Quantified Resolution Proofs (Q-Res)

• Quantified Resolution Proofs (Q-Res) are a resolution-based proof system for QBFs that utilize inference rules like existential clause-resolution and universal term-resolution to align with the quantifier structure.
• The system transforms QBFs in prenex normal form into proof trees, providing a framework for both refutation and model proof constructions essential for solver strategies and complexity evaluations.
• Extensions such as symmetry rules and generalized axioms enhance Q-Res by enabling efficient clause learning, zero-knowledge verification, and integration of external decision procedures.

Quantified Resolution Proofs (Q-Res) are a resolution-based proof system for Quantified Boolean Formulas (QBFs), forming the theoretical and practical backbone of modern QBF solvers and complexity analyses. Q-Res operates on formulas in prenex, clause or term normal form, leveraging specialized inference rules aligned with the quantifier structure. Its variants, extensions, and duals (such as term resolution, generalized axioms, and symmetry augmentation) enable concise theoretical characterizations as well as efficient practical algorithms for QBF evaluation.

1. Formal Definitions and Core Inference Rules

Q-Res applies to QBFs of the form $\Phi \equiv Q_1 x_1 Q_2 x_2 \dots Q_n x_n\,.\,\psi$, where the quantifier prefix $Q_i \in \{\exists, \forall\}$ imposes a linear variable order. The matrix $\psi$ must be in a normal form: conjunctive normal form (CNF) for clause-based resolution or disjunctive normal form (DNF) for term-based duals. A literal $\ell$ is either $x$ or $\neg x$ for $x \in P$, and inherits the level induced by the quantifier prefix.

The two principal inference rules in clause-variant Q-Res (Giunchiglia et al., 2011, Beyersdorff et al., 2016, Chen, 2014, Kauers et al., 2018, Karthikeyan et al., 19 Nov 2025) are:

• Existential Clause-resolution: For clauses $C_1 \vee \ell$ and $C_2 \vee \neg\ell$ where $\ell$ is existential and both clauses are minimal (i.e., all minimum-level literals are existential),

$$\frac{C_1 \vee \ell \quad C_2 \vee \neg\ell}{\text{minimal}((C_1 \cup C_2) \setminus \{\ell, \neg\ell\})}$$

Minimality is achieved by dropping universal literals whose levels exceed all remaining existential literals' levels.

• Universal Term-resolution (dual): For terms $T_1 \wedge \ell$ and $T_2 \wedge \neg\ell$ where $\ell$ is universal and both terms are minimal (all minimum-level literals universal),

$$\frac{T_1 \wedge \ell \quad T_2 \wedge \neg\ell}{\text{minimal}((T_1 \cup T_2) \setminus \{\ell, \neg\ell\})}$$

In both cases, the side condition ensures minimality and correct alignment with the quantifier structure.

Universal Reduction permits the removal of universal literals from a clause when those literals appear rightmost in the quantifier prefix among literals present in the clause (Beyersdorff et al., 2016, Chen, 2014).

2. Proof Objects, Trees, and Search-theoretic Correspondence

A Quantified Resolution proof for a CNF QBF is a finite tree with leaves as input clauses, inner nodes from clause-resolution, and the root as the empty clause $\emptyset$ (refutational completeness) (Giunchiglia et al., 2011, Beyersdorff et al., 2016). For term resolution in DNF, the dual proof delivers the empty term (conjunction over zero literals) as a certificate of truth. Search procedures such as DLL alternate between existential splits (corresponding to clause-resolution) and universal splits (term-resolution), with backtracking precisely reflecting Q-Res inferences (Giunchiglia et al., 2011). This establishes a bijection between search trees and alternating Q-Res refutations or model proofs.

3. Complexity, Size Measures, and Lower Bounds

Q-Res proofs are measured by:

• Size $\|\Pi\|$: The number of inference nodes (clauses or terms) in the proof tree.
• Width $w(\Pi)$: The maximal cardinality of literal sets in any clause or term.

The system is sound and complete for QBFs in CNF (Theorem 2) and DNF (Theorem 4), admitting super-polynomial and, in many important cases, exponential lower bounds on proof size (Giunchiglia et al., 2011, Beyersdorff et al., 2016, Chen, 2014). Proof checking is polynomial-time but constructing short proofs is generally computationally hard (PSPACE-complete).

A strict hierarchy exists among QBF proof calculi: $\text{Q-Res} < \text{IR-calc} < \text{IRM-calc}$, with IR-calc p-simulating Q-Res and IRM-calc p-simulating IR-calc, but each separation is exponential in worst-case formula families (Beyersdorff et al., 2016). Q-Res lacks completeness when lifted to DQBF; only IR-calc and its annotated variants remain sound and complete for full DQBF (Beyersdorff et al., 2016).

4. Extensions: Symmetry, Generalized Axioms, and Dualization

Symmetry Rule

The Q-Res calculus is significantly strengthened on symmetric QBF families by introducing the symmetry rule $S$, enabling derivations of the form $\sigma(C)$ for any clause $C$ and permissible symmetry $\sigma$ of the formula (Kauers et al., 2018). For structured formula families such as KBKF$_n$ and QUPARITY$_n$, Q-Res+S collapses exponential refutation lengths to linear, outperforming IR-calc and LQU$^+$. However, Q-Res+S and stronger expansions are incomparable: formulas without exploitable symmetry foil the S-rule but remain efficiently refutable in IR-calc.

Generalized Axioms

Q-resolution axioms are generalized in (Lonsing et al., 2016) as follows:

• gen-cl-init: For any QCDCL assignment $A$, if the restricted formula $[\Phi]_A$ is unsatisfiable, derive the clause $C = \bigvee_{l \in A} \neg l$.
• gen-cu-init: If $[\Phi]_A$ is satisfiable, derive the cube $C = \bigwedge_{l \in A} l$.

Embedding arbitrary decision procedures (SAT, QBF or preprocessing) into the proof system via generalized axioms exponentially increases Q-Res proof strength, enabling the integration of external methods (e.g., Bloqqer, PicoSAT) in search-based solvers such as DepQBF. This interface facilitates the learning of much stronger clauses and cubes earlier within search, yielding dramatic reductions in proof and search sizes on application benchmarks (Lonsing et al., 2016).

5. Applications: DLL-Based Solving, Learning, and Zero-Knowledge

Search-based QBF solvers, notably those using the Davis-Logemann-Loveland (DLL) procedure, naturally instantiate Q-Res trees during their operation (Giunchiglia et al., 2011). Nogood learning (unsatisfiable backtracks) and good learning (prime implicant cubes) correspond exactly to Q-Res derivations, facilitating efficient pruning. Recent advances employ Q-Res as the underlying soundness certificate in practical zero-knowledge protocols for PSPACE statements. By encoding Q-Res proofs via low-degree polynomials over finite fields, one enables efficient, low-overhead verification of QBF evaluation in zero knowledge. Empirical results demonstrate success rates of 72% for QBF evaluation via Q-Res proofs within 100 s, covering thousands of steps and clause widths (Karthikeyan et al., 19 Nov 2025).

Extension	Theoretical Impact	Implementation/Benchmarks
Symmetry Rule (S)	Collapses exponential to linear size in symmetric families	KBKF$_n$, QUPARITY$_n$ (Kauers et al., 2018)
Generalized Axioms	Enables proof system p-simulation of IR/expansion; exponential shortcuts	DepQBF, Bloqqer, PicoSAT (Lonsing et al., 2016)
ZK Polynomial Encoding	Certification and privacy for all PSPACE QBFs	ZKQRES protocol, QBFEVAL (Karthikeyan et al., 19 Nov 2025)

6. Limitations, Hierarchical Embedding, and Research Directions

Q-Res is PSPACE-complete and structurally tailored to prenex CNF/term forms. It lacks completeness for DQBF and multi-sorted or non-prenex quantifier structures. Embeddings into QCSP “judgement systems” allow generalization well beyond QBF, accommodating arbitrary quantifier alternations and formulas that do not admit clause normal forms (Chen, 2014). Block-wise and hierarchical refinements, dynamic integration of symmetry reasoning, and asynchronous parallel invocation of external decision procedures remain active research areas. The strict separation between Q-Res and higher-calculus variants motivates hybrid solvers that leverage generalized axioms or symmetry as opportunities arise.

7. Representative Examples and Empirical Findings

Canonical examples, such as $$\exists x \forall y \exists z : (x \vee y) \wedge (\neg x \vee \neg z) \wedge (z \vee \neg y)$$, illustrate tree construction and the alternation between existential and universal pivots. Experimental work on symmetric formulas and practical zero-knowledge verification for QBFs demonstrates that Q-Res, when appropriately extended, yields performance and certification unattainable by either pure search or pure expansion approaches alone (Giunchiglia et al., 2011, Kauers et al., 2018, Karthikeyan et al., 19 Nov 2025). This convergence of theoretical and applied advances situates Q-Res as central within QBF proof complexity, solver engineering, and cryptographic certification for PSPACE-complete reasoning.