Herbrand interpretation

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In mathematical logic, a Herbrand interpretation is an interpretation in which all constants and functions symbols are assigned trivial meanings. In other words, every constant is mapped to itself, and every function symbol is mapped to the function that applies it. The interpretation also defines predicate symbols as denoting a subset of the relevant Herbrand base, effectively specifying which ground terms are true in the interpretation. This allows the symbols in a set of clauses to be interpreted in a purely syntactic way, separated from any real instantiation.

The importance of Herbrand interpretations is that, if any interpretation satisfies a given set of clauses S then there is a Herbrand interpretation that satisfies them. Moreover, Herbrand's theorem states that if S is unsatisfiable then there is a finite unsatisfiable set of ground instances from the Herbrand universe defined by S. Since this set is finite, its unsatisfiability can be verified in finite time. However there may be an infinite number of such sets to check.