Turing Machines CS 105: Introduction to Computer Science

Some Interesting Questions u What is a computer? What is computation? u Are there problems that some computers can solve but others can’t? u Are there problems that no computer can solve?

What is a Computer? u That’s a tough question. u We want to talk about the fundamental properties of computation. u We need a model that captures the essential and throws out the rest.

What is Computation? u Mapping an input to an output. u For the sake of concreteness: mapping a binary input to a binary output. u Problems describe a particular mapping we are interested in. u Decision problems are a subset of all problems - map from inputs to 0 or 1.

First model of computation: Finite State Automata… A B

Finite State Automata u A fixed number of states. u The machine moves from one state to another in response to inputs. u If it ends in an ACCEPT state it accepts, otherwise it rejects. (We’re just considering decision problems for now.) u Let’s see some examples…

FSA Example A B Triangle indicates start state. Double circle indicates accepting state. What happens on input ?

FSA Example A B

FSA Example A B

FSA Example A B

FSA Example A B

FSA Example A B

FSA Example A B ACCEPT!

The Turing Machine u Problems with the FSA as a model of computation?? –Does an input string have matched parentheses? E.g., 01(0(10)101)110(1 u With a slight modification we get the Turing machine. u First described in 1937 by Alan Turing

How does it work?... Tape Read/Write Head Control Device Current State: Rules:

What can it do? u Write a character to the current tape cell u Move Read/Write Head one cell to the left or right u Go into a new state... Tape Read/Write Head Control Device Current State: Rules:

How does it decide what to do? Behavior is based on: u Input: the character in the current tape cell u State: the current state... Tape Read/Write Head Control Device Current State: Rules:

Let’s see one in action... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 1 Rules: 1001

What are current state and input?... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 1 Rules: 1001

Which rule will the TM follow?... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules: | 1 R 1 1

Write to the tape... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules: | 1 R 1 1

Move right or left... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules: | 1 R 1 1

Go to new state: Done with rule... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules: | 1 R 1 1

What are current state and input?... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 1 Rules: 1001

Which rule will the TM follow?... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules: | 1 R

Write to the tape... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules: | 1 R 2 11

Move right or left... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules: | 1 R 2 11

Go to new state: Done with rule... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules: | 1 R 2 11

What are current state and input?... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules:

Which rule will the TM follow?... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules: | 1 R

Write to the tape... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules: | 1 R

Move right or left... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules: | 1 R

Go to new state: Done with rule... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules: | 1 R

What are current state and input?... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules:

Which rule will the TM follow?... Tape Read/Write Head Control Device Current State: State Symbol | Write Move State 1 1 | 1 R | 1 R | 1 R 2 Rules:

Is the Turing Machine The Right Model? u The Church Turing Thesis: –Any reasonable model of computation is equivalent to the Turing machine. u “Equivalence” here refers to what can be computed, not how fast it can be done. u Not something that can be proved.

Why do we believe it? u Every TM extension to make it “more powerful” (e.g., multiple-tape TMs) has been shown to be equivalent to the basic Turing Machine. u Other models of computation have been shown to be weaker (e.g., Finite State Machines) or equivalent.

What is the significance of the Church-Turing Thesis? u Church-Turing Thesis: Anything you can do on any computer you can do with a Turing Machine. u So, no computer is more “powerful” (can compute more) than a TM.

Is my computer LESS powerful than a Turing Machine? u No. u Given an infinite input device (infinite tape, infinite number of floppies, infinite keyboard input), you can simulate a TM. u Therefore, anything that can be computed on a TM can be computed on your computer.

The Important Implication u If there’s anything that we CAN’T do with a Turing Machine then... we can’t do it on any computer now or in the future.

Are there things we can’t compute on a Turing Machine? YES An Example: The Halting Problem

Last Words. u Turing machines are –Hard to program. –Easy to prove things about. u If we can prove something about the capabilities of Turing machines, we can immediately apply it to all computers.

Regular Expressions u Another formalism for selecting strings to accept or reject. u Intimately related to FSAs u Practical applications in searching through strings.

Three Operators u Union: (0+1) –accepts/matches either 0 or 1. u Concatenation: 01 –matches 0 followed by 1 u Repetition: 0* –Matches any number of zeros (including 0)

Examples u These operators can be applied to regular expressions: –(0+1)*1 (matches any string that ends in a 1) –(0+1)(0+1))* (matches any even length string)