On the complexity of enumerating pseudo-intents

Abstract

a b s t r a c tWe investigate whether the pseudo-intents of a given formal context can efficiently be enumerated. We show that they cannot be enumerated in a specified lexicographic order with polynomial delay unless P = NP. Furthermore we show that if the restriction on the order of enumeration is removed, then the problem becomes at least as hard as enumerating minimal transversals of a given hypergraph. We introduce the notion of minimal pseudo-intents and show that recognizing minimal pseudo-intents is polynomial. Despite their less complicated nature, surprisingly it turns out that minimal pseudo-intents cannot be enumerated in output-polynomial time unless P = NP.

Some Complexity Results about Essential Closed Sets

“…In this respect, the computational behaviour of essential closed sets is worse than that of pseudo-intents, since the lectically first pseudo-intent can be computed in polynomial time [3]. Because not even the lectically first essential closed set can be computed in polynomial time it is obviously not possible to enumerate essential closed sets in the lectic order with polynomial delay.…”

“…According to (3) and (4) this is equivalent to saying that -there is some p r ∈ S φ , where p r / ∈ A i , or -there is some ¬p r ∈ S φ , where ¬p r / ∈ A i . This is equivalent to S φ ⊆ A i .…”

“…It is impossible to compute the lectically first essential closed set unless P = NP. The lectically first pseudo-intent can be computed in polynomial time [3].…”

Computing the Duquenne–Guigues basis: an algorithm for choosing the order

“…The complexity of the problem of enumerating pseudo-intents has been studied by Babin and Kuznetsov (2013), Kuznetsov (2004), Kuznetsov and Obiedkov (2008) and Distel and Sertkaya (2011) and it was found that it is impossible to enumerate them in the lectic or reverse lectic order with a polynomial delay (i.e. being able to find two consecutive pseudo-intents in polynomial time).…”

On minimal sets of graded attribute implications

“…Unlike the instance-based methods which belong to hard problems [10] even for the classic (non-graded) rules, the minimization method presented in this paper is polynomial and therefore tractable.…”