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Lecture 24: Metalinguistics

This lecture discusses the concepts of decidability, NP-completeness, and metalinguistic abstraction in computer science. It also explores the implications of quantum computing and the challenges of solving problems with real computers.

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Lecture 24: Metalinguistics

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  1. Lecture 24: Metalinguistics David Evans http://www.cs.virginia.edu/evans CS200: Computer Science University of Virginia Computer Science

  2. Menu • Theory and Reality • Nondeterministic Computing • PS7: Quantum Computing • Making New Languages CS 200 Spring 2003

  3. Problem Classification To show a problem is decidable/in NP/in P, you need to show it is easy enough to be solved with a procedure in that class: • Decidable: it is easy enough to be solved by some procedure that always terminates • NP: it is easy enough to be solved by a nondeterministic polynomial procedure: try an exponential number of guesses, but takes only P-time to check one if correct • P: it is easy enough to be solved by a polynomial time procedure – O (nk) harder to show, means problem is easier CS 200 Spring 2003

  4. To show a problem is undecidable or NP-complete, you need to show it is as hard as a problem you know enough to be solved with a procedure in that class: • Undecidable: if you had a procedure that solves this problem, you could use it to solve a known undecidable problem (e.g., the halting problem) • NP-Complete: if you had a procedure that solves this problem, you could use it to solve a known NP-Complete problem (e.g., SAT, Smiley Puzzle) • Subtlety: the transformation of the problem and answer must be in P CS 200 Spring 2003

  5. Intellectual Computability Discussion on TV CS 200 Spring 2003

  6. Ali G Multiplication Problem • Input: a list of n numbers • Output: the produce of all the numbers Is it decidable? Yes – a straightforward algorithm solves it. Is it tractable? Yes – it is in P Can real computers solve it? CS 200 Spring 2003

  7. What about C++? int main (void) { int alig = 999999999; printf ("Value: %d\n", alig); alig = alig * 99; printf ("Value: %d\n", alig); alig = alig * 99; printf ("Value: %d\n", alig); alig = alig * 99; printf ("Value: %d\n", alig); } Results from SunOS 5.8: Value: 999999999 Value: 215752093 Value: -115379273 Value: 1462353861 CS 200 Spring 2003

  8. Ali G was Right! • Theory assumes ideal computers: • Unlimited memory • Unlimited power • Unlimited (finite) time • We will formalize this later… • Real computers have: • Limited memory, time, power outages, flaky programming languages, etc. • There are many decidable problems we cannot solve with real computer: the numbers do matter CS 200 Spring 2003

  9. Quantum Computing CS 200 Spring 2003

  10. Quantum Mechanics for Dummies • Light behaves like both a wave and a particle at the same time • A single photon is in many states at once • Observing its state forces it into one state Books Search Results: we were unable to find exact matches for your search forscheme for dummies CS 200 Spring 2003

  11. Qubit • Quantum Bit • A superposition of both 0 and 1 • Represents both 0 and 1 at the same time! • Some probability it will be 0 when it is observed • Some probability it will be 1 when it is observed • Two qubits can represent 0, 1, 2 and 3 all at the same time CS 200 Spring 2003

  12. Metalinguistic Abstraction CS 200 Spring 2003

  13. Solving Problems: Recap • PS1-4: • Divide a problem into procedures that can be combined to solve it • PS5: • Divide a problem into procedures and state that can be combined to solve it • PS6: • Divide a problem into objects that can be used to model it CS 200 Spring 2003

  14. Solving Problems • PS7: • Divide a problem into creating a good language for solving the problem, and defining a solution using that language Languages change the way we think. Sometimes the best way to solve a problem is to invent a new language first. CS 200 Spring 2003

  15. PS7: Quantum Scheme Extend Mini-Scheme evaluator to support special forms: (quist ExpressionList) Abstraction for an infinite number of qubits Represents all of the values the expressions evaluate to at once. (observe ProcedureQuist) Turns a quist into a normal value. Picks one of the quist values for which (Procedure value) is true. CS 200 Spring 2003

  16. Schrödinger’s Cat • Put a live cat in a box with cyanide vial that opens depending on quantum state • Cat is both dead and alive at the same time until you open the box (qeval ‘(define cat (quist “alive” “dead”))) (qeval ‘(observe (lambda (q) #t) cat) “dead” or“alive” Its only a thought experiment, not necessary to try with real cat! CS 200 Spring 2003

  17. “Jamais Jamais Jamais” from Harmonice Musices Odhecaton A. Printed by Ottaviano Dei Petrucci in 1501 (first music with movable type) CS 200 Spring 2003

  18. “Jamais Jamais Jamais” from Harmonice Musices Odhecaton A. (1501) J S Bach, “Coffee Cantata”, BWV 211 (1732) www.npj.com/homepage/teritowe/jsbhand.html CS 200 Spring 2003

  19. Inventing a Language • Design the grammar • What strings are in the language? • Use BNF to describe all the strings in the language • Make up the evaluation rules • Describe what everything the grammar can produce means • Build an evaluator • A procedure that evaluates expressions in the language CS 200 Spring 2003

  20. Is this an exaggeration? (SICP, p. 360) It is no exaggeration to regard this as the most fundamental idea in programming: The evaluator, which determines the meaning of expressions in the programming language, is just another program. To appreciate this point is to change our images of ourselves as programmers. We come to see ourselves as designers of languages, rather than only users of languages designed by others. CS 200 Spring 2003

  21. Programming an Evaluator If a language is just a program, what language should we program the language in? CS 200 Spring 2003

  22. The Metacircular Evaluator CS 200 Spring 2003

  23. Environmental Model of Evaluation • To evaluate a combination, evaluate all the subexpressions and apply the value of the first subexpression to the values of the other subexpressions. • To apply a compound procedure to a set of arguments, evaluate the body of the procedure in a new environment. To construct this environment, make a new frame with an environment pointer that is the environment of the procedure that contains places with the formal parameters bound to the arguments. CS 200 Spring 2003

  24. Eval Eval and Apply are defined in terms of each other. Apply CS 200 Spring 2003

  25. meval (define (meval expr env) (cond ((self-evaluating? expr) expr) ((variable? expr) (environment-lookup-name expr env)) ((lambda? expr) (make-procedure (lambda-parameters expr) (lambda-body expr) env)) ((application? expr) (mapply (meval (application-operator expr) env) (map (lambda (subexpr) (meval subexpr env)) (application-operands expr)))) (else (error "Unknown expression: " exp)))) CS 200 Spring 2003

  26. mapply (define (mapply procedure operands) (cond ((primitive-procedure? procedure) (apply-primitive procedure operands)) ((compound-procedure? procedure) (meval-sequence (procedure-body procedure) (extend-environment (procedure-parameters procedure) operands (procedure-environment procedure)))) (else (error “Can’t apply: " procedure)))) CS 200 Spring 2003

  27. > (meval 3 the-global-environment) |(meval 3 (((+ primitive-procedure #<primitive:+>) (* primitive-procedure #<primitive:*>)))) |3 3 > (meval '(+ 2 2) the-global-environment) |(meval (+ 2 2) the-global-environment) | (meval + the-global-environment) | (primitive-procedure #<primitive:+>) | (meval 2 the-global-environment) | 2 | (meval 2 the-global-environment) | 2 | (mapply (primitive-procedure #<primitive:+>) (2 2)) | 4 |4 4 CS 200 Spring 2003

  28. Charge • Defining eval and apply is the guts of it • Wednesday we will see the details • This is powerful: once we have an metacircular evaluator, we can easily make changes to the language! CS 200 Spring 2003

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