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David Evans cs.virginia/~evans

Class 38: Fixed Points and Biological Computing. David Evans http://www.cs.virginia.edu/~evans. CS200: Computer Science University of Virginia Computer Science. Menu. Making Recursive Definitions without define Computing with DNA How Biology Programs. Lambda Calculus.

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David Evans cs.virginia/~evans

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  1. Class 38: Fixed Points and Biological Computing David Evans http://www.cs.virginia.edu/~evans CS200: Computer Science University of Virginia Computer Science

  2. Menu • Making Recursive Definitions without define • Computing with DNA • How Biology Programs CS 200 Spring 2002

  3. Lambda Calculus term ::= variable |term term | (term)|  variable .term -reduction (renaming) y. M v. (M [yv]) where v does not occur in M. -reduction (substitution) (x. M)N   M [ xN ] CS 200 Spring 2002

  4. 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 ), X, L ), #, R (, #, L 2: look for ( 1 • Read/Write Infinite Tape • Mutable Lists • Finite State Machine • Numbers to keep track of state • Processing • Way of making decisions (if) • Way to keep going Start (, X, R HALT #, 0, - #, 1, - Finite State Machine We have this, but we cheated using  to make recursive definitions! CS 200 Spring 2002

  5. Fixed Point Theorem • The fixed point of a function f, is a value x such that f(x) = x • If we can find the fixed point of our Turing Machine simulator, then we have something that keeps going until it halts! fixed-point TM input   result of running TM on input CS 200 Spring 2002

  6. All Lambda Calculus Terms have Fixed Points! • For any Lambda Calculus term F, there exists a Lambda Calculus Term X such thatFX = X • Proof: LetW =  x.F(xx)andX = WW. X = WW = ( x.F(xx))W   F (WW) = FX We can make F a parameter! CS 200 Spring 2002

  7. Why of Y? • Y is  f. WW: Y   f. ( x.f (xx))( x. f (xx)) • Y calculates a fixed point of any lambda term! • Hence: we don’t need define to do recursion! • Works in Scheme too - check the “lecture” from the Adventure Game CS 200 Spring 2002

  8. Lambda Calculus is Turing Universal! • All you need is beta-reduction and you can compute anything • This is just one way of representing numbers, if, etc. – many others are possible • Integers, booleans, if, while, +, *, =, <, classes, define, inheritance, etc. are for wimps! Real programmers only use . CS 200 Spring 2002

  9. Models of Computation • Mechanical: Turing Machine • Symbolic: Lambda Calculus • Next: Biological CS 200 Spring 2002

  10. Computing with DNA Leonard Adleman (Mathematical Consultant for Sneakers), 1995 CS 200 Spring 2002

  11. DNA G • Sequence of nucleotides: adenine (A), guanine (G), cytosine (C), and thymine (T) • Two strands, A must attach to T and G must attach to C C A T CS 200 Spring 2002

  12. Hamiltonian Path Problem • Input: a graph, start vertex and end vertex • Output: either a path from start to end that touches each vertex in the graph exactly once, or false indicating no such path exists RIC start: CHO end: BWI BWI CHO Hamiltonian Path is NP-Complete IAD CS 200 Spring 2002

  13. Encoding The Graph • Make up a two random 4-nucleotide sequences for each city: CHO: CHO1 = ACTT CHO2 = gcagRIC: RIC1 = TCGG RIC2 = actg IAD: IAD1 = GGCT IAD2 = atgt BWI: BWI1 = GATC BWI2 = tcca • If there is a link between two cities (AB), create a nucleotide sequence: A2B1 CHORIC gcagTCGG RICCHO actgACTT Based on Fred Hapgood’s notes on Adelman’s talk http://www.mitre.org/research/nanotech/hapgood_on_dna.html CS 200 Spring 2002

  14. Encoding The Problem • Each city nucleotide sequence binds with its complement (A  T, G  C) : CHO: CHO1 = ACTT CHO2 = gcag CHO’: TGAAcgtcRIC: TCGGactg RIC’: AGCCtgac IAD: GGCTatgt IAD’ = CCGAtaca BWI: GATCtcca BWI’ = CTAGaggt • Mix up all the link and complement DNA strands – they will bind to show a path! CS 200 Spring 2002

  15. Path Binding BWI’ CTAGaggt RIC’ AGCCtgac IAD’ CCGAtaca CHO’ TGAAcgtc gcagGGCT CHOIAD atgtTCGG IADRIC actgGATC RICBWI TCGGactg RIC BWI CHO GATCtcca ACTTgcag IAD GGCTatgt CS 200 Spring 2002

  16. Getting the Solution • Extract DNA strands starting with CHO and ending with BWI • Easy way is to remove all strands that do not start with CHO, and then remove all strands that do not end with BWI • Measure remaining strands to find ones with the right weight (7 * 8 nucleotides) • Read the sequence from one of these strands CS 200 Spring 2002

  17. Why don’t we solve NP-Complete problems this way? • Speed: shaking up the DNA strands does 1014 operations per second ($400M supercomputer does 1010) • Memory: we can store information in DNA at 1 bit per cubic nanometer • How much DNA would you need? • Volume of DNA needed grows exponentially with input size • To solve ~45 vertices, you need ~20M gallons CS 200 Spring 2002

  18. DNA-Enhanced PC CS 200 Spring 2002

  19. How does Nature program? CS 200 Spring 2002

  20. How Big is the Make-a-Human Program? • 3 Billion Base Pairs • Each nucleotide is 2 bits (4 possibilities) • 3 B pairs * 1 byte/4 pairs = 750 MB 1 CD ~ 650 MB CS 200 Spring 2002

  21. Encoding is Redundant • DNA encodes proteins • Every sequence of 3 base pairs one of 20 amino acids (or stop codon) • 21 possible codons, but 43 = 64 possible values • So, really only 750MB * (21/64) ~ 250 MB CS 200 Spring 2002

  22. People are almost all the Same • Genetic code for 2 humans differs in only 2.1 million bases • 4 million bits = 0.5 MB CS 200 Spring 2002

  23. How big is .5 MB? • 1/3 of a floppy disk • <1% of Windows 2000 • ~22 times the size of the PS6 adventure game code CS 200 Spring 2002

  24. Is DNA Really a Programming Language? CS 200 Spring 2002

  25. Nerdy Linguist’s Definition A description of pairs (S, M), where S stands for sound, or any kind of surface forms, and M stands for meaning. A theory of language must specify the properties of S and M, and how they are related. CS 200 Spring 2002

  26. Programming Language(Definition from Lecture 1) A description of pairs (S, M), where S stands for sound, or any kind of surface forms, and M stands for meaning intended to be read and written by humans and processed by machines. CS 200 Spring 2002

  27. Stuff Programming Languages are Made Of • Primitives • Means of Combination • Means of Abstraction codons (sequence of 3 nucleotides that encodes a protein) ?? Morphogenesis? Not well understood (by anyone). This is where most of the expressiveness comes from! DNA itself – separate proteins from their encoding Genes – group DNA by function (sort of) Chromosomes – package Genes together Organisms – packages for reproducing Genes CS 200 Spring 2002

  28. Biology is (becoming) a subfield of Computer Science • Biological mechanisms are mostly understood (proteomics still has a way to go) • What is not understood is how those are combined to create meaning CS 200 Spring 2002

  29. Charge • Noon (now): President Casteen’s State of the University in Old Cabal Hall • Extra credit question: “Given that Computer Science is the most liberal art, how come UVa College students are not able to major in Computer Science?” • Friday: review • Chance to ask questions about anything you want CS 200 Spring 2002

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