In database theory and knowledge representation, the certain answers is the set of answers to a given query consisting of the intersection of all the complete databases that are consistent with a given knowledge base.^[1] The notion of certain answer, investigated in database theory since the 1970s,^[1] is indeed defined in the context of open world assumption, where the given knowledge base is assumed to be incomplete.

Intuitively, certain answers are the answers that are always returned when querying a given knowledge base, considering both the extensional knowledge that the possible implications inferred by automatic reasoning, regardless of the specific interpretation.

In literature, the set of certain answers is usually defined as follows:^[1]

${\displaystyle cert_{\cap }(Q,D)=\bigcap \left\{Q(D')|D'\!\in [\![D]\!]\right\}}$

where:

• ${\displaystyle Q}$ is a query
• ${\displaystyle D}$ is an incomplete database
• ${\displaystyle D'}$ is any complete database consistent with ${\displaystyle D}$
• ${\displaystyle [\![D]\!]}$ is the semantics of ${\displaystyle D}$

In description logics, such set may be defined in a similar way as follows:^[2]

Given an ontology ${\displaystyle {\mathcal {K}}=\langle {\mathcal {T}},{\mathcal {A}}\rangle }$ and a query ${\displaystyle q({\vec {x}})}$ on ${\displaystyle {\mathcal {K}}}$, ${\displaystyle cert(q,{\mathcal {K}})}$ is the set of tuples ${\displaystyle {\vec {a}}\subseteq \Gamma }$ such that, for each model ${\displaystyle {\mathcal {I}}}$ of ${\displaystyle {\mathcal {K}}}$, we have that ${\displaystyle {\mathcal {I}}\models q[{\vec {a}}]}$.

Where:

• ${\displaystyle {\mathcal {T}}}$ and ${\displaystyle {\mathcal {A}}}$ are respectively a Tbox and an Abox;
• ${\displaystyle \Gamma }$ is the alphabet of constants (individuals or values) of the ontology;
• ${\displaystyle q[{\vec {a}}]}$ is obtained by replacing all the free variables in ${\displaystyle q({\vec {x}})}$ with constants of ${\displaystyle {\vec {a}}}$.

• Open world assumption
• Closed world assumption
• Completeness (knowledge bases)

• Poggi, Antonella; Lembo, Domenico; Calvanese, Diego; De Giacomo, Giuseppe; Lenzerini, Maurizio; Rosati, Riccardo (2008). "Linking Data to Ontologies" (PDF). J. On Data Semantics. Lecture Notes in Computer Science. Vol. 10. pp. 133–173. doi:10.1007/978-3-540-77688-8_5. ISBN 978-3-540-77687-1.
• Kontchakov, R.; Lutz, C.; Toman, D.; Wolter, F.; Zakharyaschev, M. (2011). The Combined Approach to Ontology-Based Data Access (PDF). Proocedings of the 22th International Joint Conference on Artificial Intelligence (IJCAI 2011). Barcelona: The AAAI Press. pp. 2656–2661.
• Libkin, Leonid (January 2011). Incomplete information and certain answers in general data models (PDF). Proocedings of the ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems. pp. 59–70. doi:10.1145/1989284.1989294.
• Ortiz, Magdalena (January 2013). Ontology Based Query Answering: The Story So Far (PDF). Proocedings of the CEUR Workshop. Vol. 1087.
• Libkin, Leonid (2015). How to Define Certain Answers (PDF). Proocedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI 2015). The AAAI Press. pp. 4282–4288.