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Lecture 2: Modeling Computers

David Evans

http://www.cs.virginia.edu/evans

cs302: Theory of Computation

University of Virginia

Computer Science

- Modeling Computers
- Course Organization
- Finite Automata

What can computers do?

Colossus (1944)

Cray-1 (1976)

Apollo Guidance Computer (1969)

Turing invented his model in 1936. What “computer” was he modeling?

IBM 5100 (1975)

...

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#

C

S

S

A

7

2

3

How long should the tape be?

“Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child’s arithmetic book.”

Alan Turing, On computable numbers, with an

application to the Entscheidungsproblem, 1936

- Rules for steps
- Remember a little

“For the present I shall only say that the justification lies in the fact that the human memory is necessarily limited.”

Alan Turing

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Input: 0

Write: 1

Move:

Start

A

B

Input: 0

Write: 1

Move:

Input: 1

Write: 1

Move: Halt

H

Input: 1

Write: 1

Move:

Copernicus

F = GM1M2 / R2

Newton

Ptolomy

- How well does it match “real” computers?
- Can it do everything they can do?
- Can they do everything it can do?

- Does it help us understand and reason about computing?
- What problems can computers solve?
- How long will it take?

Input: 2

Write: 1

Move:

Input: 0

Write: 1

Move:

Input: 1

Write: 2

Move:

Start

A

Input: 2

Write: 1

Move:

B

Input: 1

Write: 2

Move:

Input: 0

Write: 2

Move:

- Reading: mostly from Sipser, some additional readings later
- Problem Sets (6 – first is due in 1 week)
- Exams (2 + final)
- Extra credit:
- Challenge Problems
- Communication Efforts

- David Evans
- Office hours (Olsson 236A):
Mondays, 2-3pm

- Coffee Hours (Wilsdorf):
Wednesdays, 9:30-10:30am

- Other times: open office door, or send email to arrange

- Office hours (Olsson 236A):
- Assistants: Suzanne Collier, Qi Mi, Joe Talbott, Wuttisak Trongsiriwat
- Problem-Solving Sessions (Olsson 226D)
- Mondays 5:30-6:30pm, Wednesdays 6-7pm

First coffee hours and problem-solving session tomorrow

- Please don’t cheat!
- If you’re not sure if what you are about to do is cheating, ask first

- On most problem sets: “Gilligan’s Island” collaboration policy
- Encourages discussion in groups, but ensures you understand everything yourself
- Don’t use found solutions

- On most exams: work alone, one page of notes allowed

What problems can particular machines solve?

What is a problem?

What is a machine?

What does it mean for a machine to solve a problem?

Uninteresting: can be solved by a lookup machine

Finite

Problems

Problems with a finite number of possible inputs

Except for trick questions, all problems we

are interested in in this class have infinitely many possible inputs.

- How many possible outputs do you need for an interesting problem?

2 – “Yes” or “No”

Most problems can be framed as decision problems:

What is 1+1?

vs.

Is 1+1 = 3?

Undecidable Problems

Decidable problems

(problems that can be

solved by some TM)

Tractable problems

(problems that can be solved by some TM in reasonable time)

Regular Languages

(can be recognized by a DFA)

Context-Free Languages

(can be recognized by a PDA)

- Recognize binary strings with an even number of “1”s

What is a language?

What does it mean to recognize a language?

- Example: design a DFA that recognizes the language of binary strings that are divisible by 3
- Design tips:
- Think about what the states represent (e.g., what is the current remainder)
- Walk through what the machine should do on example inputs

A finite automaton is a 5-tuple:

Qfinite set (“states”)

- finite set (“alphabet”)
- Q x Q(“transition function”)
q0 Qstart state

F Qset of accepting states

- Define * as the “extended transition function”
*: Q x * Q

Basis: *(q, ε) = q

Induction:

w = ax a , x *

*(q, w) = *((q, a), x)

- w L(A) iff *(q0, w) F

code example:

State nextState(State q, String w) {

if (w.length() == 0) return q;

else return (nextState (transition (q, w.charAt(0)),

w.substring(1));

}

- Definition: A language is a regular language if there is some Finite Automaton that recognizes it.

- Prove: the set of regular languages is closed under complement.

- Remember to submit registration survey
- PS1 is posted on course website: due 1 week (- 73 minutes) from now
- Coffee hours tomorrow (9:30am)
- Problem-solving session tomorrow (6pm)