Enhanced Turing Machines
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1 Enhanced Turing Machines Lecture 28 Sections Robb T. Koether Hampden-Sydney College Wed, Nov 2, 2016 Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
2 1 Variants of Turing Machines A Stay Option One-Way Infinite Tape Multiple Tapes Examples 2 Assignment Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
3 Outline 1 Variants of Turing Machines A Stay Option One-Way Infinite Tape Multiple Tapes Examples 2 Assignment Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
4 Increasing the Power of a Turing Machine It is hard to believe that something as simple as a Turing machine could be powerful enough to solve complicated problems. We can imagine a number of improvements. A Stay option Multiple tapes One-way infinite tape Two-dimensional tape (n-dimensional tape) Addressable memory Nondeterminism etc. Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
5 Outline 1 Variants of Turing Machines A Stay Option One-Way Infinite Tape Multiple Tapes Examples 2 Assignment Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
6 A Stay Option Rather more left or right on every transition, we could allow the Turing machine to stay at its current tape position. Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
7 A Stay Option Theorem Any Turing machine with a Stay option is equivalent to some standard Turing machine. Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
8 Outline 1 Variants of Turing Machines A Stay Option One-Way Infinite Tape Multiple Tapes Examples 2 Assignment Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
9 One-way Infinite Tape Would a Turing machine with a one-way infinite tape be more powerful than a standard Turing machine? Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
10 One-way Infinite Tape Theorem Any Turing machine with a one-way infinite tape is equivalent to a standard Turing machine. Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
11 One-way Infinite Tape We can use a one-tape machine to simulate the two-way infinite tape. Establish a tape position that marks the left edge of the one-way tape. Symbols that would be to the right of that mark will occupy the odd positions. Symbols that would be to the left of that mark will occupy the even positions. Modity the transitions accordingly. Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
12 Outline 1 Variants of Turing Machines A Stay Option One-Way Infinite Tape Multiple Tapes Examples 2 Assignment Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
13 Multiple Tapes Would a Turing machine with k tapes, k > 1, be more powerful than a standard Turing machine? Each tape could be processed independently of the others. In other words, each transition would read each tape, write to each tape, and move left or right independently on each tape. Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
14 Multiple Tapes Theorem Any language that is accepted by a multitape Turing machine is also accepted by a standard Turing machine. Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
15 Multiple Tapes Proof (sketch): Let the tape contents be: Tape 1: x 11 x 12 x x 1n1 Tape 2: x 21 x 22 x x 2n2.. Tape k: x k1 x k2 x k3... x knk Then we would write on a single tape #x 11 x x 1n1 #x 21 x x 2n2 #... #x k1 x k2... x knk #. Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
16 Multiple Tapes Proof (sketch): To show the current location on each tape, put a special mark on one of that tape s symbols: #x 11 x x 1n1 #x 21 x x 2n2 #... #x k1 x k2... x knk # Begin with #x 11 x x 1n1 # #... # # Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
17 Multiple Tapes Proof (sketch): The Turing machine scans the tape, locating the current symbol on each tape. It then makes the appropriate transition. Write a symbol in each of the current positions. Move the location of the current symbol one space left or right for each tape. Of course, the devil is in the details. Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
18 Outline 1 Variants of Turing Machines A Stay Option One-Way Infinite Tape Multiple Tapes Examples 2 Assignment Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
19 Binary Addition Example (Binary Addition) Binary addition is much simpler if we have a three-tape machine. Write x on tape 1 and write y on tape 2. Write the sum on tape 3. For simplicity, assume fixed-length integers. Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
20 Outline 1 Variants of Turing Machines A Stay Option One-Way Infinite Tape Multiple Tapes Examples 2 Assignment Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
21 Assignment Homework Use JFLAP to do the following: Design a 3-tape machine that will do subtraction of fixed-length integers. Allow the results to wrap around. That is, if y > x, then the result of x y will be 2 n + (x y). Design a 3-tape machine that will accept the language {a n b n c n n 0}. Robb T. Koether (Hampden-Sydney College) Enhanced Turing Machines Wed, Nov 2, / 21
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