ECS 120 Lesson 17 – Church’s Thesis, The
Universal Turing Machine
Oliver Kreylos
Monday, May 7th, 2001
In the last lecture, we looked at the computation of Turing Machines,
and also at some variants of Turing Machines – nondeterministic Turing
Machines and enumerating Turing Machines.
From now on, we will shift
our attention away from machines and languages, and will start reasoning
about algorithms and problems instead.
Before we can do so, we have to
understand how different models of computation – one of them being Turing
Machines – are related to each other.
This will allow us to reason about
algorithms without necessarily having to use the lowlevel models that we
have seen so far.
1
The ChurchTuring Thesis
Parallel to the development of the Turing Machine by Alan Turing, the math
ematician Alonzo Church invented a different model of computation, the
λ

calculus
. As opposed to Turing Machines, this model is not “machinebased,”
but relies on the application of mathematical functions to their arguments.
λ
calculus inspired development of the LISP programming language by John
McCarthy in the late 1950’s. LISP, in fact, is an almost direct implementa
tion of Alonzo Church’s ideas. Though these two models are very different
in their structure, Church and Turing were later able to prove that their
models are equivalent – any computation that can be described by one can
also be described by the other. Later, other models of computation were in
vented (random access machine, register machine, higherlevel programming
languages,. . . ); all of these could be proven to be equivalent to each other.
1
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This somewhat surprising observation led computer scientists to believe
that all these models are in fact describing the same thing – algorithms, in
the intuitive meaning that they are “recipes for computation.” Recall: An
algorithm is a strategy for solving a problem, which satisfies the following
three properties:
1. An algorithm is a
clearly
and
unambiguously defined
.
2. An algorithm is
effective
, in the sense that each of its steps is
executable
.
3. An algorithm is
finite
, in the sense that the textual description itself is
of finite length, and that it always terminates after a finite number of
steps.
The equivalence of all known models of computation led Alonzo Church to
formulate the following hypothesis:
Hypothesis 1 (Church–Turing Thesis)
Every conceivable model of com
putation is equivalent to the intuitive notion of computation, defined by algo
rithms.
The reason for this hypothesis not being called a theorem is that it can
not be proven true. Since the “intuitive notion of algorithm” does not have
an exact mathematical definition, one cannot reason about it with mathe
matical means. Though it is not a theorem, the Church–Turing thesis is very
valuable for reasoning about algorithms. It has an analogous impact to the
theorem that for every regular language there must be a finite state machine
that accepts it.
This fact, for example, led us to discovery of the Pump
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 Spring '07
 Filkov
 Turing Machines

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