Integrity Constraints Revisited: From Exact to Approximate Implication

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

Integrity Constraints Revisited: From Exact to Approximate Implication

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

Optimized Framework based on Rough Set Theory for Big Data Pre-processing in Certain and Imprecise Contexts” -- Marie Sklodowska-Curie Project: Open Problems’

Peer reviewe

The Budget-Constrained Functional Dependency

Armstrong's axioms of functional dependency form a well-known logical system that captures properties of functional dependencies between sets of database attributes. This article assumes that there are costs associated with attributes and proposes an extension of Armstrong's system for reasoning about budget-constrained functional dependencies in such a setting. The main technical result of this article is the completeness theorem for the proposed logical system. Although the proposed axioms are obtained by just adding cost subscript to the original Armstrong's axioms, the proof of the completeness for the proposed system is significantly more complicated than that for the Armstrong's system

Fuzzy Logic

Fuzzy Logic is becoming an essential method of solving problems in all domains. It gives tremendous impact on the design of autonomous intelligent systems. The purpose of this book is to introduce Hybrid Algorithms, Techniques, and Implementations of Fuzzy Logic. The book consists of thirteen chapters highlighting models and principles of fuzzy logic and issues on its techniques and implementations. The intended readers of this book are engineers, researchers, and graduate students interested in fuzzy logic systems

The Doxastic Interpretation of Team Semantics

We advance a doxastic interpretation for many of the logical connectives considered in Dependence Logic and in its extensions, and we argue that Team Semantics is a natural framework for reasoning about beliefs and belief updates

Proceedings of SAT Competition 2021 : Solver and Benchmark Descriptions

Non peer reviewe

A Bunched Logic for Conditional Independence

Independence and conditional independence are fundamental concepts for reasoning about groups of random variables in probabilistic programs. Verification methods for independence are still nascent, and existing methods cannot handle conditional independence. We extend the logic of bunched implications (BI) with a non-commutative conjunction and provide a model based on Markov kernels; conditional independence can be directly captured as a logical formula in this model. Noting that Markov kernels are Kleisli arrows for the distribution monad, we then introduce a second model based on the powerset monad and show how it can capture join dependency, a non-probabilistic analogue of conditional independence from database theory. Finally, we develop a program logic for verifying conditional independence in probabilistic programs

Mitigating Insider Threat in Relational Database Systems

The dissertation concentrates on addressing the factors and capabilities that enable insiders to violate systems security. It focuses on modeling the accumulative knowledge that insiders get throughout legal accesses, and it concentrates on analyzing the dependencies and constraints among data items and represents them using graph-based methods. The dissertation proposes new types of Knowledge Graphs (KGs) to represent insiders\u27 knowledgebases. Furthermore, it introduces the Neural Dependency and Inference Graph (NDIG) and Constraints and Dependencies Graph (CDG) to demonstrate the dependencies and constraints among data items. The dissertation discusses in detail how insiders use knowledgebases and dependencies and constraints to get unauthorized knowledge. It suggests new approaches to predict and prevent the aforementioned threat. The proposed models use KGs, NDIG and CDG in analyzing the threat status, and leverage the effect of updates on the lifetimes of data items in insiders\u27 knowledgebases to prevent the threat without affecting the availability of data items. Furthermore, the dissertation uses the aforementioned idea in ordering the operations of concurrent tasks such that write operations that update risky data items in knowledgebases are executed before the risky data items can be used in unauthorized inferences. In addition to unauthorized knowledge, the dissertation discusses how insiders can make unauthorized modifications in sensitive data items. It introduces new approaches to build Modification Graphs that demonstrate the authorized and unauthorized data items which insiders are able to update. To prevent this threat, the dissertation provides two methods, which are hiding sensitive dependencies and denying risky write requests. In addition to traditional RDBMS, the dissertation investigates insider threat in cloud relational database systems (cloud RDMS). It discusses the vulnerabilities in the cloud computing structure that may enable insiders to launch attacks. To prevent such threats, the dissertation suggests three models and addresses the advantages and limitations of each one. To prove the correctness and the effectiveness of the proposed approaches, the dissertation uses well stated algorithms, theorems, proofs and simulations. The simulations have been executed according to various parameters that represent the different conditions and environments of executing tasks

Learning Markov Random Fields for Combinatorial Structures via Sampling through Lov\'asz Local Lemma

Learning to generate complex combinatorial structures satisfying constraints will have transformative impacts in many application domains. However, it is beyond the capabilities of existing approaches due to the highly intractable nature of the embedded probabilistic inference. Prior works spend most of the training time learning to separate valid from invalid structures but do not learn the inductive biases of valid structures. We develop NEural Lov\'asz Sampler (Nelson), which embeds the sampler through Lov\'asz Local Lemma (LLL) as a fully differentiable neural network layer. Our Nelson-CD embeds this sampler into the contrastive divergence learning process of Markov random fields. Nelson allows us to obtain valid samples from the current model distribution. Contrastive divergence is then applied to separate these samples from those in the training set. Nelson is implemented as a fully differentiable neural net, taking advantage of the parallelism of GPUs. Experimental results on several real-world domains reveal that Nelson learns to generate 100\% valid structures, while baselines either time out or cannot ensure validity. Nelson also outperforms other approaches in running time, log-likelihood, and MAP scores.Comment: accepted by AAAI 2023. The first two authors contribute equall