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The Turing Machine A definition of computability by Noah Richards

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Computability In the 1930s, mathematicians were looking for a definition of what it means to be able to compute a function. They came up with: A function is computable if it can be computed by a Turing machine.

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What is a Turing Machine? Invented by Alonzo Church and Alan Turing, it is an incredibly simple machine with all of the power of any digital computer.

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Parts of a Turing Machine 1) An infinite tape that is bi-directional and made up of squares containing a symbol from a finite alphabet. It contains only finitely many non-blank squares on the tape.

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Parts, cont. 2) A read/write head positioned at some square on the tape. 3) A finite number of internal states at which it can be at exactly one at any given time.

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Using a Turing Machine The general form of instruction for a touring machine is: (current_state, current_symbol, new_state, new_symbol, left/right)

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Using, cont. An example of this is: (s1, 1, s2, 0, R) This means when the machine is in state s1 and reads a 1, it transfers to state s2, writes a 0 (to the same cell), and moves to the right one cell

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Multiply Lets assume we want to write a program to multiply two numbers. We define this function as: multiply(n1, n2) = n1 * n2 As per common Turing convention, we will be representing numbers in unary (strings of 1s) separated by spaces (_s)

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Heres the code: Multiply function Turing Machine code Step by step output for the multiply function

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Virtual Turing Machine This code was created and run on the Virtual Turing Machine Virtual Turing Machine

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Other sources of information The Turing Machine, written in JavaScriptThe Turing Machine, written in JavaScript with examples in unary addition, divisibility, and primality testing You guessed it: a Turing Machine written in schemea Turing Machine written in scheme

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Moral of this story: Enjoy writing code in Scheme and Haskell. After all, you could be writing it for a Turing Machine.

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CS 154 Formal Languages and Computability April 5 Class Meeting Department of Computer Science San Jose State University Spring 2016 Instructor: Ron Mak.

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