DatalogZ

The syntax of Datalog_ℤ extends that of Datalog with numeric terms, which are integer constants, integer variables, or terms built up from these with addition, subtraction, and multiplication. Furthermore, Datalog_ℤ allows comparison atoms, which are atoms of the form t < s or t <= s for numeric terms t, s.[2]

The semantics of Datalog_ℤ are based on the model-theoretic (Herbrand) semantics of Datalog.[2]

The undecidability of entailment of Datalog_ℤ motivates the definition of limit Datalog_ℤ. Limit Datalog_ℤ restricts predicates to a single numeric position, which is marked maximal or minimal. The semantics are based on the model-theoretic (Herbrand) semantics of Datalog. The semantics require that Herbrand interpretations be limit-closed to qualify as models, in the following sense: Given a ground atom ${\displaystyle a=r(c_{1},\ldots ,c_{n})}$ of a limit predicate ${\displaystyle r}$ where the last position is a max (resp. min) position, if ${\displaystyle a}$ is in a Herbrand interpretation ${\displaystyle I}$, then the ground atoms ${\displaystyle r(c_{1},\ldots ,k)}$ for ${\displaystyle k>c_{n}}$ (resp. ${\displaystyle k<c_{n}}$) must also be in ${\displaystyle I}$ for ${\displaystyle I}$ to be limit-closed.[3]

sp(w, 0) :- .
sp(y, m + 1) :- sp(x, m), edge(x, y).

• Constraint logic programming