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Tomás Peitl

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11 papers

1 author row

SAT Conference 2025 Conference Paper

Better Extension Variables in DQBF via Independence

Leroy Chew
Tomás Peitl

We show that extension variables in (D)QBF can be generalised by conditioning on universal assignments. The benefit of this is that the dependency sets of such conditioned extension variables can be made smaller to allow easier refutations. This simple modification instantly solves many challenges in p-simulating the QBF expansion rule, which cannot be p-simulated in proof systems that have strategy extraction [Leroy Chew and Judith Clymo, 2020]. Simulating expansion is even more crucial in DQBF, where other methods are incomplete. In this paper we provide an overview of the strength of this new independent extension rule. We find that a new version of Extended Frege called IndExtFrege + ∀red can p-simulate a multitude of difficult QBF and DQBF techniques, even techniques that are difficult to approach with eFrege + ∀red. We show five p-simulations, that IndExtFrege + ∀red p-simulates QRAT, DQBF-IR-calc, IR(𝒟^rrs)-calc, Fork-Resolution and DQRAT which together underpin most DQBF solving and preprocessing techniques. The p-simulations work despite these systems using complicated rules and our new extension rule being relatively simple. Moreover, unlike recent p-simulations by eFrege + ∀red we can simulate the proof rules line by line, which allows us to mix QBF rules more easily with other inference steps.

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SAT Conference 2024 Conference Paper

Small Unsatisfiable k-CNFs with Bounded Literal Occurrence

Tianwei Zhang 0006
Tomás Peitl
Stefan Szeider

We obtain the smallest unsatisfiable formulas in subclasses of k-CNF (exactly k distinct literals per clause) with bounded variable or literal occurrences. Smaller unsatisfiable formulas of this type translate into stronger inapproximability results for MaxSAT in the considered formula class. Our results cover subclasses of 3-CNF and 4-CNF; in all subclasses of 3-CNF we considered we were able to determine the smallest size of an unsatisfiable formula; in the case of 4-CNF with at most 5 occurrences per variable we decreased the size of the smallest known unsatisfiable formula. Our methods combine theoretical arguments and symmetry-breaking exhaustive search based on SAT Modulo Symmetries (SMS), a recent framework for isomorph-free SAT-based graph generation. To this end, and as a standalone result of independent interest, we show how to encode formulas as graphs efficiently for SMS.

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SAT Conference 2023 Conference Paper

A SAT Solver's Opinion on the Erdős-Faber-Lovász Conjecture

Markus Kirchweger
Tomás Peitl
Stefan Szeider

In 1972, Paul Erdős, Vance Faber, and Lászlo Lovász asked whether every linear hypergraph with n vertices can be edge-colored with n colors, a statement that has come to be known as the EFL conjecture. Erdős himself considered the conjecture as one of his three favorite open problems, and offered increasing money prizes for its solution on several occasions. A proof of the conjecture was recently announced, for all but a finite number of hypergraphs. In this paper we look at some of the cases not covered by this proof. We use SAT solvers, and in particular the SAT Modulo Symmetries (SMS) framework, to generate non-colorable linear hypergraphs with a fixed number of vertices and hyperedges modulo isomorphisms. Since hypergraph colorability is NP-hard, we cannot directly express in a propositional formula that we want only non-colorable hypergraphs. Instead, we use one SAT (SMS) solver to generate candidate hypergraphs modulo isomorphisms, and another to reject them by finding a coloring. Each successive candidate is required to defeat all previous colorings, whereby we avoid having to generate and test all linear hypergraphs. Computational methods have previously been used to verify the EFL conjecture for small hypergraphs. We verify and extend these results to larger values and discuss challenges and directions. Ours is the first computational approach to the EFL conjecture that allows producing independently verifiable, DRAT proofs.

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SAT Conference 2022 Conference Paper

Should Decisions in QCDCL Follow Prefix Order?

Benjamin Böhm 0001
Tomás Peitl
Olaf Beyersdorff

Quantified conflict-driven clause learning (QCDCL) is one of the main solving approaches for quantified Boolean formulas (QBF). One of the differences between QCDCL and propositional CDCL is that QCDCL typically follows the prefix order of the QBF for making decisions. We investigate an alternative model for QCDCL solving where decisions can be made in arbitrary order. The resulting system QCDCL^ANY is still sound and terminating, but does not necessarily allow to always learn asserting clauses or cubes. To address this potential drawback, we additionally introduce two subsystems that guarantee to always learn asserting clauses (QCDCL^UNI-ANI) and asserting cubes (QCDCL^EXI-ANY), respectively. We model all four approaches by formal proof systems and show that QCDCL^UNI-ANY is exponentially better than QCDCL on false formulas, whereas QCDCL^EXI-ANY is exponentially better than QCDCL on true QBFs. Technically, this involves constructing specific QBF families and showing lower and upper bounds in the respective proof systems. We complement our theoretical study with some initial experiments that confirm our theoretical findings.

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SAT Conference 2021 Conference Paper

Davis and Putnam Meet Henkin: Solving DQBF with Resolution

Joshua Blinkhorn
Tomás Peitl
Friedrich Slivovsky

Abstract Davis-Putnam resolution is one of the fundamental theoretical decision procedures for both propositional logic and quantified Boolean formulas. Dependency quantified Boolean formulas ( DQBF ) are a generalisation of QBF in which dependencies of variables are listed explicitly rather than being implicit in the order of quantifiers. Since DQBFs can succinctly encode synthesis problems that ask for Boolean functions matching a given specification, efficient DQBF solvers have a wide range of potential applications. We present a new decision procedure for DQBF in the style of Davis-Putnam resolution. Based on the merge resolution proof system, it directly constructs partial strategy functions for derived clauses. The procedure requires DQBF in a normal form called H-Form. We prove that the problem of evaluating DQBF in H-Form is NEXP-complete. In fact, we show that any DQBF can be converted into H-Form in linear time.

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SAT Conference 2020 Conference Paper

Strong (D)QBF Dependency Schemes via Tautology-Free Resolution Paths

Olaf Beyersdorff
Joshua Blinkhorn
Tomás Peitl

Abstract We suggest a general framework to study dependency schemes for dependency quantified Boolean formulas (DQBF). As our main contribution, we exhibit a new tautology-free DQBF dependency scheme that generalises the reflexive resolution path dependency scheme. We establish soundness of the tautology-free scheme, implying that it can be used in any DQBF proof system. We further explore the power of DQBF resolution systems parameterised by dependency schemes and show that our new scheme results in exponentially shorter proofs in comparison to the reflexive resolution path dependency scheme when used in the expansion DQBF system \(\mathsf {\forall {Exp{\text{+ }}Res}}\). On QBFs, we demonstrate that our new scheme is exponentially stronger than the reflexive resolution path dependency scheme when used in Q-resolution, thus resulting in the strongest QBF dependency scheme known to date.

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SAT Conference 2019 Conference Paper

Combining Resolution-Path Dependencies with Dependency Learning

Tomás Peitl
Friedrich Slivovsky
Stefan Szeider

Abstract We present the first practical implementation of the reflexive resolution-path dependency scheme in a QBF solver. Unlike in DepQBF, which uses the less general standard dependency scheme, we do not compute the dependency relation upfront, but instead query relevant dependencies on demand during dependency conflicts, when the solver is about to learn a missing dependency. Thus, our approach is fundamentally tied to dependency learning, and shows that the two techniques for dependency analysis can be fruitfully combined. As a byproduct, we propose a quasilinear-time algorithm to compute all resolution-path dependencies of a given variable. Experimental results on the QBF library confirm the viability of our technique and identify families of formulas where the speedup is particularly promising.

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SAT Conference 2019 Conference Paper

Proof Complexity of Fragments of Long-Distance Q-Resolution

Tomás Peitl
Friedrich Slivovsky
Stefan Szeider

Abstract Q-resolution is perhaps the most well-studied proof system for Quantified Boolean Formulas (QBFs). Its proof complexity is by now well understood, and several general proof size lower bound techniques have been developed. The situation is quite different for long-distance Q-resolution (LDQ-resolution). While lower bounds on LDQ-resolution proof size have been established for specific families of formulas, we lack semantically grounded lower bound techniques for LDQ-resolution. In this work, we study restrictions of LDQ-resolution. We show that a specific lower bound technique based on bounded-depth strategy extraction does not work even for reductionless Q-resolution by presenting short proofs of the QParity formulas. Reductionless Q-resolution is a variant of LDQ-resolution that admits merging but no universal reduction. We also prove a lower bound on the proof size of the completion principle formulas in reductionless Q-resolution. This shows that two natural fragments of LDQ-resolution are incomparable: Q-resolution, which allows universal reductions but no merging, and reductionless Q-resolution, which allows merging but no universal reductions. Finally, we develop semantically grounded lower bound techniques for fragments of LDQ-resolution, specifically tree-like LDQ-resolution and regular reductionless Q-resolution.

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SAT Conference 2018 Conference Paper

Polynomial-Time Validation of QCDCL Certificates

Tomás Peitl
Friedrich Slivovsky
Stefan Szeider

Abstract Quantified Boolean Formulas (QBFs) offer compact encodings of problems arising in areas such as verification and synthesis. These applications require that QBF solvers not only decide whether an input formula is true or false but also output a witnessing certificate, i. e. a representation of the winning strategy. State-of-the-art QBF solvers based on Quantified Conflict-Driven Constraint Learning (QCDCL) can emit Q-resolution proofs, from which in turn such certificates can be extracted. The correctness of a certificate generated in this way is validated by substituting it into the matrix of the input QBF and using a SAT solver to check that the resulting propositional formula (the validation formula ) is unsatisfiable. This final check is often the most time-consuming part of the entire certification workflow. We propose a new validation method that does not require a SAT call and provably runs in polynomial time. It uses the Q-resolution proof from which the given certificate was extracted to directly generate a (propositional) proof of the validation formula in the RUP format, which can be verified by a proof checker such as DRAT-trim. Experiments with a prototype implementation show a robust, albeit modest, increase in the number of successfully validated certificates compared to validation with a SAT solver.

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SAT Conference 2017 Conference Paper

Dependency Learning for QBF

Tomás Peitl
Friedrich Slivovsky
Stefan Szeider

Abstract Quantified Boolean Formulas (QBFs) can be used to succinctly encode problems from domains such as formal verification, planning, and synthesis. One of the main approaches to QBF solving is Quantified Conflict Driven Clause Learning (QCDCL). By default, QCDCL assigns variables in the order of their appearance in the quantifier prefix so as to account for dependencies among variables. Dependency schemes can be used to relax this restriction and exploit independence among variables in certain cases, but only at the cost of nontrivial interferences with the proof system underlying QCDCL. We propose a new technique for exploiting variable independence within QCDCL that allows solvers to learn variable dependencies on the fly. The resulting version of QCDCL enjoys improved propagation and increased flexibility in choosing variables for branching while retaining ordinary (long-distance) Q-resolution as its underlying proof system. In experiments on standard benchmark sets, an implementation of this algorithm shows performance comparable to state-of-the-art QBF solvers.

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SAT Conference 2016 Conference Paper

Long Distance Q-Resolution with Dependency Schemes

Tomás Peitl
Friedrich Slivovsky
Stefan Szeider

Abstract Resolution proof systems for quantified Boolean formulas (QBFs) provide a formal model for studying the limitations of state-of-the-art search-based QBF solvers, which use these systems to generate proofs. In this paper, we define a new proof system that combines two such proof systems: Q-resolution with generalized universal reduction according to a dependency scheme and long distance Q-resolution. We show that the resulting proof system is sound for the reflexive resolution-path dependency scheme—in fact, we prove that it admits strategy extraction in polynomial time. As a special case, we obtain soundness and polynomial-time strategy extraction for long distance Q-resolution with universal reduction according to the standard dependency scheme. We report on experiments with a configuration of DepQBF that generates proofs in this system.

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