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

Implementing Groundness Analysis with Definite Boolean Functions

The domain of definite Boolean functions, Def, can be used to express the groundness of, and trace grounding dependencies between, program variables in (constraint) logic programs. In this paper, previously unexploited computational properties of Def are utilised to develop an efficient and succinct groundness analyser that can be coded in Prolog. In particular, entailment checking is used to prevent unnecessary least upper bound calculations. It is also demonstrated that join can be defined in terms of other operations, thereby eliminating code and removing the need for preprocessing formulae to a normal form. This saves space and time. Furthermore, the join can be adapted to straightforwardly implement the downward closure operator that arises in set sharing analyses. Experimental results indicate that the new Def implementation gives favourable results in comparison with BDD-based groundness analyses

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

Strategies for typecase optimization

International audienceWe contrast two approaches to optimizing the Common Lisp typecase macro expansion. The first approach is based on heuristics intended to estimate run time performance of certain type checks involving Common Lisp type specifiers. The technique may, depending on code size, exhaustively search the space of permutations of the type checks, intent on finding the optimal order. With the second technique, we represent a typecase form as a type specifier, encapsulating the side-effecting non-Boolean parts so as to appear compatible with the Common Lisp type algebra operators. The encapsulated expressions are specially handled so that the Common Lisp type algebra functions preserve them, and we can unwrap them after a process of Boolean reduction into efficient Common Lisp code, maintaining the appropriate side effects but eliminating unnecessary type checks. Both approaches allow us to identify un-reachable code, test for exhaustiveness of the clauses and eliminate type checks which are calculated to be redundant

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.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

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