Lecture 2: Modeling Computers

cs302: Theory of Computation University of Virginia Computer Science David Evans http://www.cs.virginia.edu/evans

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• Modeling Computers • Course Organization • Finite Automata

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What can computers do?

What is a “computer”?

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How should we model a Computer?

Colossus (1944) Cray-1 (1976) Apollo Guidance Computer (1969)

Lecture 2: Modeling Computation

IBM 5100 (1975)

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

“Computers” before WWII

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Mechanical Computing

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Modeling Pencil and Paper

...

# 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

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Modeling Brains

•Rules for steps •Remember a little

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

Lecture 2: Modeling Computation

Turing’s Model

...

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

Input: 0 Write: 1 Move:  Start

A

Input: 0 Write: 1 Move: 

B

Lecture 2: Modeling Computation

Input: 1 Write: 1 Move:  Input: 1 Write: 1 Move:

Halt

H

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What makes a good model?

Ptolomy

Lecture 2: Modeling Computation

Copernicus F = GM 1

M

2 / R 2 Newton

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Questions about Computing Model

• 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?

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Universal Turing Machine

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: 

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Course Organization

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Assignments

• 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

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Help Available

• 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 • 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

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Honor Code

• 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

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Main Question

What

problems machines

solve?

can particular What is a problem ?

What is a machine ?

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

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Finite Problems Problems with a finite number of possible inputs Uninteresting: can be solved by a lookup machine Except for trick questions, all problems we are interested in in this class have infinitely many possible inputs.

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Outputs

• 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?

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Decidable problems

(problems that can be solved by some TM)

Undecidable Problems 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)

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Finite Automata (Finite State Machines)

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Informal Example

• Recognize binary strings with an even number of “1”s What is a language?

What does it mean to recognize a language?

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Designing DFAs

• 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

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Formal Definition

A finite automaton is a 5-tuple:

Q

finite set (“states”)  

Q

x

q

0

F

 

Q Q

  finite set (“alphabet”)

Q

(“transition function”) start state set of accepting states

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Computation Model

• 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  *(q 0 , w)  F

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Inductive Definitions

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)); }

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Regular Languages

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

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Complement Proof

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

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Charge

• 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)

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