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Extra info for Formal Methods for Mining Structured Objects

Example text

We say that a rule G → S holds for a given set of sequences S ⊆ S if for all s ∈ S either G {s } or S {s }. Due to the construction of the closure operator Δ , we can argue directly that this method of constructing rules is sound, that is, all the rules of our proposed form that can be derived from an input set of sequences D do hold for each of those input sequences; we could say that our implications with order have confidence 1 (100%) in our data. Indeed, since {d} is closed for each individual input sequence d of our database D, we can consider any generator G and obtain, by monotonicity of Δ , G {d} implies Δ (G) Δ ({d}) = {d}; that is, the implication G → Δ (G) holds for {d}.

Here, by intersecting two set of sequences, S1 S2 , we understand the set of maximal sequences resulting from the cross intersection of s1 ∩ s2 , for all s1 ∈ S1 and all s2 ∈ S2 . Proof. Let S1 and S2 be two closed sets of sequences, that is we have that Δ (S1 ) = S1 and Δ (S2 ) = S2 . By monotonicity of the closure operator Δ we get: first, S1 S2 S1 implies Δ (S1 S2 ) Δ (S1 ), that is, Δ (S1 S2 ) S1 ; and second, S1 S2 S2 implies Δ (S1 S2 ) Δ (S2 ), that is, Δ (S1 S2 ) S2 . Therefore, we have that Δ (S1 S2 ) S1 S2 .

2, do not have any stable sequence with tid lists exactly O; that is, the set S˜ of the theorem is empty. 5, Algorithm 1 would not generate the closed set of sequences { (C)(A) , (B)(A) } since individually, their tid list do not coincide with the closed set of objects linked by the Galois connection. Therefore, once we get the valid pairs from the algorithm, we may still need to update the system with new concepts that ensure the closure system. This can be done by organizing the set of valid pairs into a temporary lattice, and then, checking the consistency of the dual system of closed sets of objects.

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