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

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

Aerosol climate feedback due to decadal increases in Southern Hemisphere wind speeds

Observations indicate that the westerly jet in the Southern Hemisphere troposphere is accelerating. Using a global aerosol model we estimate that the increase in wind speed of 0.45 + /- 0.2 m s(-1) decade(-1) at 50-65 degrees S since the early 1980s caused a higher sea spray flux, resulting in an increase of cloud condensation nucleus concentrations of more than 85% in some regions, and of 22% on average between 50 and 65 degrees S. These fractional increases are similar in magnitude to the decreases over many northern hemisphere land areas due to changes in air pollution over the same period. The change in cloud drop concentrations causes an increase in cloud reflectivity and a summertime radiative forcing between at 50 and 65 degrees S comparable in magnitude but acting against that from greenhouse gas forcing over the same time period, and thus represents a substantial negative climate feedback. However, recovery of Antarctic ozone depletion in the next two decades will likely cause a fall in wind speeds, a decrease in cloud drop concentration and a correspondingly weaker cloud feedback

Very Low-Noise Differential Radiometer at 30 GHz for the PLANCK LFI

The PLANCK mission of the European Space Agency is devoted to produce sky maps of the cosmic microwave background radiation. The low-frequency instrument is a wide-band cryogenic microwave radiometer array operating at 30, 44, and 70 GHz. The design, test techniques, and performance of the complete differential radiometer at 30 GHz are presented. This elegant breadboard 30-GHz radiometer is composed of a front-end module (FEM) assembled at the Jodrell Bank Observatory, Cheshire, U.K., and a back-end module assembled at the Universidad de Cantabria, Cantabria, Spain, and Telecomunicacio/spl acute/, Universitat Polite/spl acute/cnica de Catalunya, Barcelona, Spain. The system noise temperature was excellent, mainly due to the very low noise performance of the FEM amplifiers, which achieved an average noise temperature of 9.4 K.Peer Reviewe

Rust never sleeps: the continuing story of the Iron Bolt

Since 1981, Gordon Research Conferences have been held on the topic of Oxygen Radicals on a biennial basis, to highlight and discuss the latest cutting edge research in this area. Since the first meeting, one special feature of this conference has been the awarding of the so-called Iron Bolt, an award that started in jest but has gained increasing reputation over the years. Since no written documentation exists for this Iron Bolt award, this perspective serves to overview the history of this unusual award, and highlights various experiences of previous winners of this “prestigious” award and other interesting anecdotes

The association between treatment adherence to nicotine patches and smoking cessation in pregnancy: a secondary analysis of a randomised controlled trial

IntroductionIn non-pregnant ‘quitters’, adherence to nicotine replacement therapy (NRT) increases smoking cessation. We investigated relationships between adherence to placebo or NRT patches and cessation in pregnancy, including an assessment of reverse causation and whether any adherence: cessation relationship is moderated when using nicotine or placebo patches. MethodsUsing data from 1050 pregnant trial participants, regression models investigated associations between maternal characteristics, adherence and smoking cessation. ResultsAdherence during the first month was associated with lower baseline cotinine concentrations (beta -0.08, 95%CI -0.15 to -0.01) and randomisation to NRT (beta 2.59, 95%CI 1.50 to 3.68). Adherence during both treatment months was associated with being randomised to NRT (beta 0.51, 95%CI 0.29 to 0.72) and inversely associated with higher nicotine dependence. Adherence with either NRT or placebo was associated with cessation at one month (OR 1.11, 95%CI 1.08 to 1.13) and delivery (OR 1.06, 95%CI 1.03 to 1.09), but no such association was observed in the subgroup where reverse causation was not possible. Amongst all women, greater adherence to nicotine patches was associated with increased cessation (OR 2.47, 95%CI 1.32 to 4.63) but greater adherence to placebo was not (OR 0.98, 95%CI: 0.44 to 2.18). ConclusionWomen who were more adherent to NRT were more likely to achieve abstinence; more nicotine dependent women probably showed lower adherence to NRT because they relapsed to smoking more quickly. The interaction between nicotine-containing patches and adherence for cessation suggests that the association between adherence with nicotine patches and cessation may be partly causal

An Anytime Symmetry Detection Algorithm for ROBDDs

Detecting symmetries is crucial to logic synthesis, technology mapping, detecting function equivalence under unknown input correspondence, and ROBDD minimization. State-of-the-art is represented by Mishchenko's algorithm. In this paper we present an efficient anytime algorithm for detecting symmetries in Boolean functions represented as ROBDDs, that output pairs of symmetric variables until a prescribed time bound is exceeded. The algorithm is complete in that given sufficient time it is guaranteed to find all symmetric pairs. The complexity of this algorithm is in O(n^4+n|G|+|G|^3) where n is the number of variables and |G| the number of nodes in the ROBDD, and it is thus competitive with Mishchenko's |G|^3 algorithm in the worst-case since n << |G|. However, our algorithm performs significantly better because the anytime approach only requires lightweight data structure support and it offers unique opportunities for optimization

An Anytime Algorithm for Generalized Symmetry Detection in ROBDDs

Detecting symmetries has many applications in logic synthesis that include, amongst other things, technology mapping, deciding equivalence of Boolean functions when the input correspondence is unknown and finding support-reducing bound sets. Mishchenko showed how to efficiently detect symmetries in ROBDDs without the need for checking equivalence of all co-factor pairs. This work resulted in practical algorithms for detecting classical and generalized symmetries. Both the classical and generalized 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 paper we present anytime algorithms for detecting both classical and generalized 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. Anytime generality is not gained at the expense of efficiency since this approach requires only very modest data structure support and offers unique opportunities for optimization so the resulting algorithms are competitive with their monolithic counterparts

Proof of New Implicational Relationships between Generalized Symmetries (appendix for journal paper)

This note provides proof of some new implicational relationships between generalized symmetries. These relationships are formulated in terms of twelve symmetry types. Six of these symmetries are denoted Tnxi,xj where the index n?[1,6] indicates that a specific co-factor equivalence property holds between the variables xi and xj. The other six symmetries are denoted neg Tnxi,xj, and indicate that one co-factor is equivalent to the negation of the other. The relationships that are specified take the form, if Tpxi,xj and Tqxj,xk hold then Trxi,xj holds where Tp,Tq and Tr denote one of these twelve symmetry types