  The concept of uniform interpolant for a quantifier-free formula from a given
formula with a list of symbols, while well-known in the logic literature, has
been unknown to the formal methods and automated reasoning community for a long
time. This concept is precisely defined. Two algorithms for computing
quantifier-free uniform interpolants in the theory of equality over
uninterpreted symbols (EUF) endowed with a list of symbols to be eliminated are
proposed. The first algorithm is non-deterministic and generates a uniform
interpolant expressed as a disjunction of conjunctions of literals, whereas the
second algorithm gives a compact representation of a uniform interpolant as a
conjunction of Horn clauses. Both algorithms exploit efficient dedicated DAG
representations of terms. Correctness and completeness proofs are supplied,
using arguments combining rewrite techniques with model theory.
