Question

Why can we not guarantee that BCNF relation schemas will be produced by dependency-preserving decompositions of non-BCNF relation schemas? Give a counterexample to illustrate this point.

Short answer

Preserving dependencies during decomposition might not always result in BCNF schemas because the process of normalization targeting to remove redundancy might inadvertently eliminate functional dependencies. In the provided counterexample, the schema (A,B,C) with functional dependencies {A → B, B → C} is decomposed into two BCNF schemas (A,B) and (B,C). Nonetheless, the functional dependency A → C that was originally implied is no longer captured within the decomposed schemas, hence not all dependencies are preserved.

Step-by-step solution

Understand BCNF

Before finding a counterexample, it is important to comprehend the BCNF definition. A relation schema R is in Boyce-Codd Normal Form (BCNF) if for every non-trivial functional dependency X → Y that holds over R, the dependency is a superkey.

Define Dependency-preserving decomposition

A decomposition of R into R1 and R2 is a dependency-preserving decomposition if the set of functional dependencies F for R is equivalent to the union of the sets of functional dependencies F1 for R1 and F2 for R2.

Counterexample

Consider a schema R = (A, B, C) with dependencies F = {A → B, B → C}. This schema is not in BCNF because B → C is a non-trivial functional dependency, and B is not a superkey. Decomposing R into R1 = (A, B) and R2 = (B, C) produces BCNF schemas as B is a superkey in R2, and the only non-trivial functional dependencies left are superkeys. However, this decomposition is not dependency preserving as the dependency A → C, implied by F, is not implied by the dependencies of R1 and R2.

Key concepts

Boyce-Codd Normal Form

When organizing data within a relational database, structure optimization is key for ensuring both efficiency and integrity. The Boyce-Codd Normal Form (BCNF) is a level of database normalization used to reduce redundancy and dependency issues. Imagine a giant library with millions of books. In a well-organized library, every single book is uniquely identifiable by its location or a reference code; similarly, in BCNF, for every non-trivial functional dependency, like a unique reference that points to a specific information (X → Y), the element X should be a 'superkey' - a unique identifier for the row in a table. A superkey is like having a unique badge number for each employee in a big corporation, which ensures no mix-ups in identity.

For a relational schema to meet BCNF, it requires that all determinants - the data pieces on the left-hand side of the functional dependency - are candidates for unique identification. This means you can't have partial identifications; it's an all or nothing scenario. BCNF is crucial for eliminating potential anomalies during data operation processes like insertion, updating, and deletion, which ensures the data's efficiency and consistency.

Functional Dependency

The concept of functional dependency is foundational to understanding the structural relationships within database schemas. In simple terms, a functional dependency is when one piece of data (called the determinant) uniquely determines another piece of data within a database. Take, for instance, a fast-food menu: if you know the item name, 'Cheeseburger', this uniquely determines the item price. This can be expressed as 'Item Name → Price'.

The nature of this relationship is crucial for designing databases since it helps dictate how data should be structured for minimizing redundancy and ensuring data integrity. In our fast-food example, knowing just how menu items lead to preset prices, the relationship is clear – one item name will not lead to multiple prices. Keeping data points that are dependent on each other within the same table helps maintain informational integrity, but if not managed properly, could also introduce unneeded data repetition, which is why understanding functional dependencies is vital before proceeding with normalization.

Dependency-Preserving Decomposition

Taking apart a database schema into smaller, more manageable pieces without losing important information is what dependency-preserving decomposition is all about. Similar to disassembling a complex Lego structure into smaller sections for easy transport without losing any instructions for reassembly, decomposing a database schema involves breaking down tables into smaller, non-redundant tables. The goal is to maintain all the original functional dependencies - the 'instructions' - across the smaller tables.

The real challenge is making sure that when we join these decomposed tables back together, they don't lose any dependencies from the original, larger table. However, preserving dependencies while also achieving BCNF is not always possible. Like dumping a jigsaw puzzle back into its box, you can end up losing important pieces—in this case, specific data dependencies—which can render a reassembled dataset incomplete or functionally impaired. This challenge underscores the importance of careful planning and execution in database normalization.

Relational Database Schema

Think of a relational database schema like the blueprint for a building—it outlines the structure of the database and how different parts of the data relate to each other. A schema provides a logical view of the entire database by describing the tables, fields within each table, and the relationships between these fields.

Adhering to a well-thought-out schema is akin to following a recipe when cooking a complex dish. It ensures that the end result is predictable and of high quality. A well-designed schema will take into account aspects like normalization standards to maintain data integrity and efficiency. It’s an essential guide for constructing a functional and responsive database that supports the specific data needs and operations of an organization or application.

Normalization

Normalization in the database world is like organizing a cluttered desk into a set of neatly labeled drawers—each drawer contains unique items that belong together logically. The primary aim of normalization is to structure a relational database in a way that reduces data redundancy (duplicate data) and increases data integrity (accuracy and consistency).

There are several levels of normalization, called 'normal forms,' each addressing different types of redundancy and potential anomalies. Starting with the most basic, the First Normal Form (1NF), and moving up to more stringent levels like BCNF, the database becomes more streamlined and efficient with each level. However, it's a balancing act: too little normalization can leave behind redundant and unwieldy data, but too much can result in excessive complexity or loss of functional dependencies, as can sometimes happen in BCNF decomposition. It's like sorting your music playlist; you wouldn't want ten copies of the same song, but you also wouldn't want an organizational system so complex that you can't find your favorite tracks.