In what follows, we shall prove some results concerning truth functions (in § § 2-4) and apply them to the following problem (in § § 5-6). There is a set S of objects and there are n +1 subsets Z, X ± , X 2 , ..., X" of S. Let an object s(£S) be chosen arbitrarily. We are not able to decide immediately whether or not s belongs to Z; we may observe, however, the validity of any of the n relations s and we can infer to the truth of s£Z if all the relations s£X 2 , ..., s£X" are checked. We are

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... erested in deciding, whether s£Z holds or not, in such a manner that a possibly small number of the relations s£X t should be examined (successively, in a straightforward ordering). § 2. Let /C*i, x 2 , x n ) be an n-ary truth function. The rank g(f) is the number of places where/takes the value t (true); of course,/takes the value J (false) at 2 n -Q(f) places. The entropy t\{f) is defined by /K/) = min(í?(/),2"-í>(/)). We have t]{f)i=r\(J)^2"; furthermore, f?(/)=0 exactly if/is constant. Let 21 be an elementary conjunction over the set {x t ,x 2 , ..., x"}. The number of variables occuring in 21 is called the length /(21) of 21. Suppose that 21 contains (precisely) the variables x h , x h , ..., x it (/=/(21)(^1)). We denote by x jl , x j2 , ..., x jn l the elements of the set