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Free boolean algebra simplifier for propositional formulas. Rewrite connectives, push negations, fold constants, and factor common literals—with an optional exhaustive equivalence check for small variable counts.
Last updated: April 13, 2026
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Original
P AND P
Simplified (NNF-style output)
P
We expand IMPLIES, IFF, and XOR into NOT/AND/OR, push negations to variables (negation normal form), then apply constant folding, idempotent/absorption laws, and factoring OR of ANDs that share a literal. For up to five variables, the tool checks equivalence on every truth assignment.
Notes
Shape
NOT only on atoms
Prepares the tree for consistent algebraic replacement rules.
Examples
P AND P → P
Idempotent laws and absorbing TRUE/FALSE where possible.
Pattern
(A∧B)∨(A∧C) → A∧(B∨C)
Reduces repeated subexpressions when two AND clauses share a variable.
P -> Q ≡ NOT P OR Q
Standard definition used in most CS logic courses.
(P∧¬Q)∨(¬P∧Q)
Then NNF and simplification continue as usual.
Workflow
Simplify · then tabulate
Use the truth table generator to compare columns before and after rewriting.
P AND P
Simplifies to P (idempotent law).
Expressions are parsed with the same grammar as our truth table tool. Non-primitive connectives are expanded, negations are driven down to literals, and a fixed-point loop applies reductions until the tree stabilizes. Finally the tree is printed with explicit parentheses only where operator precedence would otherwise change the meaning.
NOT (P AND Q) becomes (NOT P) OR (NOT Q)NOT (P OR Q) becomes (NOT P) AND (NOT Q)After parsing, the AND node has identical children; the simplifier replaces the pair with a single P. The verifier confirms matching output on both assignments P = T and P = F.
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