  Recent work introduced Generalized First Order Decision Diagrams (GFODD) as a
knowledge representation that is useful in mechanizing decision theoretic
planning in relational domains. GFODDs generalize function-free first order
logic and include numerical values and numerical generalizations of existential
and universal quantification. Previous work presented heuristic inference
algorithms for GFODDs and implemented these heuristics in systems for decision
theoretic planning. In this paper, we study the complexity of the computational
problems addressed by such implementations. In particular, we study the
evaluation problem, the satisfiability problem, and the equivalence problem for
GFODDs under the assumption that the size of the intended model is given with
the problem, a restriction that guarantees decidability. Our results provide a
complete characterization placing these problems within the polynomial
hierarchy. The same characterization applies to the corresponding restriction
of problems in first order logic, giving an interesting new avenue for
efficient inference when the number of objects is bounded. Our results show
that for $\Sigma_k$ formulas, and for corresponding GFODDs, evaluation and
satisfiability are $\Sigma_k^p$ complete, and equivalence is $\Pi_{k+1}^p$
complete. For $\Pi_k$ formulas evaluation is $\Pi_k^p$ complete, satisfiability
is one level higher and is $\Sigma_{k+1}^p$ complete, and equivalence is
$\Pi_{k+1}^p$ complete.
