9b Recursive Functions 13'1

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9b Recursive Functions 13.1

• Gödel's Incompleteness Theorem
• Zero, Successor, Projector Functions
• Functional Composition
• Primitive Recursion Simulator
• Proving Functions are Primitive Recursive
• Ackermann's Function
• m-Recursion

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Gödel's Incompleteness Theorem

• Any interesting consistent system must be
  incomplete that is, it must contain some
  unprovable propositions.
• Hierarchy of Functions
• Primitive-Recursive Functions
• Recursive (?-recursive) Functions
• Interesting well-defined Functions but
  "unprovable"
• BB Function

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Primitive Recursive Functions

• Defined over the domain I set of all
  non-negative integers
• or domain II.
• or domain III, etc.
• Definition Functions are said to be Primitive
  Recursive if they can be built
• from the basic functions (zero, successor, and
  projection)
• using functional composition and/or primitive
  recursion.

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Zero, Successor, Projector Functions

• 1. Zero function z(x) 0, for all x ? I
• 2. Successor function s(x) x1
• 3. Projector functions p1(x1,x2) x1,
  p2(x1,x2)x2

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Functional Composition

• f(x,y) h(g1(x,y),g2(x,y))
• from previously defined functions g1, g2, and h
• e.g. min(x,y) lt(x,y)x geq(x,y)y
• h(x,y) add(x,y)
• g1(x,y) mult(lt(x,y),p1(x,y))
• h mult(), g1lt(), g2p1()
• g2(x,y) mult(geq(x,y),p2(x,y))
• h mult(), g1geq(), g2p2()

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Primitive Recursion

• f(x,0) g1(x)
• f(x,y1) h(g2(x,y),f(x,y))
• Note Last argument defined at zero and y1 only
• e.g. exp(x,0) 1 exp(x,n1) x exp(x,n)
• g1(x) s(z(x))
• h(x,y) mult(x,y)
• g2(x,y) p1(x,y)

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Constants are Primitive Recursive

• 2 s(s(z(x)))
• 3 s(s(s(z(x))))
• 5 s(s(s(s(s(z(x))))))

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Addition Multiplication

• add(x,0) xadd(x,y1) s(add(x,y))
• mult(x,0) 0mult(x,y1) add(x,mult(x,y))
• Tracing the recursion on the SIMULATOR

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Factorial Exponentiation

• fact(0) 1fact(n1) mult(s(n),fact(n))
• exp(x,0) 1exp(x,n1) mult(x,exp(x,n))

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Test for Zero(Logical Complement)

• test(0) 1test(x1) 0

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Subtraction

• pred(0) 0pred(x1) x
• monus(x,0) x // called subtr in
  textmonus(x,y1) pred(monus(x,y))
• absdiff(x,y) monus(x,y) monus(y,x)

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Relational Operators

• equal(x,y) test(absdiff(x,y))
• geq(x,y) test(monus(y,x))
• leq(x,y) test(monus(x,y))
• gt(x,y) test(leq(x,y))
• lt(x,y) test(geq(x,y))

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Minimum Maximumusing David Hannay's "Stupid
Bit Trick"(c.f. David Letterman's "Stupid Pet
Trick")

• min(x,y) lt(x,y)x geq(x,y)y
• max(x,y) geq(x,y)x lt(x,y)y

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Division

• remaind(numerator,denominator)
  rem(denominator,numerator)
• rem(x,0) 0rem(x,y1) s(rem(x,y))test(equal(x
  ,s(rem(x,y))))
• div(numerator,denominator) dv(denominator,numera
  tor)
• dv(x,0) 0dv(x,y1) dv(x,y)
  test(remaind(y1,x))

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Test for Prime

• numdiv(x) divisors_leq(x,x)
• divisors_leq(x,0) 0divisors_leq(x,y1)
  divisors_leq(x,y) test(remaind(x,y1))
• is_prime(x) equal(numdiv(x),2)

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Square Root

• sqrt(0) 0
• sqrt(x1) sqrt(x) equal(x1,(s(sqrt(x))s(sqrt
  (x))))

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a ? b mod c

• congruent(a,b,c) equal(remaind(a,c),remaind(b,c)
  )

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Greatest Common Divisor(cant use Euclidean
Algorithmnot P.R.)

• gcd(a,0) agcd(a,b1) find_gcd(a,b1,b1)
• find_gcd(a,b,0) 1find_gcd(a,b,c1)
  (c1)test_rem(a,b,c1) find_gcd(a,b,c)test(
  test_rem(a,b,c1))
• test_rem(a,b,c) test(remaind(a,c))test(re
  maind(b,c))

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Ackermann's Function

• Ackermann(x,y) A(x,y)
• A(0,y) s(y)
• A(x1,0) A(x,1)
• A(x1,y1) A(x,A(x1,y))

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Ackermann's Function is NOT Primitive Recursive

• Compute Ack(3,1) on Simulator Trace level 24
• Just because it is not defined using the
  "official" rules of primitive recursion is not a
  proof that it IS NOT primitive recursive.
• Perhaps there is another definition that uses
  primitive recursion. (NOT!) Proof is beyond the
  scope of this course

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"Meaning" of Ackermann's Function(addition,
multiplication, exponentiation, tetration)
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m -Recursion

• sqrt(x) my(yyx) // this is a partial
  function
• sqrt(x) my(s(y)s(y) gt x) // now it is
  total
• Note that m-recursion allows for partial
  functions just as do Turing machines

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m Recursive Definition ofnth_prime

• nth_prime(0) 2nth_prime(n1)
  find_prime(nth_prime(n))
• find_prime(x) my(and(less(x,y),is
  _prime(y)))

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m -Recursive Functions Equivalent to T.M.
Computable Functions

• TM for z(x)
• TM for s(x)
• TM for p1(x,y), p2(x,y)
• TM for functional composition
• TM for primitive recursion
• TM for m-recursion
• Proof in other direction is beyond the scope of
  the course...

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In-Class ExercisePrimitive-Recursive Functions

• Work in groups of 2 or 3
• Answers on next slide (no peeking)
• a) Using only initial functions, show that
  nth_odd(x) is primitive recursive.
• b) Using any functions defined in class, show
  that median(x,y,z) is primitive recursive.
• c) Using any functions defined in class, show
  that perfect(x) is primitive recursive.
• sum of all factors (including 1, not including x)
  x
• e.g. 6 (123) and 28 (124714) are
  perfect
• First group done with all 3 put a) on board, etc.

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9x Recursive Recursively Enumerable 11.1

• Enumeration Procedure Countable Sets
• Recursively Enumerable Languages
• Recursive Languages
• Existence of Languages that are not Rec.
  Enumerable
• A Language that is Rec. Enumerable but not
  Recursive

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Countable Sets

• Countable if it can be put into a 1-to-1
  correspondence with the positive integers
• You should already be familiar with the
  enumeration procedure for the set of RATIONAL
  numbers diagonalization, page 278
• Quick review
• You should already be familiar with the fact (and
  proof) that the REAL numbers are NOT countable
• Quick review

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Recursively Enumerable Languages

• A language is said to be recursively enumerable
  if there exists a Turing machine that accepts it.
• That is, if the accepting machine is started on a
  word in the language, it will halt in qf
• This says nothing about what the machine will do
  if it is presented with a word that is not in the
  language
• (i.e. whether it halts in a non-final state or
  loops)

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Recursive Languages

• A language, L, is recursive if there exists a
  Turing machine that accepts L and halts on every
  w in ?
• That is, there exists a membership decision
  procedure for L

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Existence of Languages that are not Recursively
Enumerable

• Let S be an infinite countable set. Then its
  powerset 2S is not countable. Theorem 11.1
• Proof by diagonalization page 278
• Recall the fact that the REAL numbers are not
  countable
• For any nonempty ?, there exist languages that
  are not recursively enumerable.
• Every subset of ? is a language. Therefore there
  are exactly 2? languages.
• However, there are only a countable number of
  Turing machines. Therefore there exist more
  languages than Turing machines to accept them.

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Recursively Enumerablebut not Recursive

• We can list all Turing machines that eventually
  halted on a given input tape (say blank)
• Recall the enumeration procedure for TMs from
  last period.
• Once a string of 0s and 1s was verified as a
  valid TM, we would simply run it (while
  non-deterministically continuing to list other
  machines). Note how long this would take!
• A halt on the part of the simulation (recall the
  Universal Turing Machine) would trigger adding
  the TM in question to the list of those that
  halted. (copying it to another tape?)
• However, we cannot determine (and always halt)
  whether or not a given TM will halt on a blank
  tape
• Stay tuned for the unsolvability of the Halting
  Problem...