Negation Normal Form Research Articles

Computing Energy-Optimal Tests using DNNF Graphs

The goal of testing is to discriminate between multiple hypotheses about a system – for example, different fault diagnoses of an HVAC system – by applying input patterns and verifying or falsifying the hypotheses from the observed outputs. Definitely discriminating tests (DDTs) are those input patterns that are guaranteed to discriminate between different hypotheses of non-deterministic systems. Finding DDTs is important in practice, but can be very expensive . Even more challenging is the problem of finding a DDT that minimizes the energy consumption of the testing process, i.e., an input pattern that can be enforced with minimal energy consumption and that is a DDT. This paper addresses both problems. We show how we can transform a given problem into a Boolean structure in decomposable negation normal form (DNNF), and extract from it a Boolean formula whose models correspond to DDTs. This allows us to harness recent advances in both knowledge compilation and satisfiability for efficient and scalable DDT computation in practice. Furthermore, we show how we can generate a DNNF structure compactly encoding all DDTs of the problem and use it to obtain an energy-optimal DDT in time linear in the size of the structure.

A Lower Bound on the Size of Decomposable Negation Normal Form

We consider in this paper the size of a Decomposable Negation Normal Form (DNNF) that respects a given vtree (known as structured DNNF). This representation of propositional knowledge bases was introduced recently and shown to include OBDD as a special case (an OBDD variable ordering is a special type of vtree). We provide a lower bound on the size of any structured DNNF and discuss three particular instances of this bound, which correspond to three distinct subsets of structured DNNF (including OBDD). We show that our lower bound subsumes the influential Sieling and Wegener’s lower bound for OBDDs, which is typically used for identifying variable orderings that will cause a blowup in the OBDD size. We show that our lower bound allows for similar usage but with respect to vtrees, which provide structure for DNNFs in the same way that variable orderings provide structure for OBDDs. We finally discuss some of the theoretical and practical implications of our lower bound.

Computing Cost-Optimal Definitely Discriminating Tests

The goal of testing is to discriminate between multiple hypotheses about a system - for example, different fault diagnoses - by applying input patterns and verifying or falsifying the hypotheses from the observed outputs. Definitely discriminating tests (DDTs) are those input patterns that are guaranteed to discriminate between different hypotheses of non-deterministic systems. Finding DDTs is important in practice, but can be very expensive. Even more challenging is the problem of finding a DDT that minimizes the cost of the testing process, i.e., an input pattern that can be most cheaply enforced and that is a DDT. This paper addresses both problems. We show how we can transform a given problem into a Boolean structure in decomposable negation normal form (DNNF), and extract from it a Boolean formula whose models correspond to DDTs. This allows us to harness recent advances in both knowledge compilation and satisfiability for efficient and scalable DDT computation in practice. Furthermore, we show how we can generate a DNNF structure compactly encoding all DDTs of the problem and use it to obtain a cost-optimal DDT in time linear in the size of the structure. Experimental results from a real-world application show that our method can compute DDTs in less than 1 second for instances that were previously intractable, and cost-optimal DDTs in less than 20 seconds where previous approaches could not even compute an arbitrary DDT.

The weighted Grammar constraint

We introduce the WeightedGrammar constraint and propose propagation algorithms based on the CYK parser and the Earley parser. We show that the traces of these algorithms can be encoded as a weighted negation normal form (WNNF), a generalization of NNF that allows nodes to carry weights. Based on this connection, we prove the correctness and complexity of these algorithms. Specifically, these algorithms enforce domain consistency on the WeightedGrammar constraint in time O(n 3). Further, we propose that the WNNF constraint can be decomposed into a set of primitive arithmetic constraint without hindering propagation.

Computing Optimal Tests for Non-deterministic Systems Using DNNF Graphs

The goal of testing is to distinguish between a number of hypotheses about a system—for example, different diagnoses of faults—by applying input patterns and verifying or falsifying the hypotheses from the observed outputs. Optimal distinguishing tests (ODTs) are those input patterns that are most likely to distinguish between hypotheses about non-deterministic systems. Finding ODTs is practically important, but it amounts in general to determining a ratio of model counts and is therefore computationally very expensive.In this paper, we present a novel approach to this problem, which uses structural properties of the system to limit the complexity of computing ODTs. We first construct a compact graphical representation of the testing problem via compilation into decomposable negation normal form. Based on this compiled representation, we show how one can evaluate distinguishing tests in linear time, which allows us to efficiently determine an ODT. Experimental results from a real-world application show that our method can compute ODTs for instances that were intractable for previous approaches.

A solver for QBFs in negation normal form

Various problems in artificial intelligence can be solved by translating them into a quantified boolean formula (QBF) and evaluating the resulting encoding. In this approach, a QBF solver is used as a black box in a rapid implementation of a more general reasoning system. Most of the current solvers for QBFs require formulas in prenex conjunctive normal form as input, which makes a further translation necessary, since the encodings are usually not in a specific normal form. This additional step increases the number of variables in the formula or disrupts the formula's structure. Moreover, the most important part of this transformation, prenexing, is not deterministic. In this paper, we focus on an alternative way to process QBFs without these drawbacks and describe a solver, $\ensuremath{\sf qpro}$ , which is able to handle arbitrary formulas. To this end, we extend algorithms for QBFs to the non-normal form case and compare $\ensuremath{\sf qpro}$ with the leading normal form provers on several problems from the area of artificial intelligence. We prove properties of the algorithms generalized to non-clausal form by using a novel approach based on a sequent-style formulation of the calculus.

An Extension of Deficiency and Minimal Unsatisfiability of Quantified Boolean Formulas

The notion of minimal unsatisfiability and minimal falsity will be extended to non-clausal and non-prenex quantified Boolean formulas. For quantified Boolean formulas in negation normal form we generalize the notion of deficiency to the so-called cohesion, which is 1+ the difference between the number of occurrences of the conjunction symbol ∧ and the number of existential and free variables. Further, we show that all the complexity results with respect to minimal unsatisfiability or minimal falsity known for formulas with fixed deficiency can be adapted to formulas with fixed cohesion. For example, (1) the minimal unsatisfiability of propositional formulas with fixed cohesion is still solvable in polynomial time and, (2) the minimal falsity of quantified Boolean formulas with cohesion 1 is solvable in polynomial time.

Complexity Results for Quantified Boolean Formulae Based on Complete Propositional Languages1

Several propositional fragments have been considered so far as target languages for knowledge compilation and used for improving computational tasks from major AI areas (like inference, diagnosis and planning); among them are the ordered binary decision diagrams, prime implicates, prime implicants, \formulae in decomposable negation normal form. On the other hand, the validity problem val(QPROPP S) for Quantied Boolean Formulae (QBF) has been acknowledged for the past few years as an important issue for AI, and many solvers have been designed. In this paper, the complexity of restrictions of the validity problem for QBF obtained by imposing the matrix of the input QBF to belong to propositional fragments used as target languages for compilation, is identied. It turns out that this problem remains hard (PSPACE-complete) even under severe restrictions on the matrix of the input. Nevertheless some tractable restrictions are pointed out.

Succinct definitions in the first order theory of graphs

We say that a first order sentence A defines a graph G if A is true on G but false on any graph non-isomorphic to G . Let L ( G ) (resp. D ( G ) ) denote the minimum length (resp. quantifier rank) of such a sentence. We define the succinctness function s ( n ) (resp. its variant q ( n ) ) to be the minimum L ( G ) (resp. D ( G ) ) over all graphs on n vertices. We prove that s ( n ) and q ( n ) may be so small that for no general recursive function f we can have f ( s ( n ) ) ≥ n for all n . However, for the function q ∗ ( n ) = max i ≤ n q ( i ) , which is the least nondecreasing function bounding q ( n ) from above, we have q ∗ ( n ) = ( 1 + o ( 1 ) ) log ∗ n , where log ∗ n equals the minimum number of iterations of the binary logarithm sufficient to lower n to 1 or below. We show an upper bound q ( n ) < log ∗ n + 5 even under the restriction of the class of graphs to trees. Under this restriction, for q ( n ) we also have a matching lower bound. We show a relationship D ( G ) ≥ ( 1 − o ( 1 ) ) log ∗ L ( G ) and prove, using the upper bound for q ( n ) , that this relationship is tight. For a non-negative integer a , let D a ( G ) and q a ( n ) denote the analogs of D ( G ) and q ( n ) for defining formulas in the negation normal form with at most a quantifier alternations in any sequence of nested quantifiers. We show a superrecursive gap between D 0 ( G ) and D 3 ( G ) and hence between D 0 ( G ) and D ( G ) . Despite this, for q 0 ( n ) we still have a kind of log-star upper bound: q 0 ( n ) ≤ 2 log ∗ n + O ( 1 ) for infinitely many n .

Linearity and regularity with negation normal form

Proving completeness of NC-resolution under a linear restriction has been elusive; it is proved here for formulas in negation normal form. The proof uses a generalization of the Anderson–Bledsoe excess literal argument, which was developed for resolution. That result is extended to NC-resolution with partial replacement. A simple proof of the completeness of regular, connected tableaux for formulas in conjunctive normal form is also presented. These techniques are then used to establish the completeness of regular, connected tableaux for formulas in negation normal form.

Satisfiability testing for Boolean formulas using δ-trees

The tree-based data structure of Δ-tree for propositional formulas is introduced in an improved and optimised form. The Δ-trees allow a compact representation for negation normal forms as well as for a number of reduction strategies in order to consider only those occurrences of literals which are relevant for the satisfiability of the input formula. These reduction strategies are divided into two subsets (meaning- and satisfiability-preserving transformations) and can be used to decrease the size of a negation normal form A at (at most) quadratic cost. The reduction strategies are aimed at decreasing the number of required branchings and, therefore, these strategies allow to limit the size of the search space for the SAT problem.

Decomposable negation normal form

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.

On the Tractable Counting of Theory Models and its Application to Truth Maintenance and Belief Revision

We address in this paper the problem of counting the models of a propositional theory under incremental changes to its literals. Specifcally, we show that if a propositional theory Δ is in a special form that we call smooth, deterministic, decomposable negation normal form (sd-DNNF), then for any consistent set of literals S, we can simultaneously count (in time linear in the size of Δ) the models of Δ ∪ S and the models of every theory Δ ∪ T where T results from adding, removing or flipping a literal in S. We present two results relating to the time and space complexity of compiling propositional theories into sd-DNNF. First, we show that if a conjunctive normal form (CNF) has a bounded treewidth, then it can be compiled into an sd-DNNF in time and space which are linear in its size. Second, we show that sd-DNNF is a strictly more space efficient representation than Free Binary Decision Diagrams (FBDDs). Finally, we discuss some applications of the counting results to truth maintenance systems, belief revision, and model-based diagnosis.

Cut normal forms and proof complexity

Statman and Orevkov independently proved that cut-elimination is of nonelementary complexity. Although their worst-case sequences are mathematically different the syntax of the corresponding cut formulas is of striking similarity. This leads to the main question of this paper: to what extent is it possible to restrict the syntax of formulas and — at the same time— keep their power as cut formulas in a proof? We give a detailed analysis of this problem for negation normal form (NNF), prenex normal form (PNF) and monotone formulas. We show that proof reduction to NNF is quadratic, while PNF behaves differently: although there exists a quadratic transformation into a proof in PNF too, additional cuts are needed; the elimination of these cuts requires nonelementary expense in general. By restricting the logical operators to {⋀,⋁,∀,∃}, we obtain the type of monotone formulas. We prove that the elimination of monotone cuts can be of nonelementary complexity (here generalized disjunctions in the antecedents of sequents play a central role). On the other hand, we define a large class of problems (including all Horn theories) where cut-elimination of monotone cuts is only exponential and show that this bound is tight. This implies that the elimination of monotone cuts in equational theories is at most exponential. Particularly, there are no short proofs of Statman's sequence with monotone cuts.

Model-Based Diagnosis using Structured System Descriptions

This paper presents a comprehensive approach for model-based diagnosis which includes proposals for characterizing and computing preferred diagnoses, assuming that the system description is augmented with a system structure (a directed graph explicating the interconnections between system components). Specifically, we first introduce the notion of a consequence, which is a syntactically unconstrained propositional sentence that characterizes all consistency-based diagnoses and show that standard characterizations of diagnoses, such as minimal conflicts, correspond to syntactic variations on a consequence. Second, we propose a new syntactic variation on the consequence known as negation normal form (NNF) and discuss its merits compared to standard variations. Third, we introduce a basic algorithm for computing consequences in NNF given a structured system description. We show that if the system structure does not contain cycles, then there is always a linear-size consequence in NNF which can be computed in linear time. For arbitrary system structures, we show a precise connection between the complexity of computing consequences and the topology of the underlying system structure. Finally, we present an algorithm that enumerates the preferred diagnoses characterized by a consequence. The algorithm is shown to take linear time in the size of the consequence if the preference criterion satisfies some general conditions.

CNF and DNF Considered Harmful for Computing Prime Implicants/Implicates

Several methods to compute the prime implicants and the prime implicates of a negation normal form (NNF) formula are developed and implemented. An algorithm PI is introduced that is an extension to...

Fast Subsumption Checks Using Anti-Links

The concept of anti-link is defined (an anti-link consists of two occurrences of the same literal in a formula), and useful equivalence-preserving operations based on anti-links are introduced. These operations eliminate a potentially large number of subsumed paths in a negation normal form formula. Those anti-links that directly indicate the presence of subsumed paths are characterized. The operations have linear time complexity in the size of that part of the formula containing the anti-link. The problem of removing all subsumed paths in an NNF formula is shown to be NP-hard, even though such formulas may be small relative to the size of their path sets. The general problem of determining whether there exists a pair of subsumed paths associated with an arbitrary anti-link is shown to be NP-complete. Additional techniques that generalize the concept of pure literals are introduced and are also shown to eliminate redundant subsumption checks. The effectiveness of these techniques is examined with respect to some benchmark examples from the literature.

Inference with path resolution and semantic graphs

A graphical representation of quantifier-free predicate calculus formulas in negation normal form and a new rule of inference that employs this representation are introduced. The new rule, path resolution , is an amalgamation of resolution and Prawitz analysis. The goal in the design of path resolution is to retain some of the advantages of both Prawitz analysis and resolution methods, and yet to avoid to some extent their disadvantages. Path resolution allows Prawitz analysis of an arbitrary subgraph of the graph representing a formula. If such a subgraph is not large enough to demonstrate a contradiction, a path resolvent of the subgraph may be generated with respect to the entire graph. This generalizes the notions of large inference present in hyperresolution, clash-resolution, NC-resolution, and UR-resolution. A class of subgraphs is described for which deletion of some of the links resolved upon preserves the spanning property.