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Lecture 2: Modeling Computers

This lecture explores the history and models of computers, including finite automata. It also discusses the concept of modeling brains and the importance of good models in understanding computing. The lecture covers regular languages and the design of finite automata.

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Lecture 2: Modeling Computers

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  1. Lecture 2: Modeling Computers David Evans http://www.cs.virginia.edu/evans cs302: Theory of Computation University of Virginia Computer Science

  2. Menu • Modeling Computers • Course Organization • Finite Automata

  3. What can computers do?

  4. What is a “computer”?

  5. How should we model a Computer? Colossus (1944) Cray-1 (1976) Apollo Guidance Computer (1969) Turing invented his model in 1936. What “computer” was he modeling? IBM 5100 (1975)

  6. “Computers” before WWII

  7. Mechanical Computing

  8. 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

  9. Modeling Brains • 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

  10. 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 B Input: 0 Write: 1 Move:  Input: 1 Write: 1 Move: Halt H Input: 1 Write: 1 Move: 

  11. What makes a good model? Copernicus F = GM1M2 / R2 Newton Ptolomy

  12. 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?

  13. 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: 

  14. Course Organization

  15. 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

  16. 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

  17. 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

  18. Main Question 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?

  19. 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.

  20. 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?

  21. 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)

  22. Finite Automata(Finite State Machines)

  23. Informal Example • Recognize binary strings with an even number of “1”s What is a language? What does it mean to recognize a language?

  24. 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

  25. Formal Definition A finite automaton is a 5-tuple: Qfinite set (“states”) • finite set (“alphabet”) • Q x   Q (“transition function”) q0  Q start state F  Q set of accepting states

  26. 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 *(q0, w)  F

  27. 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)); }

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

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

  30. 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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