Horn Clauses in Deductive Databases

Horn clause is clause (a disjunction of literals) with at most one positive, i.e. unnegated, literal. A clause with at most one positive (unnegated) literal is called a Horn Clause.

Deductive Database:
A type of database which can make conclusions based on sets of well-defined rules, stored in database. RDBMS and logic programming are combined using this. Deductive database designing is done with help of pure declarative programming language known as Datalog.

Types of Horn Clauses :

• Definite clause / Strict Horn clause –
  It has exactly one positive literal.
• Unit clause –
  Definite clause with no negative literals.
• Goal clause –
  Horn clause without a positive literal.

In Datalog, rules are expressed as a restricted form of clauses called Horn clauses, in which a clause can contain at most one positive literal.

A formula can have following quantifiers:

• Universal quantifier –
  It can be understood as – “For all x, P(x) holds”, meaning P(x) is true for every object x in universe. For example, all trucks has wheels.
• Existential quantifier –
  It can be understood as – “There exists an x such that P(x)”, meaning P(x) is true for at least one object x of universe. For example, someone cares for you.

Forms of a Horn Clause :

1. NOT(P1) OR NOT(P2) OR ... OR NOT(Pn) OR Q

2. NOT(P1) OR NOT(P2) OR ... OR NOT (Pn)

Transformation of Horn Clause :
The Horn clause in (1) can be transformed into clause –

P1 AND P2 AND ... AND Pn → Q 

which is written in Datalog as following rules

Q :- P1, P2, ..., Pn 

The above Datalog Rule, is hence a Horn clause.

If predicates P1 AND P2 AND … AND Pn, are all true for a particular binding to their variable arguments, then Q is also true and can hence be inferred.

The Horn clause in (2) can be transformed into –

P1 AND P2 AND ... AND Pn → 

which is written in Datalog as follows:

P1, P2, ..., Pn 

The above Datalog expression can be considered as an integrity constraint, where all predicates must be true to satisfy query.

A query in Datalog consists of two components:

• A Datalog program, which is a finite set of rules.
• A literal P(X1, X2, X3, … Xn), where each X is a variable or a constant.