Sentence examples for implication problems from inspiring English sources

Several tractable special cases of the satisfiability and implication problems for (mathsf {GFDs}) are identified in [29].

There are two classical problems associated with any class of dependencies, namely the satisfiability and implication problems, which are stated as follows.

As shown in [29], the intractability of the satisfiability and implication problems arises from subgraph isomorphism embedded in these problems, which is (mathsf {NP} -complete (cf. [102]).

(d) The validation and implication problems for (mathsf {GFDs}) have been settled [29], while matching complexity bounds for the (mathsf {FDs}) previously proposed are yet to be developed.

We characterize the associated implication problem axiomatically and algorithmically in linear input time.

We also show that the general probabilistic CI implication problem can be reduced to that for elementary CI statements.

Our second contribution is to investigate the implication problem for XFDs in the presence of a Document Type Definition (DTD).

These problems have also been settled for (mathsf {GFDs})  [29]: the satisfiability problem is (mathsf {coNP} -complete, and the implication problem is (mathsf {coNP} -complete

For (mathsf {FDs}), the satisfiability problem is in O(1) time (since all (mathsf {FDs}) are satisfiable) and the implication problem is in linear time (cf. [71]).

The implication problem for (mathsf {GFDs}) is to decide, given a set (varSigma ) of (mathsf {GFDs}) and another (mathsf {GFD}) (varphi ), whether (varSigma models varphi ).

We introduce a lattice-theoretic framework that permits the study of the conditional independence (CI) implication problem relative to the class of discrete probability measures.