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Sanjeev Arora, Boaz Barak Computational complexity a modern approach Cambridge University Press(2009) pdf

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Figure I.1. Grade-school algorithm for multiplication. Illustrated for computing 577 · 423.
Figure 1.2. The transition function of a two-tapeTM (i.e., a TM with one input tape and one work/output tape).
Figure 1.3. We can simulate a machine M using the alphabet {▷, □, a, b, . . . , z} by a machine M′ using{▷, □, 0, 1} via encoding every tape cell of M using five cells of M′.
Figure 1.4. Simulating a machine M with three tapes using a machine M˜ with a single tape.
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3 (Computing a function and running time) 9 (Efficient universal Turing machine) CHAPTER NOTES AND HISTORY 1.9 Define a RAM Turing machine to be a Turing machine that has random access EXAMPLE 2.2 ( INDSET ∈ NP) CONP , EXP , AND NEXP between eastern andwestern scientists Trakktenbrot’s survey [Tra84] describes Levin’s 10 (Error reduction for BPP) 19 ([AKL + 79] ) UPATH ∈ RL INTERACTIVE PROOF FOR THE PERMANENT

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