Chapter 10
Computable
Functions
1
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Primitive Recursive Functions
• We have defined a function to be computable if there
is a Turing machine that computes it. Most functions
are not computable (the set of functions from ℕ to ℕ
is uncountable, and the set of Turing machines is
countable). It would be desirable to have a clearer
idea of what sorts of functions can be computed
• In this chapter we present an approach due to
Kleene. We start with a set of initial functions and
develop some operations on functions that preserve
computability. We will eventually describe
operations that produce all computable functions
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Primitive Recursive Functions (cont’d.)
• Definition 10.1: The initial functions are the
following:
– Constant functions: For each k  1 and each a  0, the
constant function Cak : ℕk  ℕ is defined by the
formula Cak(X) = a for every X  ℕk
– The successor function s : ℕ  ℕ is defined by the
formula s(x) = x + 1
– Projection functions: For each k  1 and each i with
1  i  k, the projection function pik : ℕk  ℕ is defined
by the formula pik(x1, … , xi, …, xk) = xi
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Primitive Recursive Functions (cont’d.)
• Definition 10.2:
– Suppose f is a partial function from ℕk to ℕ, and
for each i with 1  i  k, gi is a partial function
from ℕm to ℕ. The partial function obtained from f
and g1, … , gk by composition is the partial function
h from ℕm to ℕ defined by the formula
h(X) = f (g1(X),…, gk(X)) for every X  ℕm
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Primitive Recursive Functions (cont’d.)
• Definition 10.2 (cont’d.)
– Suppose n  0, and g and h are functions of n and
n+2 variables, respectively
– The function obtained from g and h by the
operation of primitive recursion is the function
f : ℕn+1  ℕ defined by the formula
f (X, 0) = g(x); f(X, k+1) = h(X, k, f (X, k))
for every X  ℕn and every k  0
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Primitive Recursive Functions (cont’d.)
• A function obtained by either of these operations
from total functions is a total function
• If f is obtained from g and h by primitive recursion,
and g(X) is undefined for some X, then f (X,i) is
undefined for all X, i
• Similarly if f (X, k) is undefined for some k, say k = k0,
then f (X, k) is undefined for k  k0
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Primitive Recursive Functions (cont’d.)
• Definition 10.3: The set PR of primitive recursive
functions is defined as follows:
– All initial functions are elements of PR
– For every k  0 and m  0, if f : ℕk  ℕ and
g1, …, gk : ℕm  ℕ are elements of PR, then the function
f (g1, …, gk) obtained from f and g1, …, gk by
composition is an element of PR
– For every n  0, every function g : ℕn  ℕ in PR, and
every function h : ℕn+2 ℕ in PR, the function
f : ℕn+1  ℕ obtained from g and h by primitive
recursion is in PR
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Primitive Recursive Functions (cont’d.)
• In other words, the set PR is the smallest set of
functions that:
– Contains all the initial functions
– Is closed under the operations of composition and
primitive recursion
• The initial functions are computable
• The set of computable functions is also closed under
the operations of composition and primitive
recursion. (We can describe TMs to compute the
functions obtained by these operations)
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Primitive Recursive Functions (cont’d.)
• Theorem 10.4: Every primitive recursive function is
total and computable
– Proof: This follows from the Church-Turing thesis and
the previous observations
• We can define the arithmetic operations Add(x, y)
and Mult(x, y), and they are both primitive recursive.
So is Sub(x, y), which is defined in terms of a
predecessor function Pred(x ). We must be slightly
careful, because ordinary subtraction doesn’t always
produce natural numbers. On the slides we continue
to use the minus sign (-) for “proper subtraction”
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Primitive Recursive Functions (cont’d.)
• An n-place predicate P is a function or partial
function from ℕn to the set {true, false}
– The corresponding numeric function is the
characteristic function P defined by
P(X) = 1 if P(X) is true, 0 if P(X) is false
• By definition, the predicate P is primitive recursive,
or computable, if the characteristic function P is
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Primitive Recursive Functions (cont’d.)
• Theorem 10.6:
– The two-place predicates LT, EQ, GT, LE, GE, and NE
are primitive recursive (LT corresponds to <, EQ to =,
etc.)
– If P and Q are any primitive recursive n-place
predicates then P  Q, P  Q, and P are primitive
recursive
• Proof: the second half follows from the equations
– P1  P2 = P1 * P2
– P 1  P 2 = P 1 + P 2 - P 1  P 2
– P = 1 - P1
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Primitive Recursive Functions (cont’d.)
• To prove the first part, we introduce the function
Sg: ℕ  ℕ defined by Sg(0)=0, Sg(k+1) = 1. This is clearly
primitive recursive
• Now:
– LT(x, y) = Sg(y - x)
– EQ(x, y) = Sub(1, (Sg(x - y) + Sg(y - x))
– EQ(x, y) = Sub(1, (Sg(x - y) + Sg(y - x))
– LE = LT  EQ
– GT =  LE
– GE =  LT
– NE =  EQ
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Primitive Recursive Functions (cont’d.)
• Theorem 10.7:
– Suppose f1, … , fk are primitive recursive functions from
ℕm to ℕ; P1, … , Pk are primitive recursive n-place
predicates; and for every X  ℕn, exactly one of the
conditions P1(X), … , Pk(X) is true
– Then the function f (X), which for each i is defined to
be fi (X) if Pi(X) is true, is primitive recursive
• Proof: f = f1 * P1 + … + f1 * P because of the
definition of f and the assumption on the Pi’s
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Primitive Recursive Functions (cont’d.)
• The Div and Mod functions are primitive recursive if
we take as part of the definition that for any x,
Div(x, 0) = 0 and Mod(x, 0) = x
• This will be useful later
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Quantification, Minimalization, and
-Recursive Functions
• The operations that can be applied to predicates to
produce new ones include not only the logical
operations, but also universal and existential
quantification
• Here’s an example. If
Sq(x, y) = (x = y2)
the existential quantification of Sq would be the
predicate PerfectSquare:
PerfectSquare(x) = (there exists y such that Sq(x, y))
• Sq is a primitive recursive predicate; in this example,
the existential and universal quantifications are also.
This is not always the case
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Quantification, Minimalization, and
-Recursive Functions (cont’d.)
• Suppose that for x ≥ 0, sx denotes the xth string in
{0,1}* in canonical order, and suppose Tu is a
universal TM. Let H(x,y) be the predicate
H(x,y) = (Tu halts after y moves on input sx)
The existential quantification of H is the predicate
Halts(x) = (there exists y such that H(x,y))
• H(x,y) is computable and even primitive recursive.
However, Halts cannot be computable, because the
halting problem is undecidable
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Quantification, Minimalization, and Recursive Functions (cont’d.)
• We infer that quantification does not preserve
computability nor primitive recursiveness
• Definition 10.9: Let P be an (n+1)-place predicate
– The bounded existential quantification of P is the
(n+1)-place predicate EP defined by EP(X, k) = (there
exists y with 0  y  k such that P(X, y) is true)
– The bounded universal quantification of P is the
(n+1)-place predicate AP defined by AP(X, k) =
(for every y with 0  y  k, P(X, y) is true)
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Quantification, Minimalization, and Recursive Functions (cont’d.)
• Theorem 10.10:
– If P is a primitive recursive (n+1)-place predicate, both
the predicates EP and AP are also primitive recursive
• Proof sketch:
– The characteristic function of AP(X,k) can be expressed
as a “bounded product” of the functions P(X, i) for
0  i  k, and as a result it is primitive recursive.
– The result for the existential quantification follows
from the formula EP(X,k) = A P(X,k)
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Quantification, Minimalization, and Recursive Functions (cont’d.)
• The bounded quantifications in this section preserve
primitive recursiveness and computability
– The unbounded versions do not
• In order to separate the computable functions from
the primitive recursive ones, we need at least one
operation that preserves computability but not
primitive recursiveness
– The operation of minimalization turns out to have this
feature
– Minimalization also has a bounded and an unbounded
version
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Quantification, Minimalization, and Recursive Functions (cont’d.)
• Definition 10.11: For an (n+1)-place predicate P, the
bounded minimalization of P is the function
mP : ℕn+1  ℕ defined by
mP(X, k) = min {y | 0  y  k and P(X, y)}
if this set is empty, and k+1 otherwise
– The symbol  is often used for the minimalization
operator, and we sometimes write
mP(X, k) = k y[ P(X, y) ]
– In the special case in which P(X, y) is (f (X, y) = 0),
mP is written mf and we refer to it as the bounded
minimalization of f
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Quantification, Minimalization, and Recursive Functions (cont’d.)
• Theorem 10.12:
– If P is a primitive recursive (n+1)-place predicate, its
bounded minimalization mP is a primitive recursive
function
• Proof: by construction using primitive recursive
constructs
– See book for details.
• PrNo(n), the nth prime number, is primitive recursive
– For details, see book
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Quantification, Minimalization, and Recursive Functions (cont’d.)
• Definition 10.14: If P is an (n+1)-place predicate, the
unbounded minimalization of P is the partial function
MP : ℕn  ℕ defined by MP(X) = min{y | P(X,y) is true}
– MP(X) is undefined at any X  ℕn for which there is no y
satisfying P(X, y)
– The notation y[P(X,y)] is also used for MP(X)
– In the special case where P(X,y) = (f (X, y) = 0), we
write MP = Mf and refer to this function as the
unbounded minimalization of f
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Quantification, Minimalization, and Recursive Functions (cont’d.)
• Definition 10.15: The set M of -recursive, or simply
recursive, partial functions is defined as follows:
– Every initial function is an element of M
– Every function obtained from elements of M by
composition or primitive recursion is an element of M
– For every n  0 and every total function f : ℕn+1  ℕ in
M, the function Mf : ℕn  ℕ defined by
Mf (X) = y[f (X, y)=0] is an element of M
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Quantification, Minimalization, and Recursive Functions (cont’d.)
• Theorem 10.16: All -recursive partial functions are
computable.
• Proof: For a proof using structural induction, it is
sufficient to show that if f : ℕn+1  ℕ is a computable
total function, then its unbounded minimalization Mf
is a computable partial function
• If Tf is a Turing machine computing f, a TM T
computing Mf operates on an input X by computing
f (X, 0), f (X, 1),… until it discovers an i for which
f (X, i) = 0, and halting in ha with i as its output
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Gödel Numbering
• In 1930’s, Kurt Gödel developed a method of
“arithmetizing” a formal axiomatic system by
assigning numbers to statements and formulas
– This allowed him to describe relationships between
objects in the system using relationships between the
corresponding numbers
• This led to Gödel’s Incompleteness Theorem:
– Any formal system comprehensive enough to include
the laws of arithmetic must, if it is consistent, contain
true statements that cannot be proved within the
system
• Gödel numbering will be useful in this chapter also
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Gödel Numbering (cont’d.)
• Definition 10.17: For every n  1 and every finite
sequence x0, x1, … , xn-1 of n natural numbers, the
Gödel number of the sequence is the number
gn(x0, x1, … , xn-1)=2x03x1…(PrNo(n -1))xn-1 where
PrNo(i) is the ith prime
– Every sequence of length n is uniquely determined by
its Gödel number
– If two sequences of different length have the same
Gödel number, then they must agree except for the
number of trailing 0’s
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Gödel Numbering (cont’d.)
• Every positive integer is the Gödel number of some
sequence of integers
• For every n, the Gödel numbering determines a
function from ℕn to ℕ
– We will be imprecise and use the name gn for any of
these functions
– All of them are primitive recursive
– The function Exponent: ℕ2  ℕ, defined by
Exponent(i, x) = the exponent of PrNo(i) in x’s prime
factorization or 0 if i is 0, is primitive recursive
• See book for details
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Gödel Numbering (cont’d.)
• For many functions that are defined recursively, the
definition does not make it obvious that the function
is obtained by primitive recursion
– For example, the right side of the formula
f (n+1) = f (n) + f (n -1) is not of the form h(n, f (n))
– In general f (n+1) might depend on several, or even all,
of the terms f (0), f (1), … , f (n)
– This type of recursion is known as course-of-values
recursion
• It is related to primitive recursion as strong induction is
related to ordinary mathematical induction
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Gödel Numbering (cont’d.)
• Gödel numbering gives us a way to formulate
definitions of this type. Suppose f (n+1) depends on
f (0), f (1), … , f (n), and also directly on n. We’d like
another function f1 so that f1(n+1) depends only on n
and f1(n), and knowing f1 allows us to calculate f.
– If we could allow f1(n) to be the entire sequence
(f (0), f (1), … , f (n)), instead of a number, we’d have
what we want.
– Instead we can use the Gödel number of the sequence,
because gn(f (0), … , f (n)) has enough information to
determine each f (i)
– This allows us to use course-of-values recursion
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Gödel Numbering (cont’d.)
• Theorem 10.19: Suppose that g : ℕn  ℕ and
h : ℕn+2  ℕ are primitive recursive functions, and
f : ℕn+1  ℕ is obtained from g and h by course-ofvalues recursion; that is, f (X, 0) = g(X);
f (X, k+1) = h(X, k, gn(f (X,0),…f (X,k))), then f is
primitive recursive.
• Now we can apply Gödel numbering techniques to
Turing machines
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Gödel Numbering (cont’d.)
• A TM move can be interpreted as a transformation of
one TM configuration to another
• To arithmetize this, we need a way of characterizing
a TM configuration by a number
– Assign a number to each state
• ha gets 0, hr gets 1, and 2 represents the initial state
– The tape head position is simply the number of the
current tape square
– Tape symbols get numbers too, with  = 0
– The current tape number tn of the TM is the Gödel
number of the sequence of symbols on the tape
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Gödel Numbering (cont’d.)
• The configuration of the TM is determined by the
state, tape head position and current contents of the
tape, and we define the configuration number to be
gn(q, P, tn), where q is the number of the current
state, P is the current tape head position, and tn is the
tape number
• From the configuration number, we can reconstruct
all of the details of the TM’s configuration
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All Computable Functions are Recursive
• Suppose that f : ℕn  ℕ is a partial function
computed by the TM T
• To compute f (X), T does these things:
– It starts in the standard initial configuration
corresponding to X
– It carries out a sequence of moves
– It ends up in an accepting configuration, if X is in the
domain of f, with a string representing f (X) on the
tape
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All Computable Functions are Recursive
• Each move of the computation transforms one
configuration number to another
• Thus, f (X) = ResultT(fT(InitConfig(n)(X)))
– Here InitConfig(n)(X) is the configuration number of the
initial configuration corresponding to input X
– fT is the function that transforms the initial
configuration number to the final one, corresponding
to the computation of T
– ResultT (j) is the value y if j is the number of a
configuration in which the output is y
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All Computable Functions are Recursive (cont’d.)
• In order to show that f is -recursive, it will be
sufficient to show that each of the three functions on
the right side of the formula is
• The two outer ones, ResultT and InitConfig(n) are
actually primitive recursive
• fT is not generally primitive recursive
– It is defined in a way that involves unbounded
minimalization and is defined only at configuration
numbers corresponding to n-tuples in the domain of f
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All Computable Functions are Recursive (cont’d.)
• InitConfig(n) : ℕn  ℕ
– It is easy to see that this is primitive recursive
• IsConfigT is the one-place predicate defined by
IsConfigT(n) = (n is a configuration number for T)
– It is primitive recursive because it can be computed
using bounded universal quantification
• IsAcceptingT(m) = 0 if IsConfigT(m)∧Exponent(0,m)=0,
1 otherwise
– It is primitive recursive because IsConfigT and
Exponent are primitive recursive
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All Computable Functions are Recursive (cont’d.)
• ResultT(n) = HighestPrime(Exponent(2,n)) if
IsConfigT(n), and 0 otherwise
– It is primitive recursive because HighestPrime is
• Some helper functions that are all primitive recursive
–
–
–
–
State(m) = Exponent(0,m)
Posn(m) = Exponent(1,m)
TapeNumber(m) = Exponent(2,m)
Symbol(m) = Exponent(Posn(m), TapeNumber(m))
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All Computable Functions are Recursive (cont’d.)
• A single move may result in new values for any or all
of those four quantities
– We have corresponding functions:
– NewState, NewPosn, NewTapeNumber, and NewSymbol
– All of these are primitive recursive
• The function MoveT : ℕ  ℕ is defined by MoveT(m) =
gn(NewState(m),NewPosn(m), NewTapeNum(m)) if
IsConfigT(m), and 0 otherwise
– This is primitive recursive
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All Computable Functions are Recursive (cont’d.)
• MovesT(m,0) = m if IsConfigT(m), and 0 otherwise
• MovesT(m,k+1) = MoveT(Moves(m,k)) if IsConfigT(m),
and 0 otherwise
• MovesT is obtained by primitive recursion from two
primitive recursive functions
– Is itself primitive recursive
• Define NumberOfMovesToAcceptT : ℕ  ℕ by the
formula NumberOfMovesToAcceptT(m) =
 k[AcceptingT(MovesT(m,k))=0]
– This is -recursive, because it uses unbounded
minimalization
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All Computable Functions are Recursive (cont’d.)
• Theorem 10.20: Every Turing-compatible partial
function from ℕn to ℕ is -recursive
• Proof: use previous definitions
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Other Approaches to Computability
• Unrestricted grammars provide a way of generating
languages, and they can also be used to compute
functions
• If G=(V, , S, P) is a grammar, and f is a partial
function from * to *, G is said to compute f if there
are variables A, B, C, and D in the set V such that for
every x and y in *, f (x) = y if and only if AxB G* CyD
• It can be shown that the functions computable this
way are the same as the ones computable by TMs
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Other Approaches to Computability
(cont’d.)
• Computer programs can be viewed as computing
functions from strings to strings
• The Church-Turing thesis asserts that every function
computable by a high-level programming language is
Turing-computable
• On the other hand, we can simulate a TM in any
modern programming language
• Therefore, high-level languages and TMs have
equivalent power
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