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. We complement this by studying universal literal quantification and its applications, particularly to explainable AI. We also provide a novel semantics for quantification and discuss the interplay between variable/literal and existential/universal quantification. We further identify classes of Boolean formulas and circuits that allow efficient quantification. Literal quantification is more fine-grained than variable quantification, which leads to a refinement of quantified Boolean logic with literal quantification as its primitive.
Semantics for Conditional Literals via the SM Operator
Conditional literals are an expressive Answer Set Programming language construct supported by the solver clingo. Their semantics are currently defined by a translation to infinitary propositional logic, however, we develop an alternative characterization with the SM operator which does not rely on grounding. This allows us to reason about the behavior of a broad class of clingo programs/encodings containing conditional literals, without referring to a particular input/instance of an encoding. We formalize the intuition that conditional literals behave as nested implications, and prove the equivalence of our semantics to those implemented by clingo.
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- Award ID(s):
- 1707371
- NSF-PAR ID:
- 10379357
- Date Published:
- Journal Name:
- Logic Programming and Nonmonotonic Reasoning
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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