Common use of Generalities Clause in Contracts

Generalities. It is well-known that the class of k-testable languages in the strict sense is not closed under union. Take for instance the two 3-testable languages, represented by their DFA’s in Fig. 1a, that are generated by the following 3-test vectors: Z = ({aa}, {aa}, {aaa}, {aa}) Zj = ({ba, bb}, {ab, bb}, {baa, bab, aaa, aab}, {bb}) ∪ with Σ = a, b . The union γ3(Z) γ3(Zj) of these languages, represented by its DFA in Fig. 1a, is not a 3-testable language. Indeed, it is not a k-testable language for any value of k > 0. For k = 1, the only k-testable language that extends γ3(Z) γ3(Zj) is Σ∗. For k 2, the problem is that since ak−1 is an allowed prefix, ak−1b is an allowed segment, and ak−2b is an allowed suffix, ak−1b has to be in the language, even though it is not an element of γ3(Z) γ3(Zj). It turns out that we can generalize Theorem 14 to unions of k-TSS languages.

Generalities. It is well-known that the class of k-testable languages in the strict sense is not closed under union. Take for instance the two 3-testable languages, represented by their DFA’s in Fig. 1a, that are generated by the following 3-test vectors: Z = ({aa⟨{aa}, {aa}, {aaa}, {aa}) Zj aa}⟩ Z′ = ({ba⟨{ba, bb}, {ab, bb}, {baa, bab, aaa, aab}, {bb}) ∪ bb}⟩ with Σ = a, b . The union γ3(Z) γ3(Zjγ3(Z′) of these languages, represented by its DFA in Fig. 1a, is not a 3-testable language. Indeed, it is not a k-testable language for any value of k > 0. For k = 1, the only k-testable language that extends γ3(Z) γ3(Zjγ3(Z′) is Σ∗. For k 2, the problem is that since ak−1 is an allowed prefixprefix, ak−1b is an allowed segment, and ak−2b is an allowed suffixsuffix, ak−1b has to be in the language, even though it is not an element of γ3(Z) γ3(Zjγ3(Z′). It turns out that we can generalize Theorem 14 to unions of k-TSS languages.