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notes13 Introduction to Turing Machines

notes13 Introduction to Turing Machines - CS 373 Theory of...

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CS 373: Theory of Computation Gul Agha Mahesh Viswanathan Fall 2010 1

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1 Unrestricted Computation General Computing Machines Machines so far: DFAs, NFAs, GNFAs Limitations on how much memory they can use: fixed amount of memory Limitations on what they can compute/decide: only regular languages The complete machine? No limitations on memory usage? And maybe other ways to use computational resources that we haven’t thought of... * Come up with a model that describes all “conceivable” computation No limitation on what they can compute? * No! There are far too many languages over { 0 , 1 } than there are “machines” or programs (as long as machines can be represented digitally) General Computing Machines Alonzo Church, Emil Post, and Alan Turing (1936) Figure 1: Alonzo Church Figure 2: Emil Post 2
Figure 3: Alan Turing Church ( λ -calculus), Post (Post’s machine), Turing (Turing machine) independently came up with formal definitions of mechanical computation All equivalent! In this course: Turing Machines 2 Turing Machines 2.1 Definition The ‘aha’ moment Figure 4: Turing Machines 3

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X 1 X 2 · · · X n t t finite-state control tape head Unrestricted memory: an infinite tape A finite state machine that reads/writes symbols on the tape Can read/write anywhere on the tape Tape is infinite in one direction only (other variants possible) Initially, tape has input and the machine is reading (i.e., tape head is on) the leftmost input symbol. Transition (based on current state and symbol under head): Change control state Overwrite a new symbol on the tape cell under the head Move the head left, or right. Turing Machines Formal Definition A Turing machine is M = ( Q, Σ , Γ , δ, q 0 , q acc , q rej ) where Q is a finite set of control states Σ is a finite set of input symbols Γ Σ is a finite set of tape symbols. Also, a blank symbol
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