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the instructions in order to carry them out. An executing Turing machine just
performs meaningless string processing.
The grammar of the language is the specification of what is well-formed
programs(sets of instructions) and what constitutes well-formedinstantaneous
descriptions,in particularthe initial description. The computationalrules,also
syntactically defined, then tells us how to perform computations within the
model. The data itself has no meaning, it is just a string of symbols.
The semantics of the model enters through the interpretation of the initial
and final tape expressions as defining sets of natural numbers, or objects from
some other set of mathematical objects, functionally related via the computa-
tion. Thus, we can say that we understand a Turing machine if we understand
what it computes.
2.3.4 Decision procedures and Computation procedures
revisited
Wewillnowrefineournotionsaboutalgorithmsfordecisionsandcomputations
respectively.
Deciding recursive languages
Let L Γ be a language, i.e. a set of strings defined over the alphabet Γ.
∗
⊂
Next, let M be a Turing machine and x Γ be an input string. We say that
∗
∈
M decides the language L if the following conditions hold
M(x) q if x L
≻ y ∈ . (2.11)
(cid:26)M(x) q if x / L
n
≻ ∈
Here we use the notation M(x) q to denote the sentence: ”The machine
y
≻
M on input string x halts in the configuration q ”, and similarly in the other
y
case.
IfthelanguageLisdecidedbysomeTuringmachineM,thenLisarecursive
language.
There is also a weaker form of decision procedures.
Recognizing recursively enumerable languages
We say that M recognizes the language L if the following conditions hold
M(x) q if x L
≻ y ∈ , (2.12)
(cid:26)M(x) ⊲ if x / L
≻ ∈
30
where by M(x) ⊲ we denote the sentence: ”The machine M on input string
≻
x does not halt”.
If the language L is accepted by some Turing machine M, then L is a
recursively enumerable language. It is obvious that a recursive language is also
recursivelyenumerable. Theweakerformisstillstrongenoughtoenumeratethe
strings in the language. By judiciously employing the machine M that accepts
the language, the strings of the language can be enumerated. The intuition is
that, if a string belongs to the language, it will be found eventually. But for a
string not yet accepted, there is no way of ascertaining that it does not belong
to the language.
When a machine is used for decision problems, the output is really encoded
in the halting states q ,q and the tape contents at halting have no special
y n
{ }
significance.
Computing recursive functions
Suppose that f is a function from Γ to Γ . If there is a Turing machine that
∗ ∗
computes f as in (2.9) and (2.10), f is called a recursive function.
2.3.5 The Church-Turing Thesis
TheChurch-Turingthesisidentifiesthesetofeffectively(intuitively)computable
functions with the set of functions computable within any of the classical com-
putational models; Turing machines, λ-definable functions or general recursive
functions. It was originally formulated by Church in terms of general recursive
functions,butTuringmadesimilarremarksinreferencetohismodel, hencethe
name Church-Turing thesis (see several articles in [13]). Historically, effective
computability meant computability by a human computor who works to pre-
cise rules. Later the thesis has acquired connotations connecting it to machine
computation, in particular electronic digital computing machines. In this sense
the thesis is certainly true; what can be computed by a general purpose digital
computer can be computed by a Turing machine.11
The literature contains stronger statements to the effect that anything that
can be computed by a machine can be computed by a Turing machine (for a
discussion,see[12]). Thisisamuchstrongerstatement. Itisastatementabout
every conceivable physical system that can be harnessed to perform computa-
tions. Whether it is true or not is unknown. To determine if this statement
might be true or not, we would have to analyze the general computational
characteristics of physical systems. Such an investigation seems to require a
complete theory, or set of theories, covering all of physics. Even though many
physicist are pursuing research into finding a ”Theory of Everything”, it is far
from clear whether such a theory exists, or if it can be found in any near fu-
ture. And should such a theory exist, we know nothing of its implications for
computability.
11TheroleoftheTuringmachinemodelofcomputationforthedevelopmentofthemodern
digitalcomputer isdiscussedin[21].
31
Itisnotreallyprimarilyaquestionoftheory,butratherofphenomena. New
physicalphenomenamightverywellbediscoveredinthefuturethatrequirenew
theoretical concepts for their explication.
2.3.6 Computability
Aclassicresultin the theoryofcomputabilityis thatthere arenon-computable
functions. This follows, almost trivially, once one has accepted the following
three propositions;
(i) the set of Turing machine programs is enumerable,