INTERLEAVING LOGIC AND COUNTING

2.1 From counting to logic

Let us restrict attention to a very small fragment of the language $\mathcal {L}_\#$ described above. Given some variables $\mathsf {Var}$ and predicate symbols $\mathsf {Pred}$ , we only allow two types of atomic formulas and one operation for building complex formulas. Let $\mathcal {L}_\#^-$ be generated by the grammar:

$$ \begin{align*} \varphi ::= P(x_1,\dots,x_n) \quad\mid\quad x\neq y \quad\mid\quad \#_x\varphi \succsim \#_y\varphi. \end{align*} $$

Aside from predication and variable inequality, we can only compare cardinalities.

A first observation is that Boolean implication can already be defined in $\mathcal {L}_\#^-$ . Where x occurs free in neither $\varphi $ nor $\psi $ , we can take

(2.3) $$ \begin{align} \psi \rightarrow \varphi \equiv \#_x\varphi \succsim \#_x\psi. \end{align} $$

Boolean negation can also be defined. Where $\mathsf {0}$ is an abbreviation for the formula $\#_x (x \neq x)$ (cf. Frege), and again x is a variable that does not occur free in $\varphi $ , we can define

(2.4) $$ \begin{align} \neg \varphi \equiv \mathsf{0} \succsim \#_x\varphi. \end{align} $$

With these we recover any other Boolean connective, as well as variable equality. In some respect, count comparison already incorporates Boolean structure, and familiar Boolean laws emerge as principles of count comparisons. For instance, the pattern $\varphi \rightarrow (\psi \rightarrow \varphi )$ is encoded simply as $\#_x(\#_x\varphi \succsim \#_x\psi ) \succsim \#_x\varphi $ .

Going further, first-order quantification is expressible in $\mathcal {L}_\#^-$ :

(2.5) $$ \begin{align} \exists x. \varphi \equiv \#_x\varphi \succ \mathsf{0}. \end{align} $$

This thus brings us back to full $\mathsf {FO}(\#)$ , in which we can again define the infinity quantifier $\exists ^\infty $ in (2.1), its dual $\forall ^\infty $ , and so on. From rather austere (atomic) primitives, count comparisons already encode a significant amount of logic, provided of course that we allow iteration of comparisons within comparisons.

2.2 From logic to counting

While pure $\mathsf {FO}$ is also capable of encoding facts about counting and arithmetic, it is far less extensive. As already mentioned, first-order logic can define the simple counting quantifiers like $\exists ^{\geq n}x$ ; however, first-order logic does so by means of counting in the syntax. That is, the formula expressing that there are at least n objects satisfying a given condition achieves this by concatenating n existential quantifiers and adding n conjuncts. Basic arithmetic principles like $\exists ^{\geq m}x.\varphi \rightarrow \exists ^{\geq n}x.\varphi $ for $m\geq n$ , thus follow from elementary logical patterns like distribution of existential quantification over conjunction, applied the requisite number of times (e.g., $m-n$ times). This style of counting in the syntax also produces a case-by-case formulation of the Pigeonhole Principle:Footnote ²

Example 2.7. Suppose we have k monadic predicate symbols $P_1,\dots ,P_k$ and let $n>k$ . Then,

(2.6) $$ \begin{align} \big( \exists^{= n}x. \bigvee_{i \leq k} P_i(x) \wedge \forall x. \bigwedge_{i \neq j} \neg (P_i(x) \wedge P_j(x)) \big)\rightarrow \bigvee_{i \leq k} \exists^{\geq 2}x. P_i(x) \end{align} $$

says that if these k predicates together include n objects, then at least one must include at least two objects. This schema is of course valid for every choice of k and $n>k$ .

We will see more examples of counting in the syntax with subsequent sections (see especially Remark 9.15 and Appendix E).

Remark 2.8. The fact that $\mathsf {FO}$ can only count in the syntax reverberates in interesting ways when we consider finite variable fragments of $\mathsf {FO}$ . While the two-variable fragment is known to have the (bounded) finite-model property [Reference Mortimer89], which in turn establishes its decidability, this fragment with counting quantifiers $\exists ^{\geq n}$ can easily enforce infinite models:

$$ \begin{align*}\forall x. \exists^{=1}y. R(x,y) \wedge \forall y .\exists^{\leq 1} x. R(x,y) \wedge \exists y. \forall x. \neg R(x,y) .\end{align*} $$

Such a language is in fact decidable [Reference Grädel, Otto and Rosen45]: like the two-variable fragment without counting, its satisfiability problem is NExpTime-complete [Reference Pratt-Hartmann103]. However, the complexity analysis of this system and its extensions [Reference Kieroński, Pratt-Hartmann and Tendera69] reveals arithmetical content that does not appear in analyses of the plain two-variable fragment, witness connections to integer programming (Section 3.2.1) and to semi-linear sets (Section 4.1).

2.3 Finite models

It is natural to consider a related system in the same language, but with interpretations restricted to finite models. Call such a system $\mathsf {FO}^\phi (\#)$ . As $\mathcal {L}_\#$ extends the language of first-order logic, Trakhtenbrot’s theorem tells us that the validities of $\mathsf {FO}^\phi (\#)$ are still not computably enumerable. Nonetheless, $\mathsf {FO}^\phi (\#)$ and variations on it have also been intensely studied in the literature on finite model theory. See, e.g., [Reference Otto98] or [Reference Schweikardt116] for summaries of relevant work.

As an example of distinctive issues that come up in the finitary setting, one might ask about the asymptotic probabilities of formulas in $\mathsf {FO}^\phi (\#)$ over finite structures. It was shown in [Reference Grumbach and Tollu48] that $\mathsf {FO}$ with the Härtig quantifier in fact possesses a zero–one law, just as pure $\mathsf {FO}$ does. As possession of a zero–one law is commonly interpreted as evidence that a logic cannot formalize any non-trivial counting, this can be taken as justification for our choice of comparison rather than equality as a primitive. Indeed, $\mathsf {FO}^\phi (\#)$ lacks a zero–one law; e.g., $\#_x P(x) \succsim \#_x \neg P(x)$ has asymptotic probability $1/2$ . It is conjectured in [Reference Grumbach and Tollu48] that (an extension of) $\mathsf {FO}^\phi (\#)$ nonetheless possesses a limit law, and that the limits are all rational numbers between $0$ and $1$ .

For many purposes in finite model theory (e.g., descriptive complexity) authors have been motivated to consider proper extensions of our language $\mathcal {L}_\#$ , a notable example being fixed point logic with counting [Reference Cai, Fürer and Immerman22]. Our purpose here is different: we aim to isolate weaker fragments of this language that might further reveal the subtle interplay between logic and counting, also pinpointing differences and commonalities between finitary and infinitary patterns in counting.

2.4 Fragments of $\mathcal {L}_\#$

While full first-order logic with counting may be a natural starting point for exploring our subject, the above observations invite the search for natural fragments and weaker variants of $\mathsf {FO}(\#)$ . It may be desirable, for example, to identify decidable fragments of $\mathcal {L}_\#$ . From this perspective it is noteworthy that some familiar ways of taming complexity are less effective here. For example, finite-variable fragments do not result in decidability: as shown by Grädel et al. [Reference Grädel, Otto and Rosen46], the two-variable fragment of $\mathsf {FO}(\#)$ is still undecidable ( $\Pi ^1_1$ -complete, so we do observe a reduction in complexity, compared to Theorem 2.4). The two-variable fragment of $\mathsf {FO}^\phi (\#)$ is also undecidable. Evidently, a significant source of the complexity is the potent combination of counting and arbitrary quantificational-relational reasoning, witness Lemma 2.2. The undecidability proof in [Reference Grädel, Otto and Rosen46] for the two-variable fragment crucially involves counting successors along binary relations.

A more dramatic route would be to move to a much tamer syllogistic or propositional fragment [Reference Ding, Harrison-Trainor and Holliday32, Reference Moss and Bimbó91]. For instance, if we let $\mathcal {L}_\#^0$ be the language of propositional logic with count comparisons, the resulting system $\mathsf {PL}(\#)$ is easily shown to be decidable (e.g., it will follow immediately from our results below). This route at once eliminates relational reasoning and first-order quantification.

An alternative route is to put relational reasoning to the side, but still retain first-order quantification. The monadic fragment of $\mathcal {L}_\#$ , which we will call $\mathcal {L}_\#^1$ , does not allow counting along relations, but it otherwise preserves the counting content of $\mathsf {FO}(\#)$ . Observe, for example, that our definition of the infinity quantifier in (2.1) and our reconstruction of logical connectives from count comparisons (Section 2.1) depend in no way on the arity of available predicates. We will thus use $\mathsf {MFO}(\#)$ , monadic first-order logic with counting, as a base system to explore richer combinations (Section 3). In this context we will consider adding second-order quantification (system $\mathsf {MSO}(\#)$ in Section 4), as well as the ability to count not just individuals but sequences of individuals (systems $\mathsf {MFO}(\sharp )$ and $\mathsf {MSO}(\sharp )$ in Section 5).

Of course, counting along relations is also common and natural. We therefore explore a tractable modal fragment of $\mathcal {L}_\#$ , which we call $\mathcal {L}_\#^{\mathsf {ml}}$ , as a way of taming the interaction among counting, quantification, and relational reasoning. A summary appears in Table 1.

Table 1 A hierarchy of counting languages and logics, covered in Sections 2-7. For each logical system $\mathsf {L}(\#)$ we also have a version $\mathsf {L}^\phi (\#)$ , where we restrict to finite models. In these systems terms can only denote natural numbers

Following this work we consider a different route altogether, namely changing the semantics of $\mathcal {L}_\#$ . Relaxing either the logical interpretation (relativizing to sets of ‘admissible’ variable assignments; cf. [Reference Németi, Marx, Masuch and Pólos96]) or the numerical content of the $\#$ terms again results in systems that retain much of the character of $\mathsf {FO}(\#)$ , while gaining in tractability.

3.1 Some core principles

Both systems, $\mathsf {MFO}^\phi (\#)$ and $\mathsf {MFO}(\#)$ , are evidently invariant under automorphisms. In the monadic setting automorphisms are precisely the maps that permute elements within a region: all the points that satisfy a given state-description are indistinguishable. This means that if a property holds for one point in a region, it holds for every point in that region. This theme of permutation invariance is characteristic of counting, and it will return when we discuss generalized quantifiers in Section 9.

As demonstrated above, use of individual variables essentially allows manipulating regions—removing or adding points. We can correspondingly state a more general invariance principle. Fix some variables $\mathbf {y}$ and a fixed (finite) set $\mathbf {P}$ of predicate letters, and let $\alpha ^{\mathbf {y}}(x)$ specify a state-description for x as well as which of the variables $\mathbf {y}$ are (un)equal to x. Then, for any formula $\varphi $ (in predicates $\mathbf {P}$ ), if there is at least one x satisfying $\alpha ^{\mathbf {y}}$ and $\varphi $ , then every x satisfying $\alpha ^{\mathbf {y}}$ also satisfies $\varphi $ . Codified in a general invariance principle:

(INV) $$\begin{align} \exists x.\big(\alpha^{\mathbf{y}}(x) \wedge \varphi(x)\big) \rightarrow \#_x\big(\alpha^{\mathbf{y}}(x) \wedge \varphi(x)\big) \approx \#_x\big(\alpha^{\mathbf{y}}(x)\big). \end{align} $$

Since either none of the $\alpha $ ’s satisfy $\varphi $ or all of them do, once we have specified $\alpha $ in a count formula, reference to $\varphi $ becomes redundant. In fact, ( INV ) follows from an even stronger statement (that is, stronger provided we admit infinite models):

(INV2) $$\begin{align} \#_x\big(\alpha^{\mathbf{y}}(x) \wedge \varphi(x)\big) \succ \mathsf{0} \rightarrow \#_x\big(\alpha^{\mathbf{y}}(x) \wedge \neg \varphi(x) \big) \approx \mathsf{0}. \end{align}$$

A related observation about terms $\#_x\varphi $ is that subformulas of $\varphi $ that do not involve x do not contribute any fine-grained information to the term’s denotation. If the free variables of $\psi $ are not among the bound variables of $\#_x\varphi $ , then the following is valid:

(SUB)

Here is the result of substituting $\beta $ for every occurrence of $\gamma $ in $\alpha $ .

3.2 Normal forms

The principles recorded in ( INV ) and ( SUB ), together with basic propositional reasoning and a few other elementary principles (see Section 3.4 for the others), allow the derivation of a normal form result, which works uniformly for $\mathsf {MFO}^\phi (\#)$ and $\mathsf {MFO}(\#)$ . As a first step, we can show that any formula is equivalent to one with no embedded $\#$ -terms or quantifiers within $\#$ -terms, as these terms can always be replaced by unembedded existential quantifiers. This is already a dramatic departure from full relational $\mathsf {FO}(\#)$ , where embedding is non-trivial. (Recall that $\mathsf {FO}(\#)$ with no embedded count comparisons was $\Pi ^1_1$ -complete, in stark contrast to Theorem 2.4.)

Define depth $\mathsf {d}(\varphi )$ by recursion, with $\mathsf {d}(\alpha ) = 0$ for $\alpha $ atomic, $\mathsf {d}(\varphi \wedge \psi ) = \mbox {max}(\mathsf {d}(\varphi ),\mathsf {d}(\psi ))$ , $\mathsf {d}(\neg \varphi ) = \mathsf {d}(\varphi )$ , while $\mathsf {d}(\#_x\varphi \succsim \#_y\psi ) = \mbox {max}(\mathsf {d}(\varphi ),\mathsf {d}(\psi ))+1$ and $\mathsf {d}(\exists x. \varphi ) =\mathsf {d}(\varphi )+1$ .

Generically, a monadic formula with free variables $\mathbf {y},x$ can be written in disjunctive normal form $\bigvee _i \big (\alpha _i(\mathbf {y}) \wedge \alpha ^{\mathbf {y}}_i(x) \wedge \varphi _i(\mathbf {y},x)\big )$ , where $\alpha _i(\mathbf {y})$ specifies state-descriptions for $\mathbf {y}$ and which of these variables are (un)equal, $\alpha ^{\mathbf {y}}_i(x)$ is as in the previous subsection, and $\varphi _i(\mathbf {y},x)$ is some other formula that may in general have positive depth. We want to show that any formula

$$ \begin{align*} \#_x\bigvee_{i\in I} \big(\alpha_i(\mathbf{y}) \wedge \alpha^{\mathbf{y}}_i(x) \wedge \varphi_i(\mathbf{y},x)\big)\succsim \#_x\bigvee_{j\in J} \big(\alpha_j(\mathbf{y}) \wedge \alpha^{\mathbf{y}}_j(x) \wedge \varphi_j(\mathbf{y},x)\big)\end{align*} $$

is equivalent to one with no embedded count comparisons or quantifiers. In other words this formula is equivalent to one of depth $1$ . First, by ( SUB ) we can take the subformulas $\alpha _i(\mathbf {y}),\alpha _j(\mathbf {y})$ outside the count comparisons, which leaves

$$ \begin{align*} \#_x\bigvee_{i\in I} \big(\alpha^{\mathbf{y}}_i(x) \wedge \varphi_i(\mathbf{y},x)\big) \succsim \#_x\bigvee_{j\in J} \big( \alpha_j^{\mathbf{y}}(x) \wedge \varphi_j(\mathbf{y},x)\big) \end{align*} $$

to analyze. Let $\kappa _k$ range over formulas $\exists x.\big (\alpha _k^{\mathbf {y}}(x) \wedge \varphi _k(\mathbf {y},x)\big )$ for $k \in I \cup J$ . Then by appeal to ( INV ), we have the equivalent formula:

$$ \begin{align*} \bigvee_{K \subseteq I \cup J}\big(\bigwedge_{k \in K}\kappa_k \wedge \bigwedge_{k \notin K} \neg \kappa_k \wedge \#_x\bigvee_{i\in I\cap K} \alpha^{\mathbf{y}}_i(x) \succsim \#_x\bigvee_{j\in J \cap K} \alpha_j^{\mathbf{y}}(x) \big) \end{align*} $$

Note that we have traded one level of $\#$ embedding for one existential quantifier. Since $\alpha _i^{\mathbf {y}},\alpha _j^{\mathbf {y}}$ are of depth $0$ , this concludes the argument for:

Using Lemma 3.1, the main result of this section is:

Proof We show more generally that a formula of depth $k+1$ over predicates $\mathbf {P}$ with free variables $\mathbf {y} = y_1,\dots ,y_n$ is equivalent to a disjunction

(3.6) $$ \begin{align} \bigvee\big(\alpha(\mathbf{y})\wedge(\sigma)_{\alpha(\mathbf{y})}\big), \end{align} $$

where $\alpha (\mathbf {y})$ ranges over possible descriptions of $\mathbf {y}$ , and $\sigma $ is a complete description of the regions over $\mathbf {P}$ , i.e., specifying $T_1 {\kern-1pt}={\kern-1pt} T_2 {\kern-1pt}+{\kern-1pt}m$ or $T_1{\kern-1pt}>{\kern-1pt} T_2+m$ for all $m\leq 2(n+k)$ . The notation $(\sigma )_{\alpha (\mathbf {y})}$ denotes a formula that specifies the description $\sigma $ on the assumption of $\alpha (\mathbf {y})$ . In other words, we claim that for each disjunct of (3.6), for all variable assignments s,

(3.7) $$ \begin{align} \mathcal{M},s \vDash \alpha(\mathbf{y}) \wedge (\sigma)_{\alpha(\mathbf{y})} \Rightarrow \mathcal{M}\mbox{ satisfies the }2(n+k)\mbox{ description }\sigma. \end{align} $$

The statement in the theorem will be the special case of (3.6) with no free variables ( $n=0$ ).

Example 3.3. For an example of such a disjunct over one predicate letter P, see the formula inside the existential quantifier in Figure 1. This formula has k free variables and depth $1$ . Here $\alpha (\mathbf {y})$ is the formula $\mathsf {diff}(\mathbf {y}) \wedge \bigwedge _{i \leq k} P(y_i)$ , while $(\sigma )_{\alpha (\mathbf {y})}$ is the count comparison. Note that $(\sigma )_{\alpha (\mathbf {y})}$ has free variables and it ‘means’ that $|P| = |\neg P| + 2k$ provided $\alpha (\mathbf {y})$ holds.

To show that depth $k+1$ formulas are always equivalent to formulas (3.6) satisfying (3.7), we proceed by inducting on k, starting with the case of depth $1$ formulas ( $k=0$ ) in free variables $\mathbf {y}=y_1,\dots , y_n$ . The critical case is a count comparison:

$$ \begin{align*}\#_x \bigvee \alpha(\mathbf{y},x) \succsim \#_x \bigvee \beta(\mathbf{y},x). \end{align*} $$

As before, by ( SUB ) we can separate out the descriptions of $\mathbf {y}$ to obtain a formula

(3.8) $$ \begin{align}\bigvee \big(\gamma(\mathbf{y}) \wedge \#_x\bigvee \alpha^{\mathbf{y}}(x) \succsim \#_x\bigvee\beta^{\mathbf{y}}(x) \big), \end{align} $$

where we have a $\gamma (\mathbf {y})$ disjunct exactly when $\gamma (\mathbf {y}) = \alpha (\mathbf {y}) = \beta (\mathbf {y})$ ; that is, all disjuncts inside the $\#$ terms must agree on the characterization of variables  $\mathbf {y}$ . It is then straightforward to check, by considering all cases, that the count comparison in each disjunct of (3.8), in context $\gamma (\mathbf {y})$ , asserts $T_1 = T_2 +m$ or $T_1> T_2+m$ for $m\leq 2n$ (or a disjunction of such comparisons). So this fits the form in (3.6), and (3.7) is satisfied.

In general, the normal forms (3.6) for a fixed k and n are closed under Boolean combinations, so we only need to consider the case of depth $k+1$ and n variables. By Lemma 3.1 we can assume all count comparison subformulas have depth $1$ , so it suffices to consider an existential quantification, which by induction we assume is

$$ \begin{align*} \exists z. \bigvee \big( \alpha(\mathbf{y},z) \wedge (\sigma)_{\alpha(\mathbf{y},z)}\big). \end{align*} $$

Such a formula will be equivalent to

$$ \begin{align*}\bigvee \exists z. \big(\alpha(\mathbf{y},z) \wedge (\sigma)_{\alpha(\mathbf{y},z)}\big)\end{align*} $$

and indeed to

(3.9) $$ \begin{align} \bigvee \Big(\alpha(\mathbf{y}) \wedge \exists z. \big(\alpha^{\mathbf{y}}(z) \wedge (\sigma)_{\alpha(\mathbf{y},z)}\big)\Big). \end{align} $$

It remains to be seen that (3.9) is of the form (3.6) with each disjunct satisfying (3.7). By the inductive assumption we know that for any s, if $\mathcal {M},s \vDash \alpha (\mathbf {y},z) \wedge (\sigma )_{\alpha (\mathbf {y},z)}$ , then $\mathcal {M}$ satisfies the $2(n+k)$ description $\sigma $ . But if $\mathcal {M},s \vDash \alpha (\mathbf {y}) \wedge \exists z. \big (\alpha ^{\mathbf {y}}(z) \wedge (\sigma )_{\alpha (\mathbf {y},z)}\big )$ , then there is a z-variant $s'$ of s such that $\mathcal {M},s' \vDash \alpha (\mathbf {y},z) \wedge (\sigma )_{\alpha (\mathbf {y},z)}$ , which establishes the result.

3.3 Questions of definability

Theorem 3.2 affords a refined understanding of the numerical relations that can be defined in $\mathsf {MFO}^\phi (\#)$ , as well as $\mathsf {MFO}(\#)$ . Where T is a set of state-descriptions, let $|T|_{\mathcal {M}}$ denote the sum of cardinalities of extensions in $\mathcal {M}$ of state-descriptions in T. We will say that $\mathcal {M} \sim _k \mathcal {M}'$ if for all $T_1,T_2$ and all $m \leq k$ ,

$$ \begin{align*} |T_1|_{\mathcal{M}} \geq |T_2|_{\mathcal{M}} + m \quad \mbox{iff}\quad |T_1|_{\mathcal{M}'} \geq |T_2|_{\mathcal{M}'} + m \end{align*} $$

Then, where $\mathcal {M} \equiv _k \mathcal {M}'$ signifies that $\mathcal {M}$ and $\mathcal {M}'$ agree on all sentences up to depth k, Theorem 3.2 immediately gives:

As an initial example, we can characterize precisely the binary logical quantifiers definable in $\mathsf {MFO}^\phi (\#)$ (see Section 9 for a proof, and for further discussion of generalized quantifiers):

This includes many of the standard logical quantifiers: ‘most’, ‘all’, ‘some’, ‘all but one’, ‘at least two’, etc. The following gives an example of a statement that cannot be expressed.

For a second example, consider a natural rendering of the natural language expression ‘many’, often taken to refer to a number above some contextual threshold. On a more sophisticated, but not uncommon, reading (cf. [Reference Rett111, Reference Westerståhl128]), ‘Many Qs are P’ amounts to a comparison between the proportion of Ps among the Qs and the proportion of Ps overall, which we might symbolize as

(3.10) $$ \begin{align} \frac{\#_x\big(P(x) \wedge Q(x)\big)}{\#_x Q(x)} \succ \frac{\#_x P(x)}{\#_x \top}. \end{align} $$

We will return to more analysis of natural language constructions in Section 9. Note that Corollary 3.4 can be used to derive undefinability results in $\mathsf {MFO}(\#)$ as well:

3.3.1 Interpolation failure

Another consequence of Theorem 3.2 is a particularly simple normal form result for the ‘letterless’ fragment of $\mathcal {L}_\#^1$ , that is, the fragment with no predicate symbols, built up from atomic formulas $\top $ and $\bot $ . In fact, the normal forms are identical to those for monadic first-order logic with the infinity quantifier [Reference Carreiro, Facchini, Venema and Zanasi24]:

Lemma 3.9. Every letterless sentence is equivalent in $\mathsf {MFO}(\#)$ to a disjunction of formulas having one of the following forms $\exists ^\infty x. \top $ , $\forall ^\infty x. \bot \wedge \exists ^{\geq k} \top $ , or $\exists ^{=k} x. \top $ .

For the restriction $\mathsf {MFO}^\phi (\#)$ to finite models, this simplifies even further to include only statements of the form $\exists ^{>k}.\top $ and $\exists ^{=k} x. \top $ . As a consequence we can show:

Proposition 3.10. Neither $\mathsf {MFO}^\phi (\#)$ nor $\mathsf {MFO}(\#)$ enjoys the interpolation property.

Proof Let $\varphi (P)$ be the formula:

$$ \begin{align*}\forall^\infty x. \bot \wedge \#_x(P(x)) \approx \#_x(\neg P(x)),\end{align*} $$

which is only true in finite models of even sizes. Let $\psi (Q)$ be the formula:

$$ \begin{align*}\exists x.\#_y( y \neq x \wedge Q(y) ) \approx \#_y( y \neq x \wedge \neg Q(y) ),\end{align*} $$

which in finite models requires the domain to be odd. Evidently $\varphi (P) \vDash \neg \psi (Q)$ . Let $\chi $ be a purported interpolant: $\varphi (P) \vDash \chi \vDash \neg \psi (Q)$ . As $\chi $ must be letterless, Lemma 3.9 implies that it must be a disjunction of sentences with one of the three specified forms. Furthermore, as it is entailed by $\varphi (P)$ we can assume that $\exists ^\infty x. \top $ is not a disjunct. A straightforward case analysis shows that $\chi $ must either be true only in models up to some fixed size—in which case it cannot be entailed by $\varphi (P)$ —or it is true in all finite models from some finite size onward—in which case it cannot entail $\neg \psi (Q)$ .

A familiar way of extending a language to guarantee interpolation is to allow second-order quantification. We will turn to such an extension below in Section 4. But first, we analyze the reasoning content of our normal form analysis a bit further.

3.4 Questions of axiomatization

What is the calculus of valid reasoning suggested by our current systems? For both of our basic monadic systems, $\mathsf {MFO}^\phi (\#)$ and $\mathsf {MFO}(\#)$ , we can locate a kind of separation between two components: (a) the general, more ‘logical’ principles that allow our normal form result (Theorem 3.2) and (b) more specific numerical reasoning for solving systems of inequalities. We discuss each component in turn for the system $\mathsf {MFO}^\phi (\#)$ which allows only finite domains. The general system $\mathsf {MFO}(\#)$ involves one more component for dealing with infinite sets that we will remark on at the end.

3.4.1 Step I

The normal form principles underlying our normal form result are as follows:

1. (a) general validities of propositional and first-order predicate logic,

2. (b) the two general principles ( INV ) and ( SUB ) highlighted earlier,

3. (c) the linear order properties of the relation $\succsim $ .

Here the linearity in Principle (c), used in our case distinctions, is worth high-lighting:

(COMP) $$\begin{align} \#_x\varphi \succsim \#_x\psi \vee \#_x\psi \succsim \#_x\varphi. \end{align}$$

The soundness of ( COMP ) in $\mathsf {MFO}(\#)$ depends on the axiom of choice. Indeed, ( COMP ) is equivalent to the axiom of choice [Reference Hartogs55]. Significantly, in the generalized semantics discussed in Section 8, Principles (a) and (b) will remain valid, while the strong reasoning principle (c) is naturally replaced by just the pre-order properties for $\succsim $ .

4.1 Finitary case

We saw above that normal forms in $\mathsf {MFO}^\phi (\#)$ correspond to (disjunctions of) sets of inequality constraints, a class whose solvability problem is already NP-complete. In the general setting of integer programming, there is a close correspondence between sets of linear inequalities and quantifier-free formulas of Presburger Arithmetic, that is, first-order logic with addition over the natural numbers (see, e.g., [Reference Oppen97]). The sets of solutions to such inequalities (or equivalently, assignments satisfying Presburger formulas with free variables) are exactly the semi-linear sets [Reference Ginsburg and Spanier44], a generalization of the ‘ultimately periodic’ sets of numbers:

Definition 4.2. A set $\mathcal {V}\subseteq \mathbb {N}^n$ of n-ary vectors is called linear if there is a system of equations over variables $\mathsf {v}_1,\dots ,\mathsf {v}_n,\mathsf {u}_1,\dots ,\mathsf {u}_m$ and constants $b_1,\dots ,b_n,a_{1,1},\dots ,a_{n,m}$ ,

(4.1) $$ \begin{align} \begin{pmatrix} \;\mathsf{v}_1\; \\ \vdots \\ \mathsf{v}_n \end{pmatrix} = \begin{pmatrix} b_1 + a_{1,1}\mathsf{u}_1 + & \dots & + a_{1,m}\mathsf{u}_m \\ \vdots & & \vdots \\ b_n + a_{n,1}\mathsf{u}_1 + & \dots & + a_{n,m}\mathsf{u}_m \end{pmatrix} \end{align} $$

such that $\mathbf {x} \in \mathcal {V}$ if and only if there exist values of $\mathsf {u}_1,\dots ,\mathsf {u}_m$ for which $\mathbf {v}=\mathbf {x}$ is a solution to (4.1). We say $\mathcal {V} \subseteq \mathbb {N}^n$ is semi-linear if it is a finite union of linear sets.

Definition 4.3. Suppose $S_1,\dots ,S_n$ are some state-descriptions over predicates $\mathbf {P}$ , and that $\varphi $ is an $\mathcal {L}_\#^2$ sentence in these same predicates $\mathbf {P}$ . We say that $\varphi $ defines a set $\mathcal {V} \subseteq \mathbb {N}^n$ just in case, for any model $\mathcal {M}$ , we have

(4.2) $$ \begin{align} \mathcal{M} \vDash \varphi \quad \mbox{iff}\quad [|S_1|_{\mathcal{M}},\dots,|S_n|_{\mathcal{M}}] \in \mathcal{V}. \end{align} $$

We now want to show the other direction, that all $\mathcal {L}_\#^2$ sentences in fact define semi-linear sets. Toward this result we first note that $\mathsf {MSO}(\#)$ possesses a prenex normal form.

By Theorem 3.2 we know that $\varphi (\mathbf {P},X_1,\dots ,X_n)$ has a normal form involving expressions $T_1=T_2+m$ and $T_1> T_2+m$ , where $T_1,T_2$ are cardinalities of state-descriptions over $\mathbf {P}$ and additional predicates $X_1,\dots ,X_n$ . Such formulas are thus easily seen to be semi-linear, indeed linear. As semi-linear sets are closed under Boolean combinations, and second-order quantifiers distribute over disjunction, the main goal is to show:

Proof As $\varphi (O_1,\dots ,O_n,X)$ is linear, we can assume it defines the solutions to

(4.3) $$ \begin{align} \begin{pmatrix} \;|S_1|\; \\ \vdots \\ |S_{2^{n+1}}| \end{pmatrix} = \begin{pmatrix} b_1 + a_{1,1}\mathsf{u}_1 + & \cdots & + a_{1,m}\mathsf{u}_m \\ \vdots & & \vdots \\ b_{2^{n+1}} + a_{2^{n+1},1}\mathsf{u}_1 + & \cdots & + a_{2^{n+1},m}\mathsf{u}_m \end{pmatrix}. \end{align} $$

To show that $\exists X. \varphi (O_1,\dots ,O_n,X)$ , too, is linear, we define another linear set of equations by ‘projecting out’ the variable X.

Specifically, note for each state-description $S_k'$ , both $S_k' \wedge X$ and $S_k' \wedge \neg X$ are (equivalent to) some state-descriptions, $S_i$ and $S_j$ , and in fact $S_k'$ is equivalent to $S_i \vee S_j$ . The new linear system in $2^n$ variables is then as follows for each $k\leq 2^n$ :

(4.4) $$ \begin{align} |S^{\prime}_k| = b_i + b_j + (a_{i,1}+a_{j,1})\mathsf{u}_1+ \dots + (a_{i,m} + a_{j,m})\mathsf{u}_m. \end{align} $$

It remains only to show that:

$$ \begin{align*} \mathcal {M} & \vDash \exists X. \varphi (O_1,\dots ,O_n,X)\\ &\Leftrightarrow [|S_1'|_{\mathcal {M}},\dots ,|S_{2^n}'|_{\mathcal {M}}] \mbox{ is a solution to Equation (20)}.\end{align*} $$

$(\Rightarrow )$ : If $\mathcal {M} \vDash \exists X. \varphi (O_1,\dots ,O_n,X)$ then for some subset A of the domain, we have $\mathcal {M},s^X_A \vDash \varphi (O_1,\dots ,O_n,X)$ . Treating X now as a predicate constant, we have a model $\mathcal {M}'$ for which $X^{\mathcal {M}'}=A$ , and by assumption this gives a solution $[|S_1|_{\mathcal {M}'},\dots ,|S_{2^{n+1}}|_{\mathcal {M}'}]$ to (4.3). But each state-description $S_k'$ is equivalent to a disjunction $S_i \vee S_j$ , whose cardinality is the sum $|S_i|+|S_j|$ . Therefore $\mathcal {M}'$ will satisfy each of the constraints in (4.4). As the state-descriptions $S_1',\dots ,S_{2^n}'$ are independent of X, this means $[|S_1'|_{\mathcal {M}},\dots ,|S_{2^n}'|_{\mathcal {M}}]$ also gives a solution to (4.4).

$(\Leftarrow )$ : Suppose $[|S_1'|_{\mathcal {M}},\dots ,|S_{2^n}'|_{\mathcal {M}}]$ gives a solution to (4.4) for some particular choices $\mathsf {u}_1,\dots ,\mathsf {u}_m$ . We need to find a set A such that $\mathcal {M},s^X_A \vDash \varphi (O_1,\dots ,O_n,X)$ . Since the extensions of $S_1',\dots ,S_{2^n}'$ are all disjoint, to define A it suffices to identify subsets of each ${[\![{ {S_k'} }]\!]}_{\mathcal {M}}$ . As above, suppose $S^{\prime }_k$ is equivalent to $S_i \vee S_j$ , so that $S_i$ is equivalent to $S^{\prime }_k \wedge X$ and $S_j$ is equivalent to $S_k' \wedge \neg X$ . Then let $B_k$ be any subset of ${[\![{ {S_k'} }]\!]}_{\mathcal {M}}$ of size $b_i + a_{i,1}\mathsf {u}_1 + \dots + a_{i,m}\mathsf {u}_m$ , such that the complement ${[\![{ {S^{\prime }_k} }]\!]}_{\mathcal {M}} - B_k$ has size $b_j + a_{j,1}\mathsf {u}_1 + \dots + a_{j,m}\mathsf {u}_m$ . This is always possible since $|S^{\prime }_k|$ is simply the sum of these two numbers. Finally let $A = \bigcup _{k\leq 2^n} B_k$ . Once again absorbing X into the language and defining $\mathcal {M}'$ to be just like $\mathcal {M}$ but with $X^{\mathcal {M}'} = A$ , the tuple $[|S_1|_{\mathcal {M}'},\dots ,|S_{2^{n+1}}|_{\mathcal {M}'}]$ gives a solution to (4.3) with the same choices $\mathsf {u}_1,\dots ,\mathsf {u}_m$ . Hence, $\mathcal {M}'\vDash \varphi (O_1,\dots ,O_n,X)$ , from which it easily follows that $\mathcal {M},s^X_A \vDash \varphi (O_1,\dots ,O_n,X)$ and finally $\mathcal {M} \vDash \exists X. \varphi (O_1,\dots ,O_n,X)$ .

The foregoing thus establishes:

As in the first-order case, numerous undefinable results again follow. For example:

Indeed, the theory of definability for Presburger Arithmetic carries over exactly to $\mathsf {MSO}^\phi (\#)$ , thanks to Theorem 4.7. Moreover, since there is an algorithmic means of putting a formula of $\mathsf {MSO}^\phi (\#)$ into normal form and finding a suitable semi-linear form, decidability follows from the decidability of Presburger Arithmetic.

4.2 Infinitary case

Allowing second-order quantification does increase the expressive power of our initial system $\mathsf {MFO}^\phi (\#)$ . While the latter essentially amounts to a proper fragment of Presburger Arithmetic, $\mathsf {MSO}^\phi (\#)$ gave us precisely Presburger Arithmetic. How does this look for the system $\mathsf {MSO}(\#)$ over models of arbitrary cardinality? One immediate difference is that, in contrast to $\mathsf {MFO}(\#)$ (Fact 3.8), the successor function on cardinal numbers can now be easily expressed:

$$ \begin{align*}\forall X.\big( \#_yP(y) \succ \#_y X(y) \rightarrow \#_y Q(y) \succsim \#_yX(y)\big).\end{align*} $$

This formula states that there is no cardinality strictly in between that of P and that of Q. How much more cardinal arithmetic does $\mathsf {MSO}(\#)$ encode?

As in the case of $\mathsf {MSO}^\phi (\#)$ , we can calibrate this by appeal to additive first-order (now cardinal) arithmetic. Consider the elementary theory of the structure $\langle C_{\aleph _\omega };+ \rangle $ , addition over the cardinals numbers less than $\aleph _\omega $ . This is the theory of cardinal numbers in a first-order language with one binary function symbol, namely addition. We show in Appendix C that this theory admits quantifier elimination provided we augment the language with constants for the (definable in $\mathsf {MSO}(\#)$ ) functions and relations:

• • $\{0\}$ and $\{\aleph _0\}$ ,

• • s — the successor function,

• • $>$ — the ‘greater than’ relation,

• • $\equiv _k$ — equivalence modulo k for each $k>1$ .

Furthermore, we can derive a normal form result for this language:

• • which (ordinary first-order) variables in the disjunct are finite or infinite ( $\delta $ ),

• • a description of a linear set for the finite variables ( $\phi $ ),

• • a description of a set of infinite cardinals using $0$ , s, and $>$ over $\aleph $ -number indices, for the infinite variables ( $\iota $ ).

This can be understood as a kind of separation result. The finitary part, Presburger Arithmetic, simply describes ordinary addition. As for the infinitary part, observe that there is an isomorphism from $\langle \mathbb {N};0,s,>\rangle $ onto $\langle \{\aleph _k\}_{k\in \mathbb {N}}; \aleph _0,s,> \rangle $ , sending k to $\aleph _k$ . In other words, the additive structure of cardinals less than $\aleph _\omega $ amounts to a ‘product’ of $\langle \mathbb {N};+ \rangle $ and $\langle \mathbb {N};> \rangle $ .Footnote ³

Our aim is to show that $\mathsf {MSO}(\#)$ possesses the same normal forms as in Proposition 4.11. To see that any statement of the form $\delta \wedge \iota \wedge \phi $ can be expressed, note that $\delta $ merely requires distinguishing finite and infinite sets (recall Equation (2.1)), while definability of any linear set (specified by $\phi $ ) was shown already in Lemma 4.4. Meanwhile, $\iota $ is a conjunction of formulas of types $\mathsf {v}=s^k(\mathsf {u})$ , $\mathsf {v}>s^k(\mathsf {u})$ , $\mathsf {v} = \aleph _k$ , and $\mathsf {v}>\aleph _k$ . We noted above that successor is expressible, and, for instance, we can assert that P has cardinality $\aleph _0$ simply by stating that P is infinite and there is no infinite set with smaller cardinality. Thus, any such statement is expressible.

To show that this exhausts what is definable in $\mathsf {MSO}(\#)$ , given Lemma 4.5, it remains to observe that the $\mathcal {L}_\#^2$ -definable sets are closed under ‘projection’ by existentially quantifying one of the variables. Thus, suppose we have an $\mathcal {L}_\#^2$ formula $\varphi (\mathbf {O},Y)$ with Y a predicate variable and $\mathbf {O}=O_1,\dots ,O_n$ all either predicate variables or letters. We will assume $\varphi (\mathbf {O},Y)$ has the form $\delta (\mathbf {O},Y) \wedge \iota (\mathbf {O},Y) \wedge \phi (\mathbf {O},Y)$ , analogously to the additive language: $\delta (\mathbf {O},Y)$ describes which state descriptions over variables $O_1,\dots ,O_n,Y$ are (in)finite, $\iota (\mathbf {O},Y)$ characterizes the infinite state descriptions, while $\phi (\mathbf {O},Y)$ describes a linear set. We need to analyze $\exists Y.\big (\delta (\mathbf {O},Y) \wedge \iota (\mathbf {O},Y) \wedge \phi (\mathbf {O},Y)\big )$ .

We can replace $\delta (\mathbf {O},Y)$ with a formula $\delta '(\mathbf {O})$ specifying that a state description S over $O_1,\dots ,O_n$ is finite iff $S \wedge Y$ and $S \wedge \neg Y$ were both finite according to $\delta (\mathbf {O},Y)$ . List the finite state descriptions according to $\delta (\mathbf {O},Y)$ as $S_1,\dots ,S_k$ . The subformula $\phi (\mathbf {O},Y)$ defines a linear set over the possible (finite) cardinalities:

(4.5) $$ \begin{align} \begin{pmatrix} \;|S_1|\; \\ \vdots \\ |S_{k}| \end{pmatrix} = \begin{pmatrix} b_1 + a_{1,1}\mathsf{u}_1 + & \cdots & + a_{1,m}\mathsf{u}_m \\ \vdots & & \vdots \\ b_{k} + a_{k,1}\mathsf{u}_1 + & \cdots & + a_{k,m}\mathsf{u}_m \end{pmatrix}. \end{align} $$

Suppose $S_i = S \wedge Y$ is finite but $S \wedge \neg Y$ is infinite. Then the constraint in (4.5) on $|S_i|$ is no constraint at all: since $|S|$ must be infinite, carving out a finite portion $S \wedge Y$ of any size will always be possible. So in this case we can simply drop the equation for $S_i$ . Otherwise, if $S_i = S\wedge Y$ and $S_j=S \wedge \neg Y$ are both finite, then we can repeat the argument from Section 4.1, again combining these two equations into a single equation for $|S|$ . The result is a set of equations in (cardinalities of) state descriptions over $\mathbf {O}$ , all asserted finite in $\delta '(\mathbf {O})$ .

The subformula $\iota (\mathbf {O},Y)$ represents constraints of the form $\mathsf {v}=\mathsf {w}$ and $\mathsf {v}>\mathsf {w}$ , where $\mathsf {v}$ and $\mathsf {w}$ are either ‘infinite’ state descriptions over $\mathbf {O},Y$ , k-fold successors of such state descriptions, or aleph-numbers. In view of the isomorphism between $\langle \mathbb {N};0,s,> \rangle $ and $\langle \{\aleph _k\}_{k \in \mathbb {N}};\aleph _0,s,>\rangle $ , we can construe these conjuncts as describing relations on natural numbers. As these relations are a special case of linear sets and can thus be encoded as in (4.5), we once again run the argument to ‘merge’ the equations for $S \wedge Y$ and $S \wedge \neg Y$ into a single equation for S, provided all of these are infinite. (If only one of $S \wedge Y$ and $S \wedge \neg Y$ is infinite, then that equation remains as before since S will have the same cardinality.) The resulting formula will in general involve addition. But as discussed further in Appendix C, since all variables are infinite we can eliminate all explicit sums, using equivalences such as $\mathsf {t} = \mathsf {v}+\mathsf {w} \Leftrightarrow (\mathsf {t}=\mathsf {v} \wedge \mathsf {v} \geq \mathsf {w})\vee (\mathsf {t}=\mathsf {w} \wedge \mathsf {w} \geq \mathsf {v}) $ .

In effect, we have shown how to reduce a sentence $\varphi (\mathbf {P})$ in $\mathcal {L}_\#^2$ to an additive first-order formula $\alpha (\mathbf {x})$ , with a variable $x_i$ in $\mathbf {x}$ corresponding to each state-description over $\mathbf {P}$ . Moreover, $\varphi (\mathbf {P})$ is satisfiable if and only if $\exists \mathbf {x}.\alpha (\mathbf {x})$ is true in $\langle C_{\aleph _\omega };+ \rangle $ . Thus, from decidability of the elementary theory of $\langle C_{\aleph _\omega };+ \rangle $ (see Theorem C.3 in Appendix C) we obtain:

5.1 Diophantine inequalities

To start our discussion, consider any polynomial inequality

(5.1) $$ \begin{align} m_1(\mathbf{v}) + \dots + m_k(\mathbf{v}) \geq m_1'(\mathbf{v}) + \dots + m_j'(\mathbf{v}), \end{align} $$

where $m_1,\dots ,m_k,m^{\prime }_1,\dots ,m^{\prime }_j$ are all monomials in variables $\mathbf {v} = \mathsf {v}_1,\dots ,\mathsf {v}_n$ . Each monomial $m(\mathbf {v})$ is of the form $a \mathsf {v}_1^{e_1}\dots \mathsf {v}_n^{e_n}$ , with