By Mohamed Saeed Taha

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E. it is verifiable in a polynomial time in deterministic machine, and any NP problem can be translated to it in polynomial time. 5 Groebner Bases Complexity Groebner bases are a fundamental tool in computational algebra. They provide a multivariate generalization of the Euclidean division for the gcd computation, as well as a generalization of Gauss-Jordan elimination to higher degree. Groebner bases, as we saw in the previous two chapters, have applications to many different problems. for this reason the computation of Groebner bases is an important issue.

Mohamed Saeed Taha. Conclusion and Future Work The most well-known algorithms to calculate Groebner bases are Buchberger algorithm, introduced by Buchberger in his PhD thesis, and the recently introduced F4 and F5 algorithms. As an application of Groebner bases we considered how to determine if a graph is 3-colourable, by first translating the graph into an associated ideal and then determining if 1 is in the ideal. The interesting aspect to notice with this approach, is that one can use any monomial ordering when solving the ideal membership problem.

2 The set of all words formed by the two letters a and b. 2 Let t : N −→ N be a function. Define the time complexity class to be T I ME(t(n)) = {L| L is a language decided by an O(t(n)) time Turing machine} . 3 Let f : N −→ N be a function. Define the space complexity class to be S PACE( f (n)) = {L| L is a language decided by an O( f (n)) space Turing machine} . We list here many time and space complexity classes, giving a brief description for each class: Class Brief description P The set of language accepted by deterministic Turing machines in polynomial time.