Boolean formula

A Boolean formula is a finite expression built from Boolean variables and Boolean operations, equivalently a propositional formula with two-valued semantics. A Boolean valuation determines its truth value. In $(x\vee y)\wedge \neg x$, logical negation acts only on $x$, and the outer logical conjunction combines $x\vee y$ with $\neg x$. Occurrences of the same Boolean variable share one truth value. The Boolean satisfiability problem asks whether some Boolean valuation makes the Boolean formula true.

A Boolean formula is in disjunctive normal form when it is a disjunction of conjunctions of Boolean literals. Width bounds the number of literals in each conjunction, not the number of conjunctions. A depth-$s$ decision tree gives such a formula of width at most $s$ by taking the disjunction of its accepting leaf conditions; rejecting leaves give the dual conjunctive normal form.

A Boolean formula is in conjunctive normal form when it is a logical conjunction of Boolean clauses. It is true precisely when every Boolean clause is true under the same Boolean valuation; an empty logical conjunction is true. For example, $(x\vee \neg y)\wedge (z)$ has two Boolean clauses with widths two and one. Width counts Boolean literal occurrences within a Boolean clause, whereas the Boolean clause count counts the conjuncts. 2-SAT bounds width by two, and 3-SAT bounds width by three. A Boolean formula with an empty Boolean clause is an unsatisfiable Boolean formula.

Introduce a Boolean variable for each gate output and impose its equivalence to the gate's input function with a constant number of clauses. A final unit clause forces acceptance. For a bounded-fan-in Boolean circuit, the formula size is linear in the gate count and it is satisfiable exactly when some input makes the Boolean circuit output one. Auxiliary variables extend the satisfying assignments; this is equisatisfiability, not equality as functions of the enlarged variable set.

A Boolean clause is an expression ${\ell }_1\vee \cdots \vee {\ell }_k$ formed from Boolean literals, true when at least one Boolean literal is true. The empty Boolean clause is false. Repeating a Boolean literal does not change the truth value. A CNF Boolean formula is a logical conjunction of these Boolean clauses.

For $(\neg x\vee z)$, the only failing pair is $(x,z)=(1,0)$: its two Boolean literals then both have truth value zero. The one-Boolean literal Boolean clause $(x)$ requires $x=1$. A Boolean clause containing both $x$ and $\neg x$ is a tautology, since one of those Boolean literals is true under every Boolean valuation. These observations separate Boolean clause satisfaction from the simultaneous satisfaction required by SAT.

A Boolean literal is a Boolean variable $u$ or its logical negation $\neg u$. The two Boolean literals have opposite truth values: a Boolean valuation $v$ gives $v(\neg u)=1-v(u)$ when false and true are encoded as zero and one. Thus $u=0$ makes the positive Boolean literal $u$ false and the negative Boolean literal $\neg u$ true. A Boolean clause combines Boolean literals, and a Boolean-pair colouring gadget represents a Boolean variable and its logical negation by two vertices with opposite Boolean colours.

A Boolean variable takes one of two truth values, false or true, encoded as zero or one. It is an input to a Boolean formula; a Boolean literal uses the Boolean variable either positively or with logical negation. A Boolean valuation assigns one truth value to each distinct Boolean variable, so three distinct Boolean variables give eight possible complete assignments regardless of how often each appears.