Ramsey Quantifiers in Logic & Verification

• Ramsey quantifiers are logical extensions that express the existence of infinite cliques or anti-cliques in definable graphs.
• They enable polynomial-time elimination in linear arithmetic (LIA, LRA, LIRA), reducing complex quantified formulas to manageable existential forms.
• Applications include verifying liveness, termination, and well-foundedness in automated reasoning and model checking across finite and infinite settings.

A Ramsey quantifier is a generalized quantifier expressing the existence of arbitrarily large (infinite or unbounded finite) cliques or anti-cliques within a definable graph induced by a binary relation, typically formalized as an extension of first-order logic. This property is crucial for capturing statements about infinite combinatorial phenomena, liveness and termination properties in verification, and the structure of definable sets in logic, arithmetics, and automata-theoretic frameworks.

1. Formal Definition and Semantics

Let $\mathcal{T}$ be a first-order theory with universe $D$, and let $\varphi(\vec{x},\vec{y},\vec{z})$ be a formula (with $\vec{x},\vec{y}$ tuples of variables of the same type and $\vec{z}$ parameters). The Ramsey quantifier is written as $$\vec{x},\vec{y}\,:\,\varphi(\vec{x},\vec{y},\vec{z})$$ and asserts that there exists an infinite sequence $(\vec{a}_1, \vec{a}_2, \ldots) \subset D^k$ of pairwise-distinct $k$-tuples such that $\varphi(\vec{a}_i, \vec{a}_j, \vec{z})$ holds for every $i < j$.

In model-theoretic semantics: $$\mathcal{T} \models (\vec{x},\vec{y}:\varphi(\vec{x},\vec{y},\vec{c})) \quad\iff\quad \begin{array}{l} \text{There exists an infinite sequence }(\vec{a}_i)_{i=1}^{\infty}\subset D^k \ \text{of pairwise-distinct vectors with }\mathcal{T}\models\varphi(\vec{a}_i,\vec{a}_j,\vec{c})\ \forall i<j. \end{array}$$ Graph-theoretically, $\varphi$ defines the edges of a directed graph on $D^k$; the quantifier expresses the presence of an infinite directed clique.

In the finite model setting, the notion generalizes to quantifiers $\mathsf{R}_f$ parameterized by a function $f:\mathbb{N}\to\mathbb{N}$, with the model-theoretic statement: $$\mathcal{M} \models \mathsf{R}_f\, x\,y.\,\varphi(x,y)\quad\Longleftrightarrow\quad \exists A\subseteq M \ \bigl(|A|\geq f(|M|)\wedge \forall a,b\in A: \mathcal{M}\models \varphi(a,b)\bigr)$$ e.g., $f(n)=k$ recovers $k$-clique existence; $f(n)=\lceil r n \rceil$ yields "proportional" thresholds.

2. Quantifier Elimination in Linear Arithmetics

Given existential (or quantifier-free) formulas $\psi(\vec{x},\vec{y},\vec{z})$ in LIA (integer arithmetic), LRA (real arithmetic), or LIRA (mixed integer-real), one can eliminate the Ramsey quantifier in polynomial time, yielding an equivalent existential formula of linear size.

Elimination Process Overview

1. Pushing Inner Existentials: Any existential quantifier inside the body of the Ramsey quantifier can be restructured outside (at linear cost), yielding an equivalent formula of increased arity but no inner existential (Bergsträßer et al., 2023, Lichtner et al., 7 Nov 2025). Let $\phi(\vec{x},\vec{y},\vec{w},\vec{z})$ be quantifier-free in the background theory, then:

$$\vec{x},\vec{y} : \exists \vec{w}\; \phi(\vec{x},\vec{y},\vec{w},\vec{z})\;\iff\; (\vec{x},\vec{v}_1,\vec{v}_2),(\vec{y},\vec{w}_1,\vec{w}_2):\phi(\vec{x},\vec{y}, \vec{v}_1+\vec{w}_2, \vec{z}) \wedge \vec{x}\neq \vec{y}$$

1. Reduction to Existential Formulas: The existence of an infinite clique for quantifier-free $\phi$ is characterized by a finite system of linear (in)equalities over fresh parameters—a consequence of Ramsey-theoretic and compactness arguments. For each conjunction of literals, feasibility of the system in the background theory implies existence of the clique:
   • For LIA, the infinite clique can be modeled as an arithmetic progression $\vec{a} + t\vec{b}$.
   • For LRA, as a half-line $\vec{a} + t\vec{b}$.
   • For LIRA, as a mix of integer arithmetic progressions and real rays.
2. Complexity: The overall elimination procedure is polynomial-time and produces an output existential formula whose size is linear in the input formula.

Theory	Infinite clique canonical form	Output formula type	Elimination complexity
LIA	$\vec{a} + t\vec{b}\ (t\in\mathbb{N})$	existential LIA	$O(|\varphi|)$
LRA	$\vec{a} + t\vec{b}\ (t\geq 0)$	existential LRA	$O(|\varphi|)$
LIRA	integer + real mix; both progressions and rays	existential LIRA	$O(|\varphi|)$

This approach yields a deterministic, efficient reduction of Ramsey-quantified existential formulas in arithmetic settings to standard existential formulas, enabling practical decision procedures with off-the-shelf SMT solvers (Bergsträßer et al., 2023, Lichtner et al., 7 Nov 2025).

3. Computational Complexity and Dichotomy

Infinite Arithmetical Context

• Existential Ramsey Formula Validity: Deciding whether $x,y : \phi(x,y)$ holds, for existential formula $\phi$ in LIA, LRA, or LIRA, is NP-complete. The elimination algorithm verifies satisfiability via reduction to an existential formula and invocation of standard SMT solving.
• Well-Foundedness: For an existentially definable, transitive binary relation $R$, deciding well-foundedness (absence of infinite descending chains) reduces to Ramsey quantification, yielding NP-completeness when $R$ is defined in the target theories (Bergsträßer et al., 2023).

Finite Model Theory

Within finite models, for quantifiers $\mathsf{R}_f$ parametrized by $f$, the following dichotomy holds under ETH, for nondecreasing, polynomial-time computable $f$: $$\mathsf{R}_f \in \mathbf{P} \iff f \text{ is constant-log-bounded}$$ where $f$ is constant-log-bounded if $\exists c,n_0$ s.t. $f(n)\leq c$ or $f(n)\geq n-c\log n$ for $n>n_0$. Proportional thresholds (e.g., $f(n) = \lceil rn \rceil$ with $0 < r < 1$) correspond to NP-complete Ramsey quantifiers. Thresholds such as $f(n)=\lceil \log n\rceil$ yield NP-intermediate complexity (assuming ETH) (Haan et al., 2016).

Threshold $f(n)$	Complexity	Comment
$f(n)\leq$ constant	P	Brute-force over small sets
$f(n)\geq n - O(\log n)$	P	Clique finding in polynomial time
$f(n) = \lceil r n \rceil$	NP-complete	Reduction from CLIQUE
$f(n) = \lceil \log n \rceil$	NP-intermediate	Under ETH; strict dichotomy otherwise fails

For practical model-checking and natural-language semantics, tractability essentially coincides with having a constant-log-bounded threshold for the Ramsey quantifier (Haan et al., 2016).

4. Applications in Verification and Automated Reasoning

Well-Foundedness and Termination

By encoding the well-foundedness of transitive relations as the absence of infinite cliques, Ramsey quantifiers enable reduction of termination checking of infinite-state systems (timed automata, VASS, counter systems) to existential SMT, thereby exploiting highly optimized solvers (Bergsträßer et al., 2023, Lichtner et al., 7 Nov 2025).

Liveness Verification

Liveness (infinitely often or non-termination) reduces to asserting the existence of an infinite reachable clique, formalized by a Ramsey quantifier over the reachability relation. Recent toolchains, notably the REAL tool, extend the verification pipeline: reachability sets (e.g., from FAST(er)) are transpiled into LIA, Ramsey quantifiers are introduced to encode liveness, and elimination yields SMT-expressible existential formulas, enhancing both expressiveness and practical scalability (Lichtner et al., 7 Nov 2025).

System type	Reachability encoding	Resulting algorithm class
Timed automata ($\mathbb{R}$-clocks)	existential formula in LRA	NP-complete liveness
Continuous VASS ($\mathbb{Q}$-counters)	existential formula in LRA	NP
Reversal-bounded counter machines ($\mathbb{Z}$-counters)	LIA	NP (NEXPTIME if reachability is hard)
Succinct one-counter systems	LIA	NP

5. Ramsey Quantifiers in Automata and Automatic Structures

Ramsey quantifiers have rich interactions with automatic structures, both over finite words and trees.

• Directed Version: In automatic structures, the Ramsey quantifier expresses the existence of an infinite directed clique in a regular relation $R$ (given by a synchronized automaton). The complexity of deciding whether such a clique exists depends on both the structural assumptions on $R$ and the automaton type (Bergsträßer et al., 2022).

Context	Complexity	Notes
Word-automatic, general $R$	NL-complete
Tree-automatic, general $R$	ExpTime-complete
Tree-automatic, transitive $R$	P-complete
Tree-automatic, co-transitive $R$	P-complete

• Monadic Decomposability: Monadic decomposability (canonical for recognizability) of automatic relations can be reduced to the absence of infinite cliques in certain co-transitive relations. Consequently, this property is NL-complete (words) or P-complete (tree-automatic) in deterministic settings (Bergsträßer et al., 2022, Bergsträßer et al., 2023).
• Recurrent Reachability: Büchi and generalized Büchi model-checking for regular systems can be interpreted as checking a Ramsey quantifier over the transitive closure of the transition relation, with tight complexity outcomes across regular word and tree systems.

6. Practical Implementations and Experimental Evaluations

The REAL tool implements polynomial-time elimination of Ramsey quantifiers for existential formulas in LIA, LRA, LIRA, with integration into SMT-LIB via an extended grammar:

(ramsey (x Int) (y Int) (and (> y x) (< (+ x y) z)))

REAL parses, normalizes, computes direction constraints, synthesizes existential formulas, simplifies, and exports to SMT-LIB for downstream SMT solving. Benchmarks indicate almost linear scaling in problem size—elimination times remain practical even for formulas with thousands of atoms, while SMT solver time usually dominates (Lichtner et al., 7 Nov 2025).

In verification pipelines (e.g., FAST(er) $\to$ Armoise $\to$ Alchemist $\to$ REAL $\to$ Z3), full liveness verification on distributed protocols with complex Presburger reachability is feasible within minutes, for relations involving millions of atoms.

7. Broader Impact and Theoretical Significance

Ramsey quantifiers bridge finite and infinite combinatorial model theory, enable expressive logics for liveness/termination properties, and provide a framework for advanced decomposition properties (monadic decomposability, recognizability) in both arithmetic and automata-theoretic contexts. The dichotomy in model-checking complexity, with implications for natural language semantics and automated reasoning, underscores the fundamental importance of the constant-log-bound definition in tractability. Recent advances in elimination algorithms, especially in linear arithmetic theories, have rendered these quantifiers practically viable for large-scale verification and reasoning tasks.