In formal concept analysis (FCA) and closure system attributesare analyzed via their (simultaneous) occurrences (= objects). The implicational analysis deals with algorithms to find a minimal set of rules, by which attributes can be inferred from others (implications) .  Definition A pair of sets  is an implication. and . U is the premise and V is the conclusion  . stands for attributes. A set W respects , if . Implications by Ganter and Stumme Implications in a Concept Lattice holds in a concept lattice, if every intent respects it. This is true if%DIFDELCMD < %DIFDELCMD < %%% iff . iff . An implication follows semantically from a set L of implications, if each subset of respecting L also respects . L is closed, if it contains all implications that follow from L. A set L of implications that hold in a context is called sound. It is called complete if every implications that holds in the context follows from L. L is non-redundant if implications that follow from L are not in L.%DIFDELCMD < %DIFDELCMD < %%% If L is a set of implications, then is a closure system. Implication Basis%DIFDELCMD < %DIFDELCMD < %%% Let be a closure operator then P is pseudo-closed if The set is sound, complete and non-redundant. It is called Duquenne-Guiges-Basis (D-G-basis)or Stem basis.%DIFDELCMD < %DIFDELCMD < %%% There cannot be a implication basis with less implications than the Duquenne-Guiges-Basis, but one can choose implications that are simpler regarding .Such implications are with and .Endliche Hüllensysteme und ihre Implikationenbasen (Finite hull systems and their implications)
In formal concept analysis (FCA) implications relate sets of properties or attributes . An implication A→B holds in a given domain when every object having all attributes in A also has all attributes in B. Such implications characterize the concept hierarchy in an intuitive manner. Moreover, they are "well-behaved" with respect to algorithms. The knowledge acquisition method called attribute exploration uses implications.Ganter, Bernhard and Obiedkov, Sergei (2016) Conceptual Exploration. Springer, Definitions An implication A→B is simply a pair of sets A⊆M, B⊆M, where M is the set of attributes under consideration. A is the premise and B is the conclusion of the implication A→B . A set C respects the implication A→B when ¬(C⊆A) or C⊆B . In formal concept analysis a formal context is a triple (G,M,I), where G and M are sets (of objects and attributes, respectively), and where I⊆G×M is a relation expressing which objects have which attributes. An implication which holds in such a formal context is called a valid implication for short. That an implication is valid can be expressed by the derivation operators: A→B holds in (G,M,I) iff A′ ⊆ B′ or, equivalently, iff B⊆A". Ganter, Bernhard and Wille, Rudolf (1999) Formal Concept Analysis -- Mathematical Foundations. Springer, %DIFDELCMD < %DIFDELCMD < %%% Implications and formal concepts A set C of attributes is a concept intent if and only if C respects all valid implications. The system of all valid implications therefore suffices for constructing the closure system of all concept intents and thereby the concept hierarchy . %DIFDELCMD < %DIFDELCMD < %%% %DIFDELCMD < %DIFDELCMD < %%% The system of all valid implications of a formal context is closed under the natural inference. Formal contexts with finitely many attributes possess a canonical basis of valid implications,Guigues, J.L. and Duquenne, V. Familles minimales d'implications informatives résultant d'un tableau de données binaires. Mathématiques et Sciences Humaines 95 (1986): 5-18. i.e., an irredundant family of valid implications from with all valid implications can be inferred. This basis consists of all implications of the form P→P", where P is a pseudo-intent, i.e., a pseudo-closed %DIFDELCMD < %DIFDELCMD < %%% set in the closure system of intents. See for algorithms.
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and closure system attributesare analyzed via their (simultaneous) occurrences (= objects). The implicational analysis deals with algorithms to find a minimal set of rules, by which attributes can be inferred from others (implications)
implications relate sets of properties or attributes
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An implication A→B holds in a given domain when every object having all attributes in A also has all attributes in B. Such implications characterize the concept hierarchy in an intuitive manner. Moreover, they are "well-behaved" with respect to algorithms. The knowledge acquisition method called attribute exploration uses implications.Ganter, Bernhard and Obiedkov, Sergei (2016) Conceptual Exploration. Springer,
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Definition
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A
Definitions An implication A→B is simply a
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A⊆M, B⊆M, where M is the set of attributes under consideration. A
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an implication. and . U is
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V
B
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of the implication A→B . A set C respects the implication A→B when ¬(C⊆A) or C⊆B
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stands for attributes.
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A set W respects , if .
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Implications by Ganter and Stumme
In formal concept analysis a formal context is a triple (G,M,I), where G and M are sets (of objects and attributes, respectively), and where I⊆G×M is a relation expressing which objects have which attributes. An implication which holds in such a formal context is called a valid implication for short. That an implication is valid can be expressed by the derivation operators: A→B holds in (G,M,I) iff A′ ⊆ B′ or, equivalently, iff B⊆A". Ganter, Bernhard and Wille, Rudolf (1999) Formal Concept Analysis -- Mathematical Foundations. Springer,
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Implications in a Concept Lattice
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holds in a concept lattice, if every intent respects it. This is true if
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iff
Implications and formal concepts A set C of attributes is a concept intent if and only if C respects all valid implications. The system of all valid implications therefore suffices for constructing the closure system of all concept intents and thereby the concept hierarchy
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iff .
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An implication follows semantically from a set L of implications, if each subset of respecting L also respects . L is closed, if it contains all implications that follow from L.
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A set L of implications that hold in a context is called sound. It is called complete if every implications that holds in the context follows from L. L is non-redundant if implications that follow from L are not in L.
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If L is a set of implications, then is a closure system.
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Implication Basis
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Let be a closure operator then P is
The system of all valid implications of a formal context is closed under the natural inference. Formal contexts with finitely many attributes possess a canonical basis of valid implications,Guigues, J.L. and Duquenne, V. Familles minimales d'implications informatives résultant d'un tableau de données binaires. Mathématiques et Sciences Humaines 95 (1986): 5-18. i.e., an irredundant family of valid implications from with all valid implications can be inferred. This basis consists of all implications of the form P→P", where P is a pseudo-intent, i.e., a
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if
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The set is sound, complete and non-redundant. It is called Duquenne-Guiges-Basis (D-G-basis)or Stem basis.
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There cannot be a implication basis with less implications than the Duquenne-Guiges-Basis, but one can choose implications that are simpler regarding .Such implications are with and .Endliche Hüllensysteme und ihre Implikationenbasen (Finite hull systems and their implications)
set in the closure system of intents. See for algorithms.