By Peter Gacs, Laszlo Lovasz
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Extra resources for Complexity of Algorithms (Lecture Notes)
14 Exercise Show that the following modiﬁcation of the tiling problem is also undecidable. We use tiles marked on the corners instead of the sides and all tiles meeting in a corner must have the same mark. 15 Exercise Our proof of G¨ odel’s theorem does not seem to give a speciﬁc sentence ϕT undecidable for a given minimally adequate theory T . e. set constructed in the proof of the undecidablity of the halting problem. 1 Computation with resource bounds Introduction The algorithmic solvability of some problems can be very far from their practical solvability.
We restrict the computational tasks to yes-or-no problems; this is not too much of a restriction, and pays oﬀ in what we gain simplicity of presentation. Note that the task of computing any output can be broken down to computing its bits in any reasonable binary representation. Most of this chapter is spent on illustrating how certain computational tasks can be solved within given resource contraints. We start with the most important case, and show that most of the basic everyday computational tasks can be solved in polynomial time.
We multiply two such words by writing them after each other and repeatedly erasing any −1 pair of the form ai a−1 i or ai ai whenever they occur. It takes some, but not diﬃcult, reasoning to show that the multiplication deﬁned this way is associative. We also permit the empty word, this will be the unit element of the group. If we reverse a word and change all symbols ai in it to a−1 (and vice versa) then we obtain the inverse of the word. In this very simple structure, i the following problem is algorithmically undecidable.