introduction to complexity theory lecture notes - oded goldreich

[...]... cally, we show how the di culty of inverting one-way functions may be utilized to obtain a pseudorandom generator Finally, we state and prove that a hard -to- predict bit (called a hard-core) may be extracted from any one-way function The hard-core is fundamental in our construction of a generator Notes taken by Moshe Lewenstein and Yehuda Lindell XII Lecture 15: Derandomization of BPP We present an... the lower bound on the depth of monotone circuits computing the function st-Connectivity Notes taken by Dana Fisman and Nir Piterman Lecture 24: Average-Case Complexity We introduce a theory of average-case complexity which refers to computational problems coupled with probability distributions We start by de ning and discussing the classes of P-computable and P-samplable distributions We then de ne... communication complexity of FORK, to be given in the next lecture, will yield an analogous lower bound on the monotone circuit depth of s-t-Connectivity Notes taken by Yoav Rodeh and Yael Tauman Lecture 23: Depth Lower Bound for Monotone Circuits (cont.) We analyze the fork game, introduced in the previous lecture We give tight lower and upper bounds on the communication needed in a protocol solving... tight (as P /poly contains non-recursive languages) The e ect of introducing uniformity is discussed, and shown to collapse P /poly to P Finally, we relate the P /poly versus N P question to the question of whether NP-completeness via Cook-reductions is more powerful that NP-completeness via Karp-reductions This is done by showing, on one hand, that N P is Cook-reducible to a sparse set i N P P =poly,... to the o ine model are proved We then turn to investigate the relation between the non-deterministic and deterministic space complexity (i.e., Savitch's Theorem) Notes taken by Yoad Lustig and Tal Hassner Lecture 6: Non-Deterministic Logarithmic Space We further discuss composition lemmas underlying previous lectures Then we study the complexity class N L (the set of languages decidable within Non-Deterministic... Therefore, research on the complexity of deciding SAT relates directly to the complexity of searching RSAT In the next lecture we show that every N P -complete language has a self-reducible relation However, let us rst discuss the problem of graph isomorphism, which can be easily shown to be in N P , but is not known to be N P -hard We show that nevertheless, graph isomorphism has a self-reducible relation... preserved under multiple samples We related pseudorandom generators and one-way functions, and show how to increase the stretching of pseudorandom generators The notes are augmented by an essay of Oded Notes taken by Sergey Benditkis, Il'ya Safro and Boris Temkin Lecture 14: Pseudorandomness and Computational Di culty We continue our discus- sion of pseudorandomness and show a connection between pseudorandomness...X Lecture 5: Non-Deterministic Space We recall two basic facts about deterministic space complexity, and then de ne non-deterministic space complexity Three alternative models for measuring non-deterministic space complexity are introduced: the standard non-deterministic model, the online model and the o ine model The equivalence between the non-deterministic and online models... we present a couple of tools for proving lower bounds on the complexity of communication problems We conclude by proving a linear lower bound on the communication complexity of probabilistic protocols for computing the inner product of two vectors, where initially each party holds one vector Notes taken by Amiel Ferman and Noam Sadot Lecture 22: Circuit Depth and Communication Complexity The main result... algorithm that distinguishes between linear and far-from-linear functions Notes taken by Yoad Lustig and Tal Hassner Lecture 19: Dtime(t) contained in Dspace(t/log t) We prove that Dtime(t( )) Dspace(t( )= log t( )) That is, we show how to simulate any given deterministic multi-tape Turing Machine (TM) of time complexity t, using a deterministic TM of space complexity t= log t A main ingrediant in the simulation