286 research outputs found

    Integrity Constraints Revisited: From Exact to Approximate Implication

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    Integrity constraints such as functional dependencies (FD), and multi-valued dependencies (MVD) are fundamental in database schema design. Likewise, probabilistic conditional independences (CI) are crucial for reasoning about multivariate probability distributions. The implication problem studies whether a set of constraints (antecedents) implies another constraint (consequent), and has been investigated in both the database and the AI literature, under the assumption that all constraints hold exactly. However, many applications today consider constraints that hold only approximately. In this paper we define an approximate implication as a linear inequality between the degree of satisfaction of the antecedents and consequent, and we study the relaxation problem: when does an exact implication relax to an approximate implication? We use information theory to define the degree of satisfaction, and prove several results. First, we show that any implication from a set of data dependencies (MVDs+FDs) can be relaxed to a simple linear inequality with a factor at most quadratic in the number of variables; when the consequent is an FD, the factor can be reduced to 1. Second, we prove that there exists an implication between CIs that does not admit any relaxation; however, we prove that every implication between CIs relaxes "in the limit". Finally, we show that the implication problem for differential constraints in market basket analysis also admits a relaxation with a factor equal to 1. Our results recover, and sometimes extend, several previously known results about the implication problem: implication of MVDs can be checked by considering only 2-tuple relations, and the implication of differential constraints for frequent item sets can be checked by considering only databases containing a single transaction

    Incorporating record subtyping into a relational data model

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    Most of the current proposals for new data models support the construction of heterogeneous sets. One of the major challenges for such data models is to provide strong typing in the presence of heterogenity. Therefore the inclusion of as much as possible information concerning legal structural variants is needed. We argue that the shape of some part of a heterogeneous scheme is often determined by the contents of some other part of the scheme. This relationship can be formalized by a certain type of integrity constraint we have called attribute dependency. Attribute dependencies combine the expressive power of general sums with a notation that fits into relational models. We show that attribute dependencies can be used, besides their application in type and integrity checking, to incorporate record subtyping into a relational model. Moreover, the notion of attribute dependency yields a stronger assertion than the traditional record subtyping rule as it considers some refinements to be caused by others. To examine the differences between attribute dependencies and traditional record subtyping and to be able to predict how attribute dependencies behave under transformations like query language operations we develop an axiom system for their derivation and prove it to be sound and complete. We further investigate the interaction between functional and attribute dependencies and examine an extended axiom system capturing both forms of dependencies

    Acta Cybernetica : Volume 12. Number 2.

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    Record Subtyping in Flexible Relations by means of Attribute Dependencies

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    The model of flexible relations supports heterogeneous sets of tuples in a strongly typed way. The elegance of the standard relational model is preserved by using a single, generic scheme constructor.In each model supporting structural variants the shape of some part of a heterogeneous scheme may be determined by the contents of some other part of the scheme. We formalize this relationship by a certain kind of integrity constraint we have called "attribute dependency" (AD). We motivate how ADs can be used, besides their application in type and integrity checking, to incorporate record subtyping into our extended relational model Moreover, we show that ADs yield a stronger assertion than the traditional record subtyping rule as they consider interdependencies among refinements. We discuss how ADs are related to query processing and how they may help to identify redundant operations

    Measure-Based Inconsistency-Tolerant Maintenance of Database Integrity

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    [EN] To maintain integrity, constraint violations should be prevented or repaired. However, it may not be feasible to avoid inconsistency, or to repair all violations at once. Based on an abstract concept of violation measures, updates and repairs can be checked for keeping inconsistency bounded, such that integrity violations are guaranteed to never get out of control. This measure-based approach goes beyond conventional methods that are not meant to be applied in the presence of inconsistency. It also generalizes recently introduced concepts of inconsistency-tolerant integrity maintenance.Partially supported by FEDER and the Spanish grants TIN2009-14460-C03 and TIN2010-17139Decker, H. (2013). Measure-Based Inconsistency-Tolerant Maintenance of Database Integrity. Lecture Notes in Computer Science. 7693:149-173. https://doi.org/10.1007/978-3-642-36008-4_7S1491737693Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. 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Plenum Press (1978)Curino, C., Moon, H., Deutsch, A., Zaniolo, C.: Update Rewriting and Integrity Constraint Maintenance in a Schema Evolution Support System: PRISM++. PVLDBΒ 4, 117–128 (2010)Dawson, J.: The compactness of first-order logic: From GΓΆdel to LindstrΓΆm. History and Philosophy of LogicΒ 14(1), 15–37 (1993)Decker, H.: The Range Form of Databases and Queries or: How to Avoid Floundering. In: Proc. 5th Γ–GAI. Informatik-Fachberichte, vol.Β 208, pp. 114–123. Springer (1989)Decker, H.: Drawing Updates From Derivations. In: Kanellakis, P.C., Abiteboul, S. (eds.) ICDT 1990. LNCS, vol.Β 470, pp. 437–451. Springer, Heidelberg (1990)Decker, H.: Extending Inconsistency-Tolerant Integrity Checking by Semantic Query Optimization. In: Bhowmick, S.S., KΓΌng, J., Wagner, R. (eds.) DEXA 2008. LNCS, vol.Β 5181, pp. 89–96. Springer, Heidelberg (2008)Decker, H.: Answers That Have Integrity. In: Schewe, K.-D., Thalheim, B. (eds.) SDKB 2010. LNCS, vol.Β 6834, pp. 54–72. Springer, Heidelberg (2011)Decker, H.: Causes of the Violation of Integrity Constraints for Supporting the Quality of Databases. In: Murgante, B., Gervasi, O., Iglesias, A., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2011, Part V. LNCS, vol.Β 6786, pp. 283–292. Springer, Heidelberg (2011)Decker, H.: Inconsistency-tolerant Integrity Checking based on Inconsistency Metrics. In: KΓΆnig, A., Dengel, A., Hinkelmann, K., Kise, K., Howlett, R.J., Jain, L.C. (eds.) KES 2011, Part II. LNCS, vol.Β 6882, pp. 548–558. Springer, Heidelberg (2011)Decker, H.: Partial Repairs that Tolerate Inconsistency. In: Eder, J., Bielikova, M., Tjoa, A.M. (eds.) ADBIS 2011. LNCS, vol.Β 6909, pp. 389–400. Springer, Heidelberg (2011)Decker, H.: Consistent Explanations of Answers to Queries in Inconsistent Knowledge Bases. In: Roth-Berghofer, T., Tintarev, N., Leake, D. (eds.) Explanation-aware Computing, Proc. IJCAI 2011 Workshop ExaCt 2011, pp. 71–80 (2011), http://exact2011.workshop.hm/index.phpDecker, H., Martinenghi, D.: Classifying integrity checking methods with regard to inconsistency tolerance. In: Proc. PPDP 2008, pp. 195–204. ACM Press (2008)Decker, H., Martinenghi, D.: Modeling, Measuring and Monitoring the Quality of Information. In: Heuser, C.A., Pernul, G. (eds.) ER 2009. LNCS, vol.Β 5833, pp. 212–221. Springer, Heidelberg (2009)Decker, H., Martinenghi, D.: Inconsistency-tolerant Integrity Checking. IEEE TKDEΒ 23(2), 218–234 (2011)Decker, H., MuΓ±oz-EscoΓ­, F.D.: Revisiting and Improving a Result on Integrity Preservation by Concurrent Transactions. In: Meersman, R., Dillon, T., Herrero, P. (eds.) OTM 2010 Workshops. LNCS, vol.Β 6428, pp. 297–306. Springer, Heidelberg (2010)Dung, P., Kowalski, R., Toni, F.: Dialectic Proof Procedures for Assumption-based Admissible Argumentation. Artificial IntelligenceΒ 170(2), 114–159 (2006)Ebbinghaus, H.-D., Flum, J.: Finite Model Theory, 2nd edn. Springer (2006)Embury, S., Brandt, S., Robinson, J., Sutherland, I., Bisby, F., Gray, A., Jones, A., White, R.: Adapting integrity enforcement techniques for data reconciliation. Information SystemsΒ 26, 657–689 (2001)Enderton, H.: A Mathematical Introduction to Logic, 2nd edn. Academic Press (2001)Eiter, T., Fink, M., Greco, G., Lembo, D.: Repair localization for query answering from inconsistent databases. ACM TODS 33(2), article 10 (2008)Furfaro, F., Greco, S., Molinaro, C.: A three-valued semantics for querying and repairing inconsistent databases. Ann. Math. Artif. Intell.Β 51(2-4), 167–193 (2007)Grant, J., Hunter, A.: Measuring the Good and the Bad in Inconsistent Information. In: Proc. 22nd IJCAI, pp. 2632–2637 (2011)Greco, G., Greco, S., Zumpano, E.: A logical framework for querying and repairing inconsistent databases. 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AAAI Press (2008)Hunter, A., Konieczny, S.: On the measure of conflicts: Shapley Inconsistency Values. Artificial IntelligenceΒ 174, 1007–1026 (2010)Kakas, A., Mancarella, P.: Database updates through abduction. In: Proc. 16th VLDB, pp. 650–661. Morgan Kaufmann (1990)Kakas, A., Kowalski, R., Toni, F.: The role of Abduction in Logic Programming. In: Gabbay, D., Hogger, C., Robinson, J.A. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, vol.Β 5, pp. 235–324. Oxford University Press (1998)Lee, S.Y., Ling, T.W.: Further improvements on integrity constraint checking for stratifiable deductive databases. In: Proc. VLDB 1996, pp. 495–505. Morgan Kaufmann (1996)Lehrer, K.: Relevant Deduction and Minimally Inconsistent Sets. Journal of PhilosophyΒ 3(2,3), 153–165 (1973)Mu, K., Liu, W., Jin, Z., Bell, D.: A Syntax-based Approach to Measuring the Degree of Inconsistency for Belief Bases. J. Approx. ReasoningΒ 52(7), 978–999 (2011)Lloyd, J., Sonenberg, L., Topor, R.: Integrity constraint checking in stratified databases. J. Logic ProgrammingΒ 4(4), 331–343 (1987)Lozinskii, E.: Resolving contradictions: A plausible semantics for inconsistent systems. J. Automated ReasoningΒ 12(1), 1–31 (1994)Ma, Y., Qi, G., Hitzler, P.: Computing inconsistency measure based on paraconsistent semantics. J. Logic ComputationΒ 21(6), 1257–1281 (2011)Martinenghi, D., Christiansen, H.: Transaction Management with Integrity Checking. In: Andersen, K.V., Debenham, J., Wagner, R. (eds.) DEXA 2005. LNCS, vol.Β 3588, pp. 606–615. Springer, Heidelberg (2005)Martinenghi, D., Christiansen, H., Decker, H.: Integrity Checking and Maintenance in Relational and Deductive Databases and Beyond. In: Ma, Z. (ed.) Intelligent Databases: Technologies and Applications, pp. 238–285. IGI Global (2006)Martinez, M.V., Pugliese, A., Simari, G.I., Subrahmanian, V.S., Prade, H.: How Dirty Is Your Relational Database? An Axiomatic Approach. In: Mellouli, K. (ed.) ECSQARU 2007. LNCS (LNAI), vol.Β 4724, pp. 103–114. Springer, Heidelberg (2007)Meyer, J., Wieringa, R. (eds.): Deontic Logic in Computer Science. Wiley (1994)Nicolas, J.M.: Logic for improving integrity checking in relational data bases. Acta InformaticaΒ 18, 227–253 (1982)Plexousakis, D., Mylopoulos, J.: Accommodating Integrity Constraints During Database Design. In: Apers, P.M.G., Bouzeghoub, M., Gardarin, G. (eds.) EDBT 1996. LNCS, vol.Β 1057, pp. 495–513. Springer, Heidelberg (1996)Rahm, E., Do, H.: Data Cleaning: Problems and Current Approaches. Data Engineering BulletinΒ 23(4), 3–13 (2000)Sadri, F., Kowalski, R.: A theorem-proving approach to database integrity. In: Minker, J. (ed.) Foundations of Deductive Databases and Logic Programming, pp. 313–362. Morgan Kaufmann (1988)Thimm, M.: Measuring Inconsistency in Probabilistic Knowledge Bases. In: Proc. 25th UAI, pp. 530–537. 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    Augmented Post Systems: Syntax, Semantics, and Applications

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    Augmented Post systems (APS) are string-operating Prolog-like knowledge representation, affiliated with the β€œSet of Strings” Framework (SSF). APS descriptive and logical inference capabilities provide natural integration of Big Data with online analytic processing. This chapter is dedicated to strict formal definition of APS syntax, mathematical and operational semantics, and to its most valuable implementational issues, as well as to APS application to Big Data, Internet of Things, cyberphysical industry, and cybersecurity areas

    ΠžΠ±ΠΎΠ±Ρ‰Π΅Π½Π½Ρ‹Π΅ Ρ‚ΠΈΠΏΠΈΠ·ΠΈΡ€ΠΎΠ²Π°Π½Π½Ρ‹Π΅ зависимости Π²ΠΊΠ»ΡŽΡ‡Π΅Π½ΠΈΡ с Π½Π΅ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½Π½Ρ‹ΠΌΠΈ значСниями Π² Π±Π°Π·Π°Ρ… Π΄Π°Π½Π½Ρ‹Ρ…

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    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’ ΡΡ‚Π°Ρ‚ΡŒΠ΅ рассматриваСтся Π½ΠΎΠ²Ρ‹ΠΉ Π²ΠΈΠ΄ зависимостСй Π² Π±Π°Π·Π°Ρ… Π΄Π°Π½Π½Ρ‹Ρ…, ΡΠ²Π»ΡΡŽΡ‰ΠΈΠΉΡΡ ΠΎΠ±ΠΎΠ±Ρ‰Π΅Π½ΠΈΠ΅ΠΌ зависимостСй Π²ΠΊΠ»ΡŽΡ‡Π΅Π½ΠΈΡ. Π’Ρ€Π°Π΄ΠΈΡ†ΠΈΠΎΠ½Π½ΠΎ Ρ‚Π°ΠΊΠΈΠ΅ зависимости Π½Π° ΠΏΡ€Π°ΠΊΡ‚ΠΈΠΊΠ΅ ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΡƒΡŽΡ‚ΡΡ для обСспСчСния ссылочной цСлостности. ΠŸΡ€ΠΈ этом, ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½ΠΈΠ΅ устанавливаСтся Ρ‚ΠΎΠ»ΡŒΠΊΠΎ ΠΌΠ΅ΠΆΠ΄Ρƒ ΠΏΠ°Ρ€ΠΎΠΉ ΠΎΡ‚Π½ΠΎΡˆΠ΅Π½ΠΈΠΉ, ΠΏΠ΅Ρ€Π²ΠΎΠ΅ ΠΈΠ· ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Ρ… называСтся Π³Π»Π°Π²Π½Ρ‹ΠΌ, Π²Ρ‚ΠΎΡ€ΠΎΠ΅ β€” внСшним. На ΠΏΡ€Π°ΠΊΡ‚ΠΈΠΊΠ΅ ΡΡΡ‹Π»ΠΎΡ‡Π½ΡƒΡŽ Ρ†Π΅Π»ΠΎΡΡ‚Π½ΠΎΡΡ‚ΡŒ часто трСбуСтся ΡƒΡΡ‚Π°Π½ΠΎΠ²ΠΈΡ‚ΡŒ для большСго числа ΠΎΡ‚Π½ΠΎΡˆΠ΅Π½ΠΈΠΉ, Π³Π΄Π΅ Π² ΠΎΠ΄Π½ΠΎΠΌ ΠΎΠ³Ρ€Π°Π½ΠΈΡ‡Π΅Π½ΠΈΠΈ ΡƒΡ‡Π°ΡΡ‚Π²ΡƒΡŽΡ‚ нСсколько Π³Π»Π°Π²Π½Ρ‹Ρ… ΠΈ нСсколько ΠΏΠΎΠ΄Ρ‡ΠΈΠ½Π΅Π½Π½Ρ‹Ρ… ΠΎΡ‚Π½ΠΎΡˆΠ΅Π½ΠΈΠΉ. Вакая структура соотвСтствуСт ΡƒΠ»ΡŒΡ‚Ρ€Π°Π³Ρ€Π°Ρ„Ρƒ. Π’ Ρ€Π°Π±ΠΎΡ‚Π΅ ΠΏΡ€ΠΈΠ²Π΅Π΄Π΅Π½ΠΎ обоснованиС ΠΎΠ±ΠΎΠ±Ρ‰Π΅Π½Π½Ρ‹Ρ… зависимостСй Π²ΠΊΠ»ΡŽΡ‡Π΅Π½ΠΈΡ, ΡƒΡ‡ΠΈΡ‚Ρ‹Π²Π°ΡŽΡ‰ΠΈΡ… Π½Π°Π»ΠΈΡ‡ΠΈΠ΅ Π½Π΅ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½Π½Ρ‹Ρ… Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ Π²ΠΎ Π²Π½Π΅ΡˆΠ½ΠΈΡ… ΠΎΡ‚Π½ΠΎΡˆΠ΅Π½ΠΈΡΡ…. На основС исслСдования свойств Ρ‚ΠΈΠΏΠΈΠ·ΠΈΡ€ΠΎΠ²Π°Π½Π½Ρ‹Ρ… зависимостСй ΠΏΠΎΠ»ΡƒΡ‡Π΅Π½Π° систСма аксиом, для ΠΊΠΎΡ‚ΠΎΡ€ΠΎΠΉ Π΄ΠΎΠΊΠ°Π·Π°Π½Π° Π½Π΅ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΡ€Π΅Ρ‡ΠΈΠ²ΠΎΡΡ‚ΡŒ (Π½Π°Π΄Π΅ΠΆΠ½ΠΎΡΡ‚ΡŒ) ΠΈ ΠΏΠΎΠ»Π½ΠΎΡ‚Π°

    Анализ Ρ‚ΠΈΠΏΠΈΠ·ΠΈΡ€ΠΎΠ²Π°Π½Π½Ρ‹Ρ… зависимостСй Π²ΠΊΠ»ΡŽΡ‡Π΅Π½ΠΈΡ с Π½Π΅ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½Π½Ρ‹ΠΌΠΈ значСниями

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    Null values have become an urgent problem since the creation of the relational dataΒ model. The impact of the uncertainty affects 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 definition of untyped inclusion dependencies, which leads to permutations of the attributes,Β although, the attributes in database technology are identified 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.НСопрСдСлСнныС значСния стали Π°ΠΊΡ‚ΡƒΠ°Π»ΡŒΠ½ΠΎΠΉ ΠΏΡ€ΠΎΠ±Π»Π΅ΠΌΠΎΠΉ с ΠΌΠΎΠΌΠ΅Π½Ρ‚Π° создания рСляционной ΠΌΠΎΠ΄Π΅Π»ΠΈ Π΄Π°Π½Π½Ρ‹Ρ…. ВлияниС нСопрСдСлСнностСй сказываСтся Π½Π° всСх Π²ΠΈΠ΄Π°Ρ… зависимостСй,Β ΠΈΡΠΏΠΎΠ»ΡŒΠ·ΡƒΠ΅ΠΌΡ‹Ρ… ΠΏΡ€ΠΈ ΠΏΡ€ΠΎΠ΅ΠΊΡ‚ΠΈΡ€ΠΎΠ²Π°Π½ΠΈΠΈ ΠΈ эксплуатации Π±Π°Π·Ρ‹ Π΄Π°Π½Π½Ρ‹Ρ…. Π’ ΠΏΠΎΠ»Π½ΠΎΠΉ ΠΌΠ΅Ρ€Π΅ это относится ΠΈΒ ΠΊ зависимостям Π²ΠΊΠ»ΡŽΡ‡Π΅Π½ΠΈΡ, ΠΊΠΎΡ‚ΠΎΡ€Ρ‹Π΅ ΡΠ²Π»ΡΡŽΡ‚ΡΡ тСорСтичСской основой ссылочной цСлостности Π½Π°Β Π΄Π°Π½Π½Ρ‹Π΅. ΠŸΠΎΠΏΡ‹Ρ‚ΠΊΠΈ Ρ€Π΅ΡˆΠ΅Π½ΠΈΡ ΡƒΠΊΠ°Π·Π°Π½Π½ΠΎΠΉ ΠΏΡ€ΠΎΠ±Π»Π΅ΠΌΡ‹ содСрТат нСточности ΠΊΠ°ΠΊ Π² постановкС Π·Π°Π΄Π°Ρ‡ΠΈ,Β Ρ‚Π°ΠΊ ΠΈ Π² самом Π΅Π΅ Ρ€Π΅ΡˆΠ΅Π½ΠΈΠΈ. К постановочным ошибкам ΠΌΠΎΠΆΠ½ΠΎ отнСсти использованиС Π² ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½ΠΈΠΈ Π½Π΅Ρ‚ΠΈΠΏΠΈΠ·ΠΈΡ€ΠΎΠ²Π°Π½Π½Ρ‹Ρ… зависимостСй Π²ΠΊΠ»ΡŽΡ‡Π΅Π½ΠΈΡ, Ρ‡Ρ‚ΠΎ ΠΏΡ€ΠΈΠ²ΠΎΠ΄ΠΈΡ‚ ΠΊ пСрСстановкам Π°Ρ‚Ρ€ΠΈΠ±ΡƒΡ‚ΠΎΠ², хотя в тСхнологиях Π±Π°Π· Π΄Π°Π½Π½Ρ‹Ρ… Π°Ρ‚Ρ€ΠΈΠ±ΡƒΡ‚Ρ‹ ΠΈΠ΄Π΅Π½Ρ‚ΠΈΡ„ΠΈΡ†ΠΈΡ€ΡƒΡŽΡ‚ΡΡ ΠΏΠΎ ΠΈΠΌΠ΅Π½ΠΈ, Π° Π½Π΅ ΠΏΠΎ ΠΈΡ… ΠΏΠΎΠ·ΠΈΡ†ΠΈΠΈ. ΠšΡ€ΠΎΠΌΠ΅ Ρ‚ΠΎΠ³ΠΎ, связываниС Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡ‚ΡŒΡŽ Π²ΠΊΠ»ΡŽΡ‡Π΅Π½ΠΈΡ Ρ€Π°Π·Π½ΠΎΡ€ΠΎΠ΄Π½Ρ‹Ρ…, ΠΏΡƒΡΡ‚ΡŒ Π΄Π°ΠΆΠ΅ ΠΎΠ΄Π½ΠΎΡ‚ΠΈΠΏΠ½Ρ‹Ρ…, Π°Ρ‚Ρ€ΠΈΠ±ΡƒΡ‚ΠΎΠ² являСтся признаком потСрянной Ρ„ΡƒΠ½ΠΊΡ†ΠΈΠΎΠ½Π°Π»ΡŒΠ½ΠΎΠΉ зависимости ΠΈ ΠΏΡ€ΠΈΠ²ΠΎΠ΄ΠΈΡ‚ ΠΊ Π²Π·Π°ΠΈΠΌΠΎΠ΄Π΅ΠΉΡΡ‚Π²ΠΈΡŽ Π½Π΅Ρ‚Ρ€ΠΈΠ²ΠΈΠ°Π»ΡŒΠ½Ρ‹Ρ… зависимостСй Π²ΠΊΠ»ΡŽΡ‡Π΅Π½ΠΈΡ ΠΈ Ρ„ΡƒΠ½ΠΊΡ†ΠΈΠΎΠ½Π°Π»ΡŒΠ½Ρ‹Ρ… зависимостСй. Зависимости Π²ΠΊΠ»ΡŽΡ‡Π΅Π½ΠΈΡ Π΄ΠΎΠ»ΠΆΠ½Ρ‹Β ΠΎΠΏΡ€Π΅Π΄Π΅Π»ΡΡ‚ΡŒ количСствСнноС соотнСсСниС ΠΎΠ±ΡŠΠ΅ΠΊΡ‚ΠΎΠ² Π΄Ρ€ΡƒΠ³ с Π΄Ρ€ΡƒΠ³ΠΎΠΌ, Π° Π½Π΅ Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ Π°Ρ‚Ρ€ΠΈΠ±ΡƒΡ‚ΠΎΠ². НСточности Π² Ρ€Π΅ΡˆΠ΅Π½ΠΈΠΈ ΡƒΠΊΠ°Π·Π°Π½Π½ΠΎΠΉ ΠΏΡ€ΠΎΠ±Π»Π΅ΠΌΡ‹ содСрТатся Π² Ρ„ΠΎΡ€ΠΌΡƒΠ»ΠΈΡ€ΠΎΠ²ΠΊΠ°Ρ… аксиом ΠΈ Π΄ΠΎΠΊΠ°Π·Π°Ρ‚Π΅Π»ΡŒΡΡ‚Π²Π΅ их свойств, Π² Ρ‚ΠΎΠΌ числС ΠΏΠΎΠ»Π½ΠΎΡ‚Ρ‹. Π’ этой ΡΡ‚Π°Ρ‚ΡŒΠ΅ прСдлагаСтся ΠΎΡ€ΠΈΠ³ΠΈΠ½Π°Π»ΡŒΠ½ΠΎΠ΅ Ρ€Π΅ΡˆΠ΅Π½ΠΈΠ΅ этой ΠΏΡ€ΠΎΠ±Π»Π΅ΠΌΡ‹ Ρ‚ΠΎΠ»ΡŒΠΊΠΎ для Ρ‚ΠΈΠΏΠΈΠ·ΠΈΡ€ΠΎΠ²Π°Π½Π½Ρ‹Ρ… зависимостСй Π²ΠΊΠ»ΡŽΡ‡Π΅Π½ΠΈΡ ΠΏΡ€ΠΈ Π½Π°Π»ΠΈΡ‡ΠΈΠΈ Π½Π΅ΠΎΠΏΡ€Π΅Π΄Π΅Π»Π΅Π½Π½Ρ‹Ρ… Π·Π½Π°Ρ‡Π΅Π½ΠΈΠΉ:Β ΠΏΡ€Π΅Π΄Π»ΠΎΠΆΠ΅Π½Π° систСма аксиом, Π΄ΠΎΠΊΠ°Π·Π°Π½Π° Π΅Π΅ ΠΏΠΎΠ»Π½ΠΎΡ‚Π° ΠΈ Π½Π΅ΠΏΡ€ΠΎΡ‚ΠΈΠ²ΠΎΡ€Π΅Ρ‡ΠΈΠ²ΠΎΡΡ‚ΡŒ. На основС ΠΏΡ€Π°Π²ΠΈΠ» Π²Ρ‹Π²ΠΎΠ΄Π° Ρ€Π°Π·Ρ€Π°Π±ΠΎΡ‚Π°Π½ Π°Π»Π³ΠΎΡ€ΠΈΡ‚ΠΌ построСния Π½Π΅ ΠΈΠ·Π±Ρ‹Ρ‚ΠΎΡ‡Π½ΠΎΠ³ΠΎ мноТСства Ρ‚ΠΈΠΏΠΈΠ·ΠΈΡ€ΠΎΠ²Π°Π½Π½Ρ‹Ρ… Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡ‚Π΅ΠΉΒ Π²ΠΊΠ»ΡŽΡ‡Π΅Π½ΠΈΡ. Π”ΠΎΠΊΠ°Π·Π°Π½Π° ΠΊΠΎΡ€Ρ€Π΅ΠΊΡ‚Π½ΠΎΡΡ‚ΡŒ этого Π°Π»Π³ΠΎΡ€ΠΈΡ‚ΠΌΠ°
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