Lecture 8 Notes Polynomial Time - Lecture 8 Notes...

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Unformatted text preview: Lecture 8 Notes: Polynomial Time 1 A dminist rivia There will be two scribes notes per person for the term. In addition, you may now use the book to review anything not clear from lecture since we’re on complexity. 2 Recap In most cases of interest to us, the real question is not what’s computable; it’s what’s computable with a reasonable amount of time and other resources. Almost every problem in science and industry is computable, but not all are e ffi ciently computable. And even when we come to more speculative matters – like whether it’s possible to build a machine that passes the Turing Test – we seethat with unlimited resources, it’s possible almost trivially (just create a giant lookup table). The real question is, can we build a thinking machine that operates within the time and space constraints of the physical universe? This is part of our motivation for switching focus from computability to complexity. 2.1 Early Result s 2.1.1 Claude Shannon’s count ing argument Most Boolean functions require enormous circuits to compute them, i.e. the number of gates is exponential in the number of inputs. The bizarre thing is we know this, yet we still don’t have a good example of a function that has this property. 2.1.2 T ime H ierarchy T heorem Is it possible that everything that is computable at all is computable in linear time? No. There is so much that we don’t know about the limits of feasible computation, so we must savor what we do know. There are more problems that we can solve in n 3 than in n 2 steps. Similarly, there are more problems that we can solve in 3 n than in 2 n steps. The reason this is true is that we can consider a problem like: “ Given a Turing Machine, does it halt in ≤ n 3 steps?” Supposing it were solvable in fewer than n 3 steps, we could take the program that would solve the problem and feed it itself as input to create a contradiction. This is the Fnite analog of the halting problem....
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