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The theoretical underpinnings of computing form a standard part of almost every computer science curriculum. But the classic treatment of this material isolates it from the myriad ways in which the theory influences the design of modern hardware and software systems. The goal of this book is to change that. The book is organized into a core set of chapters (that cover the standard material suggested by the title), followed by a set of appendix chapters that highlight application areas including programming language design, compilers, software verification, networks, security, natural language processing, artificial intelligence, game playing, and computational biology. The core material includes discussions of finite state machines, Markov models, hidden Markov models (HMMs), regular expressions, context-free grammars, pushdown automata, Chomsky and Greibach normal forms, context-free parsing, pumping theorems for regular and context-free languages, closure theorems and decision procedures for regular and context-free languages, Turing machines, nondeterminism, decidability and undecidability, the Church-Turing thesis, reduction proofs, Post Correspondence problem, tiling problems, the undecidability of first-order logic, asymptotic dominance, time and space complexity, the Cook-Levin theorem, NP-completeness, Savitch's Theorem, time and space hierarchy theorems, randomized algorithms and heuristic search. Throughout the discussion of these topics there are pointers into the application chapters. So, for example, the chapter that describes reduction proofs of undecidability has a link to the security chapter, which shows a reduction proof of the undecidability of the safety of a simple protection framework.  

What is a proof? There�s a formal de�nition of proof where each statement must follow from previous statements according to speci�ed rules. This is a de�nition we will study in this course, but it�s not the relevant de�nition for when you�re doing your homework. For this class, a proof is an argument that can withstand all criticism from a highly ca_einated adversary.

Quantum states of n qubits: If you have an object that can be in two perfectly distinguishable states |0> or |1>, then it can also be in a superposition of the |0> and |1> states: _ |0> + _ |1> where _ and _ are complex numbers such that: |_| 2 + |_| 2 = 1

What we really need is a machine that can not only move backwards and forwards, but also write to the tape and halt at any time of its choosing. And that�s what a Turing machine is. The ability to write essentially gives Turing machines an unlimited memory, since any information that can�t �t in the machine�s internal state can always be written to the tape. The ability to halt at discretion means that Turing machines aren�t �tied to the input� the way �nite automata are, but can do as much auxiliary computation as they need.

What we really need is a machine that can not only move backwards and forwards, but also write to the tape and halt at any time of its choosing. And that�s what a Turing machine is. The ability to write essentially gives Turing machines an unlimited memory, since any information that can�t �t in the machine�s internal state can always be written to the tape. The ability to halt at discretion means that Turing machines aren�t �tied to the input� the way �nite automata are, but can do as much auxiliary computation as they need.

In this lecture, we�ll simply assume that we have a short random seed, and consider the problem of how to expand it into a long �random-looking� sequence.

RSA: RSA (together with its variants) is probably the most widely-used cryptographic protocol in modern electronic commerce. Much like Di_e-Hellman, it is built on modular arithmetic.

Of course the real question is: can quantum computers actually do something more e_ciently than classical computers? In this lecture, we�ll see why the modern consensus is that they can.

Our �rst example of a public-key cryptosystem, in which two people exchanging messages did not have to meet beforehand, was Di_e-Hellman. We then talked about the RSA cryptosystem, which is probably the most widely used today. Here are the basics of how it works:

Our �rst example of a public-key cryptosystem, in which two people exchanging messages did not have to meet beforehand, was Di_e-Hellman. We then talked about the RSA cryptosystem, which is probably the most widely used today. Here are the basics of how it works:

Our �rst example of a public-key cryptosystem, in which two people exchanging messages did not have to meet beforehand, was Di_e-Hellman. We then talked about the RSA cryptosystem, which is probably the most widely used today. Here are the basics of how it works: