Boolean Expression Simplifier

About this tool

Boolean Expression Simplifier minimizes complex digital logic equations and boolean algebraic propositions to their irredundant, mathematically optimal Sum of Products (SOP) and Product of Sums (POS) representations. Powered by the Quine-McCluskey algorithmic reduction method, it automatically parses variable juxtaposition (implicit AND), algebraic prime notation ($A'$), De Morgan’s inversions, and standard logic gate operators. The tool computes total literal reductions, canonical minterm/maxterm forms, and prime implicant coverage sets without sending data to external servers.

When to use it

• Reduce transistor and logic gate counts in digital circuit design (ASIC, FPGA, and TTL logic)
• Simplify compound conditional branches in programming software to accelerate branch prediction
• Verify algebraic simplification steps in Discrete Mathematics and Digital Logic homework
• Compare Minimal Sum of Products (SOP) with Minimal Product of Sums (POS) to choose the cheapest hardware gate layout
• Eliminate redundant logic variables and verify whether an expression is a tautology ($1$) or contradiction ($0$)

How to use it

To simplify a boolean expression

1. Enter your logic formula in the Boolean expression to simplify field, or click any of the variable and operator helper buttons:
   • Variables: Single-letter identifiers such as $A, B, C, D$.
   • Conjunction (AND): Enter *, &, AND, or write variables consecutively (e.g. AB is automatically interpreted as $A \land B$).
   • Disjunction (OR): Enter +, |, or OR.
   • Negation (NOT): Use postfix primes (e.g. A') or prefix operators (NOT A, !A).
   • Grouping: Use parentheses (...) to enforce precedence.
2. You can also select any of the Sample equations to quickly inspect standard algebraic identities.
3. The simplifier computes the results instantly:
   • Minimal Sum of Products (SOP): The fewest product terms with the minimum number of literals.
   • Minimal Product of Sums (POS): The optimal dual form composed of multiplied OR terms.
   • Literal Reduction: Compares original literals against simplified literals and displays percentage savings.
   • Canonical Forms: Shows the canonical minterm list ($\Sigma m$) and maxterm list ($\Pi M$).
4. Press Copy SOP or Copy POS to copy the formatted equation to your clipboard.

Questions

Which fundamental boolean algebra theorems are used during simplification?

Boolean expressions can be simplified using standard algebraic laws:

• Identity Laws:

$A \cdot 1 = A \quad\text{and}\quad A + 0 = A$

• Null (Domination) Laws:

$A \cdot 0 = 0 \quad\text{and}\quad A + 1 = 1$

• Idempotent Laws:

$A \cdot A = A \quad\text{and}\quad A + A = A$

• Complement Laws:

$A \cdot A' = 0 \quad\text{and}\quad A + A' = 1$

• De Morgan’s Laws:

$(A + B)' = A' \cdot B' \quad\text{and}\quad (A \cdot B)' = A' + B'$

• Absorption Laws:

$A + A \cdot B = A \quad\text{and}\quad A \cdot (A + B) = A$

• Adjacency (Uniting) Theorem:

$A \cdot B + A \cdot B' = A(B + B') = A(1) = A$

What is the difference between SOP and POS forms?

• Sum of Products (SOP): An OR-combination of multiple AND-terms (for example: $F = AB + C'D$). This maps directly to two-level AND-OR or NAND-NAND gate logic.
• Product of Sums (POS): An AND-combination of multiple OR-terms (for example: $F = (A + B)(C' + D)$). This maps directly to two-level OR-AND or NOR-NOR gate logic.

Depending on the distribution of $1$s and $0$s in the truth table, one form frequently requires fewer total gates or input pins than the other.