Negation As Failure (NAF, for short) is a non-monotonic inference rule in logic programming, used to derive
is assumed not to hold) from failure to derive
, 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 Planner, Negation As Failure could be implemented as follows: which says that if an exhaustive search to prove p fails, then assert ¬p.
[3] 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
can occur in the body of clauses and can be used to derive other NAF literals.
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
For example, the completion of the four clauses above is The NAF inference rule simulates reasoning explicitly with the completion, where both sides of the equivalence are negated and negation on the right-hand side is distributed down to atomic formulae.
, NAF simulates reasoning with the equivalences In the non-propositional case, the completion needs to be augmented with equality axioms, to formalize the assumption that individuals with distinct names are distinct.
NAF simulates this by failure of unification.
The completion of the program is augmented with unique names axioms and domain closure axioms.
The completion semantics justifies interpreting the result
of a NAF inference as the classical negation
However, in 1987, Michael Gelfond showed that it is also possible to interpret
The autoepistemic interpretation was developed further by Gelfond and Lifschitz in 1988, and is the basis of answer set programming.
The autoepistemic semantics of a pure Prolog program P with NAF literals is obtained by "expanding" P with a set of ground (variable-free) NAF literals Δ that is stable in the sense that In other words, a set of assumptions Δ about what can not be shown is stable if and only if Δ is the set of all sentences that truly can not be shown from the program P expanded by Δ.
Here, because of the simple syntax of pure Prolog programs, "implied by" can be understood very simply as derivability using modus ponens and universal instantiation alone.
The autoepistemic interpretation of NAF can be combined with classical negation, as in extended logic programming and answer set programming.