A Survey on Applications of Quantified Boolean Formulas

Shukla, Ankit; Biere, Armin; Pulina, Luca; Seidl, Martina

doi:10.1109/ictai.2019.00020

Cited by 45 publications

(23 citation statements)

References 37 publications

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“…This is equivalent to saying that each addition of a block of quantifiers makes the validity problem computationally harder since the validity problem for prenex and closed QBFs is Σ p n -complete if Q 1 = ∃ and Π p n -complete if Q 1 = ∀. While QBFs can be harder to decide compared to standard Boolean formulas, they do allow for exponentially more succinct encodings of problems in domains such as planning, non-monotonic reasoning, formal verification and the synthesis of computing systems (e.g., Giunchiglia et al, 2009;Shukla et al, 2019). This explains the extensive efforts dedicated to developing QBF solvers in the past few decades (see http://www.qbflib.org).…”

Section: Variable Quantificationmentioning

confidence: 99%

On Quantifying Literals in Boolean Logic and its Applications to Explainable AI

Darwiche

Marquis²

2021

jair

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Quantified Boolean logic results from adding operators to Boolean logic for existentially and universally quantifying variables. This extends the reach of Boolean logic by enabling a variety of applications that have been explored over the decades. The existential quantification of literals (variable states) and its applications have also been studied in the literature. In this paper, we complement this by introducing and studying universal literal quantification and its applications, particularly to explainable AI. We also provide a novel semantics for quantification, discuss the interplay between variable/literal and existential/universal quantification, and identify some classes of Boolean formulas and circuits on which quantification can be done efficiently. Literal quantification is more fine-grained than variable quantification as the latter can be defined in terms of the former, leading to a refinement of quantified Boolean logic with literal quantification as its primitive.

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Section: Variable Quantificationmentioning

confidence: 99%

On Quantifying Literals in Boolean Logic and its Applications to Explainable AI

Darwiche

Marquis²

2021

jair

View full text Add to dashboard Cite

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“…While QBFs can be harder to decide compared to standard (non-quantified) Boolean formulas, they do allow for exponentially more succinct encodings of various problems in domains such as planning, non-monotonic reasoning, formal verification and the synthesis of computing systems; see e.g., (Giunchiglia, Marin, & Narizzano, 2009;Shukla, Biere, Pulina, & Seidl, 2019). This explains the extensive efforts dedicated to developing QBF solvers in the past few decades (see http://www.qbflib.org).…”

Section: Variable Quantificationmentioning

confidence: 99%

On Quantifying Literals in Boolean Logic and Its Applications to Explainable AI

Darwiche,

Marquis

2021

Preprint

View full text Add to dashboard Cite

Quantified Boolean logic results from adding operators to Boolean logic for existentially and universally quantifying variables. This extends the reach of Boolean logic by enabling a variety of applications that have been explored over the decades. The existential quantification of literals (variable states) and its applications have also been studied in the literature. In this paper, we complement this by studying universal literal quantification and its applications, particularly to explainable AI. We also provide a novel semantics for quantification, discuss the interplay between variable/literal and existential/universal quantification. We further identify some classes of Boolean formulas and circuits on which quantification can be done efficiently. Literal quantification is more fine-grained than variable quantification as the latter can be defined in terms of the former. This leads to a refinement of quantified Boolean logic with literal quantification as its primitive.

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“…This is not a coincidence, but is guaranteed by the definition of SAUNF. Hence, at recursion level 5 of SkGen, any vector of functions f 3 (i), f 4 (i) can be returned in line 2 of Algorithm 1 as a Skolem function vector for (p 3 , p 4 ) in H (2) .…”

Section: B Generating Skolem Functions From Saunf Circuitsmentioning

confidence: 99%

“…Skolem functions (and their counterparts, called Herbrand functions) can be thought of as "certificates" that help us independently verify the results of satisfiability checking for Quantified Boolean Formulas, as done in [1]. QBFsatisfiability solving is used today in diverse applications [2], from planning to program repair to reactive synthesis and the like. Having certificates not only helps to verify correctness of QBF-satisfiability checks, but also has other benefits like providing a feasible plan in a planning problem.…”

Section: Introductionmentioning

confidence: 99%

A Normal Form Characterization for Efficient Boolean Skolem Function Synthesis

Shah¹,

Bansal²,

Akshay³

et al. 2021

Preprint

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Boolean Skolem function synthesis concerns synthesizing outputs as Boolean functions of inputs such that a relational specification between inputs and outputs is satisfied. This problem, also known as Boolean functional synthesis, has several applications, including design of safe controllers for autonomous systems, certified QBF solving, cryptanalysis etc. Recently, complexity theoretic hardness results have been shown for the problem, although several algorithms proposed in the literature are known to work well in practice. This dichotomy between theoretical hardness and practical efficacy has motivated the research into normal forms or representations of input specifications that permit efficient synthesis, thus explaining perhaps the efficacy of these algorithms.In this paper we go one step beyond this and ask if there exists a normal form representation that can in fact precisely characterize "efficient" synthesis. We present a normal form called SAUNF that precisely characterizes tractable synthesis in the following sense: a specification is polynomial time synthesizable iff it can be compiled to SAUNF in polynomial time. Additionally, a specification admits a polynomial-sized functional solution iff there exists a semantically equivalent polynomial-sized SAUNF representation. SAUNF is exponentially more succinct than well-established normal forms like BDDs and DNNFs, used in the context of AI problems, and strictly subsumes other more recently proposed forms like SynNNF. It enjoys compositional properties that are similar to those of DNNF. Thus, SAUNF provides the right trade-off in knowledge representation for Boolean functional synthesis.

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A Survey on Applications of Quantified Boolean Formulas

Cited by 45 publications

References 37 publications

On Quantifying Literals in Boolean Logic and its Applications to Explainable AI

On Quantifying Literals in Boolean Logic and its Applications to Explainable AI

On Quantifying Literals in Boolean Logic and Its Applications to Explainable AI

A Normal Form Characterization for Efficient Boolean Skolem Function Synthesis

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