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42
Decomposable negation normal form
 Journal of the ACM
, 2001
"... 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 offline compilation phase and an online quer ..."
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Cited by 133 (19 self)
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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 offline compilation phase and an online queryanswering phase. In the offline phase, the propositional theory is compiled into some target language, which is typically a tractable one. In the online 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 offline phase, in order to amortize that overhead over all online queries. Another motivation behind compilation is to produce very simple online reasoning systems, which can be embedded costeffectively 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
AND/OR Search Spaces for Graphical Models
, 2004
"... The paper introduces an AND/OR search space perspective for graphical models that include probabilistic networks (directed or undirected) and constraint networks. In contrast to the traditional (OR) search space view, the AND/OR search tree displays some of the independencies present in the gr ..."
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Cited by 126 (47 self)
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The paper introduces an AND/OR search space perspective for graphical models that include probabilistic networks (directed or undirected) and constraint networks. In contrast to the traditional (OR) search space view, the AND/OR search tree displays some of the independencies present in the graphical model explicitly and may sometime reduce the search space exponentially. Indeed, most
On the Tractable Counting of Theory Models and its Application to Truth Maintenance and Belief Revision
 Journal of Applied NonClassical Logics
, 2000
"... We address the problem of counting the models of a propositional theory, under incremental changes to the theory. Specifically, we show that if a propositional theory is in a special form that we call smooth, deterministic, decomposable negation normal form (sdDNNF), then for any consistent set of ..."
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Cited by 60 (19 self)
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We address the problem of counting the models of a propositional theory, under incremental changes to the theory. Specifically, we show that if a propositional theory is in a special form that we call smooth, deterministic, decomposable negation normal form (sdDNNF), then for any consistent set of literals S, we can simultaneously count, in time linear in the size of , the models of: [ S; [ S [ flg: for every literal l 62 S; [ S n flg: for every literal l 2 S; [ S n flg [ f:lg: for every literal l 2 S.
Boolean Expression Diagrams
, 1997
"... This paper presents a new data structure called Boolean Expression Diagrams (BEDs) for representing and manipulating Boolean functions. BEDs are a generalization of Binary Decision Diagrams (BDDs) which can represent any Boolean circuit in linear space and still maintain many of the desirable proper ..."
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Cited by 54 (5 self)
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This paper presents a new data structure called Boolean Expression Diagrams (BEDs) for representing and manipulating Boolean functions. BEDs are a generalization of Binary Decision Diagrams (BDDs) which can represent any Boolean circuit in linear space and still maintain many of the desirable properties of BDDs. Two algorithms are described for transforming a BED into a reduced ordered BDD. One is a generalized version of the BDD applyoperator while the other can exploit the structural information of the Boolean expression. This ability is demonstrated by verifying that two di erent circuit implementations of a 16bit multiplier implement the same Boolean function. Using BEDs, this veri cation problem is solved in less than a second, while using standard BDD techniques this problem is infeasible. Generally, BEDs are useful in applications, for example tautology checking, where the endresult as a reduced ordered BDD is small.
Efficient Boolean Manipulation with OBDD's Can be Extended to FBDD's
, 1993
"... OBDD's are the stateoftheart data structure for Boolean function manipulation since basic tasks of Boolean manipulation such as testing equivalence, satisfiability, or tautology, and performing single Boolean synthesis steps can be done efficiently. In the following we show that the efficien ..."
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Cited by 36 (0 self)
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OBDD's are the stateoftheart data structure for Boolean function manipulation since basic tasks of Boolean manipulation such as testing equivalence, satisfiability, or tautology, and performing single Boolean synthesis steps can be done efficiently. In the following we show that the efficient manipulation of OBDD's can be extended to a more general data structure, socalled FBDD's. In detail, the advantages of using FBDD's instead of OBDD's are ffl FBDD's are generally more (sometimes even exponentially more) succinct than OBDD's, ffl FBDD's provide, similarly to OBDD's, canonical representations of Boolean functions, and ffl in terms of FBDD's basic tasks of Boolean manipulation can be performed similarly efficient as in terms of OBDD's. The power of the FBDDconcept is demonstrated by showing that the verification of the benchmark circuit design for the hidden weighted bit function HWB proposed by Bryant can be carried out efficiently in terms of FBDD's while, for princip...
Binary decision diagrams in theory and practice
, 2001
"... Decision diagrams (DDs) are the stateoftheart data structure in VLSI CAD and have been successfully applied in many other fields.DDs are widely used and are also integrated in commercial tools.This special section comprises six contributed articles on various aspects of the theory and application ..."
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Cited by 31 (7 self)
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Decision diagrams (DDs) are the stateoftheart data structure in VLSI CAD and have been successfully applied in many other fields.DDs are widely used and are also integrated in commercial tools.This special section comprises six contributed articles on various aspects of the theory and application of DDs.As preparation for these contributions, the present article reviews the basic definitions of binary decision diagrams (BDDs). We provide a brief overview and study theoretical and practical aspects.Basic properties of BDDs are discussed and manipulation algorithms are described.Extensions of BDDs are investigated and by this we give a deeper insight into the basic data structure.Finally we outline several applications of BDDs and their extensions and suggest a number of articles and books for those who wish to pursue the topic in more depth.
On the Relation Between BDDs and FDDs
 INFORMATION AND COMPUTATION
, 1995
"... Data structures for Boolean functions build an essential component of design automation tools, especially in the area of logic synthesis. The state of the art data structure is the ordered binary decision diagram (OBDD), which results from general binary decision diagrams (BDDs), also called bran ..."
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Cited by 30 (13 self)
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Data structures for Boolean functions build an essential component of design automation tools, especially in the area of logic synthesis. The state of the art data structure is the ordered binary decision diagram (OBDD), which results from general binary decision diagrams (BDDs), also called branching programs, by ordering restrictions. In the context of EXORbased logic synthesis another type of decision diagram (DD), called (ordered) functional decision diagram ((O)FDD) becomes increasingly important. We study the relation between (ordered, free) BDDs and FDDs. Both, BDDs and FDDs, result from DDs by defining the represented function in different ways. If the underlying DD is complete, the relation between both types of interpretation can be described by a Boolean transformation . This allows us to relate the FDDsize of f and the BDDsize of (f) also in the case that the corresponding DDs are free or ordered, but not (necessarily) complete. We use this property to derive...
BDDbased cryptanalysis of keystream generators
 Advances in Cryptology – EUROCRYPT’02, LNCS 1462
, 2002
"... Abstract. Many of the keystream generators which are used in practice are LFSRbased in the sense that they produce the keystream according to a rule y = C(L(x)), where L(x) denotes an internal linear bitstream, produced by a small number of parallel linear feedback shift registers (LFSRs), and C de ..."
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Cited by 29 (1 self)
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Abstract. Many of the keystream generators which are used in practice are LFSRbased in the sense that they produce the keystream according to a rule y = C(L(x)), where L(x) denotes an internal linear bitstream, produced by a small number of parallel linear feedback shift registers (LFSRs), and C denotes some nonlinear compression function. We present an n O(1) 2 (1−α)/(1+α)n time bounded attack, the FBDDattack, against LFSRbased generators, which computes the secret initial state x ∈ {0, 1} n from cn consecutive keystream bits, where α denotes the rate of information, which C reveals about the internal bitstream, and c denotes some small constant. The algorithm uses Free Binary Decision Diagrams (FBDDs), a data structure for minimizing and manipulating Boolean functions. The FBDDattack yields better bounds on the effective key length for several keystream generators of practical use, so a 0.656n bound for the selfshrinking generator, a 0.6403n bound for the A5/1 generator, used in the GSM standard, a 0.6n bound for the E0 encryption standard in the one level mode, and a 0.8823n bound for the twolevel E0 generator used in the Bluetooth wireless LAN system. 1
Equivalence Checking of Combinational Circuits using Boolean Expression Diagrams
 IEEE Transactions on Computer Aided Design
, 1999
"... The combinational logiclevel equivalence problem is to determine whether two given combinational circuits implement the same Boolean function. This problem arises in a number of CAD applications, for example when checking the correctness of incremental design changes (performed either manually or b ..."
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Cited by 18 (3 self)
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The combinational logiclevel equivalence problem is to determine whether two given combinational circuits implement the same Boolean function. This problem arises in a number of CAD applications, for example when checking the correctness of incremental design changes (performed either manually or by a design automation tool). This paper introduces a data structure called Boolean Expression Diagrams (BEDs) and two algorithms for transforming a BED into a Reduced Ordered Binary Decision Diagram (OBDD). BEDs are capable of representing any Boolean circuit in linear space and can exploit structural similarities between the two circuits that are compared. These properties make BEDs suitable for verifying the equivalence of combinational circuits. BEDs can be seen as an intermediate representation between circuits (which are compact) and OBDDs (which are canonical). Based on a large number of combinational circuits, we demonstrate that BEDs either outperform or achieve results comparable to...
Knowledge compilation meets database theory : Compiling queries to decision diagrams. (under review
, 2010
"... The goal of Knowledge Compilation is to represent a Boolean expression in a format in which it can answer a range of onlinequeries in PTIME. The onlinequery of main interest to us is model counting, because of its application to query evaluation on probabilistic databases, but other onlinequeries ..."
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Cited by 17 (4 self)
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The goal of Knowledge Compilation is to represent a Boolean expression in a format in which it can answer a range of onlinequeries in PTIME. The onlinequery of main interest to us is model counting, because of its application to query evaluation on probabilistic databases, but other onlinequeries can be supported as well such as testing for equivalence, testing for implication, etc. In this paper we study the following problem. Given a database query q, decide whether its lineage can be compiled efficiently into a given target language. We consider four target languages, of strictly increasing expressive power(when the size of compilation is constrained to be polynomial in the input size): ReadOnce Boolean formulae, OBDD, FBDD and dDNNF. For each target, we study the class of database queries that admit polynomial size representation: these queries can also be evaluated in PTIME over probabilistic databases. When queries are restricted to conjunctive queries without selfjoins, it was known that these four classes collapse to the class of hierarchical queries, which is also the class of PTIME queries over probabilistic databases. Our main result in this paper is that, in the case of Unions of Conjunctive Queries (UCQ), these classes form a strict hierarchy. Thus, unlike conjunctive queries without selfjoins, the expressive power of UCQ differs considerably w.r.t. these target compilation languages. Moreover, we give a complete characterization of the first two target languages, based on the query’s syntax.