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Therefore we seek to conceive a principle of separation allowing us to partition the set Ω(S) into q(S) subsets. In practice, the principle of separation can be obtained in various ways, for example by testing one after the other all the values that we can assign to the variables that describe the problem, which means that the number of possible values has to be finite and the values must be easy to determine (one variant consists of forbidding certain values, another in partitioning the set of all the possible values of a variable into subsets that are not necessarily singletons).

Being a – If not all the terms are zero, there is a strictly negative term α ˜ i pi . α distribution of probability, α ˜ i is positive, which implies that pi is strictly negative. n From the fixed point definition, α ˜ i k=1 max{pk , 0} = max{pi , 0}, which is zero since pi < 0. The terms of the zero sum of positive terms max{pk , 0} are all zero, so pk 0 for all k. The qj are all shown to be negative or zero in a similar way. The point (ii) of ˜ which proves the theorem. 3. The Loomis lemma and the Yao principle The Loomis lemma is the essential argument for extending the minimax theorem to the Yao principle.

2. Probabilistic complexity of a problem To prove a lower bound to the complexity of a problem, we must be able to consider all the possible algorithms. To this end, the algorithms are represented in the form of trees, where each node is an instruction, each branch is an execution, and each leaf is a result: this is the decision tree model. – A questionnaire is a tree whose leaves are labeled by classes and whose internal nodes are labeled by tests. If the test has k possible results, the internal node has k threads, and the k arcs linking the node to its threads are labeled by these results.

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