A first-order interpretation I satisfies a rule r, I |= r, iff I |= Γr, where Γr = ∀~x(βr →
αr), ~x are the free variables in r, αr is the disjunction of H(r), and βr is the conjunction of B(r). It satisfies a program Π, symbolically I |= Π, iff it satisfies every r ∈ Π, i.e., if I |= ΓΠ, where ΓΠ =Sr∈Π Γr.

A first-order interpretation I is called a *generalized answer set* of Π iff it satisfies grd U (Π, I|LF )
Iand it is subset minimal among the interpretations of L on U with this property.

Traditionally, only *Herbrand interpretations* are considered as the answer sets of a logic program. The set of all (object) constants occurring in Π is called the *Herbrand universe* of Π, symbolically H. If no constant appears in Π, then H = {c}, for an arbitrary constant c.

A Herbrand interpretation is any interpretation I of LH = hH,Pi on H interpreting object constants by identity, id, i.e., I(c) = id(c) = c for all c ∈ H. A Herbrand interpretation I
is an *ordinary answer set* of Π iff it is subset minimal among the interpretations of LH on H satisfying grdH(Π, id)
I.

Furthermore, an *extended Herbrand interpretation* is an interpretation of L on **U ⊇ F**
interpreting object constants by identity. An extended Herbrand interpretation I is an *open* answer set (Heymans et al. 2007) of Π iff it is subset minimal among the interpretations of L on U satisfying grd U

(Π, id)
I.

Note that since we consider programs without equality, we semantically resort to the logic QHTs, which results from QHTs= by dropping the axioms for equality. Concerning Kripke models, however, in slight abuse of notation, we reuse QHT-models as defined for the general case. A QHT-interpretation M = h**I, J, K**i is called an (extended) QHT Herbrand interpretation, if h**I, K**i is an (extended) Herbrand interpretation. Given a program Π, h**I, K**i is a generalized answer set of Π iff h**I, K, K**i is a QEL-model of ΓΠ, and h**I, K**i is an open, respectively ordinary, answer set of Π iff h**I, K, K**i is an extended Herbrand, respectively Herbrand, QEL-model of ΓΠ. Notice that the static interpretation of constants introduced by Item (i) of the grounding process is essential for this correspondences in terms of QHTs. Abusing notation, we further on identify Π and ΓΠ.

As already mentioned for propositional programs, uniform equivalence is usually understood wrt. sets of *ground facts* (i.e., ground atoms). Obviously, uniform equivalence wrt. factual theories implies uniform equivalence wrt. ground atoms. We show the converse direction (lifting Theorem 2 in (Pearce and Valverde 2004), for a proof see (Fink 2009)).

## Proposition 10

Given two programs Π1, Π2, then Π1 ≡u Π2 iff (Π1 ∪ A) ≡a (Π2 ∪ A), for any set of ground atoms A.

Thus, there is no difference whether we consider uniform equivalence wrt. sets of ground facts or factual theories. Since one can also consider sets of clauses, i.e. disjunctions of atomic formulas and their negations, which is a more suitable representation of facts according to the definition of program rules in this article, we adopt the following terminology. A rule r is called a *fact* if B(r) = ∅, and a *factual program* is a set of facts. Then, by our result Π1 ≡u Π2 holds for programs Π1, Π2 iff (Π1 ∪ Π) ≡a (Π2 ∪ Π), for any factual program Π.