Boolean function, what is the purpose of DNF and CNF?

Question

Boolean functions can be expressed in Disjunctive normal form (DNF) or Conjunctive normal form (CNF). Can anyone explain why these forms are useful?

Answer 1

CNF is useful because this form directly describes the Boolean SAT problem, which while NP-complete, has many incomplete and heuristic exponential time solvers. CNF has been further standardized into a file format called the "DIMACS CNF file format", from which most solvers can operate on. Thus for example, the chip industry can verify their circuit designs by converting to DIMACS CNF, and feeding in to any of the solvers available. The Tseitin Transformation can convert circuits to an equisatisfiable CNF.

DNF is not as useful practically, but converting a formula to DNF means one can see a list of possible assignments that would satisfy the formula. Unfortunately, converting a formula to DNF in general, is hard, and can lead to exponential blow up (very large DNF), which is evident, because there can be exponentially number of satisfying assignments to a formula.

While CNF can be succinct compared with DNF, it is sometimes hard to reason with, because it can lose structure when converted from a circuit for example, and so another succinct form would be useful. The and-inverter graph data structure was designed for this purpose; it can more closely model the structure of circuits, and is thus much easier to reason with in some types of formulas. However there are not many solvers that operate on it.

Other forms include:

• Binary Decision Diagram
• Propositional directed acyclic graph
• Negation normal form
• Others (wikipedia)

Answer 2

It helps to express the functions in some standard way. With that, it's easier to automatically go through many algorithms.

Both forms can be used, for example, in automated theorem solving, namely CNF in the resolution method.