ||+ 1encoding of the string.

If = a, b, c, = 4.

We fix d1 = a, d2 = b, d3 = c

gn() = 0 gn(a) = gn(d1) = 1

gn(ba) = gn(d2d1) = 41 2 + 1 = 9

We add d0 so that numbers such as correspond to some

string. Let D = ( d0)

gn1 is a well defined function from IN to D.



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gn1(0) =

gn1(1) = d1 = a

gn

1

(3) = d3 = c gn1(4) = d1d0 = ad0 gn1(5) = d1d1 = aa

...

References


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References

Elements of the Theory of Computation

Harry R. Lewis Christos H. Papadimitriou

Thank You


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Thank You

Questions

recursive functions and their turing-computability

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