# Boolean Algebra Simplifier (Karnaugh Map) Minimal SOP via Quine–McCluskey with essential prime implicants, exact minimum cover, Gray-coded Karnaugh map, and full-assignment verification. > Canonical page: https://elysiatools.com/en/tools/boolean-algebra-simplifier - **Category:** Math & Numbers - **Keywords:** boolean algebra, karnaugh map, quine mccluskey, prime implicants, minimal sop, logic simplification, minterms, digital logic, switching algebra, gray code ## Overview The Boolean Algebra Simplifier minimizes switching expressions and minterm sets using the Quine–McCluskey tabular method. It computes prime implicants, determines essential prime implicants for an exact minimal Sum of Products (SOP), displays a Gray-coded Karnaugh map, and verifies the simplified output across all variable assignments. ## Inputs - **Input mode** (select) - **Boolean expression (A–D)** (text): Classic switching-algebra notation: + OR, · or juxtaposition AND, ' or ! NOT, ^ XOR, parentheses. At most 4 distinct variables. - **Minterm list (comma-separated)** (text): Used in Minterms mode: the indices where the function is 1, e.g. 0,1,2,4,5,6. - **Number of variables** (select) ## When to use - Minimizing combinatorial logic equations up to 4 variables for digital circuit designs. - Checking manual Karnaugh map groupings against an automated Quine–McCluskey prime implicant table. - Converting minterm index lists into simplified sum-of-products formulas with verification across all truth assignments. ## How it works - Choose between expression mode (using boolean variables A through D) or minterms mode (entering decimal indices). - The tool parses operators including OR (+), AND (· or juxtaposition), NOT (' or !), and XOR (^), or evaluates the selected minterm list against the chosen variable count. - Quine–McCluskey computes prime implicants and extracts an exact minimal SOP cover alongside canonical Σm notation. - Results output a formatted Gray-coded Karnaugh map table and run an automated evaluation across all 2^N truth assignments to verify equivalence. ## Use cases - Engineering students verifying homework solutions for digital logic design and switching algebra. - Hardware developers minimizing gate counts and literal counts in combinational logic blocks. - Software engineers optimizing multi-condition branching logic and Boolean flags into reduced expressions. ## Frequently asked questions ### Which Boolean operators are supported in expression mode? You can use + for OR, · or variable juxtaposition for AND, ' or ! for NOT, ^ for XOR, and standard parentheses for grouping. ### How many variables can this tool handle? The tool supports functions with 2, 3, or 4 distinct variables (A, B, C, and D). ### What is the difference between expression and minterms input modes? Expression mode accepts symbolic switching algebra equations, while minterms mode takes a comma-separated list of decimal ON-set indices (Σm). ### How does the tool verify the simplification? It tests both the original function and the minimized SOP across every possible binary truth assignment (4, 8, or 16 states) to ensure exact equivalence. ### Does the Karnaugh map follow standard Gray code ordering? 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