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An introduction to Universal Turing Machines (UTMs), their description, enumeration, and the equivalence with Lambda Calculus. UTMs are abstract computing machines that can simulate any other Turing Machine, making them universally capable of performing any mechanical computation. Lambda Calculus is a system of symbolic computation, which can be given meaning to correspond to computations. Both UTMs and Lambda Calculus provide precise and formal rules for manipulating symbols, enabling reasoning about programs and computations.
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David Evans http://www.cs.virginia.edu/evans CS200: Computer Science University of Virginia Computer Science
5 April 2004 CS 200 Spring 2004 4
TuringMachine ::= < Alphabet , Tape , FSM > FSM ::= < States , TransitionRules , InitialState , HaltingStates > States ::= { StateName* } InitialState ::= StateName must be element of States HaltingStates ::= { StateName* } all must be elements of States TransitionRules ::= { TransitionRule* } TransitionRule ::= < StateName , ;; Current State OneSquare, ;; Current square StateName, ;; Next State OneSquare , ;; Write on tape Direction > ;; Move tape Direction ::= L , R , # Start 1 HAL T ), X, L 2: look for ( ), #, R (, #, L (, X, R #, 1, #^ #, 0, # Transition Rule is a procedure: StateName X OneSquare StateName X OneSquare X Direction
Number of TM
also, just a number! Output Tape for running TM- P in tape I Can we make a Universal Turing Machine?
Alonzo Church, 1940 (LISP was developed from -calculus, not the other way round.) term = variable | term term | ( term ) | variable. term
y. M v. ( M [ y v ]) where v does not occur in M. M M M N PM PN M N MP NP M N x. M x. N M N and N P M P