Negation Normal Form Research Articles

A Compiler for Weak Decomposable Negation Normal Form

This paper integrates weak decomposable negation normal form (wDNNF) circuits, introduced by Akshay et al. in 2018, into the knowledge compilation map. This circuit type generalises decomposable negation normal form (DNNF) circuits in such a way that they allow a restricted form of sharing variables among the inputs of a conjunction node. We show that wDNNF circuits have the same properties as DNNF circuits regarding the queries and transformations presented in the knowledge compilation map, whilst being strictly more succinct than DNNF circuits (that is, they can represent Boolean functions compactly). We also present and evaluate a knowledge compiler, called Bella, for converting CNF formulae into wDNNF circuits. Our experiments demonstrate that wDNNF circuits are suitable for configuration instances.

Cut elimination for coherent theories in negation normal form

We present a cut-free sequent calculus for a class of first-order theories in negation normal form which include coherent and co-coherent theories alike. All structural rules, including cut, are admissible.

On the role of logical separability in knowledge compilation

Knowledge compilation is an alternative solution to address demanding reasoning tasks with high complexity via converting knowledge bases into a suitable target language. The notion of logical separability, proposed by Levesque, offers a general explanation for the tractability of clausal entailment for two remarkable languages: decomposable negation normal form and prime implicates. It is interesting to explore what role logical separability plays in problem tractability. In this paper, we apply the notion of logical separability to a number of reasoning problems within the context of propositional logic: satisfiability checking (CO), clausal entailment checking (CE), model counting (CT), model enumeration (ME) and forgetting (FO), as well as their dual tasks, contributing to several recursive procedures. We provide the corresponding logical separability based properties: CO-logical separability, CE-logical separability, CT-logical separability, ME-logical separability and their duals. Based on these properties, we then identify four novel normal forms: CO-LSNNF, CE-LSNNF, CT-LSNNF and ME-LSNNF, as well as their dual languages. We show that each of them is the necessary and sufficient condition under which the corresponding procedure is correct. We finally integrate the above normal forms into the knowledge compilation map.

Constraint Optimization over Semirings

Interpretations of logical formulas over semirings (other than the Boolean semiring) have applications in various areas of computer science including logic, AI, databases, and security. Such interpretations provide richer information beyond the truth or falsity of a statement. Examples of such semirings include Viterbi semiring, min-max or access control semiring, tropical semiring, and fuzzy semiring. The present work investigates the complexity of constraint optimization problems over semirings. The generic optimization problem we study is the following: Given a propositional formula phi over n variable and a semiring (K,+, . ,0,1), find the maximum value over all possible interpretations of phi over K. This can be seen as a generalization of the well-known satisfiability problem (a propositional formula is satisfiable if and only if the maximum value over all interpretations/assignments over the Boolean semiring is 1). A related problem is to find an interpretation that achieves the maximum value. In this work, we first focus on these optimization problems over the Viterbi semiring, which we call optConfVal and optConf. We first show that for general propositional formulas in negation normal form, optConfVal and optConf are in FP^NP. We then investigate optConf when the input formula phi is represented in the conjunctive normal form. For CNF formulae, we first derive an upper bound on the value of optConf as a function of the number of maximum satisfiable clauses. In particular, we show that if r is the maximum number of satisfiable clauses in a CNF formula with m clauses, then its optConf value is at most 1/4^(m-r). Building on this we establish that optConf for CNF formulae is hard for the complexity class FP^NP[log]. We also design polynomial-time approximation algorithms and establish an inapproximability for optConfVal. We establish similar complexity results for these optimization problems over other semirings including tropical, fuzzy, and access control semirings.

Characterizing Tseitin-Formulas with Short Regular Resolution Refutations

Tseitin-formulas are systems of parity constraints whose structure is described by a graph. These formulas have been studied extensively in proof complexity as hard instances in many proof systems. In this paper, we prove that a class of unsatisfiable Tseitin-formulas of bounded degree has regular resolution refutations of polynomial length if and only if the treewidth of all underlying graphs G for that class is in O(log |V (G)|). It follows that unsatisfiable Tseitin-formulas with polynomial length of regular resolution refutations are completely determined by the treewidth of the underlying graphs when these graphs have bounded degree. To prove this, we show that any regular resolution refutation of an unsatisfiable Tseitin-formula with graph G of bounded degree has length 2Ω(tw(G))/|V (G)|, thus essentially matching the known 2O(tw(G))poly(|V (G)|) upper bound. Our proof first connects the length of regular resolution refutations of unsatisfiable Tseitin-formulas to the size of representations of satisfiable Tseitin-formulas in decomposable negation normal form (DNNF). Then we prove that for every graph G of bounded degree, every DNNF-representation of every satisfiable Tseitin-formula with graph G must have size 2Ω(tw(G)) which yields our lower bound for regular resolution.

Knowledge Compilation Meets Logical Separability

Knowledge compilation is an alternative solution to address demanding reasoning tasks with high complexity via converting knowledge bases into a suitable target language. Interestingly, the notion of logical separability, proposed by Levesque, offers a general explanation for the tractability of clausal entailment for two remarkable languages: decomposable negation normal form and prime implicates. It is interesting to explore what role logical separability on earth plays in problem tractability. In this paper, we apply the notion of logical separability in three reasoning problems within the context of propositional logic: satisfiability check (CO), clausal entailment check (CE) and model counting (CT), contributing to three corresponding polytime procedures. We provide three logical separability based properties: CO- logical separability, CE-logical separability and CT-logical separability. We then identify three novel normal forms: CO-LSNNF, CE-LSNNF and CT-LSNNF based on the above properties. Besides, we show that every normal form is the necessary and sufficient condition under which the corresponding procedure is correct. We finally integrate the above four normal forms into the knowledge compilation map.

Tractable Explanations for d-DNNF Classifiers

Compilation into propositional languages finds a growing number of practical uses, including in constraint programming, diagnosis and machine learning (ML), among others. One concrete example is the use of propositional languages as classifiers, and one natural question is how to explain the predictions made. This paper shows that for classifiers represented with some of the best-known propositional languages, different kinds of explanations can be computed in polynomial time. These languages include deterministic decomposable negation normal form (d-DNNF), and so any propositional language that is strictly less succinct than d-DNNF. Furthermore, the paper describes optimizations, specific to Sentential Decision Diagrams (SDDs), which are shown to yield more efficient algorithms in practice.

Lower Bounds on Intermediate Results in Bottom-Up Knowledge Compilation

Bottom-up knowledge compilation is a paradigm for generating representations of functions by iteratively conjoining constraints using a so-called apply function. When the input is not efficiently compilable into a language - generally a class of circuits - because optimal compiled representations are provably large, the problem is not the compilation algorithm as much as the choice of a language too restrictive for the input. In contrast, in this paper, we look at CNF formulas for which very small circuits exists and look at the efficiency of their bottom-up compilation in one of the most general languages, namely that of structured decomposable negation normal forms (str-DNNF). We prove that, while the inputs have constant size representations as str-DNNF, any bottom-up compilation in the general setting where conjunction and structure modification are allowed takes exponential time and space, since large intermediate results have to be produced. This unconditionally proves that the inefficiency of bottom-up compilation resides in the bottom-up paradigm itself.

Propagation complete encodings of smooth DNNF theories

We investigate conjunctive normal form (CNF) encodings of a function represented with a decomposable negation normal form (DNNF). Several encodings of DNNFs and decision diagrams were considered by (Abío et al., 2016). The authors differentiate between encodings which implement consistency or domain consistency by unit propagation from encodings which are unit refutation complete or propagation complete. The difference is that in the former case we do not care about propagation strength of the encoding with respect to the auxiliary variables while in the latter case we treat all variables (the main and the auxiliary ones) in the same way. The currently known encodings of DNNF theories implement domain consistency. Building on these encodings we generalize the result of (Abío et al., 2016) on a propagation complete encoding of decision diagrams and present a propagation complete encoding of a DNNF and its generalization for variables with finite domains.

On the Relation Between Structured d-DNNFs and SDDs

Structured d-DNNFs and SDDs are restricted negation normal form circuits used in knowledge compilation as target languages into which propositional theories are compiled. Structuredness is imposed by so-called vtrees. By definition SDDs are restricted structured d-DNNFs. Beame and Liew (2015) as well as Bova and Szeider (2017) mentioned the question whether structured d-DNNFs are really more general than SDDs w.r.t. polynomial-size representations (w.r.t. the number of Boolean variables the represented functions are defined on.) The main result in the paper is the proof that a function can be represented by SDDs of polynomial size if the function and its complement have polynomial-size structured d-DNNFs that respect the same vtree.

On Limitations of Structured (Deterministic) DNNFs

The study of representations for propositional theories has been a central subject in knowledge compilation. Many known representations of propositional knowledge bases are restricted negation normal form circuits (NNFs) or binary decision diagrams. Sentential decision diagrams (SDDs) are one of them, by definition they are restricted structured deterministic DNNFs and more general than ordered binary decision diagrams (OBDDs). In this paper limitations of structured (deterministic) DNNFs and in particular SDDs are considered. The main result in the paper is a quasipolynomial simulation of structured (deterministic) DNNFs by equivalent (unambiguous) nondeterministic OBDDs. The result is tight in the sense that there cannot be a polynomial simulation due to a known separation result. One may ask whether there are simulations in smaller size for restricted structured (deterministic) DNNFs. Answering a question posed by Bova and Szeider (2017) in the negative it is shown that not every SDD with disjunctions of constant fan-in can be transformed into equivalent OBDDs of polynomial size.

Reducing Interpolant Circuit Size Through SAT-Based Weakening

We address the problem of reducing the size of Craig’s interpolants (ITPs) used in SAT-based model checking. Whereas it is well known that ITPs are highly redundant, their compaction is typically tackled by reducing the proof graph and/or by exploiting standard logic synthesis techniques. Furthermore, strengthening and weakening have been studied as options to control ITP quality. In this paper, 1 we propose an SAT-based ITP weakening/strengthening technique, for ITP compaction, where the UNSAT core extracted from an additional SAT query is used to obtain a gate-level abstraction of the ITP. The abstraction introduces fresh new variables at gate cuts that must be quantified out in order to obtain a valid ITP. We show how to efficiently quantify them out, by working on a negation normal form representation of the circuit. This paper includes an experimental evaluation, showing the benefits of the proposed approach, on a set of benchmark ITPs arising from hardware model checking problems. 1 Some of the techniques here described were presented in a preliminary version at FMCAD’16 [1] .

Compiling CP subproblems to MDDs and d-DNNFs

Modeling discrete optimization problems is not straightforward. It is often the case that precompiling a subproblem that involves only a few tightly constrained variables as a table constraint can improve solving time. Nevertheless, enumerating all the solutions of a subproblem into a table can be costly in time and space. In this work we propose using Multivalued Decision Diagrams (MDDs) and formulas in Deterministic Decomposable Negation Normal Form (d-DNNFs) rather than tables to compute and store all solutions of a subproblem. This, in turn, can be used to enhance the solver thanks to stronger propagation via specific propagators for these structures. We show how to precompile part of a problem into both these structures, which can then be injected back in the model by substituting the constraints it encodes, or simply adding it as a redundant constraint. Furthermore, in the case of MDDs, they can also be used to create edge-valued MDDs for optimization problems with an appropriate form. From our experiments we conclude that all three techniques are valuable in their own right, and show when each one should be chosen over the others.

Exact Model Counting of Query Expressions

We prove exponential lower bounds on the running time of the state-of-the-art exact model counting algorithms—algorithms for exactly computing the number of satisfying assignments, or the satisfying probability, of Boolean formulas. These algorithms can be seen, either directly or indirectly, as building Decision-Decomposable Negation Normal Form (decision-DNNF) representations of the input Boolean formulas. Decision-DNNFs are a special case of d -DNNFs where d stands for deterministic . We show that any knowledge compilation representations from a class (called DLDDs in this article) that contain decision-DNNFs can be converted into equivalent Free Binary Decision Diagrams (FBDDs) , also known as Read-Once Branching Programs , with only a quasi-polynomial increase in representation size. Leveraging known exponential lower bounds for FBDDs, we then obtain similar exponential lower bounds for decision-DNNFs, which imply exponential lower bounds for model-counting algorithms. We also separate the power of decision-DNNFs from d -DNNFs and a generalization of decision-DNNFs known as AND-FBDDs. We then prove new lower bounds for FBDDs that yield exponential lower bounds on the running time of these exact model counters when applied to the problem of query evaluation in tuple-independent probabilistic databases—computing the probability of an answer to a query given independent probabilities of the individual tuples in a database instance. This approach to the query evaluation problem, in which one first obtains the lineage for the query and database instance as a Boolean formula and then performs weighted model counting on the lineage, is known as grounded inference . A second approach, known as lifted inference or extensional query evaluation , exploits the high-level structure of the query as a first-order formula. Although it has been widely believed that lifted inference is strictly more powerful than grounded inference on the lineage alone, no formal separation has previously been shown for query evaluation. In this article, we show such a formal separation for the first time. In particular, we exhibit a family of database queries for which polynomial-time extensional query evaluation techniques were previously known but for which query evaluation via grounded inference using the state-of-the-art exact model counters requires exponential time.

A Type-Directed Negation Elimination

In the modal mu-calculus, a formula is well-formed if each recursive variable occurs underneath an even number of negations. By means of De Morgan's laws, it is easy to transform any well-formed formula into an equivalent formula without negations -- its negation normal form. Moreover, if the formula is of size n, its negation normal form of is of the same size O(n). The full modal mu-calculus and the negation normal form fragment are thus equally expressive and concise. In this paper we extend this result to the higher-order modal fixed point logic (HFL), an extension of the modal mu-calculus with higher-order recursive predicate transformers. We present a procedure that converts a formula into an equivalent formula without negations of quadratic size in the worst case and of linear size when the number of variables of the formula is fixed.

Experimental analyses on phase transitions in compiling satisfiability problems

In the past decade, a kind of well-known phenomena in many complex combinatorial problems such as satisfiability problem, called phase transition, have been widely studied. In this paper, the phase transition phenomena are investigated during compiling k-satisfiability problems into tractable languages with empirical methods. Ordered binary decision diagram and deterministic-decomposable negation normal form are selected as the tractable target languages for the compilation. Via intensive experiments, it can be concluded that an easy-hard-easy pattern exists during the compilations, which is only related to the ratio of the number of clauses to that of variables if we set k to a fixed value, rather than to the target languages. Moreover, it can be concluded that the space exhausted during the compilations grows exponentially with the number of variables growing, whereas there is also a phase transition separating the polynomial-increment region from the exponential-increment region. Additionally, it can be observed that there is a phase transition of prime implicants around peak points of the easy-hard-easy pattern and the ratios of random instances whose average lengths of prime implicants are larger than the threshold 0.5 change sharply. From these analyses, it can be concluded that prime implicant length and solution interchangeability are crucial impacts on sizes of compilation results.

Beyond vacuity: towards the strongest passing formula

Given an LTL formula phi in negation normal form, it can be strengthened by replacing some of its literals with FALSE. Given such a formula and a model M that satisfies it, vacuity and mutual vacuity attempt to find one or a maximal set of literals, respectively, with which phi can be strengthened while still being satisfied by M. We study the problem of finding the strongest LTL formula that satisfies M and is in the Boolean closure of strengthened versions of phi as defined above. This formula is stronger or equally strong to any formula that can be obtained by vacuity and mutual vacuity. We present our algorithms in the framework of lattice automata.

A Novel Preprocessing Technique for Session Construction using Propositional DAGs

This paper continues the line of research on Web access log analysis. Web access log analysis is to analyze the patterns of web site usage and the features of users’ behavior. It is the fact that the normal Log data is very noisy and unclear and it is vital to preprocess the log data for efficient web usage mining process. Preprocessing is the process comprises of three phases which includes data cleaning, user identification and session construction. Session construction is very vital and numerous real world problems can be modeled as traversals on graph and mining from these traversals would provide the requirement for preprocessing phase. On the other hand, the traversals on unweighted graph have been taken into consideration in existing works. This paper oversimplifies this to the case where vertices of graph are given weights to reflect their significance. The proposed method constructs sessions as a Propositional Directed Acyclic Graph (PDAGs) which contains pages with calculated weights. We identify a new property called simplenegation, which is an implicit restriction of all Negation Normal Form (NNFs) and Binary Decision Diagram(BDDs). The removal of this restriction leads to Propositional Directed Acyclic Graphs (PDAG), a more general family of graph-based languages for representing Boolean functions or propositional theories. This will help site administrators to find the interesting pages for users and to redesign their web pages. After weighting each page according to browsing time a PDAGs structure is constructed for each user session. Existing system in which there is a problem of learning with the Boolean function and the problem can be overcome by the proposed method.

A Tableau Based Decision Procedure for an Expressive Fragment of Hybrid Logic with Binders, Converse and Global Modalities

In this paper we provide the first (as far as we know) direct calculus deciding satisfiability of formulae in negation normal form in the fragment of FHL (full hybrid logic with the binder, including the global and converse modalities), where no occurrence of a universal operator is in the scope of a binder. By means of a satisfiability preserving translation of formulae, the calculus can be turned into a satisfiability decision procedure for the fragment $\textsf{FHL}\setminus\Box \mathord\downarrow\Box$ , i.e. formulae in negation normal form where no occurrence of the binder is both in the scope of and contains in its scope a universal operator. The calculus is based on tableaux and termination is achieved by means of a form of anywhere blocking with indirect blocking. Direct blocking is a relation between nodes in a tableau branch, holding whenever the respective labels (formulae) are equal up to (a proper form of) nominal renaming. Indirect blocking is based on a partial order on the nodes of a tableau branch, which arranges them into a tree-like structure.

Equivalences et forme prénexe pour les formules booléennes quantifiées

Many problems in artificial intelligence and in formal verification can be reduced to a validity test of a quantified boolean formula (QBF). To perform this test, existing QBF solvers need the formula to be in a restrictive prenex form, as conjunctive normal form or negation normal form. This prior prenexing process can lead to an exponential growth of the formula in presence of bi-implications and exclusive disjunctions. In this work, we describe a new method of prenexing with a particular pattern. Our experimental results show the interest of such a method with actual QBF solvers.