Fixed-Parameter Tractability of Dependency QBF with Structural Parameters

Ganian, Robert; Peitl, Tomáš; Slivovsky, Friedrich; Szeider, Stefan

doi:10.24963/kr.2020/40

Cited by 6 publications

(6 citation statements)

References 36 publications

(49 reference statements)

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“…While theoretical analysis is not the primary focus of this work, we observe that our reduction to DQBF allows one to lift the recently obtained results for DQBF to obtain Fixed Parameter Tractability (FPT) analysis for synthesis [15]. We leave a detailed theoretical analysis to future work.…”

Section: Solving Dqf(bv)mentioning

confidence: 99%

“…The expressiveness of DBQF comes at the cost of the hardness from a complexity-theoretic perspective: in particular, DQBF is NEXPTIME-complete [31]. Nevertheless, as noted by Ganian et al [15], motivated by the progress in QBF solving, the past few years have seen a surge of interest from diverse viewpoints such as the development of DQBF proof systems, the study of restricted fragments to development of efficient DQBF solvers [26,35,40]. In this work, we will focus on DQBF and its generalization, Dependency Quantified Formulas modulo Theory, henceforth referred to as DQF(T).…”

Section: Introductionmentioning

confidence: 99%

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Program Synthesis as Dependency Quantified Formula Modulo Theory

Golia¹,

Roy²,

Meel³

2021

Preprint

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Given a specification ϕ(X, Y ) over inputs X and output Y , defined over a background theory T, the problem of program synthesis is to design a program f such that Y = f (X) satisfies the specification ϕ. Over the past decade, syntax-guided synthesis (SyGuS) has emerged as a dominant approach for program synthesis where in addition to the specification ϕ, the end-user also specifies a grammar L to aid the underlying synthesis engine. This paper investigates the feasibility of synthesis techniques without grammar, a sub-class defined as T-constrained synthesis. We show that T-constrained synthesis can be reduced to DQF(T), i.e., to the problem of finding a witness of a Dependency Quantified Formula Modulo Theory. When the underlying theory is the theory of bitvectors, the corresponding DQF(BV) problem can be further reduced to Dependency Quantified Boolean Formulas (DQBF). We rely on the progress in DQBF solving to design DQBFbased synthesizers that outperform the domain-specific program synthesis techniques, thereby positioning DQBF as a core representation language for program synthesis. Our empirical analysis shows that T-constrained synthesis can achieve significantly better performance than syntax-guided approaches. Furthermore, the general-purpose DQBF solvers perform on par with domain-specific synthesis techniques.

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Section: Solving Dqf(bv)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Program Synthesis as Dependency Quantified Formula Modulo Theory

Golia¹,

Roy²,

Meel³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In fact, while being based on entirely different ideas, our reduction and Atserias' and Oliva's construction for QBF also show intractability for a slight restriction of treewidth called pathwidth, but do not exclude tractability under a related parameter treedepth (Nesetril and de Mendez 2012). Investigating the complexity of our problems under the parameter treedepth is the natural next choice, not only because it lies at the very boundary of intractability, but also because of its successful applications for a variety of other problems (Ganian et al 2020;Ganian and Ordyniak 2018;Gutin, Jones, and Wahlström 2016) and its close connection to the maximum path length in the network 3 . While the complexity of QBF parameterized by treedepth remains a prominent open problem, as our second main technical contribution we show: Theorem 2.…”

Section: Introductionmentioning

confidence: 95%

A Structural Complexity Analysis of Synchronous Dynamical Systems

Eiben

Ganian

Hamm

et al. 2023

AAAI

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Synchronous dynamical systems are well-established models that have been used to capture a range of phenomena in networks, including opinion diffusion, spread of disease and product adoption. We study the three most notable problems in synchronous dynamical systems: whether the system will transition to a target configuration from a starting configuration, whether the system will reach convergence from a starting configuration, and whether the system is guaranteed to converge from every possible starting configuration. While all three problems were known to be intractable in the classical sense, we initiate the study of their exact boundaries of tractability from the perspective of structural parameters of the network by making use of the more fine-grained parameterized complexity paradigm. As our first result, we consider treewidth - as the most prominent and ubiquitous structural parameter - and show that all three problems remain intractable even on instances of constant treewidth. We complement this negative finding with fixed-parameter algorithms for the former two problems parameterized by treedepth, a well-studied restriction of treewidth. While it is possible to rule out a similar algorithm for convergence guarantee under treedepth, we conclude with a fixed-parameter algorithm for this last problem when parameterized by treedepth and the maximum in-degree.

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“…Evaluating DQBF is NEXPTIME complete [3] in general, but some tractable subclasses have been identified in recent work [38,17].…”

Section: Related Workmentioning

confidence: 99%

Certified DQBF Solving by Definition Extraction

Reichl¹,

Slivovsky²,

Szeider³

2021

Preprint

Self Cite

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We propose a new decision procedure for dependency quantified Boolean formulas (DQBFs) that uses interpolation-based definition extraction to compute Skolem functions in a counter-example guided inductive synthesis (CEGIS) loop. In each iteration, a family of candidate Skolem functions is tested for correctness using a SAT solver, which either determines that a model has been found, or returns an assignment of the universal variables as a counterexample. Fixing a counterexample generally involves changing candidates of multiple existential variables with incomparable dependency sets. Our procedure introduces auxiliary variables-which we call arbiter variables-that each represent the value of an existential variable for a particular assignment of its dependency set. Possible repairs are expressed as clauses on these variables, and a SAT solver is invoked to find an assignment that deals with all previously seen counterexamples. Arbiter variables define the values of Skolem functions for assignments where they were previously undefined, and may lead to the detection of further Skolem functions by definition extraction.A key feature of the proposed procedure is that it is certifying by design: for true DQBF, models can be returned at minimal overhead. Towards certification of false formulas, we prove that clauses can be derived in an expansion-based proof system for DQBF. In an experimental evaluation on standard benchmark sets, a prototype implementation was able to match (and in some cases, surpass) the performance of state-of-the-art-solvers. Moreover, models could be extracted and validated for all true instances that were solved.

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Fixed-Parameter Tractability of Dependency QBF with Structural Parameters

Cited by 6 publications

References 36 publications

Program Synthesis as Dependency Quantified Formula Modulo Theory

Program Synthesis as Dependency Quantified Formula Modulo Theory

A Structural Complexity Analysis of Synchronous Dynamical Systems

Certified DQBF Solving by Definition Extraction

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