In this paper, we present a statistical criterion for accepting/rejecting the pairwise reciprocal comparison matrices in the analytic hierarchy process. We have studied the consistency in random matrices of different sizes. We do not agree with the traditional criterion of accepting matrices due to their inflexibility and because it is too restrictive when the size of the matrix increases. Our system is capable of adapting the acceptance requirements to different scopes and consistency necessities. The advantages of our consistency system are the introduction of adaptability in the acceptance criterion and the simplicity of the index we have used, the eigenvalue (λmax) and the simplicity of the criterion.

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A theoretical result [10] that relates tautological consequence in many-valued logics to the ideal membership problem in algebra is revisited. The intended use of the approach in this article and its implementation is the verification of consistency and the automated extraction of knowledge in rule-based knowledge systems. Programs are written in the CoCoA language. Four RBS are studied to illustrate the implementation.

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Abstract. In this paper, we present a statistical criterion for accepting/rejecting the pairwise reciprocal comparison matrices in the Analytic Hierarchy Process. We have studied statistically the consistency in random matrices of different sizes. We are not agree with the traditional criterion of accepting matrices due to their inflexibility and because it is too restrictive when the size of the matrix increases. Our system is capable to adapt the acceptance requirements to different scopes and consistency's necessities. The advantages of our consistency system is the introduction of statistical relativity in the acceptance criterion and the simplicity of the used index, the eigenvalue (λ max ).

We present a case study using ACL2 to verify a nontrivial algorithm that uses efficient data structures. The algorithm receives as input two first-order terms, and it returns a most general unifier of these terms if they are unifiable, failure otherwise. The verified implementation stores terms as directed acyclic graphs by means of a pointer structure. Its time complexity is O(n 2 ) and its space complexity O(n), and it can be executed in ACL2 at a speed comparable to a similar C implementation. We report the main issues encountered to achieve this formally verified implementation.

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