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Satisfiability of conjunctive normal form formulas whose clauses have at most two unnegated variables. It is NP-complete: binary AND/OR and unary NOT gate equivalences in a Tseitin transformation already satisfy that restriction. Clauses may contain arbitrarily many negative literals; the total clause width is not restricted to two.

Every NP verifier can be compiled into a polynomial-size Boolean circuit whose inputs encode its certificate, and then into an equisatisfiable conjunctive normal form by a Tseitin transformation. This gives a polynomial-time many-one reduction from every NP language to SAT. Checking a guessed assignment proves membership, so SAT is NP-complete.

The Cook-Levin theorem states that the Boolean satisfiability problem is NP-complete under polynomial-time many-one reductions. Membership in NP follows by guessing an assignment and evaluating the formula in time polynomial in its description length.

For hardness, let $L\in \mathrm{NP}$ have a deterministic polynomial-time verifier $M(x,y)$, with a polynomial witness-length bound. Use a fixed polynomial certificate length: a short certificate is encoded by its length followed by padded data, and the verifier checks this encoding. Pad the computation to exactly $T=T(|x|)$ steps, with accepting and rejecting states absorbing. Enlarge $T$ polynomially if necessary to cover input/witness initialization. A standard polynomial slowdown permits a single-tape Turing machine, so it suffices to handle that model.

Encode a tape cell by a fixed number of bits recording its alphabet symbol and either no head or the head's finite control state. Starting with one head, a cell's next label depends only on its own label and the two neighboring labels: a head can change the symbol where it sits and can enter only a neighbor. Each such finite local function has a constant-size Boolean circuit. The initial row fixes $x$, blanks and the starting head, leaving only the witness bits as Boolean circuit inputs. There are $O(T)$ relevant cells with blank margins beyond every possible head position, and $T$ updates. Repeating these local Boolean circuits produces a Boolean circuit $C_x(y)$ of size $O(T^2)$, with an output detecting an accepting head in the final row. The construction is computable in polynomial time. Induction on rows shows that every assignment to $y$ produces exactly the verifier's valid computation; invalid local encodings can be assigned arbitrary Boolean circuit behavior because they never arise from the valid initial row.

Convert this Boolean circuit to conjunctive normal form using a Tseitin transformation. Introduce one variable for each wire, and encode each gate output $z$ by:

Unit clauses fix constant sources and assert the final output. Each input assignment has exactly one extension to its gate values, so the resulting formula $F_x$ is satisfiable precisely when some witness makes $M(x,y)$ accept. It has $O(T^2)$ clauses of bounded length and is produced in polynomial time. ThusThis proof also gives hardness for clauses of at most three literals, without needing a separate satisfiability assumption.

A proposed assignment for 2UN-SAT can be checked in polynomial time, including checking the syntactic restriction, so the language belongs to NP.

For an explicit reduction, parse any Boolean formula into a Boolean circuit over binary AND (logical conjunction), binary OR (logical disjunction) and unary NOT (negation), and apply the gate equivalences in part (a), asserting its output. Every gate clause has at most two positive literals: the AND clauses have respectively one, one and one; the OR clauses have one, one and two; and the NOT clauses have two and zero. The unit output/constant clauses also obey the restriction. Hence the resulting formula is an instance of 2UN-SAT.

The Tseitin transformation is linear in the gate description, and its auxiliary variables enforce the gate values rather than relaxing their relation to the inputs. Therefore the original formula is satisfiable if and only if the transformed restricted formula is satisfiable. This is a polynomial-time many-one reduction from SAT, whose hardness follows from the Cook-Levin theorem. ConsequentlyThe condition limits positive literals, without limiting total clause length.