Solving Set Constraint Satisfaction Problems using ROBDDs

In this paper we present a new approach to modeling finite set domain constraint problems using Reduced Ordered Binary Decision Diagrams (ROBDDs). We show that it is possible to construct an efficient set domain propagator which compactly represents many set domains and set constraints using ROBDDs. We demonstrate that the ROBDD-based approach provides unprecedented flexibility in modeling constraint satisfaction problems, leading to performance improvements. We also show that the ROBDD-based modeling approach can be extended to the modeling of integer and multiset constraint problems in a straightforward manner. Since domain propagation is not always practical, we also show how to incorporate less strict consistency notions into the ROBDD framework, such as set bounds, cardinality bounds and lexicographic bounds consistency. Finally, we present experimental results that demonstrate the ROBDD-based solver performs better than various more conventional constraint solvers on several standard set constraint problems

Conditional Transition Systems with Upgrades

We introduce a variant of transition systems, where activation of transitions depends on conditions of the environment and upgrades during runtime potentially create additional transitions. Using a cornerstone result in lattice theory, we show that such transition systems can be modelled in two ways: as conditional transition systems (CTS) with a partial order on conditions, or as lattice transition systems (LaTS), where transitions are labelled with the elements from a distributive lattice. We define equivalent notions of bisimilarity for both variants and characterise them via a bisimulation game. We explain how conditional transition systems are related to featured transition systems for the modelling of software product lines. Furthermore, we show how to compute bisimilarity symbolically via BDDs by defining an operation on BDDs that approximates an element of a Boolean algebra into a lattice. We have implemented our procedure and provide runtime results

Lex-Partitioning: A New Option for BDD Search

For the exploration of large state spaces, symbolic search using binary decision diagrams (BDDs) can save huge amounts of memory and computation time. State sets are represented and modified by accessing and manipulating their characteristic functions. BDD partitioning is used to compute the image as the disjunction of smaller subimages. In this paper, we propose a novel BDD partitioning option. The partitioning is lexicographical in the binary representation of the states contained in the set that is represented by a BDD and uniform with respect to the number of states represented. The motivation of controlling the state set sizes in the partitioning is to eventually bridge the gap between explicit and symbolic search. Let n be the size of the binary state vector. We propose an O(n) ranking and unranking scheme that supports negated edges and operates on top of precomputed satcount values. For the uniform split of a BDD, we then use unranking to provide paths along which we partition the BDDs. In a shared BDD representation the efforts are O(n). The algorithms are fully integrated in the CUDD library and evaluated in strongly solving general game playing benchmarks.Comment: In Proceedings GRAPHITE 2012, arXiv:1210.611

A Decision Diagram Operation for Reachability

Saturation is considered the state-of-the-art method for computing fixpoints with decision diagrams. We present a relatively simple decision diagram operation called REACH that also computes fixpoints. In contrast to saturation, it does not require a partitioning of the transition relation. We give sequential algorithms implementing the new operation for both binary and multi-valued decision diagrams, and moreover provide parallel counterparts. We implement these algorithms and experimentally compare their performance against saturation on 692 model checking benchmarks in different languages. The results show that the REACH operation often outperforms saturation, especially on transition relations with low locality. In a comparison between parallelized versions of REACH and saturation we find that REACH obtains comparable speedups up to 16 cores, although falls behind saturation at 64 cores. Finally, in a comparison with the state-of-the-art model checking tool ITS-tools we find that REACH outperforms ITS-tools on 29% of models, suggesting that REACH can be useful as a complementary method in an ensemble tool

Anytime Algorithms for ROBDD Symmetry Detection and Approximation

Reduced Ordered Binary Decision Diagrams (ROBDDs) provide a dense and memory efficient representation of Boolean functions. When ROBDDs are applied in logic synthesis, the problem arises of detecting both classical and generalised symmetries. State-of-the-art in symmetry detection is represented by Mishchenko's algorithm. Mishchenko showed how to detect symmetries in ROBDDs without the need for checking equivalence of all co-factor pairs. This work resulted in a practical algorithm for detecting all classical symmetries in an ROBDD in O(|G|3) set operations where |G| is the number of nodes in the ROBDD. Mishchenko and his colleagues subsequently extended the algorithm to find generalised symmetries. The extended algorithm retains the same asymptotic complexity for each type of generalised symmetry. Both the classical and generalised symmetry detection algorithms are monolithic in the sense that they only return a meaningful answer when they are left to run to completion. In this thesis we present efficient anytime algorithms for detecting both classical and generalised symmetries, that output pairs of symmetric variables until a prescribed time bound is exceeded. These anytime algorithms are complete in that given sufficient time they are guaranteed to find all symmetric pairs. Theoretically these algorithms reside in O(n3+n|G|+|G|3) and O(n3+n2|G|+|G|3) respectively, where n is the number of variables, so that in practice the advantage of anytime generality is not gained at the expense of efficiency. In fact, the anytime approach requires only very modest data structure support and offers unique opportunities for optimisation so the resulting algorithms are very efficient. The thesis continues by considering another class of anytime algorithms for ROBDDs that is motivated by the dearth of work on approximating ROBDDs. The need for approximation arises because many ROBDD operations result in an ROBDD whose size is quadratic in the size of the inputs. Furthermore, if ROBDDs are used in abstract interpretation, the running time of the analysis is related not only to the complexity of the individual ROBDD operations but also the number of operations applied. The number of operations is, in turn, constrained by the number of times a Boolean function can be weakened before stability is achieved. This thesis proposes a widening that can be used to both constrain the size of an ROBDD and also ensure that the number of times that it is weakened is bounded by some given constant. The widening can be used to either systematically approximate an ROBDD from above (i.e. derive a weaker function) or below (i.e. infer a stronger function). The thesis also considers how randomised techniques may be deployed to improve the speed of computing an approximation by avoiding potentially expensive ROBDD manipulation

Solving hard industrial combinatorial problems with SAT

The topic of this thesis is the development of SAT-based techniques and tools for solving industrial combinatorial problems. First, it describes the architecture of state-of-the-art SAT and SMT Solvers based on the classical DPLL procedure. These systems can be used as black boxes for solving combinatorial problems. However, sometimes we can increase their efficiency with slight modifications of the basic algorithm. Therefore, the study and development of techniques for adjusting SAT Solvers to specific combinatorial problems is the first goal of this thesis. Namely, SAT Solvers can only deal with propositional logic. For solving general combinatorial problems, two different approaches are possible: - Reducing the complex constraints into propositional clauses. - Enriching the SAT Solver language. The first approach corresponds to encoding the constraint into SAT. The second one corresponds to using propagators, the basis for SMT Solvers. Regarding the first approach, in this document we improve the encoding of two of the most important combinatorial constraints: cardinality constraints and pseudo-Boolean constraints. After that, we present a new mixed approach, called lazy decomposition, which combines the advantages of encodings and propagators. The other part of the thesis uses these theoretical improvements in industrial combinatorial problems. We give a method for efficiently scheduling some professional sport leagues with SAT. The results are promising and show that a SAT approach is valid for these problems. However, the chaotical behavior of CDCL-based SAT Solvers due to VSIDS heuristics makes it difficult to obtain a similar solution for two similar problems. This may be inconvenient in real-world problems, since a user expects similar solutions when it makes slight modifications to the problem specification. In order to overcome this limitation, we have studied and solved the close solution problem, i.e., the problem of quickly finding a close solution when a similar problem is considered

Reduced Ordered Binary Decision Diagram with Implied Literals: A New knowledge Compilation Approach

Knowledge compilation is an approach to tackle the computational intractability of general reasoning problems. According to this approach, knowledge bases are converted off-line into a target compilation language which is tractable for on-line querying. Reduced ordered binary decision diagram (ROBDD) is one of the most influential target languages. We generalize ROBDD by associating some implied literals in each node and the new language is called reduced ordered binary decision diagram with implied literals (ROBDD-L). Then we discuss a kind of subsets of ROBDD-L called ROBDD-i with precisely i implied literals (0 \leq i \leq \infty). In particular, ROBDD-0 is isomorphic to ROBDD; ROBDD-\infty requires that each node should be associated by the implied literals as many as possible. We show that ROBDD-i has uniqueness over some specific variables order, and ROBDD-\infty is the most succinct subset in ROBDD-L and can meet most of the querying requirements involved in the knowledge compilation map. Finally, we propose an ROBDD-i compilation algorithm for any i and a ROBDD-\infty compilation algorithm. Based on them, we implement a ROBDD-L package called BDDjLu and then get some conclusions from preliminary experimental results: ROBDD-\infty is obviously smaller than ROBDD for all benchmarks; ROBDD-\infty is smaller than the d-DNNF the benchmarks whose compilation results are relatively small; it seems that it is better to transform ROBDDs-\infty into FBDDs and ROBDDs rather than straight compile the benchmarks.Comment: 18 pages, 13 figure