MuRecursive Functions

1
Chapter 13

• Mu-Recursive Functions

2
Recursive Functions

• Functions that can be computed by at least one
  computational system, regardless of the
  implementation execution of the system.
• Recursive function theory
• i) Start w/ a collection of initial functions,
  which are computable
• ii) Combine the initial functions to form more
  functions which are also computable

3
Partial Functions Total Functions

• A partial function f X ? Y is a relation on X ?
  Y such that y1 y2 whenever (x, y1) ? f (x,
  y2) ? f.
• x ? X is said to be defined in f if ?y ?Y such
  that (x, y) ? f o.w., x is undefined in f.
• If the domain of f is the entire set X, then f is
  a total function on X.
• A total function can be a one-to-one, onto, or
  bijection function.

4
Primitive Recursive Functions

• Are more complex functions than the basic
  primitive recursive functions, which include
• i) the successor function S S(X) X 1
• ii) the zero function Z Z(X) 0
• iii) the projection functions Pi(n) Pi(n)(X1,
  ..., Xn) Xi, 1 ? i ? n.
• These functions form the foundation of the
  hierarchy of computable functions.
• Can be constructed from the basic functions by
  applications of
• i) composition the composition of two
  computable functions g h, i.e., f g
  ? h, is a (computable) function (that
  applies the outputs of h as inputs of g )
• ii) primitive recursion

5
Primitive Recursive Functions

• Defn. 13.1.1 (Primitive Recursion). Let g h be
  total functions w/ n n2 variables,
  respectively. The n1-variable function f is
  defined by
• i) f(X1, ..., Xn, 0) g(X1, ..., Xn)
• ii) f(X1, ..., Xn, y1) h(X1, ..., Xn, y,
  f (X1, ..., Xn, y))
• is said to be obtained from g h by primitive
  recursion, where Xi s (1 ? i ? n) are called the
  parameters y the recursive variable.
• By definition, f (X1, ..., Xn, y1) is obtained
  by the sequence of computations
• f (X1, ..., Xn, 0) g(X1, ..., Xn)
• f (X1, ..., Xn, 1) h(X1, ..., Xn, 0, f (X1,
  ..., Xn, 0))
• f (X1, ..., Xn, 2) h(X1, ..., Xn, 1, f (X1,
  ..., Xn, 1))
• f (X1, ..., Xn, y1) h(X1, ..., Xn, y, f
  (X1, ..., Xn, y))

6
Primitive Recursive Function

• A function is primitive recursive (P.R.) if it
  can be obtained from the successor, zero
  projection functions by a finite number of
  applications of composition primitive
  recursion.
• e.g., let add be the function defined by P.R.
  from g(x) x h(x, y, z) z 1. Then
• add(x, 0) g(x) x
• add(x, y1) h(x, y, add(x, y))
  add(x, y) 1
• Hence, add is P.R. since g P1(1) h S ?
  P3(3)

7
Primitive Recursive Function

• e.g., add(2, 4) add(2, 3) 1
• (add(2, 2) 1) 1
• ((add(2, 1) 1) 1) 1
• (((add(2, 0) 1) 1) 1) 1
• 2 1 1 1 1
• 6
• Example. 13.1.3 Let g h be the total
  functions g z h add ? (P3(3), P1(3)),
  mult, i.e., ?, can be defined by primitive
  recursion from g h as
• mult(x, 0) g(x) 0
• mult(x, y1) h(x, y, mult(x, y))
  mult(x, y) x

8
Primitive Recursive Function

• Example 13.1.4 Let fact be the factorial
  function defined by
• 1 if y 0
• fact(y) o.w.
• fact is p.r. Let h(x, y) mult ? (P2(2),
  S?P1(2)) y ? (x1).
• Then, fact can be defined using p.r.
  from h by
• fact(0) 1 fact(y1) h(y, fact(y))
  fact(y) ? (y1)
• hence, fact(0) 1,
• fact(1) fact(0) ? 1 1
• fact(2) fact(1) ? 2 2,
• fact(3) fact(2) ? 3 6
• fact(4) fact(3) ? 4 24,
• h, the function used in a p.r. function, can be
  any function that has previously be shown to be
  p.r.

9
Table 13.2.1 Primitive Recursive Arithmetic
Functions

• Description Function Definition
• Addition add(x, y) add(x, 0) x
• x y add(x, y1) add(x, y) 1
• Multiplication mult(x, y) mult(x,0)
  0
• x ? y mult(x, y1) mult(x, y) x
• Predecessor pred(y) pred(0) 0
• pred(y1) y
• Proper sub(x, y) sub(x, 0) x
• Subtraction x - y sub(x, y1)
  pred(sub(x, y))
• Exponentation exp(x, y) exp(x, 0) 1
• xy exp(x, y1) exp(x, y) ? x

.
10

• TABLE 13.2.2 Primitive Recursive Predicates
• Description Predicate Definition
• Sign sg(x) sg(0) 0
• sg(y 1) 1
• Sign Complement cosg(x) cosg(0) 1
• cosg(y 1) 0
• Less than lt(x, y) sg(y - x)
• Greater than gt(x, y) sg(x -
  y)
• Equal to eq(x, y) cosg(lt(x,
  y) gt(x, y))
• Not equal to ne(x, y)
  cosg(eq(x, y))

.
.
11
Primitive Recursive Function

• Table 13.2.1. Primitive Recursive Arithmetic
  Functions
• Theorem 13.1.3. Every p.r. function is Turing
  computable
• Example Let f be the function defined by
• x 1 if x is even
• x - 1 o.w.
• a) Give the state transitive diagram of a TM M
  that computes f
• b) Show that f is p.r.

b) oe(x) add(mult(ev(x), s(x)),
mult(cosq(ev(x)), pred(x)))
where ev(0) 1 s z
ev(x1) h(x, ev(x))
eq(P2(2)(x, ev(x)), 0) (eq is defined
in Table 13.2.2)
12
Primitive Recursive Function

• Theorem 13.2.1 Let g be a p.r. function f a
  total function that is identical to g for all
  but a finite of input values. Then f is p.r.
• Proof Let f be defined by
• y1 if x n1
• y2 if x n2
• f(x)
• yk if x nk
• g(x) o.w.
• f is p.r. since f can be written as
• f(x) eq(x, n1) y1 eq(x, n2) y2 ...
  eq(x, nk) yk ne(x, n1) ne(x, n2)
  ... ne(x, nk) g(x)

13
Tabular Functions

• Output values are listed along w/ their
  corresponding input values, and a single common
  value is associated w/ all other inputs, i.e.,
• g1(x1, ..., xn) if P1 (x1, ...,
  xn)
• f(x1, ..., xn)
• gm(x1, ..., xn) if Pm (x1, ...,
  xn)
• gm1(x1, ..., xn) o.w.
• f is p.r. because f can be expressed in the form
• f(x1, ..., xn) g1(x1, ..., xn) C1 (x1, ...,
  xn) ...
• gm(x1, ..., xn) Cm (x1,
  ..., xn)
• gm1(x1,...,xn) cosq(C1 (x1,..., xn) ...
  Cm(x1, ..., xn))
• where
• 1 if Pp(x1,
  ..., xn) holds
• 0 o.w.
• Cp is called the characteristic function

Cp(x1, ..., xn)
14
Computable Partial Functions

• All primitive recursive functions are total, but
  not vice versa.
• Example The Ackermans function is computable
  total, but not p.r.
• A(0, y) y 1
  (basis)
• A(x1, 0) A(x, 1)
• A(x1, y1) A(x, A(x1, y))
• To illustrate, consider
• A(2,1) A(1, A(2,0))
• A(1, A(1,1)) A(1,1) A(0, A(1,
  0))
• A(1,3) A(0, A(0,
  1))
• A(0, A(1,2)) A(0,
  2)
• A(0, A(0, A(1,1)))
  3
• A(0, A(0, 3))
• A(0, 4)
• 5

15
Computable Partial Functions

• Theorem There is a computable total function
  from N to N that is not primitive recursive.
• Proof. Each p.r. function from N to N can be
  defined by a finite string of symbols. Thus, we
  can assign an order to the p.r. functions by
  arranging their definitions according to their
  lengths (short strings first) w/ the strings of
  the same length by alphabetical order, i.e., f1,
  f2, ..., fn, ...
• Let f(n) fn(n) 1 be a function from N
  to N. f is total computable. However, f is
  not p.r. (If it were, f would have to be fm, for
  some m ? N. But then f(m) would be equal to
  fm(m), which cannot be true since f(m) fm(m)
  1.)

16
Partial Recursive Functions

• The class of partial recursive functions is the
  class of partial functions that can be
  constructed from the initial (i.e., basic)
  functions by applying a finite number of
  compositions, primitive recursions,
  minimalizations.
• minimalization allows us to construct a function
  f Nn ? N from function g Nn1 ? N .?. f( ) is
  the smallest y ? N, g( , y) 0 g( , z) is
  defined for all z (? N) lt y, written f( )
  µyg( , y) 0.
• e.g., g(0,0) 2 g(1,0) 3 g(2,0)
  8 g(3,0) 2
• g(0,1) 3 g(1,1) 4 g(2,1) 3
  g(3,1) 6
• g(0,2) 1 g(1,2) 0 g(2,2)
  undefined g(3,2) 7
• g(0,3) 5 g(1,3) 2 g(2,3) 6
  g(3,3) 2
• g(0,4) 0 g(1,4) 0 g(2,4) 0
  g(3,4) 8
• g(0,5) 1 g(1,5) 0 g(2,5) 1
  g(3,5) 4
• ? f(0) 4 ? f(1) 2 ? f(2)
  is undefined

17
Partial Recursive Functions

• Example. Let f N ? N be defined as f(x)
  µyplus(x, y) 0. Thus f(0) 0, but f(x) is
  undefined for ?x gt 0
• Example. Let div N2 ? N be defined as
• integer portion
  of x/y, if y ? 0
• undefined, if y 0
• using minimalization, div can be constructed as
• div(x, y) µt((x 1) - (mult(t, y) y)) 0

.
18
?-Recursive Functions

• ?-recursive functions yield a family of
  computable partial recursive functions.
• Defn.13.6.3. The family of ?-recursive functions
  is defined as
• i) The successor, zero, projection
  functions are ?-recursive
• ii) If h is an n-variable ?-recursive function
  g1, ..., gn are k- variable
  ?-recursive functions, then f h (g1, ...,
  gn) is
  ?-recursive.
• iii) If g h are n n2 variable ?-recursive
  functions, then f defined from g h by
  primitive recursion is ?-recursive.
• iv) If p(x1, ..., xn, y) is a total ?-recursive
  predicate, then f ?z p(x1, ..., xn,
  z) is ?-recursive
• v) Each ?-recursive function can be obtained
  from i) by a finite number of
  applications of the rules in ii), iii) iv)
• Theorem 13.6.4. Every ?-recursive function is
  Turing computable
• Theorem 13.7.1. Every Turing computable function
  is ?-recursive