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Asked: 2026-05-11T07:19:29+00:00

I know that boolean satisfiability is NP-Complete, but is the minimization/simplification of a boolean

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I know that boolean satisfiability is NP-Complete, but is the minimization/simplification of a boolean expression, by which I mean taking a given expression in symbolic form and producing an equivalent but simplified expression in symbolic form, NP-Complete? I’m not sure that there’s a reduction from satisfiability to minimization, but I feel like there probably is. Does anyone know for sure?

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  1. Well, look at it this way: using a minimisation algorithm, you can compact any non-satisfiable expression to the literal false, right? This effectively solves SAT. So at least a complete minimisation algorithm must be NP-hard.

    As per the comment by Para Black, the problem (which is also known as the minimum equivalent DNF problem) was actually shown to be non-NP and Σ2P-complete by Chris Umans in 1998.

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