In propositional logic, complex logical statements are often converted into standard forms to simplify their analysis and manipulation. Normal forms and principal forms are the standardized ways of writing logical expressions so they are easier to analyze, compare, or implement.
Basic Terminologies
- Variables or their negations (e.g., 𝐴 or ¬𝐴).
- Literals connected using the logical OR (∨) operator (e.g., 𝐴∨¬𝐵).
- A minterm is a conjunction (AND) of all variables in the expression, where each variable appears exactly once, either in its original form or negated (e.g., A ∧ B ∧ C).
- A maxterm is a disjunction (OR) of all variables in the expression, where each variable appears exactly once, either in its original form or negated (e.g., A ∨ B ∨ C).
Normal Forms
Normal Forms are structured representations of logical expressions where the formula is broken down into a combination of literals (variables or their negations) connected by logical operators such as AND, OR, and NOT.
The two most common normal forms are the following:
1. Disjunctive Normal Forms (DNF): A Disjunctive Normal Form (DNF) is a logical expression consisting of one or more conjunctions (AND) of literals joined together by disjunctions (OR).
Example : (P ∧ ~ Q) ∨ (Q ∧ R) ∨ (~ P ∧ Q ∧~ R)
The DNF of the formula is not unique.
2. Conjunctive Normal Form (CNF) : A Conjunctive Normal Form (CNF) is a logical expression consisting of one or more disjunctions (OR) of literals joined together by conjunctions (AND).
Example : (P~ ∨ Q) ∧ (Q ∨ R) ∧ (~ P ∨ Q ∨ ~ R)
The CNF of the formula is not unique.
If every elementary sum in CNF is a tautology, then the given formula is also a tautology.
Conversion to Normal Forms
1. Conversion to CNF
1. Eliminate Bi-conditional and Implication: Convert any bi-conditional (↔) and implication (→) into their logical equivalents. Example: A → B becomes ¬A ∨ B.
2. Move Negations Inward: Apply De Morgan’s laws to push negations inside and eliminate double negations. Example: ¬(A ∧ B) becomes ¬A ∨ ¬B.
3. Distribute OR over AND: Apply distributive laws to achieve a conjunction of disjunctions. Example: (A ∨ (B ∧ C)) becomes (A ∨ B) ∧ (A ∨ C).
Example: Convert (A → B) ∧ ¬C to CNF: (¬A ∨ B) ∧ ¬C
1. (A → B) ∧ ¬C ⇒ (¬A ∨ B) ∧ ¬C
2. (¬A ∨ B) ∧ ¬C has no compound negations
3. No distribution is required since there is no disjunction over a conjunction
Therfore, CNF: (¬A ∨ B) ∧ ¬C
2. Conversion to DNF
1. Eliminate Bi-conditional and Implication: Similar to CNF conversion, start by eliminating any biconditional and implication. Example: A → B becomes ¬A ∨ B.
2. Move Negations Inward: Apply De Morgan’s laws to push negations inside. Example: ¬(A ∧ B) becomes ¬A ∨ ¬B.
3. Distribute AND over OR: Apply distributive laws to achieve a disjunction of conjunctions. Example: (A ∧ (B ∨ C)) becomes (A ∧ B) ∨ (A ∧ C).
Example: Convert (A ∨ B) ∧ (C ∨ D) to DNF: (A ∧ ¬ B) ∨ (C ∧ D)
1. The expression doesn't containimplications or bi-conditionals.
2. No negations in the expression.
3. (A ∨ B) ∧ (C ∨ D) = [(A ∨ B) ∧ C] ∨ [(A ∨ B) ∧ D]
= [(A ∧ C) ∨ (B ∧ C)] ∨ [(A ∧ D) ∨ (B ∧ D)]
= (A ∧ C) ∨ (B ∧ C) ∨ (A ∧ D) ∨ (B ∧ D)
Principal Forms
A Principal Form (PNF) is a canonical representation of a propositional formula in which every term contains all the variables of the expression exactly once, either in its original form or negated. Principal normal forms are constructed using minterms or maxterms obtained from the truth table of the logical expression.
There are two types of principal normal forms:
1. Principal Disjunctive Normal Form (PDNF): A Principal Disjunctive Normal Form (PDNF) is a logical expression consisting of the disjunction (OR) of minterms only. It is also known as the sum-of-products canonical form.
General Structure: m1 ∨ m2 ∨ ⋯ ∨ mn , where each mi is a minterm.
Example : (P ∧ ~ Q ∧ ~ R) ∨ (P ∧ ~ Q ∧ R) ∨ (~ P ∧ ~ Q ∧ ~ R)
- The minterm consists of conjunctions in which each statement variable or its negation, but not both, appears only once.
- The minterms are written down by including the variable if its truth value is T and its negation if its truth value is F.
2. Principal Conjunctive Normal Form (PCNF) : A Principal Conjunctive Normal Form (PCNF) is a logical expression consisting of the conjunction (AND) of maxterms only. It is also known as the product-of-sums canonical form.
General Structure: M1 ∨ M2 ∨ ⋯ ∨Mn , where each mi is a minterm.
Example : (P ∨ ~ Q ∨ ~ R) ∧ (P ∨ ~ Q ∨ R) ∧ (~ P ∨ ~ Q ∨ ~ R)
- The maxterm consists of disjunctions in which each variable or its negation, but not both, appears only once.
- The dual of a minterm is called a maxterm.
- Each of the maxterm has the truth value F for exactly one combination of the truth values of the variables.
- The maxterms are written down by including the variable if its truth value is F and its negation if its truth value is T.
Conversion to Principal Forms
1. Conversion to Principal Disjunctive Normal Form (PDNF)
1. Construct the Truth Table: Write the truth table for the given logical expression.
2. Identify the True Rows: Select all rows where the expression evaluates to True (T).
3. Write the Corresponding Minterms: For each True row, write a minterm by:
- Using the variable itself if its value is True (T).
- Using its negation if its value is False (F).
4. Combine the Minterms: Join all the minterms using the OR (∨) operator.
Example: Convert the expression A → B into PDNF.
Step 1: Replace the implication: A → B ≡ ¬A ∨ B
Step 2: Construct the truth table.
A B A → B T T T T F F F T T F F T Step 3: Write the minterms for the True rows.
- TT → A ∧ B
- FT → ¬A ∧ B
- FF → ¬A ∧ ¬B
so, (A ∧ B) ∨ (¬A ∧ B) ∨ (¬A ∧ ¬B)
2. Conversion to Principal Conjunctive Normal Form (PCNF)
1. Construct the Truth Table: Write the truth table for the given logical expression.
2. Identify the False Rows: Select all rows where the expression evaluates to False (F).
3. Write the Corresponding Maxterms: For each False row, write a maxterm by:
- Using the variable itself if its value is False (F).
- Using its negation if its value is True (T).
4. Combine the Maxterms: Join all the maxterms using the AND (∧) operator.
Example: Convert the expression A → B into PCNF.
Step 1: Replace the implication: A→B ≡ ¬A ∨ B
Step 2: Construct the truth table.
A B A → B T T T T F F F T T F F T Step 3: Write the maxterm for the False row: TF → (¬A∨B) so, (¬A ∨ B)
Properties
1. Logical Equivalence: Normal forms are logically equivalent to the original expression, meaning they produce the same truth values under all possible interpretations.
- Ensures that the logical meaning is preserved during the transformation.
2. Minimality: Principal normal forms aim to use the minimal number of literals and clauses necessary to represent the logical expression.
- Reduces complexity and improves efficiency in computational processes.
3. Canonical Representation: Principal forms provide a unique representation for a logical expression, which is especially useful in automated reasoning and digital logic design.
- Ensures consistency in logical analysis and processing.
4. Simplification: Normal forms simplify complex logical expressions, making them easier to manipulate and analyze.
- Facilitates logical reasoning, problem-solving, and optimization.
➢Practice: Solved Examples
