### Book Review: The decision problem: Solvable classes of quantificational formulas

Yuri Gurevich
1982 Bulletin of the American Mathematical Society
Reading, Mass., 1979, xii + 271 pp., \$27.50 . Unsolvable classes of quantificational formulas, by Harry R. Lewis, Addison-Wesley, Reading, Mass., 1979, xv + 198 pp., \$13.50. What is the decision problem in question? The original problem was to find an algorithm deciding logical validity of sentences of the usual first-order logic. To illustrate the strength of the desired decision procedure let us show that it would decide Fermat's Great Theorem. Choose a finitely axiomatizable firstorder
more » ... etic or set theory T such that (i) you believe that the axioms of T are true and (ii) T is strong enough for the argument below. (The Bernays-Gödel set theory would do for most mathematicians.) Let A be the conjunction of the axioms of T, and let F be a sentence saying in the language of T that there is a quadruple 2. If A -> F is logically valid then, by (i), F is true and Fermat's Great Theorem fails. If Fermat's Great Theorem fails and a, b, c, n form a failing quadruple then T proves a n + b n = c n and n > 2; hence T proves F i.e. A -» F is logically valid. Historically it appeared more convenient to speak about satisfiability than logical validity. A sentence S is called satisfiable (respectively finitely satisfiable) if there is a model (respectively a finite model) satisfying S. A sentence S is satisfiable iff NOT S is not logically valid, S is logically valid iff NOT S is not satisfiable. Thus decision problems for logical validity and satisfiability are reducible to each other. In order to give a feeling of the field let me quote some early results. In 1915 Löwenheim gave a decision procedure for satisfiability of monadic sentencesthose using only one-place predicates. He proved also that dyadic sentencesthose using only two-place predicates-form a reduction class for satisfiability. (A class C of sentences is called a reduction class for satisfiability if there is an algorithm assigning a sentence 5" in C to any first-order sentence S in such a way that S' is satisfiable if and only if S is so. Reduction classes for finite satisfiability are defined similarly.) In 1931 Herbrand sharpened the latter result showing that just three dyadic predicates suffice to get a reduction class for satisfiability. In 1936 Kalmar achieved the result that even one dyadic predicate suffices. Recall that every first-order sentence can be effectively rewritten in a so-called prenex form: with quantifiers in front. E.g. Vx(3yR(x, y) and 3zR(z, x)) 273