In this paper, we define a sound and complete inference system for triadic implications generated from a formal triadic context K:=⟨G,M,B,I⟩, where G, M, and B are object, attribute, and condition sets, respectively, and I is a ternary relation I⊆G×M×B. The inference system is expressed as a set of axioms “à la Armstrong.” The type of triadic implications we are considering in this paper is called conditional attribute implication (CAI) and has the following form: X→scriptCY, where X and Y are subsets of M, and scriptC is a subset of B. Such implication states that Ximplies Y under all conditions in scriptC and any subset of it. Moreover, we propose a method to compute CAIs from Biedermann's implications. We also introduce an algorithm to compute the closure of an attribute set X w.r.t. a set Σ of CAIs given a set scriptC of conditions.
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