Closed-world assumption
The closed-world assumption (CWA), in a
Negation as failure is related to the closed-world assumption, as it amounts to believing false every predicate that cannot be proved to be true.
Example
In the context of knowledge management, the closed-world assumption is used in at least two situations: (1) when the knowledge base is known to be complete (e.g., a corporate database containing records for every employee), and (2) when the knowledge base is known to be incomplete but a "best" definite answer must be derived from incomplete information. For example, if a database contains the following table reporting editors who have worked on a given article, a query on the people not having edited the article on Formal Logic is usually expected to return "Sarah Johnson".
Edit | |
---|---|
Editor | Article |
John Doe | Formal Logic |
Joshua A. Norton | Formal Logic |
Sarah Johnson | Introduction to Spatial Databases |
Charles Ponzi | Formal Logic |
Emma Lee-Choon | Formal Logic |
In the closed-world assumption, the table is assumed to be
Formalization in logic
The first formalization of the closed-world assumption in
entails neither nor .
Adding the negation of these two literals to the knowledge base leads to
which is inconsistent. In other words, this formalization of the closed-world assumption sometimes turns a consistent knowledge base into an inconsistent one. The closed-world assumption does not introduce an inconsistency on a knowledge base exactly when the intersection of all
Alternative formalizations not suffering from this problem have been proposed. In the following description, the considered knowledge base is assumed to be propositional. In all cases, the formalization of the closed-world assumption is based on adding to the negation of the formulae that are “free for negation” for , i.e., the formulae that can be assumed to be false. In other words, the closed-world assumption applied to a knowledge base generates the knowledge base
- .
The set of formulae that are free for negation in can be defined in different ways, leading to different formalizations of the closed-world assumption. The following are the definitions of being free for negation in the various formalizations.
- CWA (closed-world assumption)
- is a positive literal not entailed by ;
- GCWA (generalized CWA)
- is a positive literal such that, for every positive clause such that , it holds ;[2]
- EGCWA (extended GCWA)
- same as above, but is a conjunction of positive literals;
- CCWA (careful CWA)
- same as GCWA, but a positive clause is only considered if it is composed of positive literals of a given set and (both positive and negative) literals from another set;
- ECWA (extended CWA)
- similar to CCWA, but is an arbitrary formula not containing literals from a given set.[3] [4]
The ECWA and the formalism of
In situations where it is not possible to assume a closed world for all predicates, yet some of them are known to be closed, the
See also
- Open-world assumption
- Partial-closed world assumption
- Non-monotonic logic
- Circumscription (logic)
- Negation as failure
- Default logic
- Stable model semantics
- Unique name assumption
References
- ISBN 9780306400605.
- ISBN 978-3-540-11558-8
- ^ Eiter, Thomas; Gottlob, Georg (June 1993). "Propositional circumscription and extended closed-world reasoning are Π 2 p ". Theoretical Computer Science. 114 (2): 231–245. . ISSN 0304-3975.
- . ISSN 0004-3702.
- . ISSN 0022-0000.
- ^ Razniewski, Simon; Savkovic, Ognjen; Nutt, Werner (2015). "Turning The Partial-closed World Assumption Upside Down" (PDF).
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