Algebraic Decision Diagrams Research Articles

Engineering an Exact Pseudo-Boolean Model Counter

Model counting, a fundamental task in computer science, involves determining the number of satisfying assignments to a Boolean formula, typically represented in conjunctive normal form (CNF). While model counting for CNF formulas has received extensive attention with a broad range of applications, the study of model counting for Pseudo-Boolean (PB) formulas has been relatively overlooked. Pseudo-Boolean formulas, being more succinct than propositional Boolean formulas, offer greater flexibility in representing real-world problems. Consequently, there is a crucial need to investigate efficient techniques for model counting for PB formulas. In this work, we propose the first exact Pseudo-Boolean model counter, PBCount , that relies on knowledge compilation approach via algebraic decision diagrams. Our extensive empirical evaluation shows that PBCount can compute counts for 1513 instances while the current state-of-the-art approach could only handle 1013 instances. Our work opens up several avenues for future work in the context of model counting for PB formulas, such as the development of preprocessing techniques and exploration of approaches other than knowledge compilation.

Forest GUMP: a tool for verification and explanation

AbstractIn this paper, we present Forest GUMP (for Generalized, Unifying Merge Process) a tool for verification and precise explanation of Random forests. Besides pre/post-condition-based verification and equivalence checking, Forest GUMP also supports three concepts of explanation, the well-known model explanation and outcome explanation, as well as class characterization, i.e., the precise characterization of all samples that are equally classified. Key technology to achieve these results is algebraic aggregation, i.e., the transformation of a Random Forest into a semantically equivalent, concise white-box representation in terms of Algebraic Decision Diagrams (ADDs). The paper sketches the method and demonstrates the use of Forest GUMP along illustrative examples. This way readers should acquire an intuition about the tool, and the way how it should be used to increase the understanding not only of the considered dataset, but also of the character of Random Forests and the ADD technology, here enriched to comprise infeasible path elimination. As Forest GUMP is publicly available all experiments can be reproduced, modified, and complemented using any dataset that is available in the ARFF format.

ADDMC: Weighted Model Counting with Algebraic Decision Diagrams

We present an algorithm to compute exact literal-weighted model counts of Boolean formulas in Conjunctive Normal Form. Our algorithm employs dynamic programming and uses Algebraic Decision Diagrams as the main data structure. We implement this technique in ADDMC, a new model counter. We empirically evaluate various heuristics that can be used with ADDMC. We then compare ADDMC to four state-of-the-art weighted model counters (Cachet, c2d, d4, and miniC2D) on 1914 standard model counting benchmarks and show that ADDMC significantly improves the virtual best solver.

Symbolic Planning with Edge-Valued Multi-Valued Decision Diagrams

Symbolic representations have attracted significant attention in optimal planning. Binary Decision Diagrams (BDDs) form the basis for symbolic search algorithms. Closely related are Algebraic Decision Diagrams (ADDs), used to represent heuristic functions. Also, progress was made in dealing with models that take state-dependent action costs into account. Here, costs are represented as Edge-valued Multi-valued Decision Diagrams (EVMDDs), which can be exponentially more compact than the corresponding ADD representation. However, they were not yet considered for symbolic planning. In this work, we study EVMDD-based symbolic search for optimal planning. We define EVMDD-based representations of state sets and transition relations, and show how to compute the necessary operations required for EVMDD-A*. This EVMDD-based version of symbolic A* generalizes its BDD variant, and allows to solve planning tasks with state-dependent action costs. We prove theoretically that our approach is sound, complete and optimal. Additionally, we present an empirical analysis comparing EVMDD-A* to BDD-A* and explicit A* search. Our results underscore the usefulness of symbolic approaches and the feasibility of dealing with models that go beyond unit costs.

The Symbolic Interior Point Method

Numerical optimization is arguably the most prominent computational framework in machine learning and AI. It can be seen as an assembly language for hard combinatorial problems ranging from classification and regression in learning, to computing optimal policies and equilibria in decision theory, to entropy minimization in information sciences. Unfortunately, specifying such problems in complex domains involving relations, objects and other logical dependencies is cumbersome at best, requiring considerable expert knowledge, and solvers require models to be painstakingly reduced to standard forms. To overcome this, we introduce a rich modeling framework for optimization problems that allows convenient codification of symbolic structure. Rather than reducing this symbolic structure to a sparse or dense matrix, we represent and exploit it directly using algebraic decision diagrams (ADDs). Combining efficient ADD-based matrix-vector algebra with a matrix-free interior-point method, we develop an engine that can fully leverage the structure of symbolic representations to solve convex linear and quadratic optimization problems. We demonstrate the flexibility of the resulting symbolic-numeric optimizer on decision making and compressed sensing tasks with millions of non-zero entries.

Monadic second-order logic on finite sequences

We extend the weak monadic second-order logic of one successor on finite strings (M2L-STR) to symbolic alphabets by allowing character predicates to range over decidable quantifier free theories instead of finite alphabets. We call this logic, which is able to describe sequences over complex and potentially infinite domains, symbolic M2L-STR (S-M2L-STR). We then present a decision procedure for S-M2L-STR based on a reduction to symbolic finite automata, a decidable extension of finite automata that allows transitions to carry predicates and can therefore model symbolic alphabets. The reduction constructs a symbolic automaton over an alphabet consisting of pairs of symbols where the first element of the pair is a symbol in the original formula’s alphabet, while the second element is a bit-vector. To handle this modified alphabet we show that the Cartesian product of two decidable Boolean algebras (e.g., the formula’s one and the bit-vector’s one) also forms a decidable Boolean algebras. To make the decision procedure practical, we propose two efficient representations of the Cartesian product of two Boolean algebras, one based on algebraic decision diagrams and one on a variant of Shannon expansions. Finally, we implement our decision procedure and evaluate it on more than 10,000 formulas. Despite the generality, our implementation has comparable performance with the state-of-the-art M2L-STR solvers.

Symbolic BDD and ADD Algorithms for Energy Games

Energy games, which model quantitative consumption of a limited resource, e.g., time or energy, play a central role in quantitative models for reactive systems. Reactive synthesis constructs a controller which satisfies a given specification, if one exists. For energy games a synthesized controller ensures to satisfy not only the safety constraints of the specification but also the quantitative constraints expressed in the energy game. A symbolic algorithm for energy games, recently presented by Chatterjee et al., is symbolic in its representation of quantitative values but concrete in the representation of game states and transitions. In this paper we present an algorithm that is symbolic both in the quantitative values and in the underlying game representation. We have implemented our algorithm using two different symbolic representations for reactive games, Binary Decision Diagrams (BDD) and Algebraic Decision Diagrams (ADD). We investigate the commonalities and differences of the two implementations and compare their running times on specifications of energy games.

Efficient incremental planning and learning with multi-valued decision diagrams

In the domain of decision theoretic planning, the factored framework (Factored Markov Decision Process, fmdp) has produced optimized algorithms using structured representations such as Decision Trees (Structured Value Iteration (svi), Structured Policy Iteration (spi)) or Algebraic Decision Diagrams (Stochastic Planning Using Decision Diagrams (spudd)). Since it may be difficult to elaborate the factored models used by these algorithms, the architecture sdyna, which combines learning and planning algorithms using structured representations, was introduced. However, the state-of-the-art algorithms for incremental learning, for structured decision theoretic planning or for reinforcement learning require the problem to be specified only with binary variables and/or use data structures that can be improved in term of compactness. In this paper, we propose to use Multi-Valued Decision Diagrams (mdds) as a more efficient data structure for the sdyna architecture and describe a planning algorithm and an incremental learning algorithm dedicated to this new structured representation. For both planning and learning algorithms, we experimentally show that they allow significant improvements in time, in compactness of the computed policy and of the learned model. We then analyzed the combination of these two algorithms in an efficient sdyna instance for simultaneous learning and planning using mdds.

Low-complexity quantized switching controllers using approximate bisimulation

In this paper, we consider the problem of synthesizing low-complexity controllers for incrementally stable switched systems. For that purpose, we establish a new approximation result for the computation of symbolic models that are approximately bisimilar to a given switched system. The main advantage over existing results is that it allows us to design naturally quantized switching controllers for safety or reachability specifications; these can be pre-computed offline and therefore the online execution time is reduced. Then, we present a technique to reduce the memory needed to store the control law by borrowing ideas from algebraic decision diagrams for compact function representation and by exploiting the non-determinism of the synthesized controllers. We show the merits of our approach by applying it to a simple model of temperature regulation in a building.

Low-Complexity Switching Controllers for Safety using Symbolic Models

In this paper, we consider the problem of synthesizing low-complexity safety controllers for incrementally stable switched systems. For that purpose, we establish a new approximation result for the computation of symbolic models that are approximately bisimilar to a given switched system. The main advantage over existing results is that it allows us to design quantized switching controllers for safety specifications; these can be computed offline and therefore the online execution time is greatly reduced. Then, we present a technique to reduce the memory needed to store the control law by borrowing ideas from algebraic decision diagrams for compact function representation and by exploiting the non-determinism inherent to safety controllers. We show the merits of our approach by applying it to a simple model of temperature regulation in a building.

Probabilistic Relational Planning with First Order Decision Diagrams

Dynamic programming algorithms have been successfully applied to propositional stochastic planning problems by using compact representations, in particular algebraic decision diagrams, to capture domain dynamics and value functions. Work on symbolic dynamic programming lifted these ideas to first order logic using several representation schemes. Recent work introduced a first order variant of decision diagrams (FODD) and developed a value iteration algorithm for this representation. This paper develops several improvements to the FODD algorithm that make the approach practical. These include, new reduction operators that decrease the size of the representation, several speedup techniques, and techniques for value approximation. Incorporating these, the paper presents a planning system, FODD-Planner, for solving relational stochastic planning problems. The system is evaluated on several domains, including problems from the recent international planning competition, and shows competitive performance with top ranking systems. This is the first demonstration of feasibility of this approach and it shows that abstraction through compact representation is a promising approach to stochastic planning.

DTProbLog: A Decision-Theoretic Probabilistic Prolog

We introduce DTProbLog, a decision-theoretic extension of Prolog and its probabilistic variant ProbLog. DTProbLog is a simple but expressive probabilistic programming language that allows the modeling of a wide variety of domains, such as viral marketing. In DTProbLog, the utility of a strategy (a particular choice of actions) is defined as the expected reward for its execution in the presence of probabilistic effects. The key contribution of this paper is the introduction of exact, as well as approximate, solvers to compute the optimal strategy for a DTProbLog program and the decision problem it represents, by making use of binary and algebraic decision diagrams. We also report on experimental results that show the effectiveness and the practical usefulness of the approach.

Decision-diagram-based techniques for bounded reachability checking of asynchronous systems

Bounded reachability analysis and bounded model checking are widely believed to perform poorly when using decision diagrams instead of SAT procedures. Recent research suggests this to be untrue with regards to synchronous systems and, in particular, digital circuits. This article shows that the belief is also a myth for asynchronous systems, such as models specified by Petri nets. We propose several Bounded Saturation approaches to compute bounded state spaces using decision diagrams. These approaches are based on the established Saturation algorithm, which benefits from a non-standard search strategy that is very different from breadth-first search, but employ different flavors of decision diagrams: multi-valued decision diagrams, edge-valued decision diagrams, and algebraic decision diagrams. We apply our approaches to studying deadlock as a safety property. Our extensive benchmarking shows that our algorithms often, but not always, compare favorably against two SAT-based approaches that are advocated in the literature.

Probabilistic transfer matrices in symbolic reliability analysis of logic circuits

We propose the probabilistic transfer matrix (PTM) framework to capture nondeterministic behavior in logic circuits. PTMs provide a concise description of both normal and faulty behavior, and are well-suited to reliability and error susceptibility calculations. A few simple composition rules based on connectivity can be used to recursively build larger PTMs (representing entire logic circuits) from smaller gate PTMs. PTMs for gates in series are combined using matrix multiplication, and PTMs for gates in parallel are combined using the tensor product operation. PTMs can accurately calculate joint output probabilities in the presence of reconvergent fanout and inseparable joint input distributions. To improve computational efficiency, we encode PTMs as algebraic decision diagrams (ADDs). We also develop equivalent ADD algorithms for newly defined matrix operations such as eliminate_variables and eliminate_redundant_variables , which aid in the numerical computation of circuit PTMs. We use PTMs to evaluate circuit reliability and derive polynomial approximations for circuit error probabilities in terms of gate error probabilities. PTMs can also analyze the effects of logic and electrical masking on error mitigation. We show that ignoring logic masking can overestimate errors by an order of magnitude. We incorporate electrical masking by computing error attenuation probabilities, based on analytical models, into an extended PTM framework for reliability computation. We further define a susceptibility measure to identify gates whose errors are not well masked. We show that hardening a few gates can significantly improve circuit reliability.

Circuit Reliability Analysis Using Symbolic Techniques

Due to the shrinking of feature size and the significant reduction in noise margins, nanoscale circuits have become more susceptible to manufacturing defects, noise-related transient faults, and interference from radiation. Traditionally, soft errors have been a much greater concern in memories than in logic circuits. However, as technology continues to scale, logic circuits are becoming more susceptible to soft errors than memories. To estimate the susceptibility to errors in combinational logic, the use of binary decision diagrams (BDDs) and algebraic decision diagrams (ADDs) for the unified symbolic analysis of circuit reliability is proposed. A framework that uses BDDs and ADDs and enables the analysis of combinational circuit reliability from different aspects, e.g., output susceptibility to error, influence of individual gates on individual outputs and overall circuit reliability, and the dependence of circuit reliability on glitch duration, amplitude, and input patterns, is presented. This is demonstrated by the set of experimental results, which show that the mean output error susceptibility can vary from less then 0.1% for large circuits and short glitches (20% cycle time) to about 30% for very small circuits and long enough glitches (50% cycle time)

Timing analysis for full-custom circuits using symbolic DC formulations

Successful analysis of high-speed integrated circuits requires accurate delay computation. A number of delay models have been developed; however, none can claim to be truly robust in the face of large channel-connected regions (CCRs) with input "exclusivity" constraints. A good circuit-level delay model should: 1) consider input exclusivity constraints; 2) handle a wide range of circuit structures; and 3) have a robust underlying framework that can be applied independent of the actual device model. We present a symbolic timing analysis tool that aims to address these three goals. It uses algebraic decision diagrams (ADDs) to estimate delay within a CCR as a function of its inputs while easily handling Boolean input constraints. It starts with a simple linear resistor model for transistors and from there apply various heuristics to improve the delay estimation without altering the symbolic algorithms. It analyzes delay with simple series-parallel reduction when possible and use symbolic matrix techniques to handle more complex circuit structures. The effectiveness of our approach is demonstrated on circuits from industry used in the Alpha 21264 and 21364 instead of the usual International Symposium on Circuits and Systems (ISCAS) or Microelectronics Center of North Carolina (MCNC) benchmarks. Our delay estimates are within 10% of simulation program with integrated circuits emphasis (SPICE) for over 90% of the circuits we simulated. This difference can translate into significant savings in manpower by avoiding the need to verify many unrealizable worst case conditions with other, more costly, simulation techniques

Symbolic Reasoning with Weighted and Normalized Decision Diagrams

Several variants of Bryant's ordered binary decision diagrams have been suggested in the literature to reason about discrete functions. In this paper, we introduce a generic notion of weighted decision diagrams that captures many of them and present criteria for canonicity. As a special instance of such weighted diagrams, we introduce a new BDD-variant for real-valued functions, called normalized algebraic decision diagrams. Regarding the number of nodes and arithmetic operations like addition and multiplication, these normalized diagrams are as efficient as factored edge-valued binary decision diagrams, while several other operators, like the calculation of extrema, minimum or maximum of two functions or the switch from real-valued functions to boolean functions through a given threshold, are more efficient for normalized diagrams than for their factored counterpart.

Symbolic failure analysis of complex CMOS circuits due to excessive leakage current and charge sharing

As process geometries shrink, leakage currents and charge sharing are becoming increasingly critical problems, especially in full-custom circuit designs. Excessive leakage or charge sharing may cause functional failure at some or all operating conditions. Traditional circuit-analysis techniques may be used to verify if leakage currents are within allowable limits so as not to cause functional failures; however, unless the analysis takes into account specific input constraints for the circuit, the results may be overly pessimistic. Similar limitations exist for charge sharing. In this paper, we approach this verification problem symbolically using algebraic decision diagrams (ADDs). Using ADDs allows us to efficiently analyze leakage and charge sharing within a channel-connected region (CCR) as a function of its inputs. Exclusivity constraints are easily included in the analysis, thus allowing for more accurate (and less pessimistic) results. Our approach is general and can be applied to any arbitrary circuit structure, including a mesh. The effectiveness of our approach is demonstrated on circuits from industry used in the Alpha 21264 and 21364 instead of the usual international symposium on circuits and systems or Microelectronics Center of North Carolina benchmarks. We show that such an analysis can lead to up to a 90% difference in worst-case voltage drop. This difference can translate into significant savings in manpower by avoiding the need to verify many unrealizable worst-case conditions with other, more costly, simulation techniques.

Solving factored MDPs using non-homogeneous partitions

We present an algorithm for aggregating states in solving large MDPs (represented as factored MDPs) using search by successive refinement in the space of non-homogeneous partitions. Homogeneity is defined in terms of stochastic bisimulation and reward equivalence within blocks of a partition. Since homogeneous partitions that define equivalent reduced-state-space MDPs can have a large number of blocks, we relax the requirement of homogeneity. The algorithm constructs approximate aggregate MDPs from non-homogeneous partitions, solves the aggregate MDPs exactly, and then uses the resulting value functions as part of a heuristic in refining the current best non-homogeneous partition. We outline the theory motivating the use of this heuristic and present empirical results. In addition to investigating more exhaustive local search methods we explore the use of techniques derived from research on discretizing continuous state spaces. Finally, we compare the results from our algorithms which search in the space of non-homogeneous partitions with exact and approximate algorithms which represent homogeneous and approximately homogeneous partitions as decision trees or algebraic decision diagrams.

Application of symbolic FSM Markovian analysis to protocol verification

Computer-aided design tools for Markovian analysis and verification of digital circuits have gained much interest in the last few years. This is mainly due to the advent of effective data structures, e.g., binary and algebraic decision diagrams, for Boolean and pseudo-Boolean function representation and manipulation. The authors illustrate how those tools can be successfully exploited to analyse the behavior and verify the correctness of a communication protocol. They first consider the case of single protocol entities running in isolation, and present a simple application example (i.e. the sender entity of the alternating bit protocol). Then, informally illustrate how the analysis approach can be extended to the general case of multi-layer protocol stacks and complete communication systems.