Complexity Results for Quantified Boolean Formulae Based on Complete Propositional Languages1

Coste‐Marquis, Sylvie; Berre, Daniel Le; Letombe, Florian; Marquis, Pierre

doi:10.3233/sat190003

Cited by 5 publications

(5 citation statements)

References 38 publications

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“…Hence, the above procedures on Decision-DNNF and SDD circuits cannot be used to interleave literal quantifications of different types. This should not be surprising though since the existence of a polynomial-time algorithm for applying a sequence of quantifiers of different types to a tractable circuit (even an OBDD) would imply P = NP; see (Coste-Marquis et al, 2006). Nonetheless, it is worth noting that for both types of literal quantification, the NNF circuit that results from applying the above procedures is monotone, if we quantify a literal for each variable that appears in the input circuit.…”

Section: Tractable Literal 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: Tractable Literal Quantificationmentioning

confidence: 99%

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

Darwiche

Marquis²

2021

jair

Self Cite

View full text Add to dashboard Cite

show abstract

“…Hence, the above procedures on Decision-DNNF and SDD circuits cannot be used to interleave literal quantifications of different types. This should not be surprising though since the existence of a polynomial-time algorithm for applying a sequence of quantifiers of different types to a tractable circuit (even an OBDD) would imply P = NP; see (Coste-Marquis, Berre, Letombe, & Marquis, 2006). Nonetheless, it is worth noting that for both types of literal quantification, the NNF circuit that results from applying the 7.…”

Section: Tractable Literal Quantificationmentioning

confidence: 99%

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

Darwiche,

Marquis

2021

Preprint

Self Cite

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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 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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“…23 Any fully-expressive representation can represent any characteristic function, but different representations may require a different amount of space to represent the same game. Work on representing functions in a concise form has provided a insightful definition regarding the succinctness of representations [18,47,25,20]: a language L is at least as expressive as a language L if there is a polynomial translator that converts any input of L into an equivalent input in L (where 'equivalence' means that the value functions coincide). 24 The running time of algorithms that analyze cooperative games depends on the input size, so a polynomial algorithm that takes the input in one language may not be useful if the input is given in another languagethe input would need to be converted to the required language, which may take an exponential amount of time and may result in an exponentially long representation.…”

Section: Expressiveness and Succinctness Of Representation Languagesmentioning

confidence: 99%

“…Our model of CSGs is somewhat similar to that of CRGs. 25 CSGs define tasks to accomplish, and CRGs define goals desired by agents. Performing a task in CSGs requires a coalition to have a certain set of skills, and achieving a goal in CRGs requires certain resources.…”

Section: Coalitional Resource Gamesmentioning

confidence: 99%