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Clausal Form in Deductive Databases
A deductive database in SQL or any other database system is a tool that can draw conclusions about new facts based on the rules and information already present in the database. In deductive databases, datalog is the language commonly used to express facts, rules, and queries. The formula, when expressed in clausal form, consists of a number of clauses, each of which is made up of a number of literals joined exclusively by logical connectives marked with the OR symbol.
The following quantifiers are possible in a formula −
Universal quantifier − It may be read as "For all x, P(x) holds," which denotes that P(x) holds for all instances of x in the universe.
Trucks, for instance, all have wheels.
Existential quantifier − It means that P(x) holds for at least one item x in the universe and is expressed as "There exists an x such that P(x)".
Example: You are cared for by someone.
A clausal form formula has to be changed into a formula having the qualities listed below −
Every element of the formula has a quantification. Consequently, it is not essential to explicitly add universal quantifiers for all. All variables in the formula are implicitly quantified by the universal quantifier when the quantifiers are removed.
Given that the formula consists of a number of clauses, each of which is made up of a number of literals joined exclusively by logical connectives, OR, the formula is formed of clauses. Each sentence is a disjunction of literals as a result.
The sentences themselves are joined merely by AND logical connectives to create a formula. As a result, a formula's clausal form is a conjunction of clauses.
Any formula may be changed into clausal form, as can be demonstrated. For our purposes, the structure of the individual clauses—each of which is a disjunction of literals—is of primary concern. Remember those literals can be either positive or negative. Consider the following clause −
NOT(P1) OR NOT(P2) OR ..... OR NOT(Pn) OR Q1 OR Q2 OR ..... OR Qm
There are m positive literals and n negative literals in the preceding clause. The following similar logical formula may be used to represent this clause −
P1 AND P2 AND ..... AND Pn => Q1 OR Q2 OR ..... OR Qm
the implied sign is "=>," for example.
The second formula is true only if at least one of the Q's is true, which is the meaning of the (implies) symbol. This is true if all of the p literals i = (1, 2,...,) are true. For the first formula, if any one of the P literals i = (1, 2,..., n) is true, then all of its negations are also true. Therefore, in this situation, it can only be true if at least one of the Qs is true.
Consequently, the truth values of the aforementioned two formulae are always the same since they are comparable.
In clausal form, the formula is written as a series of sentences, each of which is made up of a number of literals joined exclusively by logical connectives of the OR variety.
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