A New Class of Automated Theorem-Proving Algorithms

A procedure is defined for deriving from any statement <italic>S</italic> an infinite sequence of statements <italic>S</italic><subscrpt>0</subscrpt>, <italic>S</italic><subscrpt>1</subscrpt>, <italic>S</italic><subscrpt>2</subscrpt>, <italic>S</italic><subscrpt>3</subscrpt>, ··· such that: (a) if there exists an <italic>i</italic> such that <italic>S<subscrpt>i</subscrpt></italic> is unsatisfiable, then <italic>S</italic> is unsatisfiable; (b) if <italic>S</italic> is unsatisfiable, then there exists an <italic>i</italic> such that <italic>S<subscrpt>i</subscrpt></italic> is unsatisfiable; (c) for all <italic>i</italic> the Herbrand universe of <italic>S<subscrpt>i</subscrpt></italic> is finite; hence, for each <italic>i</italic> the satisfiability of <italic>S<subscrpt>i</subscrpt></italic> is decidable. The new algorithms are then based on the idea of generating successive <italic>S<subscrpt>i</subscrpt></italic> in the sequence and testing each <italic>S<subscrpt>i</subscrpt></italic> for satisfiability. Each element in the class of new algorithms is complete.

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