Computability

1
Computability

• Go over homework problems. Godel numbering.
• Homework prepare for midterm.

2
TM for Neat addition

• Create function TM that adds two numbers and
  returns as answer all 1s starting from the left
  end of the tape, no blanks.
• ????

3
TM for multiplication

• Tape starts with two sets of 1s, recall n1 1 for
  number n.
• Follow example for addition.
• Consider using markers on tape.
• ????

4
TM to tidy up

• Claim we can create quintuples to add to any TM
  to re-format and modify outputs to facilitate
  composition
• output for N is N 1s anywhere on tape. Change to
  (N1) 1s.
• insert or remove marker symbols separating
  numbers to be proper inputs.

5
Reprise Recursive functions

• The recursive functions are a set of functions
  defined using a starter set and allowing any
  functions that can be defined using a finite
  number of applications of composition, primitive
  recursion, and minimalization

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Starter set

• Identity F(x) x This is special case
  ofProjections Uin(x1, x2,,xn) xi
• Successor S(x) x1
• Constant Fc(x) c

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Composition

• Given F and G, FG (x) F(G(x))

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Primitive recursion

• Motivation similar to mathematical induction,
  definition of factorial, exponentiation.
• Have definition for the zero case. Have way for
  building.
• x0 1x(n1) xn x

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Primitive recursion

• Given functions f N ? N and g (N,N,N) ?N, then
  define h (N,N) as follows h(x,0) f(x)
  h(x,y1) g(x,y,h(x,y))
• You will see variations. For example, f Nn and
  g Nn2

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Minimalization

• Better name MAY be inverse.
• If f(x) is recursive, then define g to be
• g(y) min x f(x) y if any x exists.
  Otherwise,
• g(y) is undefined.

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Multiplication

• is recursive. Can be defined by use of
  composition, primitive recursion, minimalization
  of other recursive functions.
• Can assume addition!
• ????

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Subtraction

• is recursive. Can be defined by use of
  composition, primitive recursion, minimalization
  of other recursive functions.
• Can assume addition!
• Hint use minimalization
• ????

13
Equivalence of TMs and Recursive functions

• If a function is recursive, then we can construct
  a TM for it.
• Build TM for starter sets. Assume fix-up function
  to get things in proper format for next step.
• Describe process for doing composition, primitive
  recursion, and minimalization.
• Minimalization given function f for which there
  is a TM, call it F, then build a TM that calls F
  in turn for 0, 1, 2, .

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TM to Recursive

• Need to come up with encoding for TM that can be
  decoded to 'do' the steps as a function.
• Gödel Numbering !
• Prove set of steps that are recursive that lead
  to the existence of a computation using the steps
  of TM

15
Arithmetization of theory of Turing machines

• Symbols of TM are
• symbols available to be on tape BS0, S1,
• R and L
• states q1,q2
• Assign odd numbers starting with 3 to
  R,L,S0,q1,S1,q2,S2,
• So quadruples q11Rq2 is represented by 9,11,3,13.
  The instantaneous description q11111 is
  represented by 9,11,11,11

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Arithmetization, cont.

• Gödel Number of expression represented by
  sequence of n numbers a1,a2,..,an is the product
  of the first n primes, each raised to the
  corresponding ak power!
• gn(q11Rq2) 29 311 53 713
• gn(sequence of n expressions) product of the
  first n primes each raised to the gn of each
  expression
• Note gn's are either an expression or a sequence
  of expressions. gn's are huge! gn's are
  uniquecan get back to expression or sequence.

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Arithmetization, cont.

• Take any Turing Machine. (Using quadruples in
  place of quintuples. The operation of writing is
  distinct from moving left or right).
• TM defines by its set of quadruples.
• Define a gn for a TM is the gn for a sequence of
  its quadruples. So different orders will produce
  different gn.

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Computation

• A computation is a sequence s1,sk of expressions
  representing steps of a TM operating on input
• starting with q1 followed by input
• sequence, si?si1, according to TM
• sk is terminal (final, no next step)

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Description of proof

• With a gn for TM T, show by a set of steps
  involving what are valid computations starting
  from expression representing initial state in
  front of input, that a recursive function exists
  that has same behavior as T on the input.

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Predicate

• A predicate is a function of 1 or more parameters
  that returns true or false.
• Proof shows that a certain predicate is primitive
  recursive (that is, in the set starting with the
  starting functions and with any functions added
  using primitive recursion and composition)

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Predicate

• T(z,x1, x2, xn,y) is true if z is the gn for a
  TM x1, x2, .. xn are the inputs y is a
  computation of the TM encoded by z
• T(x,x1,x2, , xn, y) is false, otherwise,
  including z or y not being gn for any TM or any
  computation.

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Claim

• By many steps, can show T to be primitive
  recursive
• Example, Yield(x,y,z) is defined as the
  predicate, such that
• if x and y are gn for instantaneous expressions
  and z is gn for a TM and x ? y for TM z, then
  true,
• Otherwise, false

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Define function in 2 steps

• Then
• Vz(x1,xn) defined as min y T (z,x1, xn,
  y) is recursive.
• Note at most one such y.

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Last step

• U(y), defined as if y is a computation, number
  of 1s in last expression of y. Undefined,
  otherwise. This can be shown to be primitive
  recursive.
• Then
• U(min y T (z,x1, xn, y) ) is recursive

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Comments

• The predicate T is very powerful.
• It is false most of the time.
• It does the job for ALL TMs.
• For any particular function computed by a
  particular TM, there would be a more efficient
  recursive function.
• The steps are based on the mechanistic feature of
  TMs.
• There are other formulations equivalent to these
  two (TMs and recursive functions).

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More Topics

• Implementation of TM (language recognizer or
  function) in code
• there exists on-line examples
• Gödel work on incompleteness of logical systems
• Other equivalent formulations for computability
• computable numbers
• Work on randomness and computability
• Greg Chaitin, others.
• others cited previously

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Homework

• Review midterm guide.
• Study charts.
• Come with questions based on your review.