Attribute-incremental construction of the canonical implication basis

Abstract. We investigate the complexity of enumerating pseudo-intents in the lectic order. We look at the following decision problem: Given a formal context and a set of n pseudo-intents determine whether they are the lectically first n pseudo-intents. We show that this problem is coNPhard. We thereby show that there cannot be an algorithm with a good theoretical complexity for enumerating pseudo-intents in a lectic order. In a second part of the paper we introduce the notion of minimal pseudointents, i. e. pseudo-intents that do not strictly contain a pseudo-intent. We provide some complexity results about minimal pseudo-intents that are readily obtained from the previous result.

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hardness of Enumerating Pseudo-intents in the Lectic Order

Distel

2010

“…Since the number of concept intents can be exponential in the number of pseudo-intents, this algorithm in general does not run in output polynomial time. Similarly, the attribute-incremental algorithm in [18] has also time complexity depending on both the number of pseudo-intents and the number of concept intents. In the light of our current knowledge, it is not even clear whether there can be an algorithm at all that enumerates pseudointents in output polynomial time.…”

Section: Complexity Of Enumerating Pseudo-intentsmentioning

confidence: 99%

“…The most well-known algorithm for this purpose is the next-closure algorithm [7]. Recently, an algorithm that computes the pseudo-intents by processing a single attribute at a single step, namely attribute-incremental algorithm, has been introduced in [18]. In [19], an algorithm for checking whether a set is pseudo-intent, has been presented.…”

Section: Introductionmentioning

confidence: 99%

Some Computational Problems Related to Pseudo-intents

Sertkaya

2009

Abstract. We investigate the computational complexity of several decision, enumeration and counting problems related to pseudo-intents. We show that given a formal context and a subset of its set of pseudo-intents, checking whether this context has an additional pseudo-intent is in conp, and it is at least as hard as checking whether a given simple hypergraph is not saturated. We also show that recognizing the set of pseudo-intents is also in conp, and it is at least as hard as identifying the minimal transversals of a given hypergraph. Moreover, we show that if any of these two problems turns out to be conp-hard, then unless p = np, pseudo-intents cannot be enumerated in output polynomial time. We also investigate the complexity of finding subsets of a given Duquenne-Guigues Base from which a given implication follows. We show that checking the existence of such a subset within a specified cardinality bound is np-complete, and counting all such minimal subsets is #p-complete.

“…It produces all concept intents and all pseudo-intents of a given formal context in a lexicographic order (called the lectic order ). During the last decade, newer algorithms have been developed [9,11].…”

Section: Introductionmentioning

confidence: 99%

Some Complexity Results about Essential Closed Sets

Distel

2011

Abstract. We examine the enumeration problem for essential closed sets of a formal context. Essential closed sets are sets that can be written as the closure of a pseudo-intent. The results for enumeration of essential closed sets are similar to existing results for pseudo-intents, albeit some differences exist. For example, while it is possible to compute the lectically first pseudo-intent in polynomial time, we show that it is not possible to compute the lectically first essential closed set in polynomial time unless P = NP. This also proves that essential closed sets cannot be enumerated in the lectic order with polynomial delay unless P = NP. We also look at minimal essential closed sets and show that they cannot be enumerated in output polynomial time unless P = NP.