CPSC 506: Complexity of Computa5on

Transcription

1 CPSC 506: Complexity of Computa5on On the founda5ons of our field, connec5ons between Science and Compu5ng, where our field might be headed in the future CPSC 506 MW 9-10:30, DMP 101 cs.ubc.ca/~condon/cpsc506/

2 Outline for today Course topics Course work Introduc5ons Website, logis5cs Course prerequisites Review of prerequisites: examples, exercises CPSC 506 MW 9-10:30, DMP 101 cs.ubc.ca/~condon/cpsc506/

3 CPSC 506: Course topics Gauging the hardness of a problem: 5me and space complexity, provably hard problems. How nondeterminism can help our thinking. The power of randomness: How does it help us compute more efficiently? The power of interac7on: Proofs that use interac5on and randomness. Approximability and APX-hardness: How to determine whether or not a problem has a good approxima5on algorithm. Quantum complexity theory: Novel ways of communica5ng and compu5ng. Chemical reac7on systems: Emerging theories of compu5ng with molecules; energy as a computa5onal resource. CPSC 506 MW 9-10:30, DMP 101 cs.ubc.ca/~condon/cpsc506/

4 CPSC 506: Course work Five homework assignments: 50% Student reading project and presenta5ons: 20% (more on this soon ) Final exam: 30% Lots of problem-solving during class Lots of room for individualized learning CPSC 506 MW 9-10:30, DMP 101 cs.ubc.ca/~condon/cpsc506/

5 Introduc5ons Your name, where you are in the program, what research areas you re interested in, why you want to take the course, what you hope to get out of it CPSC 506 MW 9-10:30, DMP 101 cs.ubc.ca/~condon/cpsc506/

6 More on prerequisites I'll assume familiarity with (or willingness to quickly gain familiarity with) Turing machine, 5me bounded Turing machine, universal Turing machine, decidable languages Illustra5on from: The Nature of Computa5on, Chris Moore

7 More on prerequisites I'll assume familiarity with (or willingness to quickly gain familiarity with) Turing machine, 5me bounded Turing machine, universal Turing machine, decidable languages, undecidable languages The classes P and NP, what it means for a problem to be NPcomplete, NP-hard. How polynomial-5me reduc5ons (L p L ) are used to establish NP-completeness results. How to derive simple reduc5ons. What is a nondeterminis5c Turing machine (NTM), what it means for a NTM to accept a language, and the defini5on of NP in terms of NTMs. CPSC 506 MW 9-10:30, DMP 101 cs.ubc.ca/~condon/cpsc506/

8 Review: exercises, examples Can you give examples of problems that are in P, in NP, NP-complete, decidable, undecidable? State whether true, false, or open ques5on: SAT is in P If language A is in P and language B is in P then A p B If A is in P and B is in NP then A p B If A is in P than A is NP-complete CPSC 506 MW 9-10:30, DMP 101 cs.ubc.ca/~condon/cpsc506/

9 Wang 5ling Illustra5on from: The Nature of Computa5on, Chris Moore

10 Wang 5ling The Wang Tiling problem: Given a set T of Wang 5les Can we 5le the infinite plane with 5les from T? What s the simplest algorithm you can think of to solve (or aiempt to solve) this problem? Where would you guess that this problem lies in our Venn diagram of language classes?

11 Wang 5ling The Wang Tiling problem: Given a set T of Wang 5les Can we 5le the infinite plane with 5les from T? Goal: Show Wang Tiling is undecidable, via a reduc5on from the Blank Tape Hal5ng Problem (or its complement)

12 Wang 5ling: simula5ng a TM What 5les would be useful for Turing machine simula5on?

13 Wang 5ling: simula5ng a TM Illustra5on from: The Nature of Computa5on, Chris Moore

14 Wang 5ling: simula5ng a TM Problem: There s no way to require that a 5ling contain a par5cular 5le (e.g. tape head on blank cell). Exercise: Suppose that any set T of Wang 5les, either allows a periodic* 5ling of the plane, or cannot 5le the plane at all. Show that then TILING would be decidable. * i.e., has an infinitely repeatable square of 5les, or period

15 Wang 5ling: simula5ng a TM Exercise: Suppose that any set T of Wang 5les, either allows a periodic 5ling of the plane, or cannot 5le the plane at all. Show that then Wang Tiling would be decidable. Idea: Find all 5lings of increasingly larger squares un5l either you find a period: output yes you find square with no 5ling: and output no Either a periodicity proof or a counterexample will eventually present itself!

16 Wang 5ling: simula5ng a TM Idea: To force a 5ling to contain an ini5al TM configura5on, leverage Robson s aperiodic 5les

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18 Wang 5ling: simula5ng a TM Expand and annotate Robson s 5les so that they encode tape symbols, or tape symbols and state The top row of each black k x k square can encode an ini5al TM configura5on on empty input (up to tape posi5on k)

19 Wang 5ling: simula5ng a TM This suggests that a k x k square could encode k steps of the TM computa5on The 5ling covers an infinite plane if and only if the TM does not halt

20 Wang 5ling: simula5ng a TM Problem: the black k x k square contains smaller black k/4 x k/4 black squares, each doing their own (smaller) TM simula5on... we need to eliminate interference

21 Wang 5ling: Lessons Algorithm design: ac5vely looking both for a yes proof and no proof is effec5ve (as in decidability of periodic 5ling) Simple ideas are important: the germ of the undecidability proof is quite simple: using 5le colours to encode tape cells and state Stumbling blocks lead to interes5ng new ques5ons, e.g. aperiodic 5lings

22 An aperiodic 5ling built from DNA DNA 5les self-assemble according to 5le colour rules Atomic force microscopy image Rothemund PWK, Papadakis N, Winfree E (2004) Algorithmic Self-Assembly of DNA Sierpinski Triangles. PLoS Biol 2(12): e424

23 Class summary Undecidable Wang Tiling - via reduc5on from Halt - inspira5on for DNA algorithmic selfassembly NP P Decidable NPC

24 Next Class We ll look at new complexity classes: co-np, EXP, NEXP, and some interes5ng problems in these classes We ll prove a 5me hierarchy theorem: more 5me means more language recogni5on power! Reading: Arora-Barak 2.6, 3.1