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# LectureNotes17 - ECS 120 Lesson 17 Churchs Thesis The...

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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 low-level models that we have seen so far. 1 The Church-Turing 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 “machine-based,” 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, higher-level 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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LectureNotes17 - ECS 120 Lesson 17 Churchs Thesis The...

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