Direct and Binary Direct Bases for One-Set Updates of a Closure System
We introduce a concept of a binary-direct implicational basis and show that the shortest binary-direct basis exists and it is known as the D-basis introduced in Adaricheva, Nation, Rand . Using this concept we approach the algorithmic solution to the Singleton Horn Extension problem, as well as the one set removal problem, when the closure system is given by the canonical direct or binary-direct basis. In this problem, a new closed set is added to or removed from the closure system forcing the re-write of a given basis. Our goal is to obtain the same type of implicational basis for the new closure system as was given for original closure system and to make the basis update an optimal process.
KeywordsClosure system Horn-to-Horn belief revision Singleton Horn Extension problem Direct basis Canonical direct basis The D-basis Ordered direct basis
The first results of the paper were presented on the poster session of ICFCA-2017 in Rennes, France, and both authors’ participation in the conference was supported by the research fund of Hofstra University. We thank Sergey Obiedkov for pointing to the important publication of Marcel Wild , and we thank Justin Cabot-Miller from Hofstra University for his support in producing valuable test cases in the implementation phase.
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