42 42-ness is something whos successor is 43-ness forty-two - PDF document

Glue, Photons, P=NP? Glue, Photons, P=NP? Summer CS Classes • CS 4630 – Defense against the Dark Arts – Aaron Bloomfield, M-F 10:30am - 12:45 • CS 2110 – Software Development Methods – Mark Sherriff, M -W 10:30am – 12:45 – You should probably take CS 2220 with Dave Evans in the fall instead, but taking CS 2110 instead will work for the CS major • CS 1010 – Introduction to Information Technology – Kinga Dobolyi, M-F 1pm – 3:15pm – Tell your friends! – Small class, easy way to satisfy some requirements DNA Helix Photomosaic from cover of Nature , 15 Feb 2001 (made by Eric Lander) #2 One-Slide Summary Lambda Calculus is a Universal Computer? • The lambda calculus is a universal, fundamental model of computation. You can view it as “the essence of Scheme”. It z z z z z z z z z z z z z z z z z z z z contains terms and rules describing variables, function abstraction, and function application. • Read/Write Infinite Tape • It is possible to encode programming concepts, such as true, Mutable Lists ), X, L ¬ ), #, R false, if, numbers, lists, etc., in the lambda calculus. Lambda ¬ (, #, L • Finite State Machine 2: 1 look for ( calculus can simulate Turing machines. Numbers Start (, X, R • Processing • Quantum computers and non-determinsitic Turing machines HAL T #, 0, - #, 1, - Way to make decisions (if) can try many options at once. They are not more powerful Finite State Machine Way to keep going than normal Turing machines. • The Complexity Class P contains tractable problems that can be solved in polynomial time. The Complexity Class NP contains problems for which solutions can be verified in polynomial time. • Does P = NP? We don't know! $1+ million if you know. #3 #4 What is 42? Meaning of Numbers 42 • “42-ness” is something who’s successor is “43-ness” forty-two • “42-ness” is something who’s predecessor is “41-ness” XLII • “Zero” is special. It has a successor “one-ness”, but no predecessor . cuarenta y dos #5 #6

Is this enough? Meaning of Numbers Can we define add with pred , succ , zero? pred (succ N ) → N and zero ? succ (pred N ) → N succ (pred (succ N )) → succ N add ≡ λ xy. if ( zero? x ) y zero? zero → T ( add ( pred x ) ( succ y )) zero? (succ zero ) → F #7 #8 Numbers are Lists... Can we define lambda terms zero? ≡ null? that behave like pred ≡ cdr zero , zero? , pred and succ ? succ ≡ λ x . cons F x Hint: what if we had cons , car and cdr ? The length of the list corresponds to the number value. #9 #10 Liberal Arts Trivia: Making Pairs Religious Studies •In Sunni Islam, the Five Pillars of (define ( make-pair x y) Islam are five duties incumbent ( lambda (selector) ( if selector x y))) on Muslims. They include the (define ( car-of-pair p) (p #t)) Profession of Faith, Formal (define ( cdr-of-pair p) (p #f)) Prayers, and Giving Alms. Name the remaining two pillars. A pair is just an if statement that chooses between the car (then) and the cdr (else). #11 #12

cons and car cdr too! cons ≡ λ x. λ y. λ z.zxy cons ≡ λ xyz.zxy Example: cons M N = ( λ x. λ y. λ z.zxy ) M N car ≡ λ p.p T → β ( λ y. λ z.z M y ) N cdr ≡ λ p.p F → β λ z.z MN T ≡ λ xy . x car ≡ λ p.p T Example: cdr (cons M N) Example: car (cons M N) ≡ car ( λ z.z MN) ≡ ( λ p.p T ) ( λ z.z MN) → β ( λ z.z MN) T → β T MN cdr λ z.z MN = ( λ p.p F) λ z.z MN → β ( λ xy . x ) MN → β ( λ z.z MN) F → β ( λ y . M)N → β FMN → β M → β N #13 #14 Null and null? Null and null? null ≡ λ x. T null ≡ λ x. T null? ≡ λ x. ( x λ y. λ z. F ) null? ≡ λ x. ( x λ y. λ z. F ) Example: Example: null? null → λ x. ( x λ y. λ z. F ) ( λ x. T ) null? (cons M N) → λ x. ( x λ y. λ z. F ) λ z.z MN → β ( λ x. T ) ( λ y. λ z. F ) → β ( λ z.z MN) ( λ y. λ z. F ) → β T → β ( λ y. λ z. F ) MN → β F #15 #16 42 = λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y Counting λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y 0 ≡ null λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. 1 ≡ cons F 0 ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y 2 ≡ cons F 1 λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) 3 ≡ cons F 2 λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y ... λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. succ ≡ λ x . cons F x ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) λ xy . y λ xy. ( λ z.z xy ) pred ≡ λ x . cdr x λ xy . y λ x. T #17 #18

Equivalent Computers! Lambda Calculus is a Universal Computer z z z z z z z z z z z z z z z z z z z z ... z z z z z z z term = variable | term term ), X, L ¬ ), #, R | ( term ) ¬ (, #, L 2: | λ variable . term 1 look for ( ), X, L • Read/Write Infinite Tape ¬ ), #, R Start ¬ (, #, L (, X, R  Mutable Lists 2: look can simulate 1 for ( Start HAL λ y . M ⇒ α λ v . ( M [ y → v ]) • Finite State Machine T #, 0, - #, 1, - (, X, R Finite State Machine  Numbers to keep track of state where v does not occur in M . HAL T #, 0, - can simulate #, 1, - • Processing Finite State Machine ( λ x . M ) N ⇒ β M [ x → N ]  Way of making decisions (if)  Way to keep going Lambda Calculus Turing Machine #19 #20 Liberal Arts Trivia: Universal Computer Biology, Chemistry • Lambda Calculus can simulate a Turing • While rendered fat obtained form pigs is Machine known as lard, rendered beef or mutton fat is – Everything a Turing Machine can compute, known as this . It is used to make animal feed Lambda Calculus can compute also and soap. Historically, it was used to make • Turing Machine can simulate Lambda candles: it provided a cheaper alternative to Calculus (we didn’t prove this) wax. Before switching to vegetable oil in – Everything Lambda Calculus can compute, a 1990, McDonald's cooked fries in 93% this and Turing Machine can compute also 7% cottonseed oil. • Church-Turing Thesis : this is true for any other mechanical computer also #21 #22 Quantum Physics What about “non-mechanical” for Dummies computers? • Light behaves like both a wave and a particle at the same time • A single photon is in many states at once • Can’t observe its state without forcing it into one state • 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 #23 #24

Quantum Computing Qubit • Feynman, 1982 • Regular bit: either a 0 or a 1 • Quantum particles are in all possible states • Quantum bit : 0, 1 or in between • Can try lots of possible computations at once – p% probability it is a 1 with the same particles • A single qubit is in 2 possible states at once • In theory, can test all possible • If you have 7 bits, you can represent any factorizations/keys/paths/etc. and get the one of 2 7 different states right one! • If you have 7 qubits, you have 2 7 different • In practice, very hard to keep states states (at once!) entangled: once disturbed, must be in just one possible state #25 #26 Quantum Computers Today Nondeterministic Computing • Several quantum algorithms ... ... # 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 # – Shor’s algorithm: factoring using a quantum computer • Actual quantum computers There can be multiple Input: 1 – 5-qubit computer built by IBM (2001) Input: 1 Write: 1 transitions on the same – Implemented Shor’s algorithm to factor: Start Write: 1 Move: → input. Nondeterministic • “World’s most complex quantum computation” Move: → 15 (= 5 * 3) TM takes all of them at – D-Wave 16-qubit quantum computer (2007) once. Each gets its 1 2 own independent copy • Solves Sudoku puzzles of the tape. If any path • To exceed practical normal computing need > 50 finds halting state, that qubits Input: # is the result. 3 Write: # – Adding another qubit is more than twice as hard Move: → #27 #28 Two Ways of Thinking about Computability Nondeterminstic Computing • Omniscient (all-knowing): machine always Is a nondeterministic TM more guesses right (the right guess is the one powerful than a deterministic TM? that eventually leads to a halting state) • Omnipotent (all-powerful): machine can split in two every step, all resulting machines execute on each step, if one of the machines halts its tape is the output #29 #30