What computers just cannot do. (Part II) COS 116, Spring 2012 Adam FinkelsteinRecap from last time Turing-Post computational model: Greatly simplified model Infinite tape, each cell contains symbol (0 or 1) Program = finite sequence of instructions (only 6 types!) Unlike pseudocode, no conditionals or loops, only “GOTO” code(P) = binary representation of program PExample: doubling program 1. PRINT 0 2. GO LEFT 3. GO TO STEP 2 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 Program halts on this input data if STOP is executed in a finite number of stepsSome facts Fact 1: Every pseudocode program can be written as a T-P program, and vice versa Fact 2: There exists a universal T-P program U 1 1 0 0 1 0 1 1 0 1 1 1 0 0 … … code(P) V U simulates P’s computation on VIs there a universal pseudocode program ? How would you write it? What are some examples of universal programs in real life? Discussion TimeHalting Problem Decide whether P halts on V or not Cannot be solved! Turing proved that no Turing-Post program can solve Halting Problem for all inputs (code(P), V). 1 1 0 0 1 0 1 1 0 1 1 1 0 0 … … code(P) VMakes precise something quite intuitive: “Impossible to demonstrate a negative” Suppose program P halts on input V. How can we detect this in finite time? Proposal: “Just simulate.” Intuitive difficulty: If P does not actually halt, no obvious way to detect this after just a finite amount of time. Turing’s proof makes this intuition concrete.Ingredients of the proof…..Fundamental assumption: A mathematical statement is either true or false “When something’s not right, it’s wrong.” Bob Dylan Ingredient 1: “Proof by contradiction”Aside: Epimenides Paradox Κρῆτες ἀεί ψεύσται “Cretans, always liars!” But Epimenides was a Cretan!’ More troubling: “This sentence is false.”Ingredient 2: Suppose you are given some T-P program P How would you turn P into a T-P program that does NOT halt on all inputs that P halts on? Suppose you have programs A and B. What is the program whose net effect is “Run A first and then B?”Finally, the proof… Suppose program H solves Halting Problem on ALL inputs of the form code(P), V. H Consider program D 1. On input V, check if it is code of a T-P program. 2. If no, HALT immediately. 3. If yes, use doubling program to create the bit string V, V and simulate H on it. 4. If H says “Doesn’t Halt”, HALT immediately. 5. If H says “Halts”, go into infinite loop Gotcha! Does D halt on the input code(D)? Either it halts or not – and both ways lead to contradictions!Lessons to take away Computation is a very simple process ( can arise in unexpected places) Universal Program No real boundary between hardware, software, and data No program that decides whether or not mathematical statements are theorems. Many tasks are uncomputable; e.g. “If we start Game of life in this configuration, will cell (100, 100) ever have a critter?”Age-old mystery: Self-reproduction. How does the seed encode the whole?Self-Reproduction False argument for impossibility: Blueprint Blueprint BlueprintM.C. Escher Print GalleryFallacy Resolved: “Blueprint” can involve some computation; need not be an exact copy! Print this sentence twice, the second time in quotes. “Print this sentence twice, the second time in quotes.”High-level description of program that self-reproduces Print 0 Print 1 . . . Print 0 . . . . . . . . . . . . . . . . . . . . . . . . } Prints binary code of B } Takes binary string on tape, and in its place prints (in English) the sequence of statements that produce it, followed by the translation of the binary string into English (ie into a T-P program). A B Fact: for every program P, there exists a program P’ that has the exact same functionality except at the end it also prints code(P’) on the tape Self-reproducing programs MyDoom Crash user’s computer! Reproduce program send to someone
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