In Its Usual Formulation, Fuzzy Computation Is, In General, NP-Hard, But a More Realistic Formulation Can Make It Feasible

In many practical situations, we cannot directly measure or estimate the desired quantity y – e.g., we cannot directly measure the distance to a star or the next week’s temperature. To provide the desired estimate, we measure or estimate easier-to-measure quantities x 1 , … , x n which are related to y, and then use the known relation to transform our estimates for x i into an estimate for y. In situations when x i are known with fuzzy uncertainty, we thus need fuzzy computation. Zadeh’s extension principle provides us with formulas for fuzzy computation. The challenge is that the resulting computational problem is NP-hard – which means that, unless P=NP (which most computer scientists consider to be impossible), it is not possible to solve all fuzzy computation problems in feasible time. To overcome this challenge, we propose a more realistic formalization of fuzzy computation – in which instead of an un-realistic requirement that the corresponding properties hold for all x i , we only require that they hold for almost all x i – in some reasonable sense. We show that under this modification, the problem of fuzzy computation becomes computationally feasible.