Decomposable negation normal form

Abstract

Knowledge compilation has been emerging recently as a new direction of research for dealing with the computational intractability of general propositional reasoning. According to this approach, the reasoning process is split into two phases: an off-line compilation phase and an on-line query-answering phase. In the off-line phase, the propositional theory is compiled into some target language, which is typically a tractable one. In the on-line phase, the compiled target is used to efficiently answer a (potentially) exponential number of queries. The main motivation behind knowledge compilation is to push as much of the computational overhead as possible into the off-line phase, in order to amortize that overhead over all on-line queries. Another motivation behind compilation is to produce very simple on-line reasoning systems, which can be embedded cost-effectively into primitive computational platforms, such as those found in consumer electronics.One of the key aspects of any compilation approach is the target language into which the propositional theory is compiled. Previous target languages included Horn theories, prime implicates/implicants and ordered binary decision diagrams (OBDDs). We propose in this paper a new target compilation language, known as decomposable negation normal form (DNNF), and present a number of its properties that make it of interest to the broad community. Specifically, we show that DNNF is universal; supports a rich set of polynomial--time logical operations; is more space-efficient than OBDDs; and is very simple as far as its structure and algorithms are concerned. Moreover, we present an algorithm for converting any propositional theory in clausal form into a DNNF and show that if the clausal form has a bounded treewidth, then its DNNF compilation has a linear size and can be computed in linear time (treewidth is a graph-theoretic parameter that measures the connectivity of the clausal form). We also propose two techniques for approximating the DNNF compilation of a theory when the size of such compilation is too large to be practical. One of the techniques generates a sound but incomplete compilation, while the other generates a complete but unsound compilation. Together, these approximations bound the exact compilation from below and above in terms of their ability to answer clausal entailment queries. Finally, we show that the class of polynomial--time DNNF operations is rich enough to support relatively complex AI applications, by proposing a specific framework for compiling model-based diagnosis systems.

Mapping Conformant Planning into SAT Through Compilation and Projection

“…This operation is known as value substitution or conditioning [6]. If equations (3) and (4) provided a correct formulation of conformant planning, we could obtain a conformant plan by finding a model for T (P ) , and extracting Plan from the value of the action variables in that model.…”

“…Provided that the original theory T (P ) is compiled in d-DNNF, all these transformations can be performed in time linear in the size of its DAG representation by a single bottom-up pass [6]. 4 The resulting theory T cf (P ), however, is in NNF but not in DNNF due to the added conjunction, and this is why it cannot be checked for consistency in linear time, and has to be fed into a SAT solver (the mapping from NNF to CNF is linear though, adding new 'implied' variables for each of the internal nodes of the DAG).…”

“…Projection is the dual of variable elimination (also called forgetting or existential quantification; [5]): the projection of a formula over a subset of its variables is the strongest formula over those variables; e.g., the projection of ((x ∧ y) ∨ z) over {x, z} is x ∨ z. While projection is intractable, it can be done efficiently provided that the theory is in a certain canonical form such as Decomposable Negated Normal Form or DNNF [6], a form akin to OBDDs [7].…”

Applications of Automated Reasoning

“…Decomposable Negation Normalform DNNF is short for decomposable negation normal form and is a special normal form developed in [Dar01]. A formula is in DNNF, if it is in negation normal form (NNF) and additionally satisfies the decomposability property.…”

“…The next very interesting feature of formulae in DNNF ist the possibility to project a DNNF on a set of atoms in linear time. A list of other important features can be found in [Dar01].…”