Local Presburger Quantifiers in Two-Variable Logics

• Local Presburger quantifiers are arithmetic-based devices that apply ultimately periodic constraints over guarded neighborhoods in two-variable relational and guarded logics.
• They enable the expression of parity, threshold, and modular conditions while maintaining decidability by restricting counting to unary, ultimately periodic sets.
• They support efficient decision procedures via graph-theoretic encodings and reductions to existential Presburger formulas, with complexity ranging from NP to EXP.

Local Presburger quantifiers are quantificational devices whose truth conditions are given by Presburger-definable arithmetic applied locally rather than by unrestricted global cardinality constraints. In two-variable relational logics, they constrain the number of guarded neighbors of an element by unary Presburger-definable sets, equivalently ultimately periodic sets; in guarded logics, they take the form of linear and modular constraints over counts of incoming and outgoing neighbors satisfying local formulas. A related, but distinct, usage appears in parametric Presburger arithmetic, where “local” refers to quantifiers bounded by parameter-dependent Presburger terms or guarded by linear equations rather than eliminated outright (Benedikt et al., 2020, Lu et al., 2022, Goodrick, 2016).

1. Semantic core and principal variants

The semantic core is Presburger arithmetic over additive structures, together with the fact that unary Presburger-definable subsets of $\mathbb{N}$ are exactly the ultimately periodic sets. For a set $S \subseteq \mathbb{N}$, two equivalent descriptions are standard: a finite union of linear sets $a+p\mathbb{N}$, and the existence of $N,p \ge 1$ such that for all $n \ge N$, $n \in S \Leftrightarrow n+p \in S$. In the two-variable setting this yields counting quantifiers $\exists^{S}x\,\varphi$, interpreted by

$$M \models \exists^S x\,\varphi \quad\Leftrightarrow\quad \exists a \in S:\ |\{u \in M : M \models \varphi[u]\}| = a,$$

with the extension to $\infty$ also admitted (Benedikt et al., 2020).

In guarded two-variable logics, the local Presburger quantifier is a linear or modular constraint over guarded neighbor counts. Its general form is

$$\mathcal P(x):= \sum_{i=1}^n \lambda_i \cdot \#_y^{\,r_i}[\varphi_i(x,y)] \ \circledast\ \delta + \sum_{i=1}^m \kappa_i \cdot \#_y^{\,s_i}[\psi_i(x,y)],$$

where each guard $S \subseteq \mathbb{N}$0 is either $S \subseteq \mathbb{N}$1 or $S \subseteq \mathbb{N}$2, and

$S \subseteq \mathbb{N}$3

Thus the quantifier does not count arbitrary tuples; it counts only the guarded neighborhood of the current element $S \subseteq \mathbb{N}$4 (Lu et al., 2022).

A second notion of locality appears in parametric Presburger arithmetic. There, local quantifiers are quantifiers whose range is bounded by parameter-dependent functions, for example

$S \subseteq \mathbb{N}$5

with $S \subseteq \mathbb{N}$6 supplied by the ambient ring of functions. This is a locality of bounded range rather than neighborhood counting (Goodrick, 2016).

Setting	Local form	Representative result
$S \subseteq \mathbb{N}$7	$S \subseteq \mathbb{N}$8	satisfiability and finite satisfiability are decidable; spectra are Presburger-definable (Benedikt et al., 2020)
$S \subseteq \mathbb{N}$9 / guarded two-variable logic	linear or modular constraints over $a+p\mathbb{N}$0	satisfiability is EXP-complete (Lu et al., 2022)
Parametric Presburger arithmetic	$a+p\mathbb{N}$1-bounded quantifiers	every first-order $a+p\mathbb{N}$2-formula is equivalent to one with $a+p\mathbb{N}$3-bounded quantifiers (Goodrick, 2016)

2. Two-variable logic with ultimately periodic counting

In the relational two-variable framework, local Presburger quantifiers arise in the logic denoted $a+p\mathbb{N}$4, which extends $a+p\mathbb{N}$5 by quantifiers $a+p\mathbb{N}$6 for ultimately periodic $a+p\mathbb{N}$7. After reduction to unary and binary relational symbols and elimination of constants via fresh unary predicates marking unique elements, every sentence is effectively transformed to Scott normal form

$a+p\mathbb{N}$8

where $a+p\mathbb{N}$9 is quantifier-free, each $N,p \ge 1$0 is atomic, and each $N,p \ge 1$1 is an ultimately periodic set. In this form, the counting is explicitly local around each element $N,p \ge 1$2: it counts neighbors related to $N,p \ge 1$3 by an atomic relation and constrains that count to lie in $N,p \ge 1$4 (Benedikt et al., 2020).

This local viewpoint yields the main decidability and spectrum results. For every $N,p \ge 1$5 sentence $N,p \ge 1$6, there are effectively computable existential Presburger sentences $N,p \ge 1$7 and $N,p \ge 1$8 such that $N,p \ge 1$9 has a finite or infinite model iff $n \ge N$0 holds in $n \ge N$1, and $n \ge N$2 has a finite model iff $n \ge N$3 holds in $n \ge N$4. The spectrum

$n \ge N$5

is definable by an existential Presburger formula $n \ge N$6, and an analogous $n \ge N$7 describes finite-or-countably-infinite model sizes (Benedikt et al., 2020).

The expressive gain over ordinary two-variable counting is substantial but controlled. $n \ge N$8 can express parity conditions such as “every vertex has even degree,” using $n \ge N$9, and threshold conditions such as “at least $n \in S \Leftrightarrow n+p \in S$0 neighbors,” using $n \in S \Leftrightarrow n+p \in S$1. The paper explicitly notes that $n \in S \Leftrightarrow n+p \in S$2 can express parity, which $n \in S \Leftrightarrow n+p \in S$3 cannot, while remaining within two-variable relational logic without function symbols. At the same time, the method is limited to unary, ultimately periodic counting sets. Allowing general semilinear conditions over tuples with arity $n \in S \Leftrightarrow n+p \in S$4 leads to undecidability (Benedikt et al., 2020).

Complexity reflects the cost of enumerating local behaviors. Satisfiability and finite satisfiability are both in $n \in S \Leftrightarrow n+p \in S$5-NEXPTIME; the blow-up comes from the exponential number of $n \in S \Leftrightarrow n+p \in S$6-types and $n \in S \Leftrightarrow n+p \in S$7-types and doubly exponential many local “behaviors” $n \in S \Leftrightarrow n+p \in S$8. Data complexity, with the sentence fixed and ground facts as input, is NP in the size of the fact set (Benedikt et al., 2020).

3. Guarded local Presburger constraints

In the guarded two-variable setting, locality is enforced by the guard relation itself. The guarded fragment $n \in S \Leftrightarrow n+p \in S$9 can be extended with local Presburger quantifiers so that counts are taken only over $\exists^{S}x\,\varphi$0-successors or $\exists^{S}x\,\varphi$1-predecessors of the current element. This allows constraints such as

$\exists^{S}x\,\varphi$2

which formalizes “the number of incoming blue edges plus twice the number of outgoing red edges is at most three times the number of incoming green edges” (Lu et al., 2022).

A normal form is available. Any $\exists^{S}x\,\varphi$3 sentence can be converted in linear time to an equisatisfiable sentence

$\exists^{S}x\,\varphi$4

where $\exists^{S}x\,\varphi$5 is quantifier-free, each $\exists^{S}x\,\varphi$6 is a guarded binary implication, each $\exists^{S}x\,\varphi$7 is atomic, and each $\exists^{S}x\,\varphi$8 is an LPQ in basic form. Universal and existential guarded quantification are themselves expressible by LPQs:

$\exists^{S}x\,\varphi$9

$$M \models \exists^S x\,\varphi \quad\Leftrightarrow\quad \exists a \in S:\ |\{u \in M : M \models \varphi[u]\}| = a,$$0

LPQs can also detect infinite branching by formulas such as

$$M \models \exists^S x\,\varphi \quad\Leftrightarrow\quad \exists a \in S:\ |\{u \in M : M \models \varphi[u]\}| = a,$$1

which holds exactly when the $$M \models \exists^S x\,\varphi \quad\Leftrightarrow\quad \exists a \in S:\ |\{u \in M : M \models \varphi[u]\}| = a,$$2-neighborhood of $$M \models \exists^S x\,\varphi \quad\Leftrightarrow\quad \exists a \in S:\ |\{u \in M : M \models \varphi[u]\}| = a,$$3 is infinite (Lu et al., 2022).

The guarded framework is also where the contrast with percentage quantifiers becomes sharp. Local percentage constraints are definable by Presburger equations, for example

$$M \models \exists^S x\,\varphi \quad\Leftrightarrow\quad \exists a \in S:\ |\{u \in M : M \models \varphi[u]\}| = a,$$4

so exact local percentages are subsumed by local Presburger arithmetic. Nevertheless, the guarded-fragment analysis distinguishes two regimes: local percentage quantifiers make finite satisfiability undecidable already for $$M \models \exists^S x\,\varphi \quad\Leftrightarrow\quad \exists a \in S:\ |\{u \in M : M \models \varphi[u]\}| = a,$$5, whereas $$M \models \exists^S x\,\varphi \quad\Leftrightarrow\quad \exists a \in S:\ |\{u \in M : M \models \varphi[u]\}| = a,$$6 extended with local Presburger quantifiers remains decidable (Bednarczyk et al., 2021).

The known upper bounds differ by method. One reduction-based analysis for $$M \models \exists^S x\,\varphi \quad\Leftrightarrow\quad \exists a \in S:\ |\{u \in M : M \models \varphi[u]\}| = a,$$7 with local Presburger constraints gives a $$M \models \exists^S x\,\varphi \quad\Leftrightarrow\quad \exists a \in S:\ |\{u \in M : M \models \varphi[u]\}| = a,$$8NExpTime upper bound by translating local Presburger conditions to semilinear sets and then reducing to $$M \models \exists^S x\,\varphi \quad\Leftrightarrow\quad \exists a \in S:\ |\{u \in M : M \models \varphi[u]\}| = a,$$9 with counting (Bednarczyk et al., 2021). A later graph-based analysis of the two-variable guarded fragment with expressive local Presburger constraints establishes that satisfiability is EXP-complete (Lu et al., 2022).

These guarded quantifiers capture description-logical and modal formalisms with inverse roles or converse modalities. The guarded-fragment treatment yields decidability for Presburger modal logics with converse and for $\infty$0-style description logics with expressive local cardinality constraints, while the $\infty$1 treatment is stated to capture various description logics with counting, including $\infty$2 and $\infty$3, but without constant symbols (Bednarczyk et al., 2021, Lu et al., 2022).

4. Decision procedures and graph-theoretic encodings

The principal algorithmic techniques reduce local Presburger quantification to finite combinatorial objects plus Presburger feasibility. In $\infty$4, the central method is the biregular graph method. One enumerates all unary $\infty$5-types $\infty$6 and binary $\infty$7-types $\infty$8, and then partitions the domain into classes $\infty$9, where $$\mathcal P(x):= \sum_{i=1}^n \lambda_i \cdot \#_y^{\,r_i}[\varphi_i(x,y)] \ \circledast\ \delta + \sum_{i=1}^m \kappa_i \cdot \#_y^{\,s_i}[\psi_i(x,y)],$$0 is a $$\mathcal P(x):= \sum_{i=1}^n \lambda_i \cdot \#_y^{\,r_i}[\varphi_i(x,y)] \ \circledast\ \delta + \sum_{i=1}^m \kappa_i \cdot \#_y^{\,s_i}[\psi_i(x,y)],$$1-type and $$\mathcal P(x):= \sum_{i=1}^n \lambda_i \cdot \#_y^{\,r_i}[\varphi_i(x,y)] \ \circledast\ \delta + \sum_{i=1}^m \kappa_i \cdot \#_y^{\,s_i}[\psi_i(x,y)],$$2 is a “good” behavior recording, for each direction, $$\mathcal P(x):= \sum_{i=1}^n \lambda_i \cdot \#_y^{\,r_i}[\varphi_i(x,y)] \ \circledast\ \delta + \sum_{i=1}^m \kappa_i \cdot \#_y^{\,s_i}[\psi_i(x,y)],$$3-type, and target $$\mathcal P(x):= \sum_{i=1}^n \lambda_i \cdot \#_y^{\,r_i}[\varphi_i(x,y)] \ \circledast\ \delta + \sum_{i=1}^m \kappa_i \cdot \#_y^{\,s_i}[\psi_i(x,y)],$$4-type, a value in

$$\mathcal P(x):= \sum_{i=1}^n \lambda_i \cdot \#_y^{\,r_i}[\varphi_i(x,y)] \ \circledast\ \delta + \sum_{i=1}^m \kappa_i \cdot \#_y^{\,s_i}[\psi_i(x,y)],$$5

Good behaviors are exactly those satisfying the local Presburger constraints demanded by the sets $$\mathcal P(x):= \sum_{i=1}^n \lambda_i \cdot \#_y^{\,r_i}[\varphi_i(x,y)] \ \circledast\ \delta + \sum_{i=1}^m \kappa_i \cdot \#_y^{\,s_i}[\psi_i(x,y)],$$6. The induced same-type digraphs and cross-type bipartite graphs are then summarized by degree matrices $$\mathcal P(x):= \sum_{i=1}^n \lambda_i \cdot \#_y^{\,r_i}[\varphi_i(x,y)] \ \circledast\ \delta + \sum_{i=1}^m \kappa_i \cdot \#_y^{\,s_i}[\psi_i(x,y)],$$7, and feasibility is expressed by existential Presburger constraints

$$\mathcal P(x):= \sum_{i=1}^n \lambda_i \cdot \#_y^{\,r_i}[\varphi_i(x,y)] \ \circledast\ \delta + \sum_{i=1}^m \kappa_i \cdot \#_y^{\,s_i}[\psi_i(x,y)],$$8

where $$\mathcal P(x):= \sum_{i=1}^n \lambda_i \cdot \#_y^{\,r_i}[\varphi_i(x,y)] \ \circledast\ \delta + \sum_{i=1}^m \kappa_i \cdot \#_y^{\,s_i}[\psi_i(x,y)],$$9 zeroes incompatible type-behavior combinations and $S \subseteq \mathbb{N}$00 invokes formulas $S \subseteq \mathbb{N}$01 and $S \subseteq \mathbb{N}$02 characterizing complete regular digraphs and biregular bipartite graphs. This yields the spectrum formula

$S \subseteq \mathbb{N}$03

which witnesses Presburger-definability of spectra (Benedikt et al., 2020).

The guarded-fragment graph method uses a different abstraction. For a normal-form sentence $S \subseteq \mathbb{N}$04, one constructs a graph $S \subseteq \mathbb{N}$05 whose vertices are the unary types compatible with $S \subseteq \mathbb{N}$06 and whose edges are compatible configurations $S \subseteq \mathbb{N}$07, where $S \subseteq \mathbb{N}$08 is a binary type between source type $S \subseteq \mathbb{N}$09 and target type $S \subseteq \mathbb{N}$10. For each vertex $S \subseteq \mathbb{N}$11 and each LPQ $S \subseteq \mathbb{N}$12, one writes a local linear constraint $S \subseteq \mathbb{N}$13 over variables $S \subseteq \mathbb{N}$14 that count neighbors of configuration $S \subseteq \mathbb{N}$15. For each edge $S \subseteq \mathbb{N}$16, one forms a system $S \subseteq \mathbb{N}$17 consisting of the realization requirement $S \subseteq \mathbb{N}$18 together with all $S \subseteq \mathbb{N}$19 relevant to $S \subseteq \mathbb{N}$20. An edge is bad if $S \subseteq \mathbb{N}$21 has no solution over $S \subseteq \mathbb{N}$22; a vertex is bad if it has no outgoing edges and its local LPQs are not satisfied by the zero solution. A non-empty symmetric subgraph with no bad edges or bad vertices is a good subgraph, and satisfiability is equivalent to the existence of such a good subgraph (Lu et al., 2022).

The corresponding algorithm repeatedly deletes bad edges and their inverses, then deletes bad vertices, and accepts iff the resulting graph is non-empty. Its complexity analysis is explicit: it uses a quantifier-free Presburger solver as a black box, the number of calls is bounded by $S \subseteq \mathbb{N}$23 when $S \subseteq \mathbb{N}$24 and $S \subseteq \mathbb{N}$25 are the numbers of unary and binary predicates, and checking a system $S \subseteq \mathbb{N}$26 has a solution over $S \subseteq \mathbb{N}$27 can be done in nondeterministic polynomial time by sparse-solution bounds derived from Carathéodory-type results (Lu et al., 2022).

A different guarded-fragment procedure converts local Presburger constraints into semilinear sets using Ginsburg–Spanier-style representation, then encodes degree vectors by $S \subseteq \mathbb{N}$28 counting formulas. There the critical move is to replace each local count by a sum of degrees over those $S \subseteq \mathbb{N}$29-types containing the relevant guard relation and satisfying the relevant local condition (Bednarczyk et al., 2021).

5. Expressiveness and boundary phenomena

Local Presburger quantifiers are expressive enough to capture parity, congruence, thresholds, differences of guarded counts, and mixed in/out-degree conditions. In $S \subseteq \mathbb{N}$30, parity is expressed by an ultimately periodic set such as $S \subseteq \mathbb{N}$31 and thresholds by sets such as $S \subseteq \mathbb{N}$32 (Benedikt et al., 2020). In guarded logics, modulo constraints and weighted sums are primitive, so one can require, for instance, an even number of $S \subseteq \mathbb{N}$33-successors, a unique $S \subseteq \mathbb{N}$34-self-loop, or a prescribed linear relation between incoming and outgoing colored edges (Lu et al., 2022).

The principal misconception is that “local” Presburger arithmetic is uniformly benign. The literature instead identifies a narrow decidable boundary. In $S \subseteq \mathbb{N}$35, the methods target only unary, ultimately periodic counting sets, and the paper explicitly states that allowing general semilinear conditions over tuples with $S \subseteq \mathbb{N}$36 inside two-variable logic leads to undecidability (Benedikt et al., 2020). In guarded logics, local Presburger arithmetic remains decidable in the two-variable guarded fragment, but local percentage quantifiers make finite satisfiability undecidable in $S \subseteq \mathbb{N}$37 and already in $S \subseteq \mathbb{N}$38 (Bednarczyk et al., 2021). In Presburger arithmetic over $S \subseteq \mathbb{N}$39, the move from unary to non-unary counting is likewise non-trivial: the full logic with non-unary modulo-, threshold-, and exact-counting quantifiers is decidable in two-fold exponential space, the threshold/exact-only fragment is in alternating two-fold exponential time with linearly many alternations, but the non-unary Härtig quantifier yields undecidability because it defines multiplication (Habermehl et al., 2022).

This boundary behavior explains why locality is usually paired with either unary ultimately periodic sets or guarded neighborhood counts. A plausible implication is that the successful cases all enforce a one-step decomposition of counting into types, neighborhoods, or bounded parameter intervals, while the unsuccessful cases admit cardinality interactions strong enough to encode multiplication or unrestricted equicardinality.

6. Related locality notions inside Presburger arithmetic

A broader Presburger tradition uses “local” for bounded or guarded quantification internal to arithmetic itself. In parametric expansions of Presburger arithmetic, Goodrick generalized Cooper’s method by proving that every first-order $S \subseteq \mathbb{N}$40-formula is logically equivalent to an $S \subseteq \mathbb{N}$41-formula with $S \subseteq \mathbb{N}$42-bounded quantifiers, where the bounds are terms $S \subseteq \mathbb{N}$43 from the ambient ring of functions. Full quantifier elimination fails in general, so bounded quantifiers become the local replacement for elimination (Goodrick, 2016).

In one-parametric Presburger arithmetic, quantifier elimination is obtained in an extended language with all integer division functions $S \subseteq \mathbb{N}$44 for $S \subseteq \mathbb{N}$45. The elimination procedure localizes quantified search to bounded witnesses, base-$S \subseteq \mathbb{N}$46 digits, and sign cases for parameter polynomials, and yields that satisfiability for the existential fragment is in NP and that the smallest solution to a satisfiable formula has polynomial bit size (Mansutti et al., 30 Jun 2025).

Interpolation theory supplies a guarded version even closer to the neighborhood-logic usage. In Presburger arithmetic with uninterpreted predicates or functions, the fragment PAID allows guarded quantifiers only of the forms

$S \subseteq \mathbb{N}$47

with $S \subseteq \mathbb{N}$48. These are local because the guard isolates at most one integer witness. They can be rewritten using divisibility and integer division:

$S \subseteq \mathbb{N}$49

$S \subseteq \mathbb{N}$50

This fragment is closed under interpolation for the predicate and function extensions considered there (Brillout et al., 2010).

Recent quantifier-elimination work on Presburger arithmetic formulates locality as an affine small-witness property. For systems $S \subseteq \mathbb{N}$51, if a parameter instantiation $S \subseteq \mathbb{N}$52 admits any integral witness $S \subseteq \mathbb{N}$53, then there exists one of the form

$S \subseteq \mathbb{N}$54

where $S \subseteq \mathbb{N}$55 and $S \subseteq \mathbb{N}$56 have bounded size depending only on the quantified block matrix $S \subseteq \mathbb{N}$57. Truth of the existential formula then depends only on finitely many such affine “local shapes” together with bounded residue information (Haase et al., 2024).

Taken together, these strands show that the phrase “local Presburger quantifier” names a family resemblance rather than a single formalism. In relational two-variable logics it denotes Presburger arithmetic on guarded neighborhoods; in parametric Presburger arithmetic it denotes bounded or guarded ranges; and in interpolation and efficient elimination it denotes quantifiers reducible to single guarded arithmetic witnesses. Across these settings, locality is the mechanism that preserves decidability while retaining linear and modular arithmetic expressive power.