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.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

On the Structure of Counterexamples to Symmetric Orderings for BDD's

AbstractBinary Decision Diagrams (BDDs) are used to represent boolean functions in a variety of applications. The size of a reduced ordered BDD depends on the ordering of variables. Several researchers have suggested grouping symmetric variables as a promising heuristic for finding good orderings. In this paper we study the conjecture which states that symmetric variables gather in at least one of the optimum variable orders. First, we prove some useful properties of partially symmetric functions. Next, we develop a faster procedure for finding counterexamples to this conjecture that exploits the partitioning of boolean functions into nn-equivalence classes. Third, we study the structure of counterexamples and devise a new and simple method to generate new counterexamples from given counterexamples. Finally, we present different kinds of counterexamples, which show that boolean functions are very diverse with respect to where symmetric orders can fall in the range from optimal orders to worst-case orders

Parameterized verification and repair of concurrent systems

In this thesis, we present novel approaches for model checking, repair and synthesis of systems that may be parameterized in their number of components. The parameterized model checking problem (PMCP) is in general undecidable, and therefore the focus is on restricted classes of parameterized concurrent systems where the problem is decidable. Under certain conditions, the problem is decidable for guarded protocols, and for systems that communicate via a token, a pairwise, or a broadcast synchronization. In this thesis we improve existing results for guarded protocols and we show that the PMCP of guarded protocols and token passing systems is decidable for specifications that add a quantitative aspect to LTL, called Prompt-LTL. Furthermore, we present, to our knowledge, the first parameterized repair algorithm. The parameterized repair problem is to find a refinement of a process implementation p such that the concurrent system with an arbitrary number of instances of p is correct. We show how this algorithm can be used on classes of systems that can be represented as well structured transition systems (WSTS). Additionally we present two safety synthesis algorithms that utilize a lazy approach. Given a faulty system, the algorithms first symbolically model check the system, then the obtained error traces are analyzed to synthesize a candidate that has no such traces. Experimental results show that our algorithm solves a number of benchmarks that are intractable for existing tools. Furthermore, we introduce our tool AIGEN for generating random Boolean functions and transition systems in a symbolic representation.In dieser Arbeit stellen wir neuartige Ans atze für das Model-Checking, die Reparatur und die Synthese von Systemen vor, die in ihrer Anzahl von Komponenten parametrisiert sein können. Das Problem des parametrisierten Model-Checking (PMCP) ist im Allgemeinen unentscheidbar, und daher liegt der Fokus auf eingeschränkten Klassen parametrisierter synchroner Systeme, bei denen das Problem entscheidbar ist. Unter bestimmten Bedingungen ist das Problem für Guarded Protocols und für Systeme, die über ein Token, eine Pairwise oder eine Broadcast-Synchronisation kommunizieren, entscheidbar. In dieser Arbeit verbessern wir bestehende Ergebnisse für Guarded Protocols und zeigen die Entscheidbarkeit des PMCP für Guarded Protocols und Token-Passing Systeme mit Spezifikationen in der temporalen Logik Prompt-LTL, die LTL einen quantitativen Aspekt hinzufügt. Darüber hinaus präsentieren wir unseres Wissens den ersten parametrisierten Reparaturalgorithmus. Das parametrisierte Reparaturproblem besteht darin, eine Verfeinerung einer Prozessimplementierung p zu finden, so dass das synchrone Systeme mit einer beliebigen Anzahl von Instanzen von p korrekt ist. Wir zeigen, wie dieser Algorithmus auf Klassen von Systemen angewendet werden kann, die als Well Structured Transition Systems (WSTS) dargestellt werden können. Außerdem präsentieren wir zwei Safety-Synthesis Algorithmen, die einen "lazy" Ansatz verwenden. Bei einem fehlerhaften System überprüfen die Algorithmen das System symbolisch, dann werden die erhaltenen "Gegenbeispiel" analysiert, um einen Kandidaten zu synthetisieren der keine solchen Fehlerpfade hat. Versuchsergebnisse zeigen, dass unser Algorithmus eine Reihe von Benchmarks löst, die für bestehende Tools nicht lösbar sind. Darüber hinaus stellen wir unser Tool AIGEN zur Erzeugung zufälliger Boolescher Funktionen und Transitionssysteme in einer symbolischen Darstellung vor

Advances in Functional Decomposition: Theory and Applications

Functional decomposition aims at finding efficient representations for Boolean functions. It is used in many applications, including multi-level logic synthesis, formal verification, and testing. This dissertation presents novel heuristic algorithms for functional decomposition. These algorithms take advantage of suitable representations of the Boolean functions in order to be efficient. The first two algorithms compute simple-disjoint and disjoint-support decompositions. They are based on representing the target function by a Reduced Ordered Binary Decision Diagram (BDD). Unlike other BDD-based algorithms, the presented ones can deal with larger target functions and produce more decompositions without requiring expensive manipulations of the representation, particularly BDD reordering. The third algorithm also finds disjoint-support decompositions, but it is based on a technique which integrates circuit graph analysis and BDD-based decomposition. The combination of the two approaches results in an algorithm which is more robust than a purely BDD-based one, and that improves both the quality of the results and the running time. The fourth algorithm uses circuit graph analysis to obtain non-disjoint decompositions. We show that the problem of computing non-disjoint decompositions can be reduced to the problem of computing multiple-vertex dominators. We also prove that multiple-vertex dominators can be found in polynomial time. This result is important because there is no known polynomial time algorithm for computing all non-disjoint decompositions of a Boolean function. The fifth algorithm provides an efficient means to decompose a function at the circuit graph level, by using information derived from a BDD representation. This is done without the expensive circuit re-synthesis normally associated with BDD-based decomposition approaches. Finally we present two publications that resulted from the many detours we have taken along the winding path of our research

A prover for the mu mu CRL toolset with applications : version 0.1

This document describes an automated theorem prover, based on an extension of binary decision diagrams. The prover transforms quantifier-free formulae into equivalent BDD-forms, w.r.t.~to some algebraic data specification. The prover is used by four tools for the symbolic analysis of distributed systems specified in $mu$CRL (i.e.~process algebra plus algebraic data types). The main techniques are invariants and confluence. Two case studies are reported: the DKR leader election protocol [13], and SPLICE [15], a coordination architecture of industrial origin. In both cases using confluence information leads to a reduced state space

Automated Synthesis of Memristor Crossbar Networks

The advancement of semiconductor device technology over the past decades has enabled the design of increasingly complex electrical and computational machines. Electronic design automation (EDA) has played a significant role in the design and implementation of transistor-based machines. However, as transistors move closer toward their physical limits, the speed-up provided by Moore\u27s law will grind to a halt. Once again, we find ourselves on the verge of a paradigm shift in the computational sciences as newer devices pave the way for novel approaches to computing. One of such devices is the memristor -- a resistor with non-volatile memory. Memristors can be used as junctional switches in crossbar circuits, which comprise of intersecting sets of vertical and horizontal nanowires. The major contribution of this dissertation lies in automating the design of such crossbar circuits -- doing a new kind of EDA for a new kind of computational machinery. In general, this dissertation attempts to answer the following questions: a. How can we synthesize crossbars for computing large Boolean formulas, up to 128-bit? b. How can we synthesize more compact crossbars for small Boolean formulas, up to 8-bit? c. For a given loop-free C program doing integer arithmetic, is it possible to synthesize an equivalent crossbar circuit? We have presented novel solutions to each of the above problems. Our new, proposed solutions resolve a number of significant bottlenecks in existing research, via the usage of innovative logic representation and artificial intelligence techniques. For large Boolean formulas (up to 128-bit), we have utilized Reduced Ordered Binary Decision Diagrams (ROBDDs) to automatically synthesize linearly growing crossbar circuits that compute them. This cutting edge approach towards flow-based computing has yielded state-of-the-art results. It is worth noting that this approach is scalable to n-bit Boolean formulas. We have made significant original contributions by leveraging artificial intelligence for automatic synthesis of compact crossbar circuits. This inventive method has been expanded to encompass crossbar networks with 1D1M (1-diode-1-memristor) switches, as well. The resultant circuits satisfy the tight constraints of the Feynman Grand Prize challenge and are able to perform 8-bit binary addition. A leading edge development for end-to-end computation with flow-based crossbars has been implemented, which involves methodical translation of loop-free C programs into crossbar circuits via automated synthesis. The original contributions described in this dissertation reflect the substantial progress we have made in the area of electronic design automation for synthesis of memristor crossbar networks

Integrating MDG variable ordering in a VHDL-MDG design verification system

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal