Ordered Binary Decision Diagrams Size Research Articles

Second-Order Finite Automata

Traditionally, finite automata theory has been used as a framework for the representation of possibly infinite sets of strings. In this work, we introduce the notion of second-order finite automata, a formalism that combines finite automata with ordered decision diagrams, with the aim of representing possibly infinite sets of sets of strings. Our main result states that second-order finite automata can be canonized with respect to the second-order languages they represent. Using this canonization result, we show that sets of sets of strings represented by second-order finite automata are closed under the usual Boolean operations, such as union, intersection, difference and even under a suitable notion of complementation. Additionally, emptiness of intersection and inclusion are decidable. We provide two algorithmic applications for second-order automata. First, we show that several width/size minimization problems for deterministic and nondeterministic ODDs are solvable in fixed-parameter tractable time when parameterized by the width of the input ODD. In particular, our results imply FPT algorithms for corresponding width/size minimization problems for ordered binary decision diagrams (OBDDs) with a fixed variable ordering. Previously, only algorithms that take exponential time in the size of the input OBDD were known for width minimization, even for OBDDs of constant width. Second, we show that for each k and w one can count the number of distinct functions computable by ODDs of width at most w and length k in time h(|Σ|,w) ⋅ kO(1), for a suitable h:mathbb {N}times mathbb {N}rightarrow mathbb {N}. This improves exponentially on the time necessary to explicitly enumerate all such functions, which is exponential in both the width parameter w and in the length k of the ODDs.

Basing Decisions on Sentences in Decision Diagrams

The Sentential Decision Diagram (SDD) is a recently proposed representation of Boolean functions, containing Ordered Binary Decision Diagrams (OBDDs) as a distinguished subclass. While OBDDs are characterized by total variable orders, SDDs are characterized by dissections of variable orders, known as vtrees. Despite this generality, SDDs retain a number of properties, such as canonicity and a polytime apply operator, that have been critical to the practical success of OBDDs. Moreover, upper bounds on the size of SDDs were also given, which are tighter than comparable upper bounds on the size of OBDDs. In this paper, we analyze more closely some of the theoretical properties of SDDs and their size. In particular, we consider the impact of basing decisions on sentences (using dissections as in SDDs), in comparison to basing decisions on variables (using total variable orders as in OBDDs). Here, we identify a class of Boolean functions where basing decisions on sentences using dissections of a variable order can lead to exponentially more compact SDDs, compared to OBDDs based on the same variable order. Moreover, we identify a fundamental property of the decompositions that underlie SDDs and use it to show how certain changes to a vtree can also lead to exponential differences in the size of an SDD.

On the Relative Succinctness of Sentential Decision Diagrams

Sentential decision diagrams (SDDs) introduced by Darwiche in 2011 are a promising representation language for propositional knowledge bases. The relative succinctness of representation languages is an important subject in knowledge compilation. The aim of the paper is to identify which kind of Boolean functions can be represented by SDDs of small size with respect to the number of variables the functions are defined on. For this reason the sets of Boolean functions representable by different representation languages in polynomial size are investigated and SDDs are compared with representation languages from the classical knowledge compilation map of Darwiche and Marquis. Ordered binary decision diagrams (OBDDs) which are a popular data structure for Boolean functions are one of them. SDDs are more general than OBDDs by definition but only recently, a Boolean function was presented with polynomial SDD size but exponential OBDD size. This result is strengthened in several ways. The main result is that a function can be represented by SDDs of small size if the function and its negation have small restricted nondeterministic OBDD representations. Moreover, for important Boolean functions called storage access function polynomial-size SDDs are presented. As a side effect an open problem about the relative succinctness between SDDs and free binary decision diagrams which are more general than OBDDs is answered.

Randomized OBDD-based graph algorithms

Implicit graph algorithms deal with the characteristic function χE of the edge set E of a graph G=(V,E). Encoding the nodes by binary vectors, χE can be represented by an Ordered Binary Decision Diagram (OBDD) which is a well known data structure for Boolean functions. OBDD-based graph algorithms solve graph optimization problems by mainly using functional operations and are a heuristic approach to cope with massive graphs. These algorithms heavily rely on a compact representation of the underlying Boolean functions which is why all previously known OBDD-based algorithms are deterministic since random functions are not compressible in general. Here, the first randomized OBDD-based algorithms are presented where random functions with limited independence are used to overcome the large representation size. On the theoretical part, we investigate the size of OBDDs representing k-wise independent random functions. In addition, we give a construction of almost k-wise independent random functions by means of a random OBDD generation and show an almost matching lower bound of the OBDD size representing such random functions. On the algorithmic part, randomization often facilitates the design of simple algorithms which in the context of OBDD-based algorithms means a small number of functional operations and as few input variables of the used Boolean functions as possible. We present two randomized implicit algorithms: A randomized minimum spanning tree algorithm and a maximal matching algorithm. The matching algorithm needs O(log3⁡|V|) functional operations in expectation and uses functions with at most 3log⁡|V| variables which is both better than the best known deterministic algorithms w.r.t. functional operations and variables. The algorithm may be of independent interest. The experimental evaluation of the algorithms shows that they outperform known OBDD-based algorithms for the maximal matching problem and the minimum spanning tree problem.

On the OBDD representation of some graph classes

Ordered Binary Decision Diagrams (OBDDs) are a popular data structure for Boolean functions. Given the rapid growth of application-based networks, an heuristic approach to deal with very large structured graphs are implicit OBDD-based graph algorithms. Vertices of an input graph are binary encoded and the edge set is represented by its characteristic function. Since OBDDs are able to take advantage of the presence of regular substructures, this approach leads sometimes to sublinear graph representations. By simple counting arguments it is easy to see that almost all graphs on N vertices cannot be represented by OBDDs of polylogarithmic size with respect to N. On the other hand, very simply structured graphs like grid graphs have small OBDD representations. Here, the investigation for which significant graph classes succinct OBDD representations are possible is continued. Using a new method how to use the structure of the adjacency matrix representation of a graph to count subfunctions of the characteristic Boolean function of its edge set, known upper bounds on the OBDD size of interval graphs are improved. Furthermore, the OBDD size for interval bigraphs, (bi)convex graphs, bipartite permutation graphs, chain graphs, threshold graphs, and trees is analyzed. Except for interval graphs, interval bigraphs, and convex graphs the presented bounds are tight. Finally, regular graphs are considered and examples are presented with small and large OBDD size. Moreover, a variable ordering often used in the implicit setting is investigated.

SDDs Are Exponentially More Succinct than OBDDs

Introduced by Darwiche (2011), sentential decision diagrams (SDDs) are essentially as tractable as ordered binary decision diagrams (OBDDs), but tend to be more succinct in practice. This makes SDDs a prominent representation language, with many applications in artificial intelligence and knowledge compilation. We prove that SDDs are more succinct than OBDDs also in theory, by constructing a family of boolean functions where each member has polynomial SDD size but exponential OBDD size. This exponential separation improves a quasipolynomial separation recently established by Razgon (2014), and settles an open problem in knowledge compilation (Darwiche, 2011).

On the Minimization of (Complete) Ordered Binary Decision Diagrams

Ordered binary decision diagrams (OBDDs) are a popular data structure for Boolean functions. Some applications work with a restricted variant called complete OBDDs which is strongly related to nonuniform deterministic finite automata. One of its complexity measures is the width which has been investigated in several areas in computer science like machine learning, property testing, and the design and analysis of implicit graph algorithms. For a given function the size and the width of a (complete) OBDD is very sensitive to the choice of the variable ordering but the computation of an optimal variable ordering for the OBDD size is known to be NP-hard. Since optimal variable orderings with respect to the OBDD size are not necessarily optimal for the complete model or the OBDD width and hardly anything about the relation between optimal variable orderings with respect to the size and the width is known, this relationship is investigated. Here, using a new reduction idea it is shown that the size minimization problem for complete OBDDs and the width minimization problem are NP-hard.

An Approach to Representing Access Control Policies for Improving DFANS

Trust negotiation is an approach that establishes mutual trust between strangers wishing to share resources or conduct business by gradually requesting and disclosing credentials in the Internet. In trust negotiation, negotiation strategies must be adopted to determine the search for a successful negotiation. Deterministic Finite Automaton Negotiation Strategy (DFANS) is a complete and highly efficient negotiation strategy proposed recently. DFANS takes advantage of ordered binary decision diagrams (OBDDs) to represent access control policies. However, the size of OBDDs is highly sensitive to the scanning order of resources in access control policies, which may result in exponentially large OBDDs in the worst case. To address this challenging issue, in this paper, a new data structure called access control policy diagrams (ACPDs) is proposed to take place OBDDs in DFANS. ACPDs are cyclic, directed graphs, whose sizes do not sensitive to the scan order of resources in access control policies, and the number of nodes in an ACPD is not more than the number of resources in the access control policy. Further, an efficient simplified algorithm is developed to reduce the number of nodes and shorten the path length when referring to redundant vertices in ACPDs.

An asymptotically optimal lower bound on the OBDD size of the middle bit of multiplication for the pairwise ascending variable order

We prove that each OBDD (ordered binary decision diagram) for the middle bit of n -bit integer multiplication for one of the variable orders which so far achieve the smallest OBDD sizes with respect to asymptotic order of growth, namely the pairwise ascending order x 0 , y 0 , … , x n − 1 , y n − 1 , requires a size of Ω ( 2 ( 6 / 5 ) n ) . This is asymptotically optimal due to a bound of the same order by Amano and Maruoka (2007) [1].

New Results on the Most Significant Bit of Integer Multiplication

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations and ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Analyzing the limits of symbolic graph algorithms for the reachability problem Sawitzki (Proc. of LATIN, LNCS, vol. 3887, pp. 781–792, Springer, Berlin, 2006) has presented the first exponential lower bound on the π-OBDD size for the most significant bit of integer multiplication according to one predefined variable order π. Since the choice of the variable order is a main issue to obtain OBDDs of small size the investigation is continued. As a result a new upper bound method and the first non-trivial upper bound on the size of OBDDs according to an arbitrary variable order is presented. Furthermore, Sawitzki’s lower bound is improved.

Exact OBDD Bounds for Some Fundamental Functions

Ordered binary decision diagrams (OBDDs) are nowadays one of the most common dynamic data structures or representation types for Boolean functions. Among the many areas of application are verification, model checking, computer aided design, relational algebra, and symbolic graph algorithms. Although many exponential lower bounds on the OBDD size of Boolean functions are known, there are only few functions where the OBDD size is asymptotically known exactly. In this paper the exact OBDD sizes of the fundamental functions multiplexer and addition of n-bit numbers are determined.

On the size of (generalized) OBDDs for threshold functions

Ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Among the many areas of application are hardware verification, model checking, and symbolic graph algorithms. Threshold functions are the basic functions for discrete neural networks and are used as building blocks in the design of some symbolic graph algorithms. In this paper the first exponential lower bound on the size of a more general model than OBDDs and the first nontrivial asymptotically optimal bound on the OBDD size for a threshold function are presented.

A note on the size of OBDDs for the graph of integer multiplication

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations and ordered binary decision diagrams (OBDDs) are the most common dynamic data structure for Boolean functions. Among the many areas of application are verification, model checking, computer-aided design, relational algebra, and symbolic graph algorithms. Analyzing the limits of symbolic graph algorithms for the all-pairs-shortest paths problem which work on OBDD-represented graph instances the so-called graph of integer multiplication has been investigated by Sawitzki [D. Sawitzki, Lower bounds on the OBDD size of graphs of some popular functions, in: Proc. of SOFSEM, LNCS, vol. 3381, 2005, pp. 298–309]. Using simple arguments his lower bound of 2 n / 768 − 1 on the size of OBDDs representing the graph of integer multiplication is improved up to 2 n / 24 .

On the OBDD size for graphs of bounded tree- and clique-width

We study the size of OBDDs (ordered binary decision diagrams) for representing the adjacency function f G of a graph G on n vertices. Our results are as follows: – for graphs of bounded tree-width there is an OBDD of size O ( log n ) for f G that uses encodings of size O ( log n ) for the vertices; – for graphs of bounded clique-width there is an OBDD of size O ( n ) for f G that uses encodings of size O ( n ) for the vertices; – for graphs of bounded clique-width such that there is a clique-width expression for G whose associated binary tree is of depth O ( log n ) there is an OBDD of size O ( n ) for f G that uses encodings of size O ( log n ) for the vertices; – for cographs, i.e. graphs of clique-width at most 2, there is an OBDD of size O ( n ) for f G that uses encodings of size O ( log n ) for the vertices. This last result complements a recent result by Nunkesser and Woelfel [R. Nunkesser, P. Woelfel, Representation of graphs by OBDDs, in: X. Deng, D. Du (Eds.), Proceedings of ISAAC 2005, in: Lecture Notes in Computer Science, vol. 3827, Springer, 2005, pp. 1132–1142] as it reduces the size of the OBDD by an O ( log n ) factor using encodings whose size is increased by an O ( 1 ) factor.

Better upper bounds on the QOBDD size of integer multiplication

We show that the middle bit of the multiplication of two n-bit integers can be computed by an ordered binary decision diagram (OBDD) of size less than 2.8 · 2 6 n / 5 . This improves the previously known upper bound of ( 7 3 ) · 2 4 n / 3 by Woelfel (New Bounds on the OBDD-size of integer multiplication via Universal Hashing, J. Comput. System Sci. 71(4) (2005) 520–534). The experimental results suggest that our exponent of 6 n / 5 is optimal or at least very close to optimal. A general upper bound of O ( 2 3 n / 2 ) on the OBDD size of each output bit of the multiplication is also presented.

Lower bounds on the OBDD size of two fundamental functions' graphs

Ordered Binary Decision Diagrams (OBDDs) are a data structure for Boolean functions which supports many useful operations. Among others it finds applications in CAD, model checking, and symbolic graph algorithms. Nevertheless, many simple functions are known to have exponential OBDD size with respect to their number of variables. In order to investigate the limits of symbolic graph algorithms which work on OBDD-represented graph instances, it is useful to have simply-structured graphs whose OBDD representation has exponential size. Therefore, we consider two fundamental functions with exponential lower bounds on their OBDD size and transfer these results to their corresponding graphs. Concretely, we consider the Indirect Storage Access function and the Hidden Weighted Bit function.

Ordered binary decision diagrams and the Shannon effect

We investigate the size and structure of ordered binary decision diagrams (OBDDs) for random Boolean functions. It was known that for most values of n, the expected OBDD size of a random Boolean function with n variables is equal to the worst-case size up to terms of lower order. Such a phenomenon is generally called strong Shannon effect. Here we show that the strong Shannon effect is not valid for all n. Instead it undergoes a certain periodic ‘phase transition’: If n lies within intervals of constant width around the values n=2 h + h, then the strong Shannon effect does not hold, whereas it does hold outside these intervals. Our analysis provides doubly exponential probability bounds and generalises to ordered Kronecker functional decision diagrams (OKFDDs).

The Nonapproximability of OBDD Minimization

The size of ordered binary decision diagrams (OBDDs) is determined by the chosen variable ordering. A poor choice may cause an OBDD to be too large to fit into the available memory. The decision variant of the variable ordering problem is known to be NP-complete. We strengthen this result by showing that, unless P=NP, for each constant c>1 there is no polynomial time approximation algorithm with the performance ratio c for the variable ordering problem, i.e., no polynomial time algorithm that guarantees the computation of a variable ordering so that the resulting OBDD size is larger than the minimum size by a factor of at most c. This result justifies, also from a theoretical point of view, the use of heuristics for the variable ordering problem.

A Comparison of Free BDDs and Transformed BDDs

Ordered binary decision diagrams (OBDDs) introduced by Bryant (IEEE Trans. on Computers, Vol. 35, pp. 677–691, 1986) have found a lot of applications in verification and CAD. Their use is limited if the OBDD size of the considered functions is too large. Therefore, a variety of generalized BDD models has been presented, among them FBDDs (free BDDs) and TBDDs (transformed BDDs). Here the quite tight relations between these models are revealed and their advantages and disadvantages are discussed.

Asymptotically Optimal Bounds for OBDDs and the Solution of Some Basic OBDD Problems

Ordered binary decision diagrams (OBDDs) are nowadays the most common dynamic data structure or representation type for Boolean functions. Among the many areas of application are verification, model checking, and computer aided design. For many functions it is easy to estimate the OBDD size but asymptotically optimal bounds are only known in simple situations. In this paper, methods for proving asymptotically optimal bounds are presented and applied to the solution of some basic problems concerning OBDDs. The largest size increase by a synthesis step of π-OBDDs followed by an optimal reordering is determined as well as the largest ratio of the size of deterministic finite automata, quasi-reduced OBDDs, and zero-suppressed BDDs compared to the size of OBDDs. Moreover, the worst case OBDD size of functions with a given number of 1-inputs is investigated.