Multi-terminal Binary Decision Diagrams Research Articles

Automatically Finding the Right Probabilities in Bayesian Networks

This paper presents alternative techniques for inference on classical Bayesian networks in which all probabilities are fixed, and for synthesis problems when conditional probability tables (CPTs) in such networks contain symbolic parameters rather than concrete probabilities. The key idea is to exploit probabilistic model checking as well as its recent extension to parameter synthesis techniques for parametric Markov chains. To enable this, the Bayesian networks are transformed into Markov chains and their objectives are mapped onto probabilistic temporal logic formulas. For exact inference, we compare probabilistic model checking to weighted model counting on various Bayesian network benchmarks. We contrast symbolic model checking using multi-terminal binary (aka: algebraic) decision diagrams to symbolic inference using proba- bilistic sentential decision diagrams, symbolic data structures that are tailored to Bayesian networks. For the parametric setting, we describe how our techniques can be used for various synthesis problems such as computing sensitivity functions (and values), simple and difference parameter tuning and ratio parameter tuning. Our parameter synthesis techniques are applicable to arbitrarily many, possibly dependent, parameters that may occur in multiple CPTs. This lifts restrictions, e.g., on the number of parametrized CPTs, or on parameter dependencies between several CPTs, that exist in the literature. Experiments on several benchmarks show that our parameter synthesis techniques can treat parameter synthesis for Bayesian networks (with hundreds of unknown parameters) that are out of reach for existing techniques.

Advances in probabilistic model checking with PRISM: variable reordering, quantiles and weak deterministic Büchi automata

The popular model checker PRISM has been successfully used for the modeling and analysis of complex probabilistic systems. As one way to tackle the challenging state explosion problem, PRISM supports symbolic storage and manipulation using multi-terminal binary decision diagrams for representing the models and in the computations. However, it lacks automated heuristics for variable reordering, even though it is well known that the order of BDD variables plays a crucial role for compact representations and efficient computations. In this article, we present a collection of extensions to PRISM. First, we provide support for automatic variable reordering within the symbolic engines of PRISM and allow users to manually control the variable ordering at a fine-grained level. Second, we provide extensions in the realm of reward-bounded properties, namely symbolic computations of quantiles in Markov decision processes and, for both the explicit and symbolic engines, the approximative computation of quantiles for continuous-time Markov chains as well as support for multi-reward-bounded properties. Finally, we provide an implementation for obtaining minimal weak deterministic Büchi automata for the obligation fragment of linear temporal logic (LTL), with applications for expected accumulated reward computations with a finite horizon given by a co-safe LTL formula.

Spotlight on Marta Kwiatkowska

Marta Kwiatkowska is a Professor of Computing Systems and a Fellow of Trinity College at the University of Oxford. Before joining the University of Oxford, she was Professor in the School of Computer Science at the University of Birmingham, Lecturer at the University of Leicester and Assistant Professor at the Jagiellonian University in Cracow, Poland. She holds a BSc/MSc in Computer Science from the Jagiellonian University, an MA from Oxford and a PhD from the University of Leicester. In 2014, she was awarded an honorary doctorate from KTH Royal Institute of Technology in Stockholm and she became an ACM Fellow in 2016. Marta Kwiatkowska has been a pioneer in the development of model checking for probabilistic and quantitative systems. She was one the proposers of the use of multiterminal binary decision diagrams for the symbolic analysis of probabilistic systems and has done fundamental work on many of the models of quantitative systems that have been studied in the literature, such as continuous time Markov chains, Markov decision processes and stochastic games, amongst others. A hallmark of Marta Kwiatkowska's research career has been the synthesis of innovative and rigorous theoretical work with significant real-world applications. In addition, she has the ability to forge successful research connections with scientists from other sciences. She has led the development of the highly influential PRISM model checker 1 , recognised through the HVC 2016 Award, and has identified applications of quantitative verification to a wide range of areas, including security protocols, nanotechnology designs, power management, planning, and systems biology. In the past few years she has focused on applying quantitative techniques to the verification of pervasive and ubiquitous computing through her ERC Advanced Grant VERIWARE "From software verification to everyware verification" In addition to her outstanding research achievements, Marta Kwiatkowska has served the theoretical computer science community by sitting on prestigious award committees, organizing conferences and workshops and, perhaps even more importantly, by acting as an inspiring mentor for young researchers and as a role model for female researchers in computer science, to whom she has provided support and guidance. To wit, Marta Kwiatkowska has been the Senior Member of OxWoCS (Oxford Women in Computer Science Society) since its foundation in 2013 and has been a source of continued support and encouragement for that student-led society, providing its leadership with advice and suggestions for events as well as mentoring OxWoCS Committee members. Another interesting initiative for women with which she was involved is Suffrage Science 2 , which deserves to be advertised more broadly (not least because it involves jewellery designs). In addition to being named as ACM Fellow for "contributions to the theory and practice of probabilistic verification", Marta Kwiatkowska is a member of Academia Europea and a Fellow of the British Computer Society. She is a true stalwart of the Logic in Computer Science research community and a great example for young researchers. It is therefore fitting to spotlight her contribution in this column of ACM SIGLOG News.

Sylvan: multi-core framework for decision diagrams

Decision diagrams, such as binary decision diagrams, multi-terminal binary decision diagrams and multi-valued decision diagrams, play an important role in various fields. They are especially useful to represent the characteristic function of sets of states and transitions in symbolic model checking. Most implementations of decision diagrams do not parallelize the decision diagram operations. As performance gains in the current era now mostly come from parallel processing, an ongoing challenge is to develop datastructures and algorithms for modern multi-core architectures. The decision diagram package Sylvan provides a contribution by implementing parallelized decision diagram operations and thus allowing sequential algorithms that use decision diagrams to exploit the power of multi-core machines. This paper discusses the design and implementation of Sylvan, especially an improvement to the lock-free unique table that uses bit arrays, the concurrent operation cache and the implementation of parallel garbage collection. We extend Sylvan with multi-terminal binary decision diagrams for integers, real numbers and rational numbers. This extension also allows for custom MTBDD leaves and operations and we provide an example implementation of GMP rational numbers. Furthermore, we show how the provided framework can be integrated in existing tools to provide out-of-the-box parallel BDD algorithms, as well as support for the parallelization of higher-level algorithms. As a case study, we parallelize on-the-fly symbolic reachability in the model checking toolset LTSmin. We experimentally demonstrate that the parallelization of symbolic model checking for explicit-state modeling languages, as supported by LTSmin, scales well. We also show that improvements in the design of the unique table result in faster execution of on-the-fly symbolic reachability.

Learning Weighted Assumptions for Compositional Verification of Markov Decision Processes

Probabilistic models are widely deployed in various systems. To ensure their correctness, verification techniques have been developed to analyze probabilistic systems. We propose the first sound and complete learning-based compositional verification technique for probabilistic safety properties on concurrent systems where each component is an Markov decision process. Different from previous works, weighted assumptions are introduced to attain completeness of our framework. Since weighted assumptions can be implicitly represented by multiterminal binary decision diagrams (MTBDDs), we give an >i<L>/i<*-based learning algorithm for MTBDDs to infer weighted assumptions. Experimental results suggest promising outlooks for our compositional technique.

Verifying concurrent probabilistic systems using probabilistic-epistemic logic specifications

In this paper, we address the problem of verifying probabilistic and epistemic properties in concurrent probabilistic systems expressed in PCTLK. PCTLK is an extension of the Probabilistic Computation Tree Logic (PCTL) augmented with Knowledge (K). In fact, PCTLK enjoys two epistemic modalities Ki for knowledge and PrźbKi$Pr_{\triangledown b}K_{i}$ for probabilistic knowledge. The approach presented for verifying PCTLK specifications in such concurrent systems is based on a transformation technique. More precisely, we convert PCTLK model checking into the problem of model checking Probabilistic Branching Time Logic (PBTL), which enjoys path quantifiers in the range of adversaries. We then prove that model checking a formula of PCTLK in concurrent probabilistic programs is PSPACE-complete. Furthermore, we represent models associated with PCTLK logic symbolically with Multi-Terminal Binary Decision Diagrams (MTBDDs), which are supported by the probabilistic model checker PRISM. Finally, an application, namely the NetBill online shopping payment protocol, and an example about synchronization illustrated through the dining philosophers problem are implemented with the MTBDD engine of this model checker to verify probabilistic epistemic properties and evaluate the practical complexity of this verification.

Leveraging Weighted Automata in Compositional Reasoning about Concurrent Probabilistic Systems

We propose the first sound and complete learning-based compositional verification technique for probabilistic safety properties on concurrent systems where each component is an Markov decision process. Different from previous works, weighted assumptions are introduced to attain completeness of our framework. Since weighted assumptions can be implicitly represented by multi-terminal binary decision diagrams (MTBDD's), we give an L*-based learning algorithm for MTBDD's to infer weighted assumptions. Experimental results suggest promising outlooks for our compositional technique.

Model-Based Contract Testing of Graphical User Interfaces

Graphical User Interfaces (GUIs) are critical for the security, safety and reliability of software systems. Injection attacks, for instance via SQL, succeed due to insufficient input validation and can be avoided if contract-based approaches, such as Design by Contract, are followed in the software development lifecycle of GUIs. This paper proposes a model-based testing approach for detecting GUI data contract violations, which may result in serious failures such as system crash. A contract-based model of GUI data specifications is used to develop test scenarios and to serve as test oracle. The technique introduced uses multi terminal binary decision diagrams, which are designed as an integral part of decision table-augmented event sequence graphs, to implement a GUI testing process. A case study, which validates the presented approach on a port scanner written in Java programming language, is presented.

Efficient calculation of the autocorrelation of Boolean functions with a large number of variables

The autocorrelation of a Boolean function is animportant mathematical concept with various applications. It is a kernel ofmany algorithms with essential applications whose efficiency is directlylimited by the time and space complexity of methods for computing theautocorrelation. These limitations, in this paper, can be overcome by computingthe autocorrelation through Shared Multi-Terminal Binary Decision Diagram(SMTBDD) that are a data structure allowing compact representations of largeBoolean functions. The computation is performed in the spectral domain byexploiting the Wiener-Khinchin theorem and the fast calculation algorithm throughSMTBDDs. It is necessary to develop a specialized decision diagram package withall the standard BDD operations that support fast calculation algorithm throughdecision diagrams and dynamically resizable terminal nodes allows to deal withlarge integers that appear in computing the autocorrelation coefficients. Anexperimental evaluation over benchmarks, confirmed favorably the efficiency ofthe proposed data structure and related algorithms.

Bounded Model Checking for Markov Decision Processes

Bounded model checking has been proven to be a powerful technique for the verification of reactive systems,since it can avoid the space blow up of BDDs(Binary Decision Diagrams).To avoid the space blow up of MTBDDs(Multi Terminal Binary Decision Diagrams),a bounded model checking technique for Markov decision processes is proposed.The biggest feature of Markov decision processes is able to describe the nondeterministic choice.For this feature,the bounded semantics of the probabilistic computation tree logic is first presented,and its correctness is proven.Second,based on the evolution trend of the sequence consisting of probability measure calculated by the bounded model checking procedure,three termination criterions are given.Third,the bounded model checking procedure of the probabilistic computation tree logic is transformed into a linear equation.Finally,the experiment results show that compared with the unbounded model checking if the length of the witness is short then the bounded model checking needs less space.

Some notes on the abstraction operation for multi-terminal binary decision diagrams

The starting point of this work are inaccurate statements found in the literature for Multi-terminal Binary Decision Diagrams (MTBDDs) (Bahar et al., Form. Methods Syst. Des. 10(2/3):171–206, 1997; Siegle, Behaviour analysis of communication systems: compositional modelling, compact representation and analysis of performability properties, 2002; Kuntz, Symbolic semantics and verification of stochastic process algebras, 2006) regarding the well-definedness of the MTBDD abstraction operation. The statements try to relate an operation ∗ on a set of terminal values M to the property that the abstraction over this operation does depend on the order of the abstracted variables. This paper gives a necessary and sufficient condition for the independence of the abstraction operation of the order of the abstracted variables in the case of an underlying monoid and it treats the more general setting of a magma.

A methodology for qualitative/quantitative analysis of weighted attack trees

Attack and Defense Trees (ADT) constitute a formal modeling technique that has become dominant in recent years in the area of qualitative and quantitative cybersecurity analysis of ICT and digital control systems. A Weighted-ADT (WADT) is augmented with cost or impact attributes to evidence the most convenient attack sequence in terms of investment budget and provoked damage and to provide an indication on how to mitigate the located breaches by means of suitable countermeasures. The original analysis technique proposed in this paper is based on the representation of a WADT by means of an extension of Binary Decision Diagrams (BDD), called Multi Terminal Binary Decision Diagrams (MTBDD). MTBDDs allow the modeler to evaluate the probability distribution function of the cost and impact related to any possible attack scenario. A running example illustrates the methodology.

Evolutionary Multivalued Decision Diagrams for Obtaining Motion Representation of Humanoid Robots

In this paper, we propose a method, using multivalued decision diagrams (MDDs), to obtain motion representation of humanoid robots. Kanoh et al. have proposed a method, which uses multiterminal binary decision diagrams (MTBDDs), to acquire robot controller. However, nonterminal vertices of MTBDDs can only treat values of 0 or 1; multiple variables are needed to represent a single joint angle. This increases the number of the non-terminal vertices and MTBDDs that represent that the controller becomes complex. Therefore, we consider using the MDD, in which its nonterminal vertices can take on multiple output values. To obtain humanoid robot motion representation, we propose evolutionary MDDs and show experimental results comparing evolutionary MDDs and evolutionary MTBDDs through simulations of acquisition of robot motion in this paper. Moreover, we verify whether the evolution of MDD using a memetic algorithm is effective.

On the Complexity and Optimization of Branching Programs for Decision Diagram Machines

Decision Diagram Machines (DDMs) are special purpose processors that evaluate decision diagrams. First, this paper derives upper bounds on the cost of multi-terminal binary decision diagrams (MTBDDs) for multiple-output logic functions. From these bounds we can estimate the size of branching programs running on various DDMs. Second, optimization of heterogeneous branching programs is undertaken that makes the area-time trade-off between the amount of memory required for a branching program and its execution time. As a case study, optimal architectures of branching programs are found for a set of benchmark tasks. Beside DDMs, the technique can also be used for micro-controllers with a support for multi-way branching running logic-intensive embedded firmware.

Path-based calculation of MTTFF, MTTFR, and asymptotic unavailability with the stochastic process algebra tool CASPA

CASPA is a stochastic process algebra tool for performance and dependability modelling, analysis, and verification. It is based entirely on the symbolic data structure of the multi-terminal binary decision diagram (MTBDD) which enables the tool to handle models with very large state space. This paper describes an extension of CASPA’s solving engine for path-based approximation of the mean time to first failure, the mean time to first recovery, and asymptotic unavailability by MTBDD algorithms. A non-trivial case study illustrates the use of path-based analysis and comparisons between the path-based unavailability calculations and results obtained from standard Markovian analysis are presented.

Determining the Number of Paths in Decision Diagrams by Using Autocorrelation Coefficients

This paper deals with the number of paths in multiterminal binary decision diagrams (MTBDDs) and shared binary decision diagrams (SBDDs) representing a set of Boolean functions. It is shown that the number of paths in an MTBDD (SBDD) can be uniquely determined by values of specific weighted-autocorrelation coefficients. An analytical expression for the number of paths as a linear function of the values of the weighted-autocorrelation coefficients is presented. Based on this expression, a method of minimization of the number of paths is proposed. The method is based on replacing the initial set of input variables with their linear combinations. By using this method, a deterministic paths-reduction procedure, which provides MTBDDs and SBDDs with a reduced number of paths, is presented. The efficiency of the suggested approach is demonstrated on benchmark functions.

Fine-grained integration of access control policies

Collaborative and distributed applications, such as dynamic coalitions and virtualized grid computing, often require integrating access control policies of collaborating parties. Such an integration must be able to support complex authorization specifications and the fine-grained integration requirements that the various parties may have. In this paper, we introduce an algebra for fine-grained integration of sophisticated policies. The algebra, which consists of three binary and two unary operations, is able to support the specification of a large variety of integration constraints. For ease of use, we also introduce a set of derived operators and provide guidelines for users to edit a policy with desired properties. To assess the expressive power of our algebra, we define notion of completeness and prove that our algebra is complete and minimal with respect to the notion. We then propose a framework that uses the algebra for the fine-grained integration of policies expressed in XACML. We also present a methodology for generating the actual integrated XACML policy, based on the notion of Multi-Terminal Binary Decision Diagrams. Experimental results have demonstrated both effectiveness and efficiency of our approach. In addition, we also discuss issues regarding obligations.

Partially-shared zero-suppressed multi-terminal BDDs: concept, algorithms and applications

Multi-Terminal Binary Decision Diagrams (MTBDDs) are a well accepted technique for the state graph (SG) based quantitative analysis of large and complex systems specified by means of high-level model description techniques. However, this type of Decision Diagram (DD) is not always the best choice, since finite functions with small satisfaction sets, and where the fulfilling assignments possess many 0-assigned positions, may yield relatively large MTBDD based representations. Therefore, this article introduces zero-suppressed MTBDDs and proves that they are canonical representations of multi-valued functions on finite input sets. For manipulating DDs of this new type, possibly defined over different sets of function variables, the concept of partially-shared zero-suppressed MTBDDs and respective algorithms are developed. The efficiency of this new approach is demonstrated by comparing it to the well-known standard type of MTBDDs, where both types of DDs have been implemented by us within the C++-based DD-package JINC. The benchmarking takes place in the context of Markovian analysis and probabilistic model checking of systems. In total, the presented work extends existing approaches, since it not only allows one to directly employ (multi-terminal) zero-suppressed DDs in the field of quantitative verification, but also clearly demonstrates their efficiency.

Application of multiroot decision diagrams for integer functions

New data structures for representation of integer functions and matrices are proposed, namely, multiroot binary decision diagrams (MRBDD). Algorithms for performing standard operations on functions and matrices represented by MRBDDs are presented. Thanks to the more efficient reuse of structural elements, representation by multiroot binary decision diagrams is more efficient than the widely used multiterminal binary decision diagrams (MTBDD). Experimental results are given, which show that multiroot binary decision diagrams are a promising alternative to multiterminal binary decision diagrams, in particular, in probabilistic verification, manipulation of probability distributions, analysis of Petri nets, and other computational models.

Firmware Optimization for Embedded Logic Control

This paper presents a new method to represent a subclass of multiple-output incompletely specified functions by means of multi-terminal binary decision diagrams (MTBDDs). Algorithm to reduce the cost and width of MTBDDs is presented. A software CAD tool makes use of iterative decomposition to obtain a MTBDD data structure that can be directly mapped to firmware in a form of chained dispatch tables. A practical example shows that there is a space-time trade-off between the amount of memory required for all dispatch tables in a control store and the speed of firmware execution. Support for multi-way branching in a micro-sequencer is assumed.