Negation as failure

Negation as failure (NAF, for short) is a non-monotonic inference rule in logic programming, used to derive ${\displaystyle \mathrm {not} ~p}$ (i.e. that ${\displaystyle p}$ is assumed not to hold) from failure to derive ${\displaystyle p}$. Note that ${\displaystyle \mathrm {not} ~p}$ can be different from the statement ${\displaystyle \neg p}$ of the logical negation of ${\displaystyle p}$, depending on the completeness of the inference algorithm and thus also on the formal logic system.

Negation as failure has been an important feature of logic programming since the earliest days of both Planner and Prolog. In Prolog, it is usually implemented using Prolog's extralogical constructs.

More generally, this kind of negation is known as weak negation, in contrast with the strong (i.e. explicit, provable) negation.

In Planner, negation as failure could be implemented as follows:

which says that if an exhaustive search to prove p fails, then assert ¬p. This states that proposition p shall be assumed as "not true" in any subsequent processing. However, Planner not being based on a logical model, a logical interpretation of the preceding remains obscure.

In pure Prolog, NAF literals of the form ${\displaystyle \mathrm {not} ~p}$ can occur in the body of clauses and can be used to derive other NAF literals. For example, given only the four clauses

NAF derives ${\displaystyle \mathrm {not} ~s}$, ${\displaystyle \mathrm {not} ~r}$ and ${\displaystyle p}$ as well as ${\displaystyle t}$ and ${\displaystyle q}$.

The semantics of NAF remained an open issue until 1978, when Keith Clark showed that it is correct with respect to the completion of the logic program, where, loosely speaking, "only" and ${\displaystyle \leftarrow }$ are interpreted as "if and only if", written as "iff" or "${\displaystyle \equiv }$".