Chapter 4: A Universal Program

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Chapter 4A Universal Program
2
Coding programs

• Example For our programs P we have variables
  that are arranged in a certain order
• Y1 X1 Z1 X2 Z2 X3 Z3
• Similarly labels are ordered
• A1 B1 C1 D1 E1 A1 B1 C1 D1
• Definition (V) is the position of a variable or
  label in the given ordering, where V is the label
  or variable.
• Example (X3) 6
• (C) 3
• (X1) (X) 2

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Coding programs

• Definition Let I be (labeled or unlabeled)
  instruction of the language L. Thus
• , where the following points are satisfied
• If I is unlabeled, then a 0 if I is labeled
  with L, then a (L).
• If the variable V is mentioned in I, then c
  (V) 1.
• If the statement in I is either
• V ? V , then b 0
• V ? V 1, b 1
• V ? V 1, b 2
• If the statement in I is IF V ? 0 GOTO L', then b
  (L') 2

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Coding programs

• Reminder from Chapter 3
• Given the following variable ordering Y X1,
  Z1, X2, Z2
• Provided the following label ordering A1 B1
  C1 D1 E1 ....
• We have a program as follows A X
  ? X 1
• IF X ? 0 GOTO A
• The instruction numbers for this program are
  thus
• (I1) lt (A) ,lt 1, (X) - 1gtgt
  lt1, lt 1,1 gtgt lt 1,5 gt 21
• (I2) lt0,lt (A) 2 , (X) - 1
  gtgt lt0, lt 3 , 1 gtgt lt 0 , 23 gt 46
• Definition The program P number is then
  calculated as
• (P) (I1), (I2) , (I3) , ... - 1
• In the program above (P) 221 346 1

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Coding programs

• In order to avoid ambiguity of unlabeled Y ? Y
  instruction, which is lt0,lt0,0gtgt an additional
  rule is imposed
• The final instruction in a program is not
  permitted to be the unlabeled statement Y ? Y.
• N.B The program number thus determines a unique
  program, which can reconstructed.
• Example Given (P) 576 575 1
• 575 52 231 0 , 0 , 2 , 0 , 0 , 0 , 0 ,
  0 , 1 9 instructions
• For 2 lt0,1gt lt0,lt1,0gtgt so unlabeled Y ? Y
  1
• For 1 lt1, 0gt lt1, lt0,0gtgt so labeled A Y
  ? Y
• For 0 lt0, lt0,0gtgt , this will be unlabeled Y ?
  Y
• Empty program has the number 0.

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The Halting Problem

• Definition For a given y, let P be the program
  such that (P) y. Then HALT(x , y) is true if
  is defined and false if
• is undefined, so
• HALT( x , y ) program number y eventually
  halts on input x
• Theorem HALT( x , x ) is not a computable
  predicate.
• Proof Construct a program P A IF HALT( X ,
  X ) GOTO A
• Let (P) y0, then HALT( x, y0)
  HALT( x , x )
• So we can set xy0, thus HALT (y0,y0)
  HALT(y0,y0)
• Contradiction!

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Insolvability of Halting Problem

• There is no algorithm that, given a program of P
  and an input to that program, can determine
  whether or not the given program will eventually
  halt on the given input (Chapter 4, page 68)
• Churchs Thesis Any algorithm for computing on
  numbers can be carried out by a program of P.
• See Golbachs conjecture (every even number
  greater or equal to 4 is the sum of two prime
  numbers) for an example of computable program for
  which it is hard to determine whether it will
  ever halt.

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Universality

• Universality Theorem For each n gt 0, where
• For each n gt0, the function
  is partially computable.
• Step-Counter Theorem For each n gt 0, the
• predicate is
  primitive recursive, where

,where (P)y
Program number y halts after t or fewer Steps on
inputs x1, , xn.
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Normal Form Theorem

• Normal Form Theorem Let f( x1 ,, xn) be a
  partially computable function. Then there is a
  primitive recursive predicate R( x1, , xn, y )
  such that
• Theorem A function is computable iff can be
  obtained from the initial functions by a finite
  number of applications of composition, recursion
  and (proper ) minimalization

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Normal Form Theorem

• Normal Form Theorem Proof
• Let y0 be the program number for f(x1,,xn).
• when the right-hand side of the equation is
  defined, then there exists a number z
• For any z, the program with number y0 has
  reached a terminal snapshot in r(z) or fewer
  steps and l(z) is in output variable Y.
• If the right side is undefined, then
  is false for all t i.e.

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Recursively Enumerable Sets

• Theorem Let the sets B , C belong to some PRC
  class C. Then so do the sets
  
• (prove using predicate union and
  intersection)
• Theorem Let C be a PRC class, and let B be a
  subset of . Then B belongs to C
  iff
• Definition The set is called
  recursively enumerable if there is a partially
  computable function g(x) such that

belongs to C
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Recursively Enumerable Sets

• If B is a recursive set, then B is recursively
  enumerable.
• The set B is recursive iff B and are both
  recursively enumerable.
• If B and C are recursively enumerable sets so are
• Enumeration Theorem A set B is recursively
  enumerable iff there is an n for which B Wn

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Recursively Enumerable Sets

• Definition K is a set of all numbers n such that
  program number n eventually halts on input n.
• K is recursively enumerable but not recursive.
• Theorem Let B be a recursively enumerable set.
  Then there is a primitive recursive predicate
  R(x,t) such that

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Recursively Enumerable Sets

• Theorem Let S be a nonempty recursively
  enumerable set. Then there is a primitive
  recursive function f(u) such that
• S f(n) n ? N f(0) , f(1) ,
• , where S is the range of f.
• Theorem Let f(x) be a partially computable
  function and let
• S f(x) f(x) ?
• , where S is the range. Then S is recursively
  enumerable.

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Recursively Enumerable Sets

• Theorem Suppose that , then the
  following statements are all equivalent
• S is recursively enumerable
• S is the range of a primitive recursive function
• S is the range of a recursive function
• S is the range of a partial recursive function

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The Parameter Theorem

• s-m-n Theorem
• For each n , m gt 0 there is a primitive
  recursive function such
  that

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The Parameter Theorem

• s-m-n proof using induction on n.
• Base case For program P take n 1, then the
  following must be shown to be true
• should be the number of the
  program with m inputs
• , which must be the same as
  program number y having m 1 inputs
  .
• (P) y
• So will be the number of the
  program, that will provide the variable
  before proceeding with P.

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The Parameter Theorem

• s-m-n proof continued
• The instruction will have
  a number associated with it as follows
  lt0,lt1,2m1gtgt 16m 10
• Thus may be defined as a primitive
  recursive function
• Induction step let n k, then

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Diagonalization and Reducibility

• Definition Let TOT be the set of all numbers p
  such that p is the number of a program that
  computes a total function f(x) of one variable.
  That is,
• , where
• Theorem TOT is not recursively enumerable.

,s
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Diagonalization and Reducibility

• Theorem TOT is not recursively enumerable.
• Proof Assume that TOT is recursively enumerable.
  Then given that , then there is
  a computable function g(x)
• and let
• Since g(x) is the number of a program that
  computes a total function, thus h is a computable
  function. Let P be the program that computes h
  and p(P). Then
• Contradiction!

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Diagonalization and Reducibility

• Theorem Suppose A m B, then
• If B is recursive, then A is recursive.
• If B is recursively enumerable, then A is as
  well.
• Definition A set A is m-complete if
• A is recursively enumerable and
• For every recursively enumerable set B, B m A

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Diagonalization and Reducibility

• If A m B and B m C, then A m C (transitivity)
• If A is m-complete, B is recursively enumerable
  and A m B then B is m-complete.
• Definition A B means that A m B and B m A
• Theorem
• K and K0 are m-complete
• K K0
• Theorem EMPTY is not recursively enumerable,
  where

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Rice's Theorem

• Examples
• ? is the set of computable functions
• ? is the set of primitive recursive functions
• ? is the set of partially computable functions
  that are defined for all but a finite number of
  values of x
• Rices Theorem Let ? be a collection of
  partially computable functions of one variable,
  such that there are functions f(x), g(x) with
  f(x) in ?, g(x) not in ?. Then R? is not
  recursive.
• Corollary There are no algorithms for testing a
  given program P of the language L to determine
  whether belongs to any of the classes
  described in the above examples.
• What does it mean? If ? is a non-trivial property
  of functions, then ? is undecidable.

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Rice's Theorem

• Proof (using recursion theorem)
• Let f(x) and g(x) be partially computable
  functions such that
• f(x) ?
• g(x) ?
• Suppose is computable. Let
• 
  and
• Then, since
• - is partially computable.

Iis not recursive
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Rice's Theorem
Proof (contd) Since is partially
computable, by recursion theorem there is a
number e s.t. As and

Contradiction!
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• The Recursion Theorem
• Recursion Theorem Let g(z, x1, , xm) be a
  partially computable function of m 1 variables.
  Then, there is a number e such that
• Discussion Given a partially computable
  function g, we must produce a program e that is
  supplied m arguments and one of the implicit
  arguments is its own encoding e.
• We may attempt to prove the theorem by first
  incrementing a variable e times. But then the
  problem is that the encoding of such a program
  will be larger than e!
• The program P with (P) e must contain a
  partial description of itself built-in so that
  that it can computer its own encoding e from that
  description.

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Let Q be the program Z Y g(Z, X1,
X2, , Xm) Now, the program P will consist of
(Q) copies of the instruction Xm1 Xm1 1,
followed by the program Q. After executing the
first (Q) increments as well as the instruction
Z , Z holds the number of the program
consisting of the (Q) copies of the instruction
followed by the program Q (Remember that
as defined in the proof of the
parameter theorem computes the of the program
consisting of (Q) copies of the increment
instruction Xm1 Xm1 1 followed by program
Q). But this is exactly the program P.
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• The Recursion Theorem

• Proof Let g be a partially computable function
  such as
• is the function in the parameter theorem. Then
  for some z0 and using s-m-n theorem
• Set v z0 and , then
• Contradiction!

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The Recursion Theorem

• Corollary There is a number e such that for all
  x
• Proof Let g be a computable function
• Using the recursion theorem
• Fixed Point Theorem Let f(z) be a computable
  function. Then there is a number e such that
• Take g(z, x)

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Computable functions that are not primitive
recursive

• The primitive recursive functions are precisely
  the functions from the initial functions (see
  Chapter 3)
• Theorem The unary primitive recursive functions
  are precisely those obtained from the initial
  functions s(x) x 1, n(x) 0, l(x) , r(x) by
  applying the following three operations on unary
  functions
• To go from f(x) and g(x) to f(g(x))
• To go from f(x) and g(x) to lt f(x) , g(x) gt
• To go from f(x) and g(x) to the function defined
  by the recursion
• Theorem The function f( x , x ) 1 is a
  computable function that is not primitive
  recursive.