Free binary decision diagrams for the computation of EAR n

Free binary decision diagrams (FBDDs) are graph-based data structures representing Boolean functions with the constraint (additional to binary decision diagram) that each variable is tested at most once during the computation. The function EAR_n is the following Boolean function defined for n × n Boolean matrices: EAR_n(M) = 1 iff the matrix M contains two equal adjacent rows. We prove that each FBDD computing EAR_n has size at least and we present a construction of such diagrams of size approximately .     

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.     

A uniform framework for weighted decision diagrams and its implementation

This paper introduces a generic framework for OBDD variants with weighted edges. It covers many boolean and multi-valued OBDD-variants that have been studied in the literature and applied to the symbolic representation and manipulation of discrete functions. Our framework supports reasoning about reducedness and canonicity of new OBDD-variants and provides a platform for the implementation and comparison of OBDD-variants. Furthermore, we introduce a new multi-valued OBDD-variant, called normalized algebraic decision diagram, which supports min/max-operations and turns out to be very useful for, e.g., integer linear programming and model checking probabilistic systems. Finally, we explain the main features of an implementation of a newly developed BDD-package, called JINC, which relies on our generic notion of weighted decision diagrams, and realizes various synthesis algorithms, reordering techniques and transformation algorithms for boolean and multi-terminal OBDDs, with or without edge-attributes, and their zero-suppressed variants.     

Polynomial-Size Binary Decision Diagrams for the Exactly Half-<Emphasis Type="Italic">d</Emphasis>-Hyperclique Problem Reading Each Input Bit Twice

A binary decision diagram (BDD) is a graph-based data structure representing Boolean functions; ℓ-BDDs are BDDs with an additional restriction that each input bit can be tested at most ℓ times. A d-uniform hypergraph H on N vertices is an exactly half-d-hyperclique if N/2 of its vertices form a hyperclique and the remaining vertices are isolated. Wegener [J. ACM 35(2), 461–471 (1988)] conjectured that there is no polynomial-size (d−1)-BDD for the Exactly half-d-hyperclique problem. We refute this conjecture by constructing polynomial-size (syntactic) 2-BDDs for the Exactly half-d-hyperclique problem for every d≥2. Institute for Theoretical Computer Science (ITI) is supported by Ministry of Education of Czech Republic as projects LN00A056 and 1M0545.     

Introduction to the OBDD algorithm for the ATP community

We describe in terms familiar to the automated reasoning community the graph-based algorithm for deciding propositional equivalence published by R. E. Bryant in 1986. Such algorithms, based onordered binary decision diagrams orOBDDs, are among the fastest known ways to decide whether two propositional expressions are equivalent and are generally hundreds or thousands of times faster on such problems than most automatic theorem-proving systems. An OBDD is a normalized IF (if-then-else) expression in which the tests down any branch are ascending in some previously chosen fixed order. Such IF expressions represent a canonical form for propositional expressions. Three coding tricks make it extremely efficient to manipulate canonical IF expressions. The first is that two canonicalized expressions can be rapidly combined to form the canonicalized form of their disjunction (conjunction, exclusive-or, etc.) by exploiting the fact that the tests are ordered. The second is that every distinct canonical IF expression should be assigned a unique integer index to enable fast recognition of identical forms. The third trick is that the operation in which one combines canonicalized subterms term should be memo-ized or cached so that if the same operation is required in the future its result can be looked up rather than recomputed.This work was supported in part at Computational Logic, Inc., by the National Computer Security Center (Contract MDA904-92-C-5167). The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of Computational Logic, Inc., the National Computer Security Center, or the U.S. Government.     

The semantics of augmented constraint diagrams

Constraint diagrams are a diagrammatic notation which may be used to express logical constraints. They generalize Venn diagrams and Euler circles, and include syntax for quantification and navigation of relations. The notation was designed to complement the Unified Modelling Language in the development of software systems.Since symbols representing quantification in a diagrammatic language can be naturally ordered in multiple ways, some constraint diagrams have more than one intuitive meaning in first-order predicate logic. Any equally expressive notation which is based on Euler diagrams and conveys logical statements using explicit quantification will have to address this problem.We explicitly augment constraint diagrams with reading trees, which provides a partial ordering for the quantifiers (determining their scope as well as their relative ordering). Alternative approaches using spatial arrangements of components, or alphabetical ordering of symbols, for example, can be seen as implicit representations of a reading tree.Whether the reading tree accompanies the diagram explicitly (optimizing expressiveness) or implicitly (simplifying diagram syntax), we show how to construct unambiguous semantics for the augmented constraint diagram.     

Set space diagrams

This paper introduces set space diagrams and defines their formal syntax and semantics. Conventional region based diagrams, like Euler circles and Venn diagrams, represent sets and their intersections by means of overlapping regions. By contrast, set space diagrams provide a certain layout that avoids overlapping geometrical entities. This enables the representation of a good deal of sets without getting diagrams which are cluttered due to overlapping regions. In particular, these diagrams can be employed for illustration purposes, e.g., for showing the laws of Boolean algebras. Additionally, cardinalities are represented and can be easily compared; inferences can be drawn to derive unknown cardinalities from a given knowledge base. The soundness of set space diagrams is shown with respect to their set-theoretic interpretation.     

Nondeterministic ordered binary decision diagrams with repeated tests and various modes of acceptance

Ordered binary decision diagrams with repeated tests are considered both in complexity theory and in applications. Bollig et al. have proved in [B. Bollig, M. Sauerhoff, D. Sieling, I. Wegener, Hierarchy theorems of kOBDDs and kIBDDs, Theoret. Comput. Sci. 205 (1998) 45-60] a tight hierarchy result for the classes of functions representable by k layers of polynomial-size deterministic ordered binary decision diagrams. In this paper the nondeterministic case is investigated, where the layers are driven by one and the same variable ordering. For k being a constant, it is shown that for the existential, the parity-, and the majority acceptance mode the analogous hierarchy collapses.     

A Unifying Approach to Edge-valued and Arithmetic Transform Decision Diagrams

This paper shows that binary decision diagrams (BDDs) and their generalizations are not only representations of switching and integer-valued functions, but also Fourier-like series expansions of them. Furthermore, it shows that edge-valued binary decision diagrams (EVBDDs) are related to arithmetic transform decision diagrams (ACDDs), which are the integer counterparts of the functional decision diagrams (FDDs). Finally, it shows that the complexity of multi-terminal binary decision diagrams (MTBDDs), EVBDDs and ACDDs of a function f depends on the structure of the truth-vector of f, partial arithmetic transform spectra of f and the arithmetic transform spectrum of f, respectively.     

Approximate steady-state analysis of large Markov models based on the structure of their decision diagram encoding

We propose a new approximate numerical algorithm for the steady-state solution of general structured ergodic Markov models. The approximation uses a state-space encoding based on multiway decision diagrams and a transition rate encoding based on a new class of edge-valued decision diagrams. The new method retains the favorable properties of a previously proposed Kronecker-based approximation, while eliminating the need for a Kronecker-consistent model decomposition. Removing this restriction allows for a greater utilization of event locality, which facilitates the generation of both the state-space and the transition rate matrix, thus extends the applicability of this algorithm to larger and more complex models.     

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 ²n/768−1 on the size of OBDDs representing the graph of integer multiplication is improved up to ²n/24.     

Symbolic Verification and Analysis of Discrete Timed Systems

This paper presents a novel approach for real-time model checking. It combines the efficiency of traditional symbolic model checking with possibilities to describe and specify real-time systems. Using multi-terminal binary decision diagrams to represent time and time intervals, it becomes possible to transfer efficient algorithms and optimization heuristics known from standard CTL model checking to real-time applications. By introducing a new variant of models called I/O-interval structures we can describe systems in a modular way. Interval structures allow model composition of real-time structures such that state explosion effects are greatly reduced. Besides model checking we also present analysis algorithms which allow to compute key properties like system latencies and minimal response times from the structures describing the system. The practical applicability is proven by experimental results, computed by the verification system RAVEN, which implements all described algorithms, including counterexample generation and waveform visualization.     

Finding robust solutions in requirements models

Solutions to non-linear requirements engineering problems may be “brittle”; i.e. small changes may dramatically alter solution effectiveness. Hence, it is not enough to just generate solutions to requirements problems- we must also assess solution robustness. The KEYS2 algorithm can generate decision ordering diagrams. Once generated, these diagrams can assess solution robustness in linear time. In experiments with real-world requirements engineering models, we show that KEYS2 can generate decision ordering diagrams in O(N ²). When assessed in terms of terms of (a) reducing inference times, (b) increasing solution quality, and (c) decreasing the variance of the generated solution, KEYS2 out-performs other search algorithms (simulated annealing, ASTAR, MaxWalkSat).     

Efficient manipulation of decision diagrams

Over the last decade significant progress has been made in the efficient implementation of algorithms for the manipulation of decision diagrams. We review the main issues involved in designing a decision diagram package, with special emphasis on the basic data structures and their management, on fast variable reordering, and on the collection of statistics to guide the tuning of the algorithms. Our analysis focuses on depth-first manipulation of reduced, ordered binary decision diagrams. Published online: 15 May 2001     

Probabilistic symbolic model checking with PRISM: a hybrid approach

In this paper we present efficient symbolic techniques for probabilistic model checking. These have been implemented in PRISM, a tool for the analysis of probabilistic models such as discrete-time Markov chains, continuous-time Markov chains and Markov decision processes using specifications in the probabilistic temporal logics PCTL and CSL. Motivated by the success of model checkers such as SMV which use BDDs (binary decision diagrams), we have developed an implementation of PCTL and CSL model checking based on MTBDDs (multi-terminal BDDs) and BDDs. Existing work in this direction has been hindered by the generally poor performance of MTBDD-based numerical computation, which is often substantially slower than explicit methods using sparse matrices. The focus of this paper is a novel hybrid technique which combines aspects of symbolic and explicit approaches to overcome these performance problems. For typical examples, we achieve a dramatic improvement over the purely symbolic approach. In addition, thanks to the compact model representation using MTBDDs, we can verify systems an order of magnitude larger than with sparse matrices, while almost matching or even beating them for speed.     

Topological Function Optimization for Continuous Shape Matching

We present a novel approach for optimizing real‐valued functions based on a wide range of topological criteria. In particular, we show how to modify a given function in order to remove topological noise and to exhibit prescribed topological features. Our method is based on using the previously‐proposed persistence diagrams associated with real‐valued functions, and on the analysis of the derivatives of these diagrams with respect to changes in the function values. This analysis allows us to use continuous optimization techniques to modify a given function, while optimizing an energy based purely on the values in the persistence diagrams. We also present a procedure for aligning persistence diagrams of functions on different domains, without requiring a mapping between them. Finally, we demonstrate the utility of these constructions in the context of the functional map framework, by first giving a characterization of functional maps that are associated with continuous point‐to‐point correspondences, directly in the functional domain, and then by presenting an optimization scheme that helps to promote the continuity of functional maps, when expressed in the reduced basis, without imposing any restrictions on metric distortion. We demonstrate that our approach is efficient and can lead to improvement in the accuracy of maps computed in practice.     

Calculation of the paired Haar transform through shared binary decision diagrams

A concept of paired Haar transform (PHT) for representation and efficient optimization of systems of incompletely Boolean functions has recently been introduced. In this article, a method to calculate PHT for incompletely specified switching functions through shared binary decision diagrams (SBDDs) is presented. The algorithm converts switching functions in the form of SBDDs into their paired Haar spectra and can operate on functions with many variables.     

Visualizing Flow Diagrams in WebSphere Studio Using SHriMP Views

This paper describes the integration of an information visualization tool, called SHriMP Views, with IBM WebSphere Studio Application Developer Integration Edition, which was developed with Eclipse technology. Although SHriMP was originally developed for visualizing programs, it is content-independent. We have re-targeted SHriMP for visualizing flow diagrams. Flow diagrams, as supported by WebSphere Studio Application Developer Integration Edition, can be hierarchically composed, thus leveraging the key features of SHriMP that allow a user to easily navigate hierarchically composed information spaces. We discuss the differences between programs and flow diagrams, in terms of their semantics and their visual representation. We also report on the main technical challenges we faced, due to the different widget sets used by SHriMP (Swing/AWT) and Eclipse (SWT).     

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.     

Exponential space complexity for OBDD-based reachability analysis

Ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Nevertheless, many basic graph problems are known to be PSPACE-hard if their input graphs are represented by OBDDs. Computing the set of nodes that are reachable from some source s∈V in a digraph G=(V,E) is an important problem in computer-aided design, hardware verification, and model checking. Until now only exponential lower bounds on the space complexity of a restricted class of OBDD-based algorithms for the reachability problem have been known. Here, the result is extended by presenting an exponential lower bound for the general reachability problem. As a by-product a general exponential lower bound is obtained for the computation of OBDDs representing all connected node pairs in a graph, the transitive closure.