5 research outputs found
On the Interaction of Inclusion Dependencies with Independence Atoms
Proceeding volume: 46Inclusion dependencies are one of the most important database constraints. In isolation their finite and unrestricted implication problems coincide, are finitely axiomatizable, PSPACE-complete, and fixed-parameter tractable in their arity. In contrast, finite and unrestricted implication problems for the combined class of functional and inclusion de- pendencies deviate from one another and are each undecidable. The same holds true for the class of embedded multivalued dependencies. An important embedded tractable fragment of embedded multivalued dependencies are independence atoms. These stipulate independence between two attribute sets in the sense that for every two tuples there is a third tuple that agrees with the first tuple on the first attribute set and with the second tuple on the second attribute set. For independence atoms, their finite and unrestricted implication problems coincide, are finitely axiomatizable, and decidable in cubic time. In this article, we study the implication problems of the combined class of independence atoms and inclusion dependencies. We show that their finite and unrestricted implication problems coincide, are finitely axiomatizable, PSPACE-complete, and fixed-parameter tractable in their arity. Hence, significant expressivity is gained without sacrificing any of the desirable properties that inclusion dependencies have in isolation. Finally, we establish an efficient condition that is sufficient for independence atoms and inclusion dependencies not to inter- act. The condition ensures that we can apply known algorithms for deciding implication of the individual classes of independence atoms and inclusion dependencies, respectively, to decide implication for an input that combines both individual classes.Peer reviewe
ΠΠ±ΠΎΠ±ΡΠ΅Π½ΠΈΠ΅ ΠΏΡΠ°Π²ΠΈΠ» Π²ΡΠ²ΠΎΠ΄Π° Π΄Π»Ρ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ ΡΠΎΠ΅Π΄ΠΈΠ½Π΅Π½ΠΈΡ Π² Π±Π°Π·Π°Ρ Π΄Π°Π½Π½ΡΡ
In this paper a generalisation of the inference rules of the join dependencies that are used in the design of database schemas that meets the requirements of the fifth normal form is considered. In the previous works devoted to this problem, attempts to construct systems of the axioms of such dependencies based on inference rules are made. However, while the justification for the consistency (soundness) of the obtained axioms does not cause difficulties, the proof of completeness in general has not been satisfactorily resolved. First of all, this is due to the limitations of the inference rules themselves. This work focuses on two original axiom systems presented in the works of Sciore and Malvestuto. For the inclusion dependencies a system of rules that generalises existing systems and has fewer restrictions has been obtained. The paper presents a proof of the derivability of known systems of axioms from the presented inference rules. In addition, evidence of the consistency (soundness) of these rules is provided. The question of the completeness of the formal system based on the presented rules did not find a positive solution. In conclusion, the theoretical and practical significance of the inference rules for the join dependencies is noted.Π ΡΠ°Π±ΠΎΡΠ΅ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½ΠΈΠ΅ ΠΏΡΠ°Π²ΠΈΠ» Π²ΡΠ²ΠΎΠ΄Π° Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ ΡΠΎΠ΅Π΄ΠΈΠ½Π΅Π½ΠΈΡ, ΠΊΠΎΡΠΎΡΡΠ΅ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ ΠΏΡΠΈ ΠΏΡΠΎΠ΅ΠΊΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΡΡ
Π΅ΠΌΡ Π±Π°Π·Ρ Π΄Π°Π½Π½ΡΡ
, ΡΠ΄ΠΎΠ²Π»Π΅ΡΠ²ΠΎΡΡΡΡΠ΅ΠΉ ΡΡΠ΅Π±ΠΎΠ²Π°Π½ΠΈΡΠΌ ΠΏΡΡΠΎΠΉ Π½ΠΎΡΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΡΠΎΡΠΌΡ. Π ΠΏΡΠ΅Π΄ΡΠ΅ΡΡΠ²ΡΡΡΠΈΡ
ΡΠ°Π±ΠΎΡΠ°Ρ
, ΠΏΠΎΡΠ²ΡΡΠ΅Π½Π½ΡΡ
Π΄Π°Π½Π½ΠΎΠΉ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ°ΡΠΈΠΊΠ΅, Π΄Π΅Π»Π°ΡΡΡΡ ΠΏΠΎΠΏΡΡΠΊΠΈ ΠΏΠΎΡΡΡΠΎΠΈΡΡ ΡΠΈΡΡΠ΅ΠΌΡ Π°ΠΊΡΠΈΠΎΠΌ ΡΠ°ΠΊΠΈΡ
Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ, ΠΎΡΠ½ΠΎΠ²Π°Π½Π½ΡΡ
Π½Π° ΠΏΡΠ°Π²ΠΈΠ»Π°Ρ
Π²ΡΠ²ΠΎΠ΄Π°. ΠΠ΄Π½Π°ΠΊΠΎ, Π΅ΡΠ»ΠΈ ΠΎΠ±ΠΎΡΠ½ΠΎΠ²Π°Π½ΠΈΠ΅ Π½Π΅ΠΏΡΠΎΡΠΈΠ²ΠΎΡΠ΅ΡΠΈΠ²ΠΎΡΡΠΈ (Π½Π°Π΄Π΅ΠΆΠ½ΠΎΡΡΠΈ) ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΡΡ
Π°ΠΊΡΠΈΠΎΠΌ Π½Π΅ Π²ΡΠ·ΡΠ²Π°Π΅Ρ Π·Π°ΡΡΡΠ΄Π½Π΅Π½ΠΈΠΉ, ΡΠΎ Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²ΠΎ ΠΏΠΎΠ»Π½ΠΎΡΡ Π² ΠΎΠ±ΡΠ΅ΠΌ ΡΠ»ΡΡΠ°Π΅ Π½Π΅ ΠΏΠΎΠ»ΡΡΠΈΠ»ΠΎ ΡΠ΄ΠΎΠ²Π»Π΅ΡΠ²ΠΎΡΠΈΡΠ΅Π»ΡΠ½ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ. ΠΡΠ΅ΠΆΠ΄Π΅ Π²ΡΠ΅Π³ΠΎ, ΡΡΠΎ ΡΠ²ΡΠ·Π°Π½ΠΎ Ρ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½Π½ΠΎΡΡΡΡ ΡΠ°ΠΌΠΈΡ
ΠΏΡΠ°Π²ΠΈΠ» Π²ΡΠ²ΠΎΠ΄Π°. Π Π΄Π°Π½Π½ΠΎΠΉ ΡΠ°Π±ΠΎΡΠ΅ Π°ΠΊΡΠ΅Π½ΡΠΈΡΠΎΠ²Π°Π½ΠΎ Π²Π½ΠΈΠΌΠ°Π½ΠΈΠ΅ Π½Π° Π΄Π²ΡΡ
ΠΎΡΠΈΠ³ΠΈΠ½Π°Π»ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌΠ°Ρ
Π°ΠΊΡΠΈΠΎΠΌ, ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½Π½ΡΡ
Π² ΡΠ°Π±ΠΎΡΠ°Ρ
Sciore ΠΈ Malvestuto. ΠΠ»Ρ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ ΠΏΠΎΠ»ΡΡΠ΅Π½Π° ΡΠΈΡΡΠ΅ΠΌΠ° ΠΏΡΠ°Π²ΠΈΠ», ΠΊΠΎΡΠΎΡΠ°Ρ ΠΎΠ±ΠΎΠ±ΡΠ°Π΅Ρ ΡΡΡΠ΅ΡΡΠ²ΡΡΡΠΈΠ΅ ΡΠΈΡΡΠ΅ΠΌΡ ΠΈ ΠΏΡΠΈ ΡΡΠΎΠΌ ΠΈΠΌΠ΅Π΅Ρ ΠΌΠ΅Π½ΡΡΠ΅ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΠΉ. Π ΡΠ°Π±ΠΎΡΠ΅ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΎ Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²ΠΎ Π²ΡΠ²ΠΎΠ΄ΠΈΠΌΠΎΡΡΠΈ ΠΈΠ·Π²Π΅ΡΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ Π°ΠΊΡΠΈΠΎΠΌ ΠΈΠ· ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½Π½ΡΡ
ΠΏΡΠ°Π²ΠΈΠ» Π²ΡΠ²ΠΎΠ΄Π°. ΠΡΠΎΠΌΠ΅ ΡΠΎΠ³ΠΎ, ΠΏΡΠΈΠ²ΠΎΠ΄ΠΈΡΡΡ Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²ΠΎ Π½Π΅ΠΏΡΠΎΡΠΈΠ²ΠΎΡΠ΅ΡΠΈΠ²ΠΎΡΡΠΈ (Π½Π°Π΄Π΅ΠΆΠ½ΠΎΡΡΠΈ) ΡΡΠΈΡ
ΠΏΡΠ°Π²ΠΈΠ». ΠΠΎΠΏΡΠΎΡ ΠΎ ΠΏΠΎΠ»Π½ΠΎΡΠ΅ ΡΠΎΡΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΡ, ΠΎΡΠ½ΠΎΠ²Π°Π½Π½ΠΎΠΉ Π½Π° ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½Π½ΡΡ
ΠΏΡΠ°Π²ΠΈΠ»Π°Ρ
, Π½Π΅ Π½Π°ΡΠ΅Π» ΠΏΠΎΠ»ΠΎΠΆΠΈΡΠ΅Π»ΡΠ½ΠΎΠ³ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΡ. Π Π·Π°ΠΊΠ»ΡΡΠ΅Π½ΠΈΠ΅ ΠΎΡΠΌΠ΅ΡΠ΅Π½Π° ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠ°Ρ ΠΈ ΠΏΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠ°Ρ Π·Π½Π°ΡΠΈΠΌΠΎΡΡΡ ΠΏΡΠ°Π²ΠΈΠ» Π²ΡΠ²ΠΎΠ΄Π° Π΄Π»Ρ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ ΡΠΎΠ΅Π΄ΠΈΠ½Π΅Π½ΠΈΡ
ΠΠ±ΠΎΠ±ΡΠ΅Π½Π½ΡΠ΅ ΡΠΈΠΏΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΠ΅ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠΈ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ Ρ Π½Π΅ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΠΌΠΈ Π·Π½Π°ΡΠ΅Π½ΠΈΡΠΌΠΈ Π² Π±Π°Π·Π°Ρ Π΄Π°Π½Π½ΡΡ
The paper discusses a new type of dependency in databases, which is a generalization of inclusion dependencies. Traditionally, such dependencies are used in practice to ensure referential integrity. In this case, the restriction is established only between a pair of relations, the first of which is called the main, the second is external. In practice, referential integrity often needs to be established for a larger number of relations, where several main and several external relations participate in the same constraint. Such a structure corresponds to an ultragraph. The paper provides a rationale for generalized inclusion dependencies that take into account the presence of null values in external relations. Based on the study of the properties of typed dependencies, a system of axioms is obtained, for which consistency (soundness) and completeness are proved.Π ΡΡΠ°ΡΡΠ΅ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ Π½ΠΎΠ²ΡΠΉ Π²ΠΈΠ΄ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ Π² Π±Π°Π·Π°Ρ
Π΄Π°Π½Π½ΡΡ
, ΡΠ²Π»ΡΡΡΠΈΠΉΡΡ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½ΠΈΠ΅ΠΌ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ. Π’ΡΠ°Π΄ΠΈΡΠΈΠΎΠ½Π½ΠΎ ΡΠ°ΠΊΠΈΠ΅ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠΈ Π½Π° ΠΏΡΠ°ΠΊΡΠΈΠΊΠ΅ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ Π΄Π»Ρ ΠΎΠ±Π΅ΡΠΏΠ΅ΡΠ΅Π½ΠΈΡ ΡΡΡΠ»ΠΎΡΠ½ΠΎΠΉ ΡΠ΅Π»ΠΎΡΡΠ½ΠΎΡΡΠΈ. ΠΡΠΈ ΡΡΠΎΠΌ, ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΠ΅ ΡΡΡΠ°Π½Π°Π²Π»ΠΈΠ²Π°Π΅ΡΡΡ ΡΠΎΠ»ΡΠΊΠΎ ΠΌΠ΅ΠΆΠ΄Ρ ΠΏΠ°ΡΠΎΠΉ ΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΠΉ, ΠΏΠ΅ΡΠ²ΠΎΠ΅ ΠΈΠ· ΠΊΠΎΡΠΎΡΡΡ
Π½Π°Π·ΡΠ²Π°Π΅ΡΡΡ Π³Π»Π°Π²Π½ΡΠΌ, Π²ΡΠΎΡΠΎΠ΅ β Π²Π½Π΅ΡΠ½ΠΈΠΌ. ΠΠ° ΠΏΡΠ°ΠΊΡΠΈΠΊΠ΅ ΡΡΡΠ»ΠΎΡΠ½ΡΡ ΡΠ΅Π»ΠΎΡΡΠ½ΠΎΡΡΡ ΡΠ°ΡΡΠΎ ΡΡΠ΅Π±ΡΠ΅ΡΡΡ ΡΡΡΠ°Π½ΠΎΠ²ΠΈΡΡ Π΄Π»Ρ Π±ΠΎΠ»ΡΡΠ΅Π³ΠΎ ΡΠΈΡΠ»Π° ΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΠΉ, Π³Π΄Π΅ Π² ΠΎΠ΄Π½ΠΎΠΌ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΠΈ ΡΡΠ°ΡΡΠ²ΡΡΡ Π½Π΅ΡΠΊΠΎΠ»ΡΠΊΠΎ Π³Π»Π°Π²Π½ΡΡ
ΠΈ Π½Π΅ΡΠΊΠΎΠ»ΡΠΊΠΎ ΠΏΠΎΠ΄ΡΠΈΠ½Π΅Π½Π½ΡΡ
ΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΠΉ. Π’Π°ΠΊΠ°Ρ ΡΡΡΡΠΊΡΡΡΠ° ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΠ΅Ρ ΡΠ»ΡΡΡΠ°Π³ΡΠ°ΡΡ. Π ΡΠ°Π±ΠΎΡΠ΅ ΠΏΡΠΈΠ²Π΅Π΄Π΅Π½ΠΎ ΠΎΠ±ΠΎΡΠ½ΠΎΠ²Π°Π½ΠΈΠ΅ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½Π½ΡΡ
Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ, ΡΡΠΈΡΡΠ²Π°ΡΡΠΈΡ
Π½Π°Π»ΠΈΡΠΈΠ΅ Π½Π΅ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ
Π·Π½Π°ΡΠ΅Π½ΠΈΠΉ Π²ΠΎ Π²Π½Π΅ΡΠ½ΠΈΡ
ΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΡΡ
. ΠΠ° ΠΎΡΠ½ΠΎΠ²Π΅ ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΡ ΡΠ²ΠΎΠΉΡΡΠ² ΡΠΈΠΏΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΡ
Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ ΠΏΠΎΠ»ΡΡΠ΅Π½Π° ΡΠΈΡΡΠ΅ΠΌΠ° Π°ΠΊΡΠΈΠΎΠΌ, Π΄Π»Ρ ΠΊΠΎΡΠΎΡΠΎΠΉ Π΄ΠΎΠΊΠ°Π·Π°Π½Π° Π½Π΅ΠΏΡΠΎΡΠΈΠ²ΠΎΡΠ΅ΡΠΈΠ²ΠΎΡΡΡ (Π½Π°Π΄Π΅ΠΆΠ½ΠΎΡΡΡ) ΠΈ ΠΏΠΎΠ»Π½ΠΎΡΠ°
ΠΠ½Π°Π»ΠΈΠ· ΡΠΈΠΏΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΡ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ Ρ Π½Π΅ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΠΌΠΈ Π·Π½Π°ΡΠ΅Π½ΠΈΡΠΌΠΈ
Null values have become an urgent problem since the creation of the relational dataΒ model. The impact of the uncertainty aο¬ects all types of dependencies used in the design and operationΒ of the database. This fully applies to the inclusion dependencies, which are the theoretical basis forΒ referential integrity on the data. Attempts to solve this problem contain inaccuracy in the statementΒ of the problem and its solution. The errors in formulation of the problem can be associated with theΒ use in the deο¬nition of untyped inclusion dependencies, which leads to permutations of the attributes,Β although, the attributes in database technology are identiο¬ed by name and not by their place. In addition, linking with the use of the inclusion dependencies of heterogeneous attributes, even of the same type, is a sign of lost functional dependencies and leads to interaction of inclusion dependencies and non-trivial functional dependencies. Inaccuracies in the solution of the problem are contained in the statements of axioms and the proof of their properties, including completeness. In this paper we propose an original solution of this problem only for typed inclusion dependencies in the presence of Null values: a new axiom system is proposed, its completeness and soundness are proved. On the basis of inference rules we developed an algorithm for the construction of a not surplus set of typed inclusion dependencies. The correctness of the algorithm is proved.ΠΠ΅ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΠ΅ Π·Π½Π°ΡΠ΅Π½ΠΈΡ ΡΡΠ°Π»ΠΈ Π°ΠΊΡΡΠ°Π»ΡΠ½ΠΎΠΉ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠΎΠΉ Ρ ΠΌΠΎΠΌΠ΅Π½ΡΠ° ΡΠΎΠ·Π΄Π°Π½ΠΈΡ ΡΠ΅Π»ΡΡΠΈΠΎΠ½Π½ΠΎΠΉ ΠΌΠΎΠ΄Π΅Π»ΠΈ Π΄Π°Π½Π½ΡΡ
. ΠΠ»ΠΈΡΠ½ΠΈΠ΅ Π½Π΅ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΠΎΡΡΠ΅ΠΉ ΡΠΊΠ°Π·ΡΠ²Π°Π΅ΡΡΡ Π½Π° Π²ΡΠ΅Ρ
Π²ΠΈΠ΄Π°Ρ
Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ,Β ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΡΡ
ΠΏΡΠΈ ΠΏΡΠΎΠ΅ΠΊΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΠΈ ΡΠΊΡΠΏΠ»ΡΠ°ΡΠ°ΡΠΈΠΈ Π±Π°Π·Ρ Π΄Π°Π½Π½ΡΡ
. Π ΠΏΠΎΠ»Π½ΠΎΠΉ ΠΌΠ΅ΡΠ΅ ΡΡΠΎ ΠΎΡΠ½ΠΎΡΠΈΡΡΡ ΠΈΒ ΠΊ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΡΠΌ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ, ΠΊΠΎΡΠΎΡΡΠ΅ ΡΠ²Π»ΡΡΡΡΡ ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΠΎΡΠ½ΠΎΠ²ΠΎΠΉ ΡΡΡΠ»ΠΎΡΠ½ΠΎΠΉ ΡΠ΅Π»ΠΎΡΡΠ½ΠΎΡΡΠΈ Π½Π°Β Π΄Π°Π½Π½ΡΠ΅. ΠΠΎΠΏΡΡΠΊΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΡΠΊΠ°Π·Π°Π½Π½ΠΎΠΉ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ ΡΠΎΠ΄Π΅ΡΠΆΠ°Ρ Π½Π΅ΡΠΎΡΠ½ΠΎΡΡΠΈ ΠΊΠ°ΠΊ Π² ΠΏΠΎΡΡΠ°Π½ΠΎΠ²ΠΊΠ΅ Π·Π°Π΄Π°ΡΠΈ,Β ΡΠ°ΠΊ ΠΈ Π² ΡΠ°ΠΌΠΎΠΌ Π΅Π΅ ΡΠ΅ΡΠ΅Π½ΠΈΠΈ. Π ΠΏΠΎΡΡΠ°Π½ΠΎΠ²ΠΎΡΠ½ΡΠΌ ΠΎΡΠΈΠ±ΠΊΠ°ΠΌ ΠΌΠΎΠΆΠ½ΠΎ ΠΎΡΠ½Π΅ΡΡΠΈ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠ΅ Π² ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΠΈ Π½Π΅ΡΠΈΠΏΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΡ
Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ, ΡΡΠΎ ΠΏΡΠΈΠ²ΠΎΠ΄ΠΈΡ ΠΊ ΠΏΠ΅ΡΠ΅ΡΡΠ°Π½ΠΎΠ²ΠΊΠ°ΠΌ Π°ΡΡΠΈΠ±ΡΡΠΎΠ², Ρ
ΠΎΡΡ Π²Β ΡΠ΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΡΡ
Π±Π°Π· Π΄Π°Π½Π½ΡΡ
Π°ΡΡΠΈΠ±ΡΡΡ ΠΈΠ΄Π΅Π½ΡΠΈΡΠΈΡΠΈΡΡΡΡΡΡ ΠΏΠΎ ΠΈΠΌΠ΅Π½ΠΈ, Π° Π½Π΅ ΠΏΠΎ ΠΈΡ
ΠΏΠΎΠ·ΠΈΡΠΈΠΈ. ΠΡΠΎΠΌΠ΅ ΡΠΎΠ³ΠΎ,Β ΡΠ²ΡΠ·ΡΠ²Π°Π½ΠΈΠ΅ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΡΡ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ ΡΠ°Π·Π½ΠΎΡΠΎΠ΄Π½ΡΡ
, ΠΏΡΡΡΡ Π΄Π°ΠΆΠ΅ ΠΎΠ΄Π½ΠΎΡΠΈΠΏΠ½ΡΡ
, Π°ΡΡΠΈΠ±ΡΡΠΎΠ² ΡΠ²Π»ΡΠ΅ΡΡΡΒ ΠΏΡΠΈΠ·Π½Π°ΠΊΠΎΠΌ ΠΏΠΎΡΠ΅ΡΡΠ½Π½ΠΎΠΉ ΡΡΠ½ΠΊΡΠΈΠΎΠ½Π°Π»ΡΠ½ΠΎΠΉ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠΈ ΠΈ ΠΏΡΠΈΠ²ΠΎΠ΄ΠΈΡ ΠΊ Π²Π·Π°ΠΈΠΌΠΎΠ΄Π΅ΠΉΡΡΠ²ΠΈΡ Π½Π΅ΡΡΠΈΠ²ΠΈΠ°Π»ΡΠ½ΡΡ
Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ ΠΈ ΡΡΠ½ΠΊΡΠΈΠΎΠ½Π°Π»ΡΠ½ΡΡ
Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ. ΠΠ°Π²ΠΈΡΠΈΠΌΠΎΡΡΠΈ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ Π΄ΠΎΠ»ΠΆΠ½ΡΒ ΠΎΠΏΡΠ΅Π΄Π΅Π»ΡΡΡ ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π΅Π½Π½ΠΎΠ΅ ΡΠΎΠΎΡΠ½Π΅ΡΠ΅Π½ΠΈΠ΅ ΠΎΠ±ΡΠ΅ΠΊΡΠΎΠ² Π΄ΡΡΠ³ Ρ Π΄ΡΡΠ³ΠΎΠΌ, Π° Π½Π΅ Π·Π½Π°ΡΠ΅Π½ΠΈΠΉ Π°ΡΡΠΈΠ±ΡΡΠΎΠ². ΠΠ΅ΡΠΎΡΠ½ΠΎΡΡΠΈ Π² ΡΠ΅ΡΠ΅Π½ΠΈΠΈ ΡΠΊΠ°Π·Π°Π½Π½ΠΎΠΉ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ ΡΠΎΠ΄Π΅ΡΠΆΠ°ΡΡΡ Π² ΡΠΎΡΠΌΡΠ»ΠΈΡΠΎΠ²ΠΊΠ°Ρ
Π°ΠΊΡΠΈΠΎΠΌ ΠΈ Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²Π΅ ΠΈΡ
Β ΡΠ²ΠΎΠΉΡΡΠ², Π² ΡΠΎΠΌ ΡΠΈΡΠ»Π΅ ΠΏΠΎΠ»Π½ΠΎΡΡ. Π ΡΡΠΎΠΉ ΡΡΠ°ΡΡΠ΅ ΠΏΡΠ΅Π΄Π»Π°Π³Π°Π΅ΡΡΡ ΠΎΡΠΈΠ³ΠΈΠ½Π°Π»ΡΠ½ΠΎΠ΅ ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ ΡΡΠΎΠΉ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ ΡΠΎΠ»ΡΠΊΠΎ Π΄Π»Ρ ΡΠΈΠΏΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΡ
Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ ΠΏΡΠΈ Π½Π°Π»ΠΈΡΠΈΠΈ Π½Π΅ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ
Π·Π½Π°ΡΠ΅Π½ΠΈΠΉ:Β ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Π° ΡΠΈΡΡΠ΅ΠΌΠ° Π°ΠΊΡΠΈΠΎΠΌ, Π΄ΠΎΠΊΠ°Π·Π°Π½Π° Π΅Π΅ ΠΏΠΎΠ»Π½ΠΎΡΠ° ΠΈ Π½Π΅ΠΏΡΠΎΡΠΈΠ²ΠΎΡΠ΅ΡΠΈΠ²ΠΎΡΡΡ. ΠΠ° ΠΎΡΠ½ΠΎΠ²Π΅ ΠΏΡΠ°Π²ΠΈΠ» Π²ΡΠ²ΠΎΠ΄Π° ΡΠ°Π·ΡΠ°Π±ΠΎΡΠ°Π½ Π°Π»Π³ΠΎΡΠΈΡΠΌ ΠΏΠΎΡΡΡΠΎΠ΅Π½ΠΈΡ Π½Π΅ ΠΈΠ·Π±ΡΡΠΎΡΠ½ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²Π° ΡΠΈΠΏΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΡ
Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉΒ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΡ. ΠΠΎΠΊΠ°Π·Π°Π½Π° ΠΊΠΎΡΡΠ΅ΠΊΡΠ½ΠΎΡΡΡ ΡΡΠΎΠ³ΠΎ Π°Π»Π³ΠΎΡΠΈΡΠΌΠ°
Π’Π΅ΡΡΠΈΡΠΎΠ²Π°Π½ΠΈΠ΅ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ ΠΈ ΠΏΡΠ°Π²ΠΈΠ» Π²ΡΠ²ΠΎΠ΄Π° Π² Π±Π°Π·Π°Ρ Π΄Π°Π½Π½ΡΡ
The process of testing dependencies and inference rules can be used in two ways. First, testing allows verification hypotheses about unknown inference rules. The main goal, in this case, is to search for the relation - a counterexample that illustrates the feasibility of the initial dependencies and contradicts the consequence. The found counterexample refutes the hypothesis, the absence of a counterexample allows searching for a generalization of the rule and conditions for its feasibility (logically imply). Testing cannot be used as a proof of the feasibility of inference rules, since the process of generalization requires the search for universal inference conditions for each rule, which cannot be programmed, since even the form of these conditions is unknown. Secondly, when designing a particular database, it may be necessary to test the feasibility of a rule for which there is no theoretical justification. Such a situation can take place in the presence of anomalies in the superkey. The solution to this problem is based on using join dependency inference rules. For these dependencies, a complete system of rules (axioms) has not yet been found. This paper discusses: 1) a technique for testing inference rules using the example of join dependencies, 2) a scheme of a testing algorithm is proposed, 3) some hypotheses are considered for which there are no counterexamples and inference rules, 4) an example of using testing when searching for a correct decomposition of a superkey is proposed.ΠΡΠΎΡΠ΅ΡΡ ΡΠ΅ΡΡΠΈΡΠΎΠ²Π°Π½ΠΈΡ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ ΠΈ ΠΏΡΠ°Π²ΠΈΠ» Π²ΡΠ²ΠΎΠ΄Π° ΠΌΠΎΠΆΠ΅Ρ Π±ΡΡΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ Π² Π΄Π²ΡΡ
Π½Π°ΠΏΡΠ°Π²Π»Π΅Π½ΠΈΡΡ
. ΠΠΎ-ΠΏΠ΅ΡΠ²ΡΡ
, ΡΠ΅ΡΡΠΈΡΠΎΠ²Π°Π½ΠΈΠ΅ ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΠΏΡΠΎΠ²Π΅ΡΠΈΡΡ Π³ΠΈΠΏΠΎΡΠ΅Π·Ρ ΠΎΡΠ½ΠΎΡΠΈΡΠ΅Π»ΡΠ½ΠΎ Π½Π΅ΠΈΠ·Π²Π΅ΡΡΠ½ΡΡ
ΠΏΡΠ°Π²ΠΈΠ» Π²ΡΠ²ΠΎΠ΄Π°. ΠΡΠ½ΠΎΠ²Π½Π°Ρ ΡΠ΅Π»Ρ ΠΏΡΠΈ ΡΡΠΎΠΌ ΠΏΠΎΠΈΡΠΊ ΡΠ΅Π°Π»ΠΈΠ·Π°ΡΠΈΠΈ ΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΡ - ΠΊΠΎΠ½ΡΡΠΏΡΠΈΠΌΠ΅ΡΠ°, ΠΊΠΎΡΠΎΡΡΠΉ ΡΠ΄ΠΎΠ²Π»Π΅ΡΠ²ΠΎΡΡΠ΅Ρ ΠΈΡΡ
ΠΎΠ΄Π½ΡΠΌ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΡΠΌ ΠΈ ΠΏΡΠΎΡΠΈΠ²ΠΎΡΠ΅ΡΠΈΡ ΡΠ»Π΅Π΄ΡΡΠ²ΠΈΡ. ΠΠ°ΠΉΠ΄Π΅Π½Π½ΡΠΉ ΠΊΠΎΠ½ΡΡΠΏΡΠΈΠΌΠ΅Ρ ΠΎΠΏΡΠΎΠ²Π΅ΡΠ³Π°Π΅Ρ Π³ΠΈΠΏΠΎΡΠ΅Π·Ρ, ΠΎΡΡΡΡΡΡΠ²ΠΈΠ΅ ΠΊΠΎΠ½ΡΡΠΏΡΠΈΠΌΠ΅ΡΠ° ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΠΏΡΠΈΡΡΡΠΏΠΈΡΡ ΠΊ ΠΏΠΎΠΈΡΠΊΡ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½ΠΈΡ ΠΏΡΠ°Π²ΠΈΠ»Π° ΠΈ ΡΡΠ»ΠΎΠ²ΠΈΠΉ Π΅Π³ΠΎ Π²ΡΠΏΠΎΠ»Π½ΠΈΠΌΠΎΡΡΠΈ (logically imply). Π’Π΅ΡΡΠΈΡΠΎΠ²Π°Π½ΠΈΠ΅ Π½Π΅ ΠΌΠΎΠΆΠ΅Ρ ΡΠ»ΡΠΆΠΈΡΡ Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²ΠΎΠΌ Π²ΡΠΏΠΎΠ»Π½ΠΈΠΌΠΎΡΡΠΈ ΠΏΡΠ°Π²ΠΈΠ» Π²ΡΠ²ΠΎΠ΄Π°, ΠΏΠΎΡΠΊΠΎΠ»ΡΠΊΡ ΠΏΡΠΎΡΠ΅ΡΡ ΠΎΠ±ΠΎΠ±ΡΠ΅Π½ΠΈΡ ΡΡΠ΅Π±ΡΠ΅Ρ ΠΏΠΎΠΈΡΠΊΠ° ΡΠ½ΠΈΠ²Π΅ΡΡΠ°Π»ΡΠ½ΡΡ
ΡΡΠ»ΠΎΠ²ΠΈΠΉ Π²ΡΠ²ΠΎΠ΄ΠΈΠΌΠΎΡΡΠΈ Π΄Π»Ρ ΠΊΠ°ΠΆΠ΄ΠΎΠ³ΠΎ ΠΏΡΠ°Π²ΠΈΠ»Π°, ΡΡΠΎ Π½Π΅Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎ Π·Π°ΠΏΡΠΎΠ³ΡΠ°ΠΌΠΌΠΈΡΠΎΠ²Π°ΡΡ, ΠΏΠΎΡΠΊΠΎΠ»ΡΠΊΡ Π΄Π°ΠΆΠ΅ Π²ΠΈΠ΄ ΡΡΠΈΡ
ΡΡΠ»ΠΎΠ²ΠΈΠΉ Π½Π΅ΠΈΠ·Π²Π΅ΡΡΠ΅Π½. ΠΠΎ-Π²ΡΠΎΡΡΡ
, ΠΏΡΠΈ ΠΏΡΠΎΠ΅ΠΊΡΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΠΊΠΎΠ½ΠΊΡΠ΅ΡΠ½ΠΎΠΉ Π±Π°Π·Ρ Π΄Π°Π½Π½ΡΡ
ΠΌΠΎΠΆΠ΅Ρ ΠΏΠΎΡΡΠ΅Π±ΠΎΠ²Π°ΡΡΡΡ ΠΏΡΠΎΠ²Π΅ΡΠΊΠ° Π²ΡΠΏΠΎΠ»Π½ΠΈΠΌΠΎΡΡΠΈ ΠΏΡΠ°Π²ΠΈΠ»Π°, Π΄Π»Ρ ΠΊΠΎΡΠΎΡΠΎΠ³ΠΎ ΠΎΡΡΡΡΡΡΠ²ΡΠ΅Ρ ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ ΠΎΠ±ΠΎΡΠ½ΠΎΠ²Π°Π½ΠΈΠ΅. Π’Π°ΠΊΠ°Ρ ΡΠΈΡΡΠ°ΡΠΈΡ ΠΌΠΎΠΆΠ΅Ρ ΠΏΡΠΎΡΠ²ΠΈΡΡΡΡ ΠΏΡΠΈ Π½Π°Π»ΠΈΡΠΈΠΈ Π°Π½ΠΎΠΌΠ°Π»ΠΈΠΉ Π² ΡΡΠΏΠ΅ΡΠΊΠ»ΡΡΠ΅. Π Π΅ΡΠ΅Π½ΠΈΠ΅ ΡΡΠΎΠΉ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ ΠΎΡΠ½ΠΎΠ²ΡΠ²Π°Π΅ΡΡΡ Π½Π° ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠΈ ΠΏΡΠ°Π²ΠΈΠ» Π²ΡΠ²ΠΎΠ΄Π° Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ ΡΠΎΠ΅Π΄ΠΈΠ½Π΅Π½ΠΈΡ. ΠΠ»Ρ ΡΡΠΈΡ
Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ ΠΏΠΎΠΊΠ° Π½Π΅ Π½Π°ΠΉΠ΄Π΅Π½Π° ΠΏΠΎΠ»Π½Π°Ρ ΡΠΈΡΡΠ΅ΠΌΠ° ΠΏΡΠ°Π²ΠΈΠ» (Π°ΠΊΡΠΈΠΎΠΌ). Π Π΄Π°Π½Π½ΠΎΠΉ ΡΡΠ°ΡΡΠ΅ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ: 1) ΠΌΠ΅ΡΠΎΠ΄ΠΈΠΊΠ° ΠΏΡΠΎΠ²Π΅Π΄Π΅Π½ΠΈΡ ΡΠ΅ΡΡΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΠΏΡΠ°Π²ΠΈΠ» Π²ΡΠ²ΠΎΠ΄Π° Π½Π° ΠΏΡΠΈΠΌΠ΅ΡΠ΅ Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΠ΅ΠΉ ΡΠΎΠ΅Π΄ΠΈΠ½Π΅Π½ΠΈΡ, 2) ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Π° ΡΡ
Π΅ΠΌΠ° Π°Π»Π³ΠΎΡΠΈΡΠΌΠ° ΡΠ΅ΡΡΠΈΡΠΎΠ²Π°Π½ΠΈΡ, 3) ΡΠ°ΡΡΠΌΠΎΡΡΠ΅Π½Ρ Π½Π΅ΠΊΠΎΡΠΎΡΡΠ΅ Π³ΠΈΠΏΠΎΡΠ΅Π·Ρ, Π΄Π»Ρ ΠΊΠΎΡΠΎΡΡΡ
ΠΎΡΡΡΡΡΡΠ²ΡΡΡ ΠΊΠΎΠ½ΡΡΠΏΡΠΈΠΌΠ΅ΡΡ ΠΈ ΠΏΡΠ°Π²ΠΈΠ»Π° Π²ΡΠ²ΠΎΠ΄Π°, 4) ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½ ΠΏΡΠΈΠΌΠ΅Ρ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΡ ΡΠ΅ΡΡΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΠΏΡΠΈ ΠΏΠΎΠΈΡΠΊΠ΅ ΠΊΠΎΡΡΠ΅ΠΊΡΠ½ΠΎΠΉ Π΄Π΅ΠΊΠΎΠΌΠΏΠΎΠ·ΠΈΡΠΈΠΈ ΡΡΠΏΠ΅ΡΠΊΠ»ΡΡΠ°