Question

Consider the following set of facts for the relation parent \((X, Y),\) where \(Y\) is the parent of \(x:\) parent(a,aa), parent(a,ab), parent(aa,aaa), parent(aa,aab), parent(aaa,aaaa), parent(aaa,aaab). Consider the rules r1: ancestor \((X, Y):-\) parent \((X, Y)\) \(r 2:\) ancestor \((X, Y):-\) parent \((X, Z),\) ancestor \((Z, Y)\) which define ancestor \(Y\) of \(X\) as above. a. Show how to solve the Datalog query ancestor \((\mathrm{aa}, \mathrm{X}) ?\) using the naive strategy. Show your work at each step. b. Show the same query by computing only the changes in the ancestor relation and using that in rule 2 each time. [This question is derived from Bancilhon and Ramakrishnan (1986).]

Short answer

The Datalog query ancestor(aa, X) would give the result X equals to {aaa, aab, aaaa, aaab}. That means, 'aa' is an ancestor to 'aaa', 'aab', 'aaaa' and 'aaab'.

Step-by-step solution

Application of rule R1

For the first part of the query, to get ancestor(aa, X), the first rule R1 is applied. According to it, the ancestor data can be inferred directly from the parent data, thus r1 will infer that aa is an ancestor of aaa and aab. At this step, the answer for X in ancestor(aa, X) would be aaa and aab.

Application of rule R2

Now the rule 2 (r2) is used: ancestor(X, Y) if parent(X, Z), ancestor(Z, Y). This rule is used to capture the indirect ancestors. By utilizing rule r2 on parent(aa,aaa) and ancestor(aaa,aaaa), it can be inferred that aa is an ancestor of aaaa. Similarly, for parent(aa,aaa) and ancestor(aaa,aaab), aa becomes an ancestor of aaab. At this step, the answer for X in ancestor(aa, X) would be aaa, aab, aaaa and aaab.

Computing only changes in ancestor relation

The second part of the exercise requires computing only the changes in the ancestor relation. This is done through an optimizing process called 'semi-naive' evaluation. Initially, the process is the same as naive evaluation where Rule 1 (R1) is applied and gives the direct parent-child relations 'aaa' and 'aab' for 'aa'. After that, our modification, instead of using all ancestor facts in the database, only those produced in the last round are considered. Therefore, we would look at 'aaa' and 'aab' only and using Rule 2 (R2), we would get 'aaaa' and 'aaab' as ancestors of 'aa'. At the end, no new ancestor facts are produced and we return the results: aaa, aab, aaaa and aaab.

Conclusion

Thus, Datalog rules r1 and r2 are applied iteratively to solve the ancestor(aa, X) query. The semi-naive evaluation optimizes the process by considering only newly derived ancestor facts in each round after the direct parent-child relation has been derived. All inferred ancestor data comprises the solution.

Key concepts

Understanding Database Systems

In the world of technology, database systems are essential for managing and storing data efficiently. A database is a collection of data that's organized in a way that makes it easy to access, manage, and update. Imagine a digital filing cabinet where all your records are neatly sorted and can be found in seconds. That's how database systems function.

They can be visualized as a vast collection of tables that interlink information in a structured format, using various data models like the hierarchical, network, relational, or object-oriented models. The relational model, for instance, uses tables to represent data and relationships between those data. This type of system is what typically underlies a Datalog query evaluation.

Deciphering Inferencing Rules

At the heart of database systems are sets of inferencing rules. These are logical statements that help us derive new facts from existing facts in the database. Like 'if-then' statements, they allow the system to deduce relationships and provide answers to complex queries.

An inferencing rule usually consists of a head and a body. The head is what you're trying to prove, while the body contains information that, if true, supports the head. For example, with the rule 'ancestor(X, Y) if parent(X, Z), ancestor(Z, Y)', we are saying 'if X is the parent of Z, and Z is an ancestor of Y, then X is an ancestor of Y', which allows us to discern extended family trees just based on parent-child relationships.

Exploring the Ancestor Relation

When we talk about ancestor relations in a database, we're essentially mapping out a family tree. This concept is central to understanding relationships extending across generations. In our example, we define an ancestor as either a direct parent or as someone who is a parent to an ancestor.

To crystallize, imagine looking up your great-great-grandparents. Your great-grandparent is an ancestor because they are a direct parent to your grandparent, who in turn is your parent's parent. Thus, the ancestor relation loop continues up the family line. The ancestor relationship is not immediately apparent and must be inferred from the basic parent-child relationships stored in the database.

The Efficiency of Semi-Naive Evaluation

With semi-naive evaluation, we take a smart shortcut in query processing that avoids redundancy and speeds up the inferencing process significantly. Unlike naive methods that reprocess all data with every iteration, semi-naive evaluation focuses only on the most recent changes.

Think of it like checking your house for changes. Instead of reviewing every room each time, you only check the rooms where changes were last made - much faster! Similarly, by only using the facts inferred in the most recent cycle to drive further inferencing, semi-naive evaluation bypasses the evaluation of unchanged data, thus reducing the computational load and enhancing performance.