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Additional resources for Complexity of Algorithms (Lecture Notes)
There is a number of variants of the previous result, asserting the undecidability of similar problems. Let T be a Turing machine. The halting problem for T is the problem to decide, for all possible inputs x, whether T halts on x. Thus, the decidability of the halting problem of T means the decidability of the set of those x for which T halts. We can also speak about the halting problem in general, which means that a pair (T, x) is given where T is a Turing machine (given by its transition table) and x is an input.
Obviously, just as its emptyness, we cannot decide any other property P of of LT either if the empty language has it and Σ∗0 has not, or vice versa. Even a more general negative result is true. We call a property of a language trivial if either all languages have it or none. 9) Rice’s Theorem For any non-trivial language-property P , it is undecidable whether the language LT of an arbitrary Turing machine T (given by its description) has this property. Thus, it is undecidable on the basis of the description of T whether LT is ﬁnite, regular, contains a given word, etc.
These problems had a great eﬀect on the development of the mathematics of the century. 5) Diophantine equation Given a polynomial p(x1 , . . , xn ) with integer coeﬃcients and n variables, decide whether the equation p = 0 has integer solutions. ) In Hilbert’s time, the notion of algorithms was not formalized but he thought that a universally acceptable and always executable procedure could eventually be found that decides for every Diophantine equation whether it is solvable. After the clariﬁcation of the notion of algorithms and the ﬁnding of the ﬁrst algorithmically undecidable problems, it became more probable that this problem is algorithmically undecidable.