Doctor of Philosophy

dissertationFormal verification of hardware designs has become an essential component of the overall system design flow. The designs are generally modeled as finite state machines, on which property and equivalence checking problems are solved for verification. Reachability analysis forms the core of these techniques. However, increasing size and complexity of the circuits causes the state explosion problem. Abstraction is the key to tackling the scalability challenges. This dissertation presents new techniques for word-level abstraction with applications in sequential design verification. By bundling together k bit-level state-variables into one word-level constraint expression, the state-space is construed as solutions (variety) to a set of polynomial constraints (ideal), modeled over the finite (Galois) field of 2^k elements. Subsequently, techniques from algebraic geometry -- notably, Groebner basis theory and technology -- are researched to perform reachability analysis and verification of sequential circuits. This approach adds a "word-level dimension" to state-space abstraction and verification to make the process more efficient. While algebraic geometry provides powerful abstraction and reasoning capabilities, the algorithms exhibit high computational complexity. In the dissertation, we show that by analyzing the constraints, it is possible to obtain more insights about the polynomial ideals, which can be exploited to overcome the complexity. Using our algorithm design and implementations, we demonstrate how to perform reachability analysis of finite-state machines purely at the word level. Using this concept, we perform scalable verification of sequential arithmetic circuits. As contemporary approaches make use of resolution proofs and unsatisfiable cores for state-space abstraction, we introduce the algebraic geometry analog of unsatisfiable cores, and present algorithms to extract and refine unsatisfiable cores of polynomial ideals. Experiments are performed to demonstrate the efficacy of our approaches

Time machines and the Principle of Self-Consistency as a consequence of the Principle of Stationary Action (II): the Cauchy problem for a self-interacting relativistic particle

We consider the action principle to derive the classical, relativistic motion of a self-interacting particle in a 4-D Lorentzian spacetime containing a wormhole and which allows the existence of closed time-like curves. In particular, we study the case of a pointlike particle subject to a `hard-sphere' self-interaction potential and which can traverse the wormhole an arbitrary number of times, and show that the only possible trajectories for which the classical action is stationary are those which are globally self-consistent. Generically, the multiplicity of these trajectories (defined as the number of self-consistent solutions to the equations of motion beginning with given Cauchy data) is finite, and it becomes infinite if certain constraints on the same initial data are satisfied. This confirms the previous conclusions (for a non-relativistic model) by Echeverria, Klinkhammer and Thorne that the Cauchy initial value problem in the presence of a wormhole `time machine' is classically `ill-posed' (far too many solutions). Our results further extend the recent claim by Novikov et al. that the `Principle of self-consistency' is a natural consequence of the `Principle of minimal action.'Comment: 39 pages, latex fil

Identities and bases in plactic, hypoplactic, sylvester, and related monoids

The ubiquitous plactic monoid, also known as the monoid of Young tableaux, has deep connections to several areas of mathematics, in particular, to the theory of symmetric functions. An active research topic is the identities satisfied by the plactic monoids of finite rank. It is known that there is no “global" identity satisfied by the plactic monoid of every rank. In contrast, monoids related to the plactic monoid, such as the hypoplactic monoid (the monoid of quasi-ribbon tableaux), sylvester monoid (the monoid of binary search trees) and Baxter monoid (the monoid of pairs of twin binary search trees), satisfy global identities, and the shortest identities have been characterized. In this thesis, we present new results on the identities satisfied by the hypoplactic, sylvester, #-sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, thus showing that the identity checking problems of these monoids are in the complexity class P, and prove that the varieties generated by these monoids have finite axiomatic rank, by giving a finite basis for them. We also give a subdirect representation ofmultihomogeneous monoids by finite subdirectly irreducible Rees factor monoids, thus showing that they are residually finite.O ubíquo monóide plático, também conhecido como o monóide dos diagramas de Young, tem ligações profundas a várias áreas de Matemática, em particular à teoria das funções simétricas. Um tópico de pesquisa ativo é o das identidades satisfeitas pelos monóides pláticos de característica finita. Sabe-se que não existe nenhuma identidade “global” satisfeita pelos monóides pláticos de cada característica. Em contraste, sabe-se que monóides ligados ao monóide plático, como o monóide hipoplático (o monóide dos diagramas quasifita), o monóide silvestre (o monóide de árvores de busca binárias) e o monóide de Baxter (o monóide de pares de árvores de busca binária gémeas), satisfazem identidades globais, e as identidades mais curtas já foram caracterizadas. Nesta tese, apresentamos novos resultados acerca das identidades satisfeitas pelos monóides hipopláticos, silvestres, silvestres-# e de Baxter. Mostramos como mergulhar estes monóides, de característica estritamente maior que 2, num produto direto de cópias do monóide correspondente de característica 2. Confirmamos assim que todos os monóides da mesma família, de característica maior ou igual a 2, satisfazem exatamente as mesmas identidades. A seguir, damos uma caracterização completa dessas identidades, mostrando assim que os problemas de verificação de identidades destes monóides estão na classe de complexidade P, e provamos que as variedades geradas por estes monóides têm característica axiomática finita, ao apresentar uma base finita para elas. Também damos uma representação subdireta de monóides multihomogéneos por monóides fatores de Rees finitos e subdiretamente irredutíveis, mostrando assim que são residualmente finitos

Faster Algorithms for Dynamic Algebraic Queries in Basic RSMs with Constant Treewidth

Interprocedural analysis is at the heart of numerous applications in programming languages, such as alias analysis, constant propagation, and so on. Recursive state machines (RSMs) are standard models for interprocedural analysis. We consider a general framework with RSMs where the transitions are labeled from a semiring and path properties are algebraic with semiring operations. RSMs with algebraic path properties can model interprocedural dataflow analysis problems, the shortest path problem, the most probable path problem, and so on. The traditional algorithms for interprocedural analysis focus on path properties where the starting point is fixed as the entry point of a specific method. In this work, we consider possible multiple queries as required in many applications such as in alias analysis. The study of multiple queries allows us to bring in an important algorithmic distinction between the resource usage of the one-time preprocessing vs for each individual query. The second aspect we consider is that the control flow graphs for most programs have constant treewidth. Our main contributions are simple and implementable algorithms that support multiple queries for algebraic path properties for RSMs that have constant treewidth. Our theoretical results show that our algorithms have small additional one-time preprocessing but can answer subsequent queries significantly faster as compared to the current algorithmic solutions for interprocedural dataflow analysis. We have also implemented our algorithms and evaluated their performance for performing on-demand interprocedural dataflow analysis on various domains, such as for live variable analysis and reaching definitions, on a standard benchmark set. Our experimental results align with our theoretical statements and show that after a lightweight preprocessing, on-demand queries are answered much faster than the standard existing algorithmic approaches

Enabling geometry-based 3-D tele-immersion with fast mesh compression and linear rateless coding

3-D tele-immersion (3DTI) enables participants in remote locations to share, in real time, an activity. It offers users interactive and immersive experiences, but it challenges current media-streaming solutions. Work in the past has mainly focused on the efficient delivery of image-based 3-D videos and on realistic rendering and reconstruction of geometry-based 3-D objects. The contribution of this paper is a real-time streaming component for 3DTI with dynamic reconstructed geometry. This component includes both a novel fast compression method and a rateless packet protection scheme specifically designed towards the requirements imposed by real time transmission of live-reconstructed mesh geometry. Tests on a large dataset show an encoding speed-up up to ten times at comparable compression ratio and quality, when compared with the high-end MPEG-4 SC3DMC mesh encoders. The implemented rateless code ensures complete packet loss protection of the triangle mesh object and a delivery delay within interactive bounds. Contrary to most linear fountain codes, the designed codec enables real-time progressive decoding allowing partial decoding each time a packet is received. This approach is compared with transmission over TCP in packet loss rates and latencies, typical in managed WAN and MAN networks, and heavily outperforms it in terms of end-to-end delay. The streaming component has been integrated into a larger 3DTI environment that includes state of the art 3-D reconstruction and rendering modules. This resulted in a prototype that can capture, compress transmit, and render triangle mesh geometry in real-time in realistic internet conditions as shown in experiments. Compared with alternative methods, lower interactive end-to-end delay and frame rates over three times higher are achieved