Chapter 8
Decidability
8.1 Decision Problems
A decision problem P is a set of questions, each of which has a yes or no answer. The
single question ''Is 8 a perfect square?'' is an example of the type of question under
consideration in a decision problem. A decision problem usually consists of an infinite
number of related questions. For example, the problem PSQ of determining whether an
arbitrary natural number is a perfect square consists of the following questions:
p0 : Is 0 a perfect square?
p1 : Is 1 a perfect square?
p2 : Is 2 a perfect square?
. .
A solution to a decision problem P is an algoritm that determines the appropriate answer
to every question p  P .
An algorithm that solves a decision problem should be
1. Complete
2. Mechanistic
3. Deterministic.
A procedure that satisfies the preceding properties is often called effective.
A problem is decidable if it has a representation in which the set of accepted input strings
form a recursive language. Since computations of deterministic multitrack and multitape
machines can be simulated on a standard Turing machine, solutions using these machines
also establishes the decidability of a problem.
8.2 The Church-Turing Thesis
The Church-Turing thesis asserts that every solvable decision problem can be
transformed into an equivalent Turing machine problem.
The Church-Turing thesis for decision problems: There is an effective procedure to
solve a decision problem if, and only if, there is a Turing machine that halts for all input
strings and solves the problem.
The extended Church-Turing thesis for decision problems A decision problem P is
partially solvable if, and only if, there is a Turing machine that accepts precisely the
elements of P whose answer is yes.
A proof by the Church-Turing thesis is a shortcut often taken in establishing the existence
of a decision algorithm. Rather than constructing a Turing machine solution to a decision
problem, we describe an intuitively effective procedure that solves the problem. The
Church-Turing thesis asserts that a decision problem P has a solution if, and only if, there
is a Turing machine that determines the answer for every p  P . If no such Turing
machine
exists, the problem is said to be undecidable.
8.3 The Halting Problem for Turing Machines
Theorem 3.1
The halting problem for Turing machines is undecidable.
Proof: The proof is by contradiction. Assume that there is a Turing machine H that
solves the halting problem. A string is accepted by H if
i.
the input consists of the representation of a
Turing machine M followed by a string w
ii.
the computation of M with input w halts.
If either of these conditions is not satisfied, H rejects the input. The operation of the
machine H is depicted by the fig 28
Figure
28: Halting Machine
The machine H is modified to construct a Turing machine H ' . The computations of H '
are the same as H except H ' loops indefinitely whenever H terminates in an accepting
state, that is, whenever M halts on input w . The transition function of H ' is constructed
from that of H by adding transitions that causes H ' to move indefinitely to the right
upon entering an accepting configuration of H .
H ' is combined with a copy machine to construct another Turing machine D . The input
to D is a Turing machine representation R (M ) . A computation of D begins by creating
the string R( M ) R( M ) from the input R (M ) . The computation continues by running H '
on R( M ) R( M ) .
Figure 29: Turing Machine D with R(M) as input
The input to the machine D may be the representation of any Turing machine with
alphabet 0,1, B . In particular, D is such a machine. Consider a computation of D with
input R (D ) . Rewriting the previous diagram with M replaced by D and R (M )
by R (D ) , we get
Figure 30: Turing Machine D with R(D) as input
Examining the preceding computation, we see that D halts with input R (D ) if, and only
if, D does not halt with input R (D ) . This is obviously a contradiction. However, the
machine D can be constructed directly from a machine H that solves the halting
problem. The assumption that the halting problem is decidable produces the preceding
contradiction. Therefore, we conclude that the halting problem is undecidable.
Corollary 3.1 The language LH = {R(M )w | R(M )} where R (M ) is the representation
of a Turing machine M and M halts with input w over {0,1}* is not recursive.
8.4 A Universal Machine
The machine U is called a universal Turing machine since the outcome of the
computation of any machine M with input w can be obtained by the computation of U
with input R ( M ) w .
Figure 31: Universal Machine
Theorem 4.1
The language LH is recursively enumerable.
Proof:
A deterministic three-tape machine U is designed to accept LH by halting. A
computation of U begins with the input on tape 1. The encoding scheme presented in
Section 8.3 is used to represent the input Turing machine. If the input string has the
form R ( M ) w , the computation of M with input w is simulated on tape 3. The universal
machine uses the information encoded in the representation R (M ) to simulate the
transitions of M . A computation of U consists of the following actions:
1. If the string does not have the form R ( M ) w for a Turing machine M and string
w , U moves indefinitely to the right.
2. The string w is written on tape 3 beginning at position one. The tape head is then
repositioned at the leftmost square of the tape. The configuration of tape 3 is the
initial configuration of a computation of M with input w .
3. A single 1, the encoding of state q0 , is written on tape 2.
4. A transition of M is simulated on tape 3. The transition of M is determined by the
symbol scanned on tape 3 and the state encoded on tape 2. Let x be the symbol
from tape 3 and qi the state encoded on tape 2.
a.
Tape 1 is scanned for a transition whose first
two components match en(qi ) and en(x ) . If there is no such transition,
U halts accepting the input.
b.
Assume tape 1 contains the encoded transition
en(qi )0en( x)0en(q j )0en( y )0en(d ) .
Then
i.
en(qi ) is replaced by en( q j ) on tape 2.
The symbol y is written on tape 3.
The tape head of tape 3 is moved in the
direction Example 4.3-2n specified by
5. The next transition of M is simulated by repeating steps 4 and 5.
ii.
iii.
The simulations of the universal machine U accepts the strings in LH . The computations
of U loop indefinitely for strings in {0,1}*  LH . Since LH = L(U ) , LH is recursively
enumerable.
Corollary 4.1 The recursive languages are a proper subset of the recursively enumerable
languages.
Proof: The acceptance of LH by the universal machine demonstrates that LH is
recursivelyenumerable while Corollary 3.1 established that LH is not recursive.
Note: A language L is recursive if both L and L are recursively enumerable.
Corollary 4.2 The language LH is not recursively enumerable.
8.5 The Post Correspondence Problem
The undecidable problems presented in the preceding sections have been concerned with
the properties of Turing machines or mathematical systems that stimulate Turing
machines. The Post correspondence problem is a combinatorial question that can be
described as a simple game of manipulating dominoes. A domino consists of two strings
from a fixed alphabet, one on the top half of the domino and the other on the bottom
The game begins when one of the dominoes is placed on a table. Another domino is then
placed to the immediate right of the domino on the table. This process is repeated,
constructing a sequence of adjacent dominoes. A Post correspondence system can be
thought of as defining a finite set of domino types. We assume that there is an unlimited
number of dominoes of each type; playing a domino does not limit the number of future
moves. A string is obtained by concatenating the strings in the top halves of a sequence
of dominoes. We refer to this as the top string. Similarly, a sequence of dominoes defines
a bottom string. The game is successfully completed by constructing a finite sequence of
dominoes in which the top and bottom strings are identical.
Consider the Post correspondence system defined by dominoes in fig 32,
Figure 32: Post Correspondence System
The sequence in fig 33
Figure 33: Post Correspondence Solution
is a solution to this Post correspondence system.
Formally, a Post Correspondence System consists of an alphabet  and a finite set of
ordered pairs [ui , vi ], i = 1,2,, n, where ui , vi    . A solution to a Post correspondence
system is a sequence i1 , i2 ,, ik such that
ui1ui 2 ui k = vi1vi 2 vi k .
The problem of determining whether a Post correspondence system has a solution is the
Post correspondence problem.
Example 5.1
The Post correspondence
system
with
alphabet
[aaa, aa ], [baa, abaaa ] has a solution.
Figure 34: Example 5.1
{a , b}
and
ordered
pairs