Highly Automated Formal Verification of Arithmetic Circuits

This dissertation investigates the problems of two distinctive formal verification techniques for verifying large scale multiplier circuits and proposes two approaches to overcome some of these problems. The first technique is equivalence checking based on recurrence relations, while the second one is the symbolic computation technique which is based on the theory of Gröbner bases. This investigation demonstrates that approaches based on symbolic computation have better scalability and more robustness than state-of-the-art equivalence checking techniques for verification of arithmetic circuits. According to this conclusion, the thesis leverages the symbolic computation technique to verify floating-point designs. It proposes a new algebraic equivalence checking, in contrast to classical combinational equivalence checking, the proposed technique is capable of checking the equivalence of two circuits which have different architectures of arithmetic units as well as control logic parts, e.g., floating-point multipliers

Decidability and complexity of equivalences for simple process algebras

In this thesis I study decidability, complexity and structural properties of strong and weak bisimilarity with respect to two process algebras, Basic Process Algebras and Basic Parallel Process Algebras. The decidability of strong bisimilarity for both algebras is an established result. For the subclasses of normed BPA-processes and BPP there even exist polynomial decision procedures. The complexity of deciding strong bisimilarity for the whole class of BPP is unsatisfactory since it is not bounded by any primitive recursive function. Here we present a new approach that encodes BPP as special polynomials and expresses strong bisimulation in terms of polynomial ideals and then uses a theorem about polynomial ideals (Hilbert's Basis Theorem) and an algorithm from computer algebra (Gröbner bases) to construct a new decision procedure. For weak bisimilarity, Hirshfeld found a decision procedure for the subclasses of totally normed BPA-processes and BPP, and Esparza demonstrated a semidecision procedure for general BPP. The remaining questions are still unsolved. Here we provide some lower bounds on the computational complexity of a decision procedure that might exist. For BPP we show that the decidability problem is NP-hard (even for the class of totally normed BPP), for BPA-processes we show that the decidability problem is PSPACE-hard. Finally we study the notion of weak bisimilarity in terms of its inductive definition. We start from the relation containing all pairs of processes and then form a non-increasing chain of relations by eliminating pairs that do not satisfy a certain expansion condition. These relations are labelled by ordinal numbers and are called approximants. We know that this chain eventually converges for some a' such that =a' = =b' = = for all a' w^w, and for BPPA, a' => w.2. For some restricted classes of BPA and BPPA we show that = = =w.2

When Less Is More: Consequence-Finding in a Weak Theory of Arithmetic

This paper presents a theory of non-linear integer/real arithmetic and algorithms for reasoning about this theory. The theory can be conceived as an extension of linear integer/real arithmetic with a weakly-axiomatized multiplication symbol, which retains many of the desirable algorithmic properties of linear arithmetic. In particular, we show that the conjunctive fragment of the theory can be effectively manipulated (analogously to the usual operations on convex polyhedra, the conjunctive fragment of linear arithmetic). As a result, we can solve the following consequence-finding problem: given a ground formula F, find the strongest conjunctive formula that is entailed by F. As an application of consequence-finding, we give a loop invariant generation algorithm that is monotone with respect to the theory and (in a sense) complete. Experiments show that the invariants generated from the consequences are effective for proving safety properties of programs that require non-linear reasoning

On the intelligent management of sepsis in the intensive care unit

The management of the Intensive Care Unit (ICU) in a hospital has its own, very specific requirements that involve, amongst others, issues of risk-adjusted mortality and average length of stay; nurse turnover and communication with physicians; technical quality of care; the ability to meet patient's family needs; and avoid medical error due rapidly changing circumstances and work overload. In the end, good ICU management should lead to an improvement in patient outcomes. Decision making at the ICU environment is a real-time challenge that works according to very tight guidelines, which relate to often complex and sensitive research ethics issues. Clinicians in this context must act upon as much available information as possible, and could therefore, in general, benefit from at least partially automated computer-based decision support based on qualitative and quantitative information. Those taking executive decisions at ICUs will require methods that are not only reliable, but also, and this is a key issue, readily interpretable. Otherwise, any decision tool, regardless its sophistication and accuracy, risks being rendered useless. This thesis addresses this through the design and development of computer based decision making tools to assist clinicians at the ICU. It focuses on one of the main problems that they must face: the management of the Sepsis pathology. Sepsis is one of the main causes of death for non-coronary ICU patients. Its mortality rate can reach almost up to one out of two patients for septic shock, its most acute manifestation. It is a transversal condition affecting people of all ages. Surprisingly, its definition has only been standardized two decades ago as a systemic inflammatory response syndrome with confirmed infection. The research reported in this document deals with the problem of Sepsis data analysis in general and, more specifically, with the problem of survival prediction for patients affected with Severe Sepsis. The tools at the core of the investigated data analysis procedures stem from the fields of multivariate and algebraic statistics, algebraic geometry, machine learning and computational intelligence. Beyond data analysis itself, the current thesis makes contributions from a clinical point of view, as it provides substantial evidence to the debate about the impact of the preadmission use of statin drugs in the ICU outcome. It also sheds light into the dependence between Septic Shock and Multi Organic Dysfunction Syndrome. Moreover, it defines a latent set of Sepsis descriptors to be used as prognostic factors for the prediction of mortality and achieves an improvement on predictive capability over indicators currently in use.La gestió d'una Unitat de Cures Intensives (UCI) hospitalària presenta uns requisits força específics incloent, entre altres, la disminució de la taxa de mortalitat, la durada de l'ingrès, la rotació d'infermeres i la comunicació entre metges amb al finalitad de donar una atenció de qualitat atenent als requisits tant dels malalts com dels familiars. També és força important controlar i minimitzar els error mèdics deguts a canvis sobtats i a la presa ràpida de deicisions assistencials. Al cap i a la fi, la bona gestió de la UCI hauria de resultar en una reducció de la mortalitat i durada d'estada. La presa de decisions en un entorn de crítics suposa un repte de presa de decisions en temps real d'acord a unes guies clíniques molt restrictives i que, pel que fa a la recerca, poden resultar en problemes ètics força sensibles i complexos. Per tant, el personal sanitari que ha de prendre decisions sobre la gestió de malalts crítics no només requereix eines de suport a la decisió que siguin fiables sinó que, a més a més, han de ser interpretables. Altrament qualsevol eina de decisió que no presenti aquests trets no és considerarà d'utilitat clínica. Aquesta tesi doctoral adreça aquests requisits mitjançant el desenvolupament d'eines de suport a la decisió per als intensivistes i es focalitza en un dels principals problemes als que s'han denfrontar: el maneig del malalt sèptic. La Sèpsia és una de les principals causes de mortalitats a les UCIS no-coronàries i la seva taxa de mortalitat pot arribar fins a la meitat dels malalts amb xoc sèptic, la seva manifestació més severa. La Sèpsia és un síndrome transversal, que afecta a persones de totes les edats. Sorprenentment, la seva definició ha estat estandaritzada, fa només vint anys, com a la resposta inflamatòria sistèmica a una infecció corfimada. La recerca presentada en aquest document fa referència a l'anàlisi de dades de la Sèpsia en general i, de forma més específica, al problema de la predicció de la supervivència de malalts afectats amb Sèpsia Greu. Les eines i mètodes que formen la clau de bòveda d'aquest treball provenen de diversos camps com l'estadística multivariant i algebràica, geometria algebraica, aprenentatge automàtic i inteligència computacional. Més enllà de l'anàlisi per-se, aquesta tesi també presenta una contribució des de el punt de vista clínic atès que presenta evidència substancial en el debat sobre l'impacte de l'administració d'estatines previ a l'ingrès a la UCI en els malalts sèptics. També s'aclareix la forta dependència entre el xoc sèptic i el Síndrome de Disfunció Multiorgànica. Finalment, també es defineix un conjunt de descriptors latents de la Sèpsia com a factors de pronòstic per a la predicció de la mortalitat, que millora sobre els mètodes actualment més utilitzats en la UCI

The geometry of Gaussian double Markovian distributions

Gaussian double Markovian models consist of covariance matrices constrained by a pair of graphs specifying zeros simultaneously in the covariance matrix and its inverse. We study the semi-algebraic geometry of these models, in particular their dimension, smoothness and connectedness as well as algebraic and combinatorial properties.Comment: 31 pages. v2: major revision; the new Theorem 3.23 unified some earlier results; the numbers in Remark 3.33 have been correcte