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Contents

I Introduction – Contributions of Behmann’s Habilitation Thesis
from the View of Computational Logic
 1 Introduction to Part I
 2 Specification of the Decision Problem
 3 Solution of the Decision Problem for Relational Monadic First and SecondOrder Formulas with Equality
 4 Specification of the Problem of SecondOrder Quantifier Elimination
 5 Solution of the SecondOrder Quantifier Elimination Problem for Relational Monadic Formulas with Equality
 6 Clarification of Schröder’s Early Results on Elimination
 7 Methodology: Computation by Equivalence Preserving Rewriting
 8 Methodology: Normal Forms
 II Behmann’s Decision and Elimination Method for Relational Monadic Formulas
1 Introduction to Part I
The Habilitation thesis of Heinrich Behmann (1891–1970), published in 1922 in Mathematische Annalen [beh:22], belongs, along with works by Löwenheim [loewenheim:15] and Skolem [skolem:19, skolem:20], to the standard references on the decision problem for relational monadic firstorder formulas (the Löwenheim class),^{1}^{1}1Relational monadic formulas are firstorder formulas with only unary predicates and no functions other than constants. and also for the extension of this class by secondorder quantification upon predicates. Early such references of [beh:22] include [hilbert:ackermann:28, p. 77], [hilbert:ackermann:second:38, p. 95], and [hilbert:bernays:34, p. 200], where also the methods by Behmann are reproduced [hilbert:bernays:34, p. 193ff and p. 200ff]. A detailed historic account is provided in Church’s Introduction to Mathematical Logic [church:book, §49, in particular p. 293]. The book by Church also presents variants of methods from [beh:22].
Behmann’s early work up to 1921 is presented in historic context by Mancosu [mancosu:behmann:99]. The focus there is the dissertation from 1918, but various issues concerning [beh:22], in particular its embedding into the context of the Hilbert school, are also documented. The historic analysis of the development of logics in the period 1917–23 by Zach [zach:99:completeness] describes Behmann’s contributions. In particular, it is observed there that Behmann’s talk on 10 May 1921 at the Mathematische Gesellschaft in Göttingen on the topic of his Habilitation thesis seems the first documented use of the term Entscheidungsproblem (decision problem) [zach:99:completeness, p. 363]. A transcript and English translation of this talk, along with a comprehensive introduction, has recently been published by Mancosu and Zach [mancosu:zach:2015]. The first published explicit statement of the decision problem seems to be in [beh:22, p. 166] (see [zach:99:completeness] for an English translation of the relevant passages).
Behmann reduces the decision problem for relational monadic formulas to the secondorder quantifier elimination problem, that is, the problem to compute for a given secondorder formula an equivalent firstorder formula. In computational logic, secondorder quantifier elimination [soqe], with variants called uniform interpolation, forgetting and projection, is today an area with a wide variety of applications and techniques.^{2}^{2}2As for example reflected in the SOQE 2017 workshop [soqe2017]. Some of today’s advanced methods for secondorder quantifier elimination are explicitly based on the socalled Ackermann’s Lemma, due to Wilhelm Ackermann [ackermann:35], and involve equivalence preserving rewriting of formulas as key technique [dls]. Although Behmann actually uses such rewriting techniques and gave with [beh:22] at that time Ackermann the impetus to investigate the elimination problem – as Ackermann courteously remarked in a letter to Behmann dated 29 Oct 1934 [Letter LABEL:corr:ab:1934:10:29]^{3}^{3}3Letters and manuscripts in Behmann’s bequest are listed in Part LABEL:partappsources. (see also Sect. LABEL:seccorrelim1934) – it appears that [beh:22] so far has been largely overlooked in the context of secondorder quantifier elimination in computational logic, such as the monograph [soqe], with the exception of historic references in [craig:2008, schmidt:2012:ackermann] and a recent paper by the present author [cwrelmon].
In this report we provide a detailed technical reconstruction of the methods and results from [beh:22] (Part II) and discuss various related issues, of which many are still today of relevance in computational logic (Part LABEL:partfurther). We summarize followup works by Behmann himself in unpublished manuscripts and in the correspondence with Wilhelm Ackermann, which mainly concerns elimination in presence of predicates with arity larger than one (Part LABEL:partpolyadic). This is supplemented by commented listings of publications by Behmann and documents in his bequest that are related to secondorder quantifier elimination (Part LABEL:partappsources). The correspondence with Wilhelm Ackermann, as far as archived in Behmann’s bequest in the Staatsbibliothek zu Berlin, is registered there completely. Part LABEL:partconclusion concludes the report.
We do not address another major concern of Behmann that is related to computational logic: his approach to resolve paradoxes, based on the idea that these emerge from unjustified elimination of shorthands (Kurzzeichen), leading to a variant of lambda conversion and restricted quantifiers [beh:31:widersprueche, beh:59:limitierte]. He discussed his approach, which is briefly mentioned by Curry and Feys in [curry:combinatoric:1, p. 4, 9, 260f], in correspondence with, among others, Ackermann, Bernays, Church, Gödel and Ramsey.
As already indicated, Behmann’s Habilitation thesis [beh:22] has so far mainly been considered in the context of the history of the decision problem. However, from the point of view of computational logic it is relevant also in various further respects, not merely for historical reasons, but there are also technical aspects that are still of significance today, for example, the successful termination of secondorder quantifier elimination methods on relational monadic formulas [cwrelmon], as well as methodical aspects, such as the roles of normal forms. The remaining sections of this part discuss these contributions.
2 Specification of the Decision Problem
As already mentioned, the first explicit statement of the decision problem seems to be in [beh:22]. For a translation of the relevant passages and discussions see [zach:99:completeness, mancosu:zach:2015].
3 Solution of the Decision Problem for Relational Monadic First and SecondOrder Formulas with Equality
As indicated above, this result was first obtained by Löwenheim, whereas Skolem and Behmann provided further proofs. Like Behmann’s method, the techniques of Löwenheim and Skolem also apply if predicate quantification is considered [church:book, p. 293]. As further noted in [church:book, p. 293], Behmann’s method to handle equality is similar to that of Skolem in some important respects, but seems to have been found independently. Behmann himself describes this in a letter dated 27 December 1927 to Heinrich Scholz [beh:nl, Kasten 3, I 63], brought to attention in [mancosu:behmann:99] with excerpts published in [mancosu:zach:2015] and below – see p. 4 and LABEL:pagescholzlengthy. Skolem’s proof is outlined from the perspective of elimination in [craig:2008]. A methodical aspect of [beh:22] seems worth mentioning: The decision problem is attacked there by investigating decidability explicitly for specific syntactically characterized formula classes (Aussagenbereiche).
4 Specification of the Problem of SecondOrder Quantifier Elimination
As described in [craig:2008], elimination problems play an important role in the works of Boole [boole:laws] and Schröder [schroeder]. It seems, however, that the problem of secondorder quantifier elimination has not been fully understood and explicitly stated accordingly before [beh:22]. The secondorder quantifier elimination problem is called there a new “elimination problem” (neue[s] „Eliminationsproblem”) and is explicated in the context of the instance that occurs first in that paper, the elimination of a unary predicate with respect to a formula of relational monadic firstorder logic without equality.^{4}^{4}4[beh:22, p. 196f]: Ich möchte dieses neue „Eliminationsproblem” in der folgenden Weise bestimmter fassen: Gegeben ist eine Aussage
Following Schröder [schroeder], Behmann calls the formula sought after resultant (Resultante). In the context of his elimination method, Behmann speaks in early manuscripts from 1921 of separation (Aussonderung) instead of Elimination. For instance, on p. 13 in [Manuscript LABEL:man:beh:21:ms:k9:37], a method description is headed Eliminationsverfahren. (Aussonderung?). In [Manuscript LABEL:man:beh:21:carbon], the manuscript for [beh:22], on p. 40, the specification of the elimination problem quoted in footnote 4 uses „Aussonderungsproblem” in place of „Eliminationsproblem”, on p. 45 the originally typed term Aussonderungshauptform is altered by a handwritten annotation to Eliminationshauptform (German for main form for elimination).
Behmann [beh:22, p. 218ff] remarks that Schröder distinguishes between “elimination problem” and “summation problem”, which are in Behmann’s view actually identical. Schröder’s elimination problem is, in Behmann’s words, to find a condition for the satisfiability of that is free from .^{5}^{5}5Another interpretation of Schröder’s concept of the elimination problem is to find a consequence of that has exactly those formulas as consequences which are consequences of and do not contain . See, for example, [schroeder:2.1, p. 200]. Schröder’s “summation problem” is, according to Behmann, to find a formula that is equivalent to and is free from . As Behmann describes, Schröder observed the equivalence of both problems in a note inserted during printing of the third volume of his Vorlesungen über die Algebra der Logik [schroeder:3, p. 489–490], whereas the actual identity of both problems escaped him through his concern for analogy with numerical algebra. Behmann concludes with commenting that this is a strange evidence for the extent in which for Schröder content receded in favor of form.^{6}^{6}6In modern view, a divergence between syntactic and semantic conceptions of elimination actually arises: For example, the quantified Boolean formula is equivalent to the propositional formula , where the Boolean quantifier has been “eliminated”. However, the formula still contains syntactically the formerly quantified atom , although, from a semantic point of view, redundantly. The characterization of such redundancy is not always evident, for example, for modern variants of secondorder quantifier elimination, where it its possible to “quantify upon” just a particular ground atom. On the other hand, for firstorder logic, both views coincide in a sense: As noted in [otto:interpolation:2000, Introduction], the construction of interpolants according to Craig’s interpolation theorem can be applied to compute for a given firstorder formula that is known to be equivalent to a formula expressed with a certain signature (predicates, functions and constants), an equivalent formula that is syntactically in that signature. The existence of an equivalent formula in a given signature can be expressed as validity.
Although Behmann explicitly formulates the problem of eliminating secondorder quantifiers and reduces the decision problem for relational monadic formulas to that elimination problem, he remains skeptical on whether the generalization to predicates with arbitrary arities and higherorder concepts can still be based on the elimination problem. His argument in [beh:22, p. 226f] is summarized in Part LABEL:partpolyadic, p. LABEL:pagecitelimitationsofelimination. In his letter to Heinrich Scholz dated 27 December 1927 [beh:nl, Kasten 3, I 63], answering Scholz’s question about who has written before him and, in particular, who has written at first, about the decision problem (Postcard from Scholz to Behmann, dated 19 December 1927, [beh:nl, Kasten 3, I 63]), he relates the decision problem to the elimination problem and remarks that the latter has been treated “first by the Americans, in particular Peirce, and later with particular love and persistence by Schröder, and eventually found a specialist in Löwenheim, who wrote several treatises about it in the Math. Annalen”. Behmann continues that the most important of these works by Löwenheim is Über Möglichkeiten im Relativkalkül [loewenheim:15] and mentions that Löwenheim came there already to important partial results of his work [beh:22], “however – in a presentation that is neither mathematically strict nor sufficiently comprehensible, such that I have construed these properly only after publication of my own paper.”^{7}^{7}7An excerpt of the German letter is quoted in Sect. LABEL:seccorrfurther, p. LABEL:pagescholzlengthy. Further parts of that letter are summarized in [mancosu:zach:2015].
5 Solution of the SecondOrder Quantifier Elimination Problem for Relational Monadic Formulas with Equality
Behmann [beh:22] presents an effective method for eliminating the secondorder quantifiers in a given relational monadic formula with equality and with predicate quantification. (The earlier decidability proofs in [loewenheim:15] and [skolem:19] mentioned above are similarly based on secondorder quantifier elimination in monadic logics.) Behmann’s method terminates after a finite number of steps with an equivalent relational monadic firstorder formula. The result formula might be with equality also in cases where the given formula is without equality. For a given formula in which all predicates are quantified, the result formula just expresses constraints on the domain cardinality: The formula is either true for all domain cardinalities with exception of a finite number or false for all domain cardinalities with exception of a finite number. Obviously, valid and unsatisfiable formulas are special cases of such formulas.
The key technique of Behmann’s procedure is to propagate quantifiers inward, also for the price of expensive operations such as distribution of conjunction over disjunction. As suggested by Behmann, we call the resulting form innex.^{8}^{8}8Die Endform meines Reduktionsverfahrens wird gelegentlich (sprachlich wenig glücklich) als "kontrapränex" bezeichnet; ich ziehe die Benennung "innex" vor. Letter from Heinrich Behmann to Alonzo Church, 30 January 1959 [beh:nl, Kasten 1, I 11]. This inward propagation is is applied to quantifiers upon individual variables as well as to quantifiers upon predicates. A detailed presentation of Behmann’s method is the topic of Part II, further aspects of the method will be discussed in Part LABEL:partfurther.
6 Clarification of Schröder’s Early Results on Elimination
Most issues solved in [beh:22] have been raised by Schröder in his Vorlesungen über die Algebra der Logik [schroeder]. Their solutions are developed by Behmann in a more modern representational framework, with a dedicated notation for logics, not obfuscated by the aim for correspondence to numeric algebra. The work by Behmann seems thus also useful as a guide to Schröder’s results, complementing the outline in [craig:2008]. We already sketched Behmann’s discussion of Schröder’s notion of elimination in Sect. 4. The precise relationship of Behmann’s core result to Schröder’s earlier partial result, in particular, his ,,crude resultant” („Resultante aus dem Rohen”), as described Behmann will be shown in Sect. LABEL:secelimnoeq.
7 Methodology: Computation by Equivalence Preserving Rewriting
With the requirement to decide a statement after a finite number of steps, the Entscheidungsproblem inherently involves some notion of computation. Computation steps are expressed in [beh:22] as equivalence preserving rewriting steps of logical formulas, justified by a collection of formula equivalences. A method starts with a given formula. At each step, a subformula occurrence is replaced with an equivalent formula, according to some computation rule (Rechenregel), that is, an equivalence from the collection, oriented either from left to right or from right to left. Computation terminates if the formula has reached a specific syntactic form. As Behmann states [beh:22, p. 167], the particular collection of equivalences he gives is motivated not by finding a small set of orthogonal axioms, but by satisfying the needs of practical computation (Bedürfnisse des praktischen Rechnens).^{9}^{9}9See [zach:99:completeness, p. 351f] for an English translation of the relevant section from [beh:22].
The foundation on equivalence, a semantic property, ensures that equivalence to the originally given formula is maintained as an invariant throughout the computation. Today, the representation by rewriting rules or transition systems that preserve semantic properties is a well established elegant way to represent computational methods such that they can be analyzed. Secondorder quantification allows to represent also notions like equisatisfiability (two formulas are either both satisfiable or unsatisfiable) as equivalence of formulas, making the preservation of equivalence a particularly useful invariant.
The modern socalled direct methods or methods of the Ackermann approach, for secondorder quantifier elimination [soqe], initiated by [dls:early, dls], typically founded on Ackermann’s Lemma [ackermann:35], quite literally follow Behmann’s template of applying equivalence preserving formula rewritings that include various formula conversions and elimination steps.
8 Methodology: Normal Forms
The methods introduced in [beh:22] essentially operate by converting given logical formulas to equivalent formulas in specific normal forms, that is, formulas with specific syntactic properties. Conjunctive and disjunctive normal forms are used dually there, the innex normal forms for quantifiers upon instance variables as well as upon predicates are developed and counting (or cardinality) quantifiers are applied (preceded in [loewenheim:15, skolem:19] – see [craig:2008]). Behmann’s method rewrites formulas to a certain intermediate form that allows predicate elimination according to a simple scheme.
The syntax for formulas used in [beh:22] is based on disjunction, conjunction and negation, exposing the symmetry and duality inherent in these operations, which is, as criticized by Behmann, obscured in notations based on implication used by Frege and in the Principia Mathematica. For monadic formulas in a certain normal form, Behmann introduces a special notation (Klassensymbolik), where individual variables are suppressed.
In modern computational logic, normal forms play various roles. Systems typically operate on inputs in conjunctive normal form, obtained from preprocessors. Normal forms that allow to perform certain operations in an inexpensive way are investigated as target formats for knowledge compilation. The preservation of a certain normal form by calculi is applied to ensure that outputs are in a certain fragment of firstorder logic or can be mapped to some other logic such as a modal or description logic (the method of [ks:2013:frocos] for secondorder quantifier elimination in description logics can be considered as an example). Normal forms can provide representations of formulas that facilitate to understand their meaning, which is useful in the development of techniques as well as to present results to end users. This aspect of getting an overview on a solution in its totality has been a continuous concern for Behmann, for example in his comments to Ackermann’s resolutionbased elimination technique, summarized below in Part LABEL:partpolyadic, or in his later work on the solution problem (Auflösungsproblem) [beh:50:aufloesungs:phil:1, beh:51:aufloesungs:phil:2].
The paradigm of “model computation” or “answer set programming” in automated reasoning and logic programming can be considered as a variant of normal form computation: Such systems enumerate data structures that represent models. Their solutions can be regarded as a normalized representation of the input. A particular special case is enumerating all conjunctive clauses of a disjunctive normal norm.
The socalled quantifier elimination approach in early model theory of the 1920s is another area where variants of normal form computation play an essential role, as will be discussed below in Sect. LABEL:secnfqemfe. The integration of such quantifier elimination methods for decidable theories into reasoning systems is currently an area of extensive research in automated deduction, motivated in particular by applications in software and hardware verification. Also the evaluation of relational database queries can be considered as elimination of firstorder quantifiers [kanellakis:95, revesz].
9 Introduction to Part Ii
This part focuses on the main technical material in [beh:22] from the point of view of secondorder quantifier elimination as considered in computational logic. Modern notation is used throughout, on occasion a concordance to Behmann’s original labeling and various notations, which have merits on their own, are given. We aim here at a general formalization, where Behmann sometimes introduces techniques only with exemplary cases that provide intuition and indicate the general case. Also the structuring of the presentation deviates from the original paper, aiming at the modern reader.
The rest of this part is structured as follows: In Sect. 10 notation and terminology are introduced and general remarks on the presentation are given. Section 11 provides an overview on Behmann’s results, proceeding in a “topdown” fashion where the more involved methods and proofs are only sketched. With a collection of equivalences and entailments for use in formula rewriting and considerations on deciding and normalizing propositional logic, Sect. 12 paves the way for the more thorough presentation of Behmann’s techniques in the subsequent sections. First, the general case of monadic formulas with equality is considered. Sect. LABEL:secccnf describes a normalization method for such formulas. The secondorder quantifier elimination method, which applies to the normalized formulas, is then shown in detail in Sect. LABEL:secelimeq. A simplified variant of the general method is then considered in Sect. LABEL:secelimnoeq. It applies just to the case without equality, but facilitates discussion of other issues, in particular the correspondence to earlier works by Schröder, as shown by Behmann.
10 Notational Conventions and Preliminary Remarks
10.1 Syntax
We briefly write predicate, function and constant for predicate symbol, function symbol and constant symbol, respectively. For secondorder logic with equality, that is firstorder logic with equality and extended by quantification upon predicates and functions, we use the following syntactic notation: An atomic formula, or briefly atom, is either of the form , where and where is a predicate of arity , or of the form . In both cases, each subscripted is a term, that is an individual variable or of the form , where , the are terms and is a function of arity .^{10}^{10}10This parenthesisfree notation for terms and atoms is used in the modern textbook [eft:german]. A nullary function is also called constant. An atom of the form is called equality atom. Formulas and classes of formulas which may contain/may not contain equality atoms are called briefly with equality or without equality, respectively. If we speak of firstorder formulas, unless explicitly indicated otherwise, we assume formulas with equality.
A formula is constructed from atoms, the constant operators (true), (false), and a finite number of applications of the unary connectives (negation), the binary connectives (conjunction) and (disjunction), as well as quantifications with (universal quantification) and (existential quantification). Negated equality , further binary operators , as well as ary versions of and can be understood as metalevel notation. For these ary versions, the cases , which are and , respectively, are included. The scope of , the quantifiers, and the ary versions of and in prefix notation is the immediate subformula to the right.
A subformula occurrence in a given formula is positive (negative) if it is in the scope of an even (odd) number of negations. A literal is an atom or a negated atom. For a formula
of the form , the complement of is , if has a form different from , then the complement is .A quantifier occurrence may be upon an individual variable (“firstorder”) or upon a predicate or function (“secondorder”). We call the former also an individual quantifier and the latter predicate quantifier. An occurrence of an individual variable, predicate or function that is not bound by a quantifier occurrence in a formula is called free in that formula. As common in discussions of firstorder logic, we distinguish between constants and free occurrences of individual variables. However, we do not make an analogous distinction for predicates and functions, since it would not be of relevance in the considered contexts. Thus, an occurrence of a predicate or function in a formula is just free or bound by a quantifier. In a firstorder formula, all predicate and function occurrences are free.
10.2 Boolean Combination of Basic Formulas
Following patterns suggested by early model theory (see e.g. [chang:keisler, Sect. 1.5]), in this presentation, several normal forms are characterized as Boolean combination of basic formulas, that is, as the formulas that are obtained from certain basic formulas, the constant operators , and repeated application of the operators , and .^{11}^{11}11This choice of operators has been made for convenience. Of course, technically it would be sufficient to just permit e.g. and , and express and as disjunction and conjunction, respectively, of an arbitrarily picked basic formula and its negation.
10.3 Considered Formula Classes
We use the following symbols for particularly considered formula classes: is the class of relational monadic formulas (also called Löwenheim class), that is, the class of firstorder formulas with nullary and unary predicates, with constants but no other functions, and without equality. is with equality. and are and , resp., extended by secondorder quantification upon predicates.
All of these classes are decidable. admits secondorder quantifier elimination, that is, there is an effective method to compute for a given formula an equivalent formula in which all predicates are unquantified predicates in , as well as all constants and free variables are also in . In this sense is closed under secondorder quantifier elimination, which does not hold for , since elimination applied to a formula might introduce equality.
A quantified Boolean formula is a formula where all predicates are nullary (called then also Boolean variables in the literature) and quantification is allowed just upon predicates.
10.4 Remarks on the Presentation of Behmann’s Results
Theorems that Assert the Existence of Effective Methods.
We formulate results often as theorem statements that assert the existence of an effective method to compute for a given formula an equivalent formula with certain properties. The proof is then typically a description of such a method. An alternative would be to just state the existence of an equivalent formula with certain properties as theorem. This course has not been followed here, since in contexts where the existence of a method is relevant, reference to the theorem alone (playing the role of a “module interface”) would not be sufficient, but the underlying proof (the “module implementation”) would have to be referenced.
Free Individual Variables and Constants.
Many of the results of [beh:22] apply to formulas – which may contain free variables – such that free variables and constants are handled in exactly the same way. While these results are presented here explicitly as properties of formulas, they are presented in [beh:22] as properties of sentences (Aussagen), that is, formulas without free individual variables, considering free individual variables just as constants (see [beh:22, footnote 25, p. 196]^{12}^{12}12Behmann’s footnote translates as: I thus call the basic components of an expression “variable” („veränderlich”) or “constant” („konstant”), depending on whether it is represented within the expression by quantifiers (Operatoren) or not. As far as I can see, this is exactly the sense in which this distinction is actually made in mathematics.).
Consideration of Duality.
Methods based on equivalence preserving transformations of classical logic formulas typically come in two dual variants, where the roles of conjunction and disjunction as well as the roles of existential and universal quantification are switched (as made more precise with Prop. LABEL:propequidual in Sect. 12.1). In [beh:22], such methods are in general explicitly developed for one of the variants and the dual variant is then indicated. In the presentation here, the discussion of dual variants is completely neglected, with exception of a few specific cases. Actually, from a technical point of view, the dual variants can be completely disregarded, since for inputs where they would seem adequate, their behavior would be just simulated by the original variant with the only difference that instead of atomic formulas (or basic formulas of other forms) their complements are used. As an example, consider the equivalence and its dual . The dual can be derived from the first variant with negated atoms in the following steps, which additionally only involve inward propagation of negation and expansion/contraction of universal quantifiers: .
11 Overview on Behmann’s Results and Methods
The final result of [beh:22] can be stated as a theorem about formulas of a certain syntactic form. It allows to derive various results on secondorder quantifier elimination and decidability of monadic formulas, including the decidability of . In this overview section we start with presenting this theorem and sketch in a “topdown” manner the techniques used in [beh:22] to prove it. We then present the derived results, proven as corollaries of the theorem. More indepth proofs of the theorem itself and discussions of the involved techniques will then be provided in a “bottomup” manner in subsequent sections.
11.1 Elimination Method
The core result of [beh:22] can be stated as follows: [Predicate Elimination for ] There is an effective method to compute from a given predicate and formula a formula such that

is a formula,

,

does not occur in ,

All free individual variables, constants and predicates in do occur in .
The proof given in [beh:22] for Theorem 11.1 resides on the conversion of arbitrary formulas to a certain syntactic normal form that only allows restricted use of quantification. In particular, the scopes of quantifier occurrences are not permitted to overlap. To achieve this property, it is utilized that equality atoms with quantified variables can be represented implicitly by counting quantifiers , which express existence of at least individuals. Formulas with counting quantifiers can be expanded into equivalent formulas of particular shapes with standard firstorder quantifiers and equality atoms, as discussed in more detail below in Sect. LABEL:seccq. The standard quantifier can be equivalently expressed as . In the considered normal form, the argument formula of a counting quantifier must be a conjunction of literals of applications of a unary predicate to the quantified individual variable. The representability of formulas in this normal form can be stated as follows: [Counting Quantifier Normal Form for ] There is an effective method to compute from a given formula a formula such that

is a Boolean combination of basic formulas of the form:

[label=(),ref=]

, where is a nullary predicate,

, where is a unary predicate and is a constant or an individual variable,

, where each of is a constant or an individual variable,

, where , and the are pairwise different and pairwise noncomplementary positive or negative literals with a unary predicate applied to the individual variable ,


,

All free individual variables, constants and predicates in do occur in .
If the given formula in Theorem 11.1 is without equality, the allowed basic formulas can be strengthened by excluding the case (1c) and restricting the case (1d) to , such that the counting quantifier can be considered as standard quantifier. The method of [beh:22] to compute the normal form according to Theorem 11.1 proceeds by applying equivalence preserving formula rewritings to move quantifiers inward such that their scopes do not overlap. All predicate occurrences in the scope of a quantifier then have exactly the quantified variable as argument.
To achieve this, aside from inexpensive transformations such as narrowing quantifier scopes to subformulas where the quantified variables actually occur, distribution of existential (universal, resp.) quantifiers over disjunction (conjunction, resp.), propagating negation inward, and rearranging binary connectives according to associativity and commutativity, also expensive transformations, in particular distribution of conjunction over disjunction and vice versa, as familiar from conversion to disjunctive and conjunctive normal form, are required. Consider the following example, where initially is in the scope of :
(1) 
Equality literals require special handling, as described in the full exposition in Sect. LABEL:secthmfoqemonadiceqfullproof below.
We now sketch a method as asserted by Theorem 11.1 for the case where the input formula is without equality. Given is the formula , where is a formula and is a predicate. We only consider unary here, nullary could be handled analogously as a particularly simple special case. The input formula is first rewritten to an equivalent formula in which all occurrences of are in subformulas of a specific syntactic form, called Eliminationshauptform (main form for elimination) in [beh:22]:
(2) 
where , , , are natural numbers and the , , , are firstorder formulas in which does not occur. This can be achieved with the following steps: Normalize with the method of Theorem 11.1 and convert to disjunctive normal form, generalized such that the role of atoms is played by the basic formulas. Rewrite occurrences of , where is a constant or a variable that is free in , first with the equivalence and then with . This step introduces equality, which thus may also be present in the result. If is without equality, the counting quantifiers are decorated with , directly corresponding to standard quantifiers. They can thus be rewritten with the equivalence , and then with . After the predicate quantifier upon is then moved inward with the techniques outlined for individual variable quantifiers in the context of Theorem 11.1, all occurrences of are in subformulas that match the Eliminationshauptform.
The Eliminationshauptform allows to move the existential individual quantifiers and the constituents and to the front of the predicate quantifier, while the occurrences of with existentially quantified arguments can be rewritten to universally quantified occurrences that match the forms or , respectively, by applying the equivalences and . (Notice that in this step again equality may be introduced also in cases where the original formula is without equality.) In this way, a formula in Eliminationshauptform can be further converted such that only occurs in a subformula which is in a restricted form of the Eliminationshauptform allowing only the two universally quantified constituents, that is, . When restricted in this way, the Eliminationshauptform matches the left side of the following Basic Elimination Lemma, which gives a firstorder equivalent for secondorder formulas matching its left side: [Basic Elimination Lemma] Let be a unary predicate and let be firstorder formulas with equality in which does not occur. It then holds that
Note that and in that proposition may contain free variables, in particular free occurrences of , which are then bound by the surrounding universal quantifiers on the left as well as on the right side of the proposition. Rewriting all subformulas headed with according to Lemma 11.1 then completes the method asserted by Theorem 11.1.
11.2 Applications to Predicate Elimination and Decidability
We now turn to applications of Theorem 11.1. Repeatedly running the method ensured by that theorem allows to eliminate all predicate quantifiers in a formula. The result is firstorder, but might be with equality even in cases where the input is without equality, since a single run of the method already might introduce equality. This transfer of Theorem 11.1 to formulas with several predicate quantifiers is made precise with the following corollary: [Elimination in ] There is an effective method to compute for a given formula of a formula such that

is a formula,

,

Predicates that just occur bound by a secondorder quantifier in do not occur in ,

All free individual variables, constants and predicates in do occur in .
Proof
Return the result of applying the following equivalence preserving rewritings to : First, exhaustively^{13}^{13}13By rewriting exhaustively we mean that in each step a subformula occurrence of the indicated form is replaced and the resulting overall formula is subjected again to rewriting, until it does no longer contain a subformula of the indicated form. rewrite subformula occurrences of the form where is a predicate with the equivalent formula . Second, exhaustively rewrite subformula occurrences of the form where is a predicate and is firstorder with (i.e. is an innermost secondorder quantification) to the equivalent firstorder formula obtained according to Theorem 11.1. ∎
Basic formulas of form (1d) in Theorem 11.1 include the case where , that is, . If is without constants, without free individual variables and such that all predicate occurrences are quantified, then the result of applying the method according to Corollary 11.2 followed by normalization according to Theorem 11.1 must be a Boolean combination of basic formulas of just the form , where is a number . A formula is satisfied by exactly those interpretations whose domain has at least distinct members. A formula by those whose domain has less than members. As observed in [beh:22], a Boolean combination of formulas of the form with is either true for all domain cardinalities with exception of a finite number or false for all domain cardinalities with exception of a finite number. It is not hard to see that validity and satisfiability of Boolean combinations of formulas of the form is decidable. We discuss this in more depth in Sect. LABEL:secqppure below.
The decidability problem for can be reduced to the elimination problem. Hence, the decidability of , and thus also of its subclasses , and , follows from Corollary 11.2, and thus indirectly from Theorem 11.1. The following corollary states this from two perspectives, validity and satisfiability.
[Decidability of ]
corrdecidevalid There is an effective method to decide whether a formula is valid.
corrdecidesat There is an effective method to decide whether a formula is satisfiable.
Proof
(LABEL:corrdecidevalid) Let be the given formula. Let be all predicates with free occurrences in , let be the free individual variables in , and let be the constants in . Let be the formula
The formula is valid if and only if is valid. When applied to , the method according to Corollary 11.2 followed by normalization according to Theorem 11.1 then yields an equivalent formula which is a Boolean combination of formulas of the form , where . Validity of such Boolean combinations can be decided.
(LABEL:corrdecidesat) Decidability of satisfiability follows trivially from the decidability of validity, since a formula is satisfiable if and only if its negation, which is also a formula, is not valid. However, it is also possible to express the involved intermediate steps directly in terms of satisfiability: Let be the given formula. Let be the formula
where the quantified predicates, variables and constants are as specified in the proof of Prop. LABEL:corrdecidevalid Then is satisfiable if and only if is satisfiable. As in the proof for the decidability of validity, when applied to , the method according to Corollary 11.2 followed by normalization according to Theorem 11.1 yields an equivalent formula which is a Boolean combination of formulas of the form . Satisfiability of such Boolean combinations can be decided. ∎
12 Starting Points: Rewrite Rules and Deciding Propositional Logic
The methodical approach of [beh:22] essentially consists in developing effective methods that operate by rewriting of formulas in an equivalence preserving way to certain normal forms. The rewriting is done according to a set of rules, that is, oriented equivalences. The involved normal forms are in particular Boolean combinations of certain basic formulas as well as disjunctive and conjunctive normal form, generalized such that the role of atoms is played by certain basic formulas. In this section, a collection of the relevant equivalences and entailments that are used as rules is presented. In addition, the relationship between clausal normal forms and decision methods is sketched for propositional logic and quantified Boolean formulas, along with some comments on history.
12.1 Equivalences and Entailments for Rewriting Formulas
Wellknow equivalences and entailments between formulas are listed below as labeled propositions, such that they can be referenced in the sequel. Their choice is mainly motivated by their role in the methods of [beh:22]. A concordance with the rule labels used in [beh:22] is provided with Table LABEL:tabconcordanceb22 at the end of the section. The following Prop. 12.1 gathers equivalences between formulas. In [beh:22], such equivalences are used as rules for reversible inferences (Regeln für umkehrbare Schlüsse): they can be applied oriented from left to right as well oriented from right to left to obtain a formula that is equivalent to a given formula, but has different syntactic properties.
[Equivalences Useful for Rewriting] We consider secondorder logic with equality. For all formulas , quantifiers , individual variables or predicates and binary connectives the following equivalences hold:
Interaction of negation with other operators
EQ 1  . 

EQ 2  . 
EQ 3  . 
EQ 4  . 
EQ 5  . 
Associativity, commutativity and idempotence of conjunction and disjunction
EQ 6  . 

EQ 7  . 
EQ 8  . 
Interaction of truth values with other operators
EQ 9  . 

EQ 10  . 
EQ 11  . . 
EQ 12  . . 
EQ 13  . . 
EQ 14  . . 
EQ 15  . 
EQ 16  . 
Cancellation of complementary formulas
EQ 17  . . 

EQ 18  . . 
Distribution among conjunction and disjunction
EQ 19  . 

.  
EQ 20  . 
. 
Quantifier shifting
EQ 21  . 

EQ 22  . 
EQ 23  , if does not occur free in G. 
, if does not occur free in F. 
Vacuous quantifiers, quantifier switching and variable renaming
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