Early Steps of Second-Order Quantifier Elimination beyond the Monadic Case: The Correspondence between Heinrich Behmann and Wilhelm Ackermann 1928-1934 (Abstract)

This presentation focuses on the span between two early seminal papers on second-order quantifier elimination on the basis of first-order logic: Heinrich Behmann’s Habilitation thesis Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem (Contributions to the algebra of logic, in particular to the decision problem), published in 1922 as [4], and Wilhelm Ackermann’s Untersuchungen über das Eliminationsproblem der mathematischen Logik (Investigations on the elimination problem of mathematical logic) from 1935 [2]. Behmann developed in [4] a method to decide relational monadic formulas (that is, first-order formulas with only unary predicates and no functions other than constants, also known as Löwenheim class) that actually proceeds by performing second-order quantifier elimination with a technique that improves Schröder’s rough-and-ready resultant (Resultante aus dem Rohen) [22,10]. If all predicates are existentially quantified, then elimination yields either a truth value constant or a formula that just expresses with counting quantifiers a cardinality constraint on the domain. Although technically related to earlier works by Löwenheim [16] and Skolem [23,24], Behmann’s presentation appears quite modern from the view of computational logic: He shows a method that proceeds by equivalence preserving formula rewriting until a normal form is achieved in which second-order subformulas have a certain shape for which the elimination result is known [27,26]. Ackermann laid in [2] the foundation for the two major modern paradigms of second-order quantifier elimination, the resolution-based approach [12], and the so-called direct or Ackermann approach [11,13,21], which is like Behmann’s method based on formula rewriting until second-order subformulas have a certain shape for which the elimination result is known, however, now based on more powerful equivalences of secondto first-order formulas, such as Ackermann’s Lemma. Another result of Ackermann’s paper was a proof that second-order quantifier elimination on the basis of first-order logic does not succeed in general. As documented by letters and manuscripts in Behmann’s scientific bequest [6], between 1922 and 1935 Behmann and Ackermann both thought about possibilities to extend elimination to formulas with predicates of arity two or more. Behmann gave in 1926 at the Jahresversammlung der Deutschen Mathematiker102