Atserias, Albert and Balcázar, José L. and Piceno, Marie Ely - Relative Entailment Among Probabilistic Implications

lmcs:3791 - Logical Methods in Computer Science, February 6, 2019, Volume 15, Issue 1
Relative Entailment Among Probabilistic Implications

Authors: Atserias, Albert and Balcázar, José L. and Piceno, Marie Ely

We study a natural variant of the implicational fragment of propositional logic. Its formulas are pairs of conjunctions of positive literals, related together by an implicational-like connective; the semantics of this sort of implication is defined in terms of a threshold on a conditional probability of the consequent, given the antecedent: we are dealing with what the data analysis community calls confidence of partial implications or association rules. Existing studies of redundancy among these partial implications have characterized so far only entailment from one premise and entailment from two premises, both in the stand-alone case and in the case of presence of additional classical implications (this is what we call "relative entailment"). By exploiting a previously noted alternative view of the entailment in terms of linear programming duality, we characterize exactly the cases of entailment from arbitrary numbers of premises, again both in the stand-alone case and in the case of presence of additional classical implications. As a result, we obtain decision algorithms of better complexity; additionally, for each potential case of entailment, we identify a critical confidence threshold and show that it is, actually, intrinsic to each set of premises and antecedent of the conclusion.


Source : oai:arXiv.org:1501.04826
DOI : 10.23638/LMCS-15(1:10)2019
Volume: Volume 15, Issue 1
Published on: February 6, 2019
Submitted on: July 17, 2017
Keywords: Computer Science - Logic in Computer Science,Computer Science - Databases,Computer Science - Machine Learning


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