What computers just cannot do.COS 116: 2/28/2008Sanjeev AroraAdministrivia• In-class midterm in midterms week; Thurs Mar 13 (closed book; ?• No lab in midterms week; review session instead.“Prof, what’s with all the negative thinking?!?”An obvious motivation: Understand the limits of technology“What computers can’t do.”The power of negative thinking….In Science….Impossibility of trisecting angle with ruler and compass (Galois)Often, impossibility result deep insightExamples Nothing travels faster than lightGroup Theory and much of modern mathRelativity and modern physicsIn Mathematics…..“Can mathematicians be replaced by machines?”Axioms – Set of statementsDerivation rules – finite set of rules for deriving new statements from axiomsTheorems – Statements that can be derived from axioms in a finite number of stepsMathematician – Person who tries to determine whether or not a statement is a theorem.[Hilbert, 1900]Math is axiomatic“Given starting configuration for the game of life, determine whether or not cell (100,100) is ever occupied by a critter.”John ConwayUnderstanding complex (or even simple) systems….Can a simple set of mathematical equations “solve” problems like:In computer science……CAPTCHA (CMU Group)Computer generated test thatComputers (at least with currentalgorithmic knowledge) seem unable to solve pass.CryptographyMore Computer Science…Automated Checking of Software?Windows XP: 40 million line programCan computers check whether or not it will ever crash?Discussion TimeWhat is a computation?How did Turing set about formalizing this age-old notion and what were the features of his model?What is a computation?Basic ElementsScratch PadStep-by-step description of what to do (“program”); should be finite!At each step:Can only scan a fixed number of symbolsCan only write a fixed number of symbolsA formalization of an age-old notionTuring’s model1 dimensional unlimited scratchpad (“infinite”)Only symbols are 0/1 (tape has a finite number of 1s)Can only scan/write one symbol per stepProgram looks like 1. PRINT 0 2. GO LEFT 3. GO TO STEP 1 IF 1 SCANNED 4. PRINT 1 5. GO RIGHT 6. GO TO STEP 5 IF 1 SCANNED 7. PRINT 1 8. GO RIGHT 9. GO TO STEP 1 IF 1 SCANNED 10. STOP The Doubling ProgramExample: What does this program do?1. PRINT 0 2. GO RIGHT 3. GO TO STEP 1 if 1 SCANNED 4. GO TO STEP 2 if 0 SCANNEDDiscussion TimeCan this computational model do every computation that pseudocode can?How do we implement arithmetic instructions, arrays,loops?Surprising facts about this “Micky-mouse” modelIt can do everything that pseudocodecanHence it can “simulate” any other physical system, and in particular simulate any otherphysically realizable “computer.”[CHURCH-TURING THESIS]THIS MODEL CAPTURES THE NOTION OF “COMPUTATION” ----TURINGRecall: Numbers and letters can be written in binary.A program can also be represented by a string of bits!“Code” for a programMany conventions possible (e.g., ASCII)Davis’s convention: P Code (P)= Binary RepresentationPrograms and DataUsual viewpoint - A False Dichotomy!ProgramDataBut can have - ProgramCode of ProgramUniversal Program UU “simulates” what P would do on that dataData U Program P(Sometimes also known as “interpreter”)DVAutomated Bug Checking RevisitedHalting ProblemLet P = program such that code(P) = V. Does P halt on data D?Trivial Idea: Simulate P using universal program U. If P halts, will eventually detect.Problem: But if P never halts, neither does the simulation.IDEAS???Next Time: Halting Problem is unsolvable by another programTurn in on Tues: A Turing-Post program that prints the bit sequence 101 infinitely often,as well as its binary code Read this proof in the Davis article, and try to understand.Ponder the meaning of “Proof by contradiction.”How convincing is such a proof?“When something’s not right its wrong…” Bob
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