EE 109 Unit 20 Theoretical Computer Science and Turing Machines 2 - - PowerPoint PPT Presentation

1

EE 109 Unit 20 – Theoretical Computer Science and Turing Machines

2

THEORETICAL COMPUTER SCIENCE

Credit: Adapted from Gaurav Sukhatme's CSCI 109 Lecture Slides

3

Computability

• Theoretical Computer Science

– The study of what can be computed and how efficiently – Uses mathematical models – Tries to generalize computers so that what is proven is applicable to ANY computer system no matter its speed, # of processors, etc.

4

State Machine Review

• Recall that a state machine is defined by a 6-tuple:

– A set of possible input values – A set of possible states – A set of possible outputs – An initial state – A transition function: {States x Inputs} -> the Next state – An output function: {States x Inputs} -> Output value(s)

S0

Out=False

S1

Out=False

S2

• ut=True

Input=1 Input=1 Input=0 Input=1 Input=0 Input=0 On Reset

5

Formal Definition

• Mathematically, a state machine consists of:

– A set of possible input values: {0, 1} – A set of possible states: {S0, S1, S2} – A set of possible outputs: {False, True} – An initial state = S0 – A transition function:

• {States x Inputs} -> the Next state

– An output function:

• {States x Inputs} -> Output value(s)

A

Out=False

B

Out=False

C

• ut=True

Input=1 Input=1 Input=0 Input=1 Input=0 Input=0 On Reset

Inputs State 1 S0 S0 S1 S1 S0 S2 S2 S0 S2

State Transition Function Output Function

Inputs State 1 S0 False False S1 False False S2 True True

6

Limits of FSM's

• Finite state machines have certain limits

– Why? Because they have a finite number of states!

• Problem: Given an arbitrary length string of A's and B's

determine if there is an equal number of A's and B's

– Key is "arbitrary length" – Show execution for "AABABBBBA"

Equal 1 Extra A 1 Extra B 2 Extra As 2 Extra B's

A A A A B B B B A A B B initial

7

Turing Machine

• A Turing Machine was a fictitious machine / mathematical

model of a machine proposed by Alan Turing

• It consisted of a controller which was an FSM and an "infinite"

roll of tape marked into distinct squares with input symbols in each square

– The tape was like memory/RAM and so we could now process an arbitrarily long input sequence with a finite set of states because we could use the tape to store results – It had a read/write "head" that could read and write the symbol in the current tape square

8

Formal Definition of a Turing Machine

• Set of input symbols
• Set of output symbols
• Set of states
• Initial state and initial square location
• State transition function {State x Input => Next State}
• Action Function {State x Input => Action}

– At each state the Turing machine could

• Read a symbol from the current square of the tape

– Based on the state and the symbol read it could then

• Erase a symbol from the tape
• Write a symbol onto the tape (erasing the old symbol)
• Do nothing

– ..and..

• Move the tape right or left or not move {R, L, N}

A A B A B B B

Turing Machine

9

Sample Turing Machine

• Turing Machine to detect equal/unequal number of A's and B's
• The machine works by making repeated passes through the

string, each time removing one A and one B. Eventually string is either empty (A’s = B’s) or something is left (A’s ≠ B’s)

• The steps the machine takes…

– Start at beginning of string (left side) – Move right looking for either an A or B or end of string

• If end of string found, number of A’s and B’s equal. Done.
• If A or B found, replace it with an X.

– Continue to move right, looking for the other character or end of string

• If other character is found replace it with an X, back to start of string, repeat.
• If end of string reached (other character was NOT found), number A’s and B’s unequal.

Done.

10

Sample Turing Machine

• Turing Machine to detect equal/unequal number of A's and B's

– Sample operation on "AAB"

Start

• A A B -

A,RT

• X

A B -

A,RT

• X

A B -

Lt

• X

A X

• Lt
• X

A X

• Lt
• X

A X

• Find
• X

A X

• Find
• X

A X

• A,Rt
• X

X X

• A,Rt
• X

X X

• UNEQ
• X

X X

• Start

EQ A,Rt B,Rt UNEQ Lt Find First

'A','X',R 'B','X',R '-','-',N '-','-',N 'B','X',L 'A','X',L '-','-',R 'A','X',R 'B','X',R '-','-',N {'A','X','B'}, '-',L 'X',-,R 'X',-,R 'X','-',R

11

TM Highlights

• Programming a Turing machine is not fun
• But anything a current computer can do, a Turing machine can do and vice versa
• We can use Turing machines to prove what current computers' can't do

– What is computable? – It a Turing Machine can't do it, a computer as we know it today can't do it

• Example of what isn't computable?

– Halting Problem: Write a program that is given another program as input and determines if that program will eventually halt. – Turing proved this is not computable in a famous paper

• Turing also proved that there is a Turing Machine that could take as input a

description of another Turing Machine and then emulate its operation

– Universal Turing Machine – Forerunner of the "Stored Program Concept" (instructions & data are treated similarly)

• If an algorithm is O(polynomial) on TM then it is also O(polynomial) on a

normal computer as well

– Modern computers just seek to increase performance

12

CENG

Pre-requisite structure

Your Future CENG Courses

EE 109L (3)

Intro to CENG

<< None >>

EE 277L (2)

Fundamentals of Digital Circuits << EE 109 >>

EE 154 (2)

• Intro. to Logic Design

<< EE 109 >>

EE 254L (4)

Digital Sys. Design

<< EE 154 >>

EE 457 (3)

Computer Sys. Org. << EE 254L >>

EE 454L (4)

• Intro. to SoC

<< EE 254L >>

EE 451 (3)

Parallel & Distributed << EE 355 or CSCI 104>>

CS 353 (3) or EE 450(3)

Networks

<< CS 350 >>

CS 350 (4)

Intro to OS

EE 477L (4)

VLSI

<< EE 277L, 254L >>

13

Congratulations!!

• You have learned a LOT this semester
• You should be proud of yourself and what you've

designed, programmed, and built

• This is just the beginning…you will learn how to

design all kinds of complex and exciting software and hardware systems

– SW and HW are integrally intertwined…the more you know about each the better engineer you will be

• Stay curious! Don't be afraid to try, fail, then

debug