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database theory Database theory encapsulates a broad range of topics related to the study and research of the theoretical realm of databases and database management systems. Theoretical aspects of data management include, among other areas, the foundations of q ...
and
knowledge representation Knowledge representation and reasoning (KRR, KR&R, KR²) is the field of artificial intelligence (AI) dedicated to representing information about the world in a form that a computer system can use to solve complex tasks such as diagnosing a medic ...
, the one of the certain answers is the set of answers to a given query consisting of the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of all the
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
database In computing, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases sp ...
s that are consistent with a given knowledge base. The notion of certain answer, investigated in database theory since the 1970s, is indeed defined in the context of
open world assumption In a formal system of logic used for knowledge representation, the open-world assumption is the assumption that the truth value of a statement may be true irrespective of whether or not it is ''known'' to be true. It is the opposite of the clos ...
, where the given knowledge base is assumed to be incomplete. Intuitively, certain answers are the answers that are always returned when quering a given
knowledge base A knowledge base (KB) is a technology used to store complex structured and unstructured information used by a computer system. The initial use of the term was in connection with expert systems, which were the first knowledge-based systems. Ori ...
, considering both the extensional knowledge that the possible implications inferred by
automatic reasoning In computer science, in particular in knowledge representation and reasoning and metalogic, the area of automated reasoning is dedicated to understanding different aspects of reasoning. The study of automated reasoning helps produce computer progr ...
, regardless of the specific interpretation.


Definition

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![_D_!.html"_;"title="D_.html"_;"title="![_D_">![_D_!">D_.html"_;"title="![_D_">![_D_!/math>_is_the_semantics_of_D In_description_logics">D_">![_D_!.html"_;"title="D_.html"_;"title="![_D_">![_D_!">D_.html"_;"title="![_D_">![_D_!/math>_is_the_semantics_of_D In_description_logics,_such_set_may_be_defined_in_a_similar_way_as_follows:. :Given_an_Ontology_(information_science)">ontology_ In_metaphysics,_ontology_is_the_philosophical_study_of_being,_as_well_as_related_concepts_such_as_existence,__becoming,_and_reality._ Ontology_addresses_questions_like_how_entities_are_grouped_into_categories_and_which_of_these_entities_exis_...
_\mathcal=\langle\mathcal,\mathcal\rangle_and_a_query_q(\vec_x)_on_\mathcal,_cert(q,\mathcal)_is_the_set_of_tuples_\vec_a_\subseteq_\Gamma_such_that,_for_each_Model_(mathematical_logic).html" "title="Ontology_(information_science).html" "title="description_logics.html" ;"title="D_">![_D_!.html" ;"title="D_.html" ;"title="![ D ">![ D !">D_.html" ;"title="![ D ">![ D !/math> is the semantics of D In description logics">D_">![_D_!.html" ;"title="D_.html" ;"title="![ D ">![ D !">D_.html" ;"title="![ D ">![ D !/math> is the semantics of D In description logics, such set may be defined in a similar way as follows:. :Given an Ontology (information science)">ontology In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exis ...
\mathcal=\langle\mathcal,\mathcal\rangle and a query q(\vec x) on \mathcal, cert(q,\mathcal) is the set of tuples \vec a \subseteq \Gamma such that, for each Model (mathematical logic)">model A model is an informative representation of an object, person or system. The term originally denoted the Plan_(drawing), plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a mea ...
\mathcal of \mathcal, we have that \mathcal\models q vec a/math>. Where: * \mathcal and \mathcal are respectively a Tbox and an Abox; * \Gamma is the alphabet of Logical constant, constants (individuals or values) of the ontology; * q vec a/math> is obtained by replacing all the
free variables In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
in q(\vec x) with constants of \vec a.


See also

*
Open world assumption In a formal system of logic used for knowledge representation, the open-world assumption is the assumption that the truth value of a statement may be true irrespective of whether or not it is ''known'' to be true. It is the opposite of the clos ...
*
Closed world assumption The closed-world assumption (CWA), in a formal system of logic used for knowledge representation, is the presumption that a statement that is true is also known to be true. Therefore, conversely, what is not currently known to be true, is false. Th ...
*
Completeness (knowledge bases) The term completeness as applied to knowledge bases refers to two different concepts. Formal logic In formal logic, a knowledge base KB is complete ''if'' there is no formula α such that KB ⊭ α and KB ⊭ ¬α. Example of knowledge base with ...


References


Further reading

* * * * * {{cite conference, url=https://www.ijcai.org/Proceedings/15/Papers/609.pdf, first=Leonid, last=Libkin, title=How to Define Certain Answers, conference=Proocedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI 2015), publisher=The AAAI Press, pages=4282–4288, year=2015 Knowledge representation