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An atomic formula containing no variables. In STRIPS, a state records true ground atoms, and the closed-world assumption treats unlisted ground atoms as false. Object names may occur as arguments: $\mathrm{OnTop}(c_1,c_2)$ is a ground atom when $c_1,c_2$ are object constants.

The set of all ground atoms formed using a Herbrand universe and the signature's predicates.

Index the coins so that $c_1$ is smallest and $c_n$ largest. Choose $s_1$ as the initial stack and $s_2$ as the destination, assuming $k\ge 2$. The static predicate facts describing size areNo size comparison with $g$ is needed: moving onto empty ground will have a separate action schema. The initial STRIPS state isUnder the closed-world assumption, all other ground ground atoms are false. In particular, the initial state contains just one predicate fact naming the top of each stack. The predicate $\mathrm{OnTop}$ names the immediate support, not every coin farther below.

An explicit positive conjunctive goal is the following set of required ground atoms:A state succeeds when it contains $G_{\mathrm{goal}}$. The static predicate facts in $S_{\mathrm{size}}$ persist automatically. Starting from $I$, the planning state invariants proved below make this goal exactly the desired completed stack on $s_2$ with all other stacks empty. If any new stack is acceptable, the choice of $s_2$ loses no generality because the initially empty stacks are interchangeable.

Fix a signature, adding a fresh constant if it otherwise has no ground terms. Its Herbrand universe $U$ is the set of all ground terms, and its Herbrand base $B$ is the set of all ground atoms over $U$. A Herbrand interpretation is identified with a subset $I\subseteq B$: precisely its members are true. Function symbols and constants have their term interpretations.

A least Herbrand model of a knowledge base $\mathcal{K}$ is a Herbrand model $M$ satisfyingIf it exists, it is unique. This is stronger than being minimal: a minimal model merely has no strictly smaller model contained in it.

For a positive definite Datalog program with facts, let $T$ add every fact and the head of every ground rule whose body is already true. Starting from the empty set and repeatedly adding consequences yieldswhere $T$ is taken inflationary, so it also retains its input. Every added ground atom belongs to every model, and the union is closed under every finite rule body. These two facts establish that it is the least Herbrand model. The intersection assertion is guaranteed here by the definite Horn form, not for arbitrary knowledge bases.

Add the ground atomTogether with the three rules above, this is the Datalog with negation knowledge base $\mathcal{K}$. There are no supplied divisibility facts and no other year facts. In particular, nothing in this knowledge base asserts $DivisibleBy(N,4)$ or $DivisibleBy(N,100)$.