Datalog - Database Systems - Lecture Slides, Slides of Database Management Systems (DBMS)

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Datalog

Logical Rules

Recursion

Logic As a Query Language

• If-then logical rules have been used in many systems. - Important example: EII (Enterprise Information Integration).
• Nonrecursive rules are equivalent to the core relational algebra.
• Recursive rules extend relational algebra and appear in SQL-99.

EII – (2)

• Approach 2: Describe how JoeMenu can be used as a view to help answer queries about Sells and other relations.

JoeMenu(b, p) <- Sells(’Joe’’s Bar’, b, p)

• More about information integration later.

A Logical Rule

• Our first example of a rule uses the relations Frequents(drinker, bar), Likes(drinker, beer), and Sells(bar, beer, price).
• The rule is a query asking for “happy” drinkers --- those that frequent a bar that serves a beer that they like.

Subgoals Are Atoms

• An atom is a predicate , or relation name with variables or constants as arguments.
• The head is an atom; the body is the AND of one or more atoms.
• Convention: Predicates begin with a capital, variables begin with lower-case.

Example: Atom

Sells(bar, beer, p)

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The predicate = name of a relation

Arguments are variables (or constants).

Example: Interpretation

Happy(d) <- Frequents(d,bar) AND

Likes(d,beer) AND Sells(bar,beer,p)

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Distinguished variable

Nondistinguished variables

Interpretation: drinker d is happy if there exist a bar, a beer, and a price p such that d frequents the bar, likes the beer, and the bar sells the beer at price p.

Applying a Rule

• Approach 1: consider all combinations of values of the variables.
• If all subgoals are true, then evaluate the head.
• The resulting head is a tuple in the result.

A Glitch (Fixed Later)

• Relations are finite sets.
• We want rule evaluations to be finite and lead to finite results.
• “Unsafe” rules like P(x)<-Q(y) have infinite results, even if Q is finite.
• Even P(x)<-Q(x) requires examining an infinity of x -values.

Applying a Rule – (2)

• Approach 2: For each subgoal, consider all tuples that make the subgoal true.
• If a selection of tuples define a single value for each variable, then add the head to the result.
• Leads to finite search for P(x)<-Q(x), but P(x)<- Q(y) is problematic.

Arithmetic Subgoals

• In addition to relations as predicates, a predicate for a subgoal of the body can be an arithmetic comparison.
• We write arithmetic subgoals in the usual way, e.g., x < y.

Example: Arithmetic

• A beer is “cheap” if there are at least two bars that sell it for under $2.

Cheap(beer) <- Sells(bar1,beer,p1) AND

Sells(bar2,beer,p2) AND p1 < 2. AND p2 < 2.00 AND bar1 <> bar

Safe Rules

• A rule is safe if:
  1. Each distinguished variable,
  2. Each variable in an arithmetic subgoal, and
  3. Each variable in a negated subgoal, also appears in a nonnegated, relational subgoal.
• Safe rules prevent infinite results.

Example: Unsafe Rules

• Each of the following is unsafe and not allowed:

1. S(x) <- R(y)
2. S(x) <- R(y) AND NOT R(x)
3. S(x) <- R(y) AND x < y

• In each case, an infinity of x ’s can satisfy the rule, even if R is a finite relation.