The Complexity of Reasoning with FODD and GFODD

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.Comment: A short version of this paper appears in AAAI 2014. Version 2 includes a reorganization and some expanded proof

Binary Decision Diagrams for First Order Predicate Logic

We present an extension of Binary Decision Diagrams (BDDs) such that they can be used for predicate logic. We present a sound and complete proof search method which we apply to a number of examples

Binary decision diagrams for first-order predicate logic

Binary decision diagrams (BDDs) are known to be a very efficient technique to handle propositional formulas. We present an extension of BDDs such that they can be used for predicate logic. We define BDDs similar to Bryant [IEEE Trans. Comp. C-35 (1986) 677–691], but with the difference that we allow predicates as labels instead of proposition symbols. We present a sound and complete proof search method for first-order predicate logic based on BDDs which we apply to a number of examplesPeer reviewe

Binary Decision Diagrams for First Order Predicate Logic

We present an extension of Binary Decision Diagrams (BDDs) such that they can be used for predicate logic. We present a sound and complete proof search method which we apply to a number of examples. Key Words & Phrases: Automatic Reasoning, Binary Decision Diagrams, First Order Predicate Logic. 1 Introduction In 1986 Randy Bryant proposed to represent propositional formulas by Ordered Binary Decision Diagrams (BDDs) [2]. A BDD is a node-labelled DAG (Directed Acyclic Graph) where in general each node has two outgoing vertices. Bryant provided straightforward algorithms to transform a formula into a BDD and moreover he proved that logically equivalent formulae have canonical BDD representations. In [2] these representations are called `reduced'. E.g. tautologies and contradictions have as associated reduced BDDs B t and B f (as depicted in Figure 1 on page 6). This yields a very simple procedure to find out whether a given formula OE belongs to one of these classes. Just calculate the..