Alexey Solovyev

A machine-checked proof of the odd order theorem

This paper reports on a six-year collaborative effort that culminated in a complete formalization of a proof of the Feit-Thompson Odd Order Theorem in the Coq proof assistant. The formalized proof is constructive, and relies on nothing but the axioms and rules of the foundational framework implemented by Coq. To support the formalization, we developed a comprehensive set of reusable libraries of formalized mathematics, including results in finite group theory, linear algebra, Galois theory, and the theories of the real and complex algebraic numbers.

A dynamic view of trauma/hemorrhage-induced inflammation in mice: principal drivers and networks.

Background Complex biological processes such as acute inflammation induced by trauma/hemorrhagic shock/ (T/HS) are dynamic and multi-dimensional. We utilized multiplexing cytokine analysis coupled with data-driven modeling to gain a systems perspective into T/HS. Methodology/Principal Findings Mice were subjected to surgical cannulation trauma (ST) ± hemorrhagic shock (HS; 25 mmHg), and followed for 1, 2, 3, or 4 h in each case. Serum was assayed for 20 cytokines and NO2 −/NO3 −. These data were analyzed using four data-driven methods (Hierarchical Clustering Analysis [HCA], multivariate analysis [MA], Principal Component Analysis [PCA], and Dynamic Network Analysis [DyNA]). Using HCA, animals subjected to ST vs. ST + HS could be partially segregated based on inflammatory mediator profiles, despite a large overlap. Based on MA, interleukin [IL]-12p40/p70 (IL-12.total), monokine induced by interferon-γ (CXCL-9) [MIG], and IP-10 were the best discriminators between ST and ST/HS. PCA suggested that the inflammatory mediators found in the three main principal components in animals subjected to ST were IL-6, IL-10, and IL-13, while the three principal components in ST + HS included a large number of cytokines including IL-6, IL-10, keratinocyte-derived cytokine (CXCL-1) [KC], and tumor necrosis factor-α [TNF-α]. DyNA suggested that the circulating mediators produced in response to ST were characterized by a high degree of interconnection/complexity at all time points; the response to ST + HS consisted of different central nodes, and exhibited zero network density over the first 2 h with lesser connectivity vs. ST at all time points. DyNA also helped link the conclusions from MA and PCA, in that central nodes consisting of IP-10 and IL-12 were seen in ST, while MIG and IL-6 were central nodes in ST + HS. Conclusions/Significance These studies help elucidate the dynamics of T/HS-induced inflammation, complementing other forms of dynamic mechanistic modeling. These methods should be applicable to the analysis of other complex biological processes.

Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Rigorous estimation of maximum floating-point round-off errors is an important capability central to many formal verification tools. Unfortunately, available techniques for this task often provide overestimates. Also, there are no available rigorous approaches that handle transcendental functions. We have developed a new approach called Symbolic Taylor Expansions that avoids this difficulty, and implemented a new tool called FPTaylor embodying this approach. Key to our approach is the use of rigorous global optimization, instead of the more familiar interval arithmetic, affine arithmetic, and/or SMT solvers. In addition to providing far tighter upper bounds of round-off error in a vast majority of cases, FPTaylor also emits analysis certificates in the form of HOL Light proofs. We release FPTaylor along with our benchmarks for evaluation.

Efficient search for inputs causing high floating-point errors

Tools for floating-point error estimation are fundamental to program understanding and optimization. In this paper, we focus on tools for determining the input settings to a floating point routine that maximizes its result error. Such tools can help support activities such as precision allocation, performance optimization, and auto-tuning. We benchmark current abstraction-based precision analysis methods, and show that they often do not work at scale, or generate highly pessimistic error estimates, often caused by non-linear operators or complex input constraints that define the set of legal inputs. We show that while concrete-testing-based error estimation methods based on maintaining shadow values at higher precision can search out higher error-inducing inputs, suit able heuristic search guidance is key to finding higher errors. We develop a heuristic search algorithm called Binary Guided Random Testing (BGRT). In 45 of the 48 total benchmarks, including many real-world routines, BGRT returns higher guaranteed errors. We also evaluate BGRT against two other heuristic search methods called ILS and PSO, obtaining better results.

Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations

We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds. We developed the verification tool as a part of the Flyspeck project (a formal proof of the Kepler conjecture). The Flyspeck project includes about 1000 nonlinear inequalities. We successfully tested our method on more than 100 Flyspeck inequalities and estimated that the formal verification procedure is about 3000 times slower than an informal verification method implemented in C++. We also describe future work and prospective optimizations for our method.

A multiscale agent-based in silico model of liver fibrosis progression

Chronic hepatic inflammation involves a complex interplay of inflammatory and mechanical influences, ultimately manifesting in a characteristic histopathology of liver fibrosis. We created an agent-based model (ABM) of liver tissue in order to computationally examine the consequence of liver inflammation. Our liver fibrosis ABM (LFABM) is comprised of literature-derived rules describing molecular and histopathological aspects of inflammation and fibrosis in a section of chemically injured liver. Hepatocytes are modeled as agents within hexagonal lobules. Injury triggers an inflammatory reaction, which leads to activation of local Kupffer cells and recruitment of monocytes from circulation. Portal fibroblasts and hepatic stellate cells are activated locally by the products of inflammation. The various agents in the simulation are regulated by above-threshold concentrations of pro- and anti-inflammatory cytokines and damage-associated molecular pattern molecules. The simulation progresses from chronic inflammation to collagen deposition, exhibiting periportal fibrosis followed by bridging fibrosis, and culminating in disruption of the regular lobular structure. The ABM exhibited key histopathological features observed in liver sections from rats treated with carbon tetrachloride (CCl4). An in silico “tension test” for the hepatic lobules predicted an overall increase in tissue stiffness, in line with clinical elastography literature and published studies in CCl4-treated rats. Therapy simulations suggested differential anti-fibrotic effects of neutralizing tumor necrosis factor alpha vs. enhancing M2 Kupffer cells. We conclude that a computational model of liver inflammation on a structural skeleton of physical forces can recapitulate key histopathological and macroscopic properties of CCl4-injured liver. This multiscale approach linking molecular and chemomechanical stimuli enables a model that could be used to gain translationally relevant insights into liver fibrosis.

SPARK: A Framework for Multi-Scale Agent-Based Biomedical Modeling

Multi-scale modeling of complex biological systems remains a central challenge in the systems biology community. A method of dynamic knowledge representation known as agent-based modeling enables the study of higher level behavior emerging from discrete events performed by individual components. With the advancement of computer technology, agent-based modeling has emerged as an innovative technique to model the complexities of systems biology. In this work, the authors describe SPARK Simple Platform for Agent-based Representation of Knowledge, a framework for agent-based modeling specifically designed for systems-level biomedical model development. SPARK is a stand-alone application written in Java. It provides a user-friendly interface, and a simple programming language for developing Agent-Based Models ABMs. SPARK has the following features specialized for modeling biomedical systems: 1 continuous space that can simulate real physical space; 2 flexible agent size and shape that can represent the relative proportions of various cell types; 3 multiple spaces that can concurrently simulate and visualize multiple scales in biomedical models; 4 a convenient graphical user interface. Existing ABMs of diabetic foot ulcers and acute inflammation were implemented in SPARK. Models of identical complexity were run in both NetLogo and SPARK; the SPARK-based models ran two to three times faster.

Hybrid Equation/Agent-Based Model of Ischemia-Induced Hyperemia and Pressure Ulcer Formation Predicts Greater Propensity to Ulcerate in Subjects with Spinal Cord Injury

Pressure ulcers are costly and life-threatening complications for people with spinal cord injury (SCI). People with SCI also exhibit differential blood flow properties in non-ulcerated skin. We hypothesized that a computer simulation of the pressure ulcer formation process, informed by data regarding skin blood flow and reactive hyperemia in response to pressure, could provide insights into the pathogenesis and effective treatment of post-SCI pressure ulcers. Agent-Based Models (ABM) are useful in settings such as pressure ulcers, in which spatial realism is important. Ordinary Differential Equation-based (ODE) models are useful when modeling physiological phenomena such as reactive hyperemia. Accordingly, we constructed a hybrid model that combines ODEs related to blood flow along with an ABM of skin injury, inflammation, and ulcer formation. The relationship between pressure and the course of ulcer formation, as well as several other important characteristic patterns of pressure ulcer formation, was demonstrated in this model. The ODE portion of this model was calibrated to data related to blood flow following experimental pressure responses in non-injured human subjects or to data from people with SCI. This model predicted a higher propensity to form ulcers in response to pressure in people with SCI vs. non-injured control subjects, and thus may serve as novel diagnostic platform for post-SCI ulcer formation.

Rigorous floating-point mixed-precision tuning

Virtually all real-valued computations are carried out using floating-point data types and operations. The precision of these data types must be set with the goals of reducing the overall round-off error, but also emphasizing performance improvements. Often, a mixed-precision allocation achieves this optimum; unfortunately, there are no techniques available to compute such allocations and conservatively meet a given error target across all program inputs. In this work, we present a rigorous approach to precision allocation based on formal analysis via Symbolic Taylor Expansions, and error analysis based on interval functions. This approach is implemented in an automated tool called FPTuner that generates and solves a quadratically constrained quadratic program to obtain a precision-annotated version of the given expression. FPTuner automatically introduces all the requisite precision up and down casting operations. It also allows users to flexibly control precision allocation using constraints to cap the number of high precision operators as well as group operators to allocate the same precision to facilitate vectorization. We evaluate FPTuner by tuning several benchmarks and measuring the proportion of lower precision operators allocated as we increase the error threshold. We also measure the reduction in energy consumption resulting from executing mixed-precision tuned code on a real hardware platform. We observe significant energy savings in response to mixed-precision tuning, but also observe situations where unexpected compiler behaviors thwart intended optimizations.

A Computational, Tissue-Realistic Model of Pressure Ulcer Formation in Individuals with Spinal Cord Injury.

People with spinal cord injury (SCI) are predisposed to pressure ulcers (PU). PU remain a significant burden in cost of care and quality of life despite improved mechanistic understanding and advanced interventions. An agent-based model (ABM) of ischemia/reperfusion-induced inflammation and PU (the PUABM) was created, calibrated to serial images of post-SCI PU, and used to investigate potential treatments in silico. Tissue-level features of the PUABM recapitulated visual patterns of ulcer formation in individuals with SCI. These morphological features, along with simulated cell counts and mediator concentrations, suggested that the influence of inflammatory dynamics caused simulations to be committed to “better” vs. “worse” outcomes by 4 days of simulated time and prior to ulcer formation. Sensitivity analysis of model parameters suggested that increasing oxygen availability would reduce PU incidence. Using the PUABM, in silico trials of anti-inflammatory treatments such as corticosteroids and a neutralizing antibody targeted at Damage-Associated Molecular Pattern molecules (DAMPs) suggested that, at best, early application at a sufficiently high dose could attenuate local inflammation and reduce pressure-associated tissue damage, but could not reduce PU incidence. The PUABM thus shows promise as an adjunct for mechanistic understanding, diagnosis, and design of therapies in the setting of PU.