Theory of Computation and Complexity Theory. COL 705! Lecture 0!

Transcription

1 Theory of Computation and Complexity Theory COL 705! Lecture 0!

2 Class Format Two Minors: 20% each! Major: 35%! 4 HWs: 20%! Class Participation: 5%

3 Resources Slides generously shared by Manoj Prabhakaran! Text: Computational Complexity, a modern approach by Barak and Arora (freely available online)! Other resources on course webpage:

4 Plagiarism Policy What constitutes cheating?! Any cheating results in F (minimum)! No bending this rule, no discussion! If you are facing any problems, come talk to me, I will do everything possible to help.! If you resort to cheating, I will take strictest action and have no sympathy

5 Other Class Tips Ask questions! Do not be shy: if you have a doubt, then others likely do too! Be selfish, take responsibility for your learning: no one else can clarify things for you if you don t! I care very much that you understand, and will be happy to answer as many questions as you need! Ask me to speed up or slow down, give feedback. Teaching and learning are fun when

6 Computation A paradigm of modern science! Theory of computation/computational complexity is to computer science what theoretical physics is to electronics

7 Computational Complexity Computation:! Problems to be solved! Algorithms to solve them! in various models of computation! Complexity of a problem (in a comp. model)! How much resource is sufficient/necessary

8 Computational Complexity of! Problems in! Models of! computatio n w.r.t! Complexity! measures

9 Problems Input represented as (say) a binary string! Given input, find a satisfactory output! Function evaluation: only one correct output! Approximate evaluation! Search problem: find one of many (if any)! Decision problem: find out if any! A Boolean function evaluation (TRUE/FALSE)

10 Decision Problems Evaluate a Boolean function (TRUE/FALSE)! i.e., Decide if input has some property! Language! Set of inputs with a particular property! e.g. L = {x x has equal number of 0s and 1s}! Decision problem: given input, decide if it is in L

11 Problems we care about? Given a graph, find shortest path between two vertices! Given two matrices, compute their product! Given an integer, find its prime factors! Given a boolean formula, find a satisfying assignment

12 Decision versions? Is a node A on the shortest path?! Is there a factor of n that is less than k?! Given a boolean formula, is it satisfiable?

13 Complexity of Languages Some languages are simpler than others! L 1 = {x x starts with 0}! L 2 = {x x has equal number of 0s and 1s}! Simpler in what way?! Fewer calculations, less memory, need not read all input, can do in an FSM

14 Complexity of Languages Relating complexities of problems! Mo = {x x has more 0s than 1s}! Eq = {x x has equal number of 0s and 1s}! Eq(x):! Eq reduces to Mo if (Mo(x0) == TRUE and Mo(x) == FALSE) then TRUE; else FALSE! Eq is not (much) more complex than Mo. i.e., Mo is at least (almost) as complex as Eq.

15 Models of Computation FSM, PDA, TM! Variations: Non-deterministic, probabilistic. Other models: quantum computation! Church-Turing thesis: TM is as powerful as it gets! Non-uniform computation: circuit families

16 Turing Machine The idea behind digital computers may be explained by saying that these machines are intended to carry out any operations which could be done by a human computer. The human computer is supposed to be following fixed rules; he has no authority to deviate from them in any detail. We may suppose that these rules are supplied in a book, which is altered whenever he is put on to a new job. He has also an unlimited supply of paper on which he does his calculations. Alan Turing

17 Complexity Measures Number of computational steps, amount of memory, circuit size/depth,...! Exact numbers very much dependent on exact specification of the model (e.g. no. of tapes in TM)! But broad trends robust! Trends: asymptotic! Broad: Log, Poly, Exp Log Poly Exp

18 Complexity Theory What is computable?! For problems that are computable, how many resources do they need?! Many fundamental open problems

19 Complexity Theory Understand complexity of problems (i.e., how much resource used by best algorithm for it)! Relate problems to each other [Reduce]! Relate computational models/complexity measures to each other [Simulate]! Calculate complexity of problems

20 Complexity Classes Collect (decision) problems with similar complexity into classes! NP PSPACE IP IP=PSPACE P vs NP? P vs BPP? Relate classes to each other! Hundreds of classes! P BPP

21 compnp AM AM[polylog] QAM IP QIP[2] AmpP-BQP DQP NIQSZK QCMA MA BPP FH N.BPP NISZK PZK TreeBQP WAPP MA N.NISZK NISZK_h SZK SBP QSZK QMA CZK NE NP ZQP RQP RBQP YP com QMA(2) Complexity Zoo! (NP-cap-coNP)/poly NP/poly PP/poly NE/poly (k>=5)-pbp NC^1 PBP L QNC^1 CSL +EXP EXPSPACE EESPACE EEXP +L +L/poly +SAC^1 AL P/poly NC^2 P BQP/poly +P ModP SF_2 AmpMP SF_3 +SAC^0 AC^0[2] QNC_f^0 ACC^0 QACC^0 NC 1NAuxPDA^p SAC^1 AC^1 2-PBP 3-PBP 4-PBP TC^0 TC^0/poly AC^0 AC^0/poly FOLL MAC^0 QAC^0 L/poly AH ALL AvgP HalfP NT P-Close P-Sel P/log UP beta_2p compnp AM AM[polylog] BPP^{NP} QAM Sigma_2P ZPP^{NP} IP Delta_3P SQG BP.PP QIP[2] RP^{NP} PSPACE MIP MIP* QIP AM_{EXP} IP_{EXP} NEXP^{NP} MIP_{EXP} EXPH APP PP P^{#P[1]} AVBPP HeurBPP EXP AWPP A_0PP Almost-PSPACE BPEXP BPEE MA_{EXP} MP AmpP-BQP BQP Sigma_3P BQP/log DQP NIQSZK QCMA YQP PH AvgE EE NEE E Nearly-P UE ZPE BH P^{NP[log]} BPP_{path} P^{NP[log^2]} BH_2 CH EXP/poly BPE MA_E EH EEE PEXP BPL PL SC NL/poly L^{DET} polyl BPP BPP/log BPQP Check FH N.BPP NISZK PZK TreeBQP WAPP XOR-MIP*[2,1] BPP/mlog QPSPACE frip MA N.NISZK NISZK_h SZK SBP QMIP_{le} BPP//log BPP/rlog BQP/mlog BQP/qlog QRG ESPACE QSZK QMA BQP/qpoly BQP/mpoly CFL GCSL NLIN QCFL Q NLINSPACE RG CZK C_=L C_=P Coh DCFL LIN NEXP Delta_2P P^{QMA} S_2P P^{PP} QS_2P RG[1] NE RPE NEEXP NEEE ELEMENTARY PR R EP Mod_3P Mod_5P NP NP/one RP^{PromiseUP} US EQP LWPP ZQP WPP RQP NEXP/poly EXP^{NP} SEH Few P^{FewP} SPP FewL LFew NL SPL FewP FewUL LogFew RP ZPP RBQP YP ZBQP IC[log,poly] QMIP_{ne} QMIP R_HL UL RL MAJORITY PT_1 PL_{infty} MP^{#P} SF_4 RNC QNC QP NC^0 PL_1 QNC^0 SAC^0 NONE PARITY TALLY REG SPARSE NP/log NT* UAP QPLIN betap compip RE QMA(2) SUBEXP YPP Collect (decision) problems with similar complexity into classes! Relate classes to each other! Hundreds of classes!

22 Complexity Classes Easy to define classes but hard to define meaningful classes!! Capture genuine computational phenomenon such as parallelism! Robust under variations of computational model! Possibly closed under natural operations

23 Central Open Questions Is finding a solution as easy as verifying one? Is P= NP?! Is every sequential algorithm parallelizable? Is P = NC?! Are time efficient algorithms the same as those that use little space? Is P=L?! Can every efficient randomised algorithm be converted to an efficient deterministic one? Is P = BPP?

24 Complexity in various settings With various models of computation: decision trees, interactive settings, probabilistic computation! Various measures: depth, width, amount of communication, number of rounds, amount of randomness, amount of non-uniformity,...! Various connections: time vs. space, randomness vs. hardness

25 Cryptography Need to prove that a scheme is secure (according to some definition)! i.e., breaking security has high complexity! Reductions: if you could break my scheme s security efficiently, I can solve a hard problem almost as efficiently! Hard problems: almost all instances hard! For most keys scheme should be secure! Contrast with Information Theory: limited cryptography if no computational constraints on adversary

26 Computational Complexity in Economics/Games Traditionally, no computational constraints for strategizing players! But many problems in game-theory are computationally hard! Rational players may not be able to play optimally! Suggests the need to redesign markets/financial instruments to be computationally tractable! Recent results...

27 All that and much more.. Welcome to COL 705!! Textbook: complexity/