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[Research on the Application of Fuzzy Logic to Systems Analysis and Control]

Research conducted with the support of NASA Grant NCC2-275 has been focused in the main on the development of fuzzy logic and soft computing methodologies and their applications to systems analysis and control. with emphasis 011 problem areas which are of relevance to NASA's missions. One of the principal results of our research has been the development of a new methodology called Computing with Words (CW). Basically, in CW words drawn from a natural language are employed in place of numbers for computing and reasoning. There are two major imperatives for computing with words. First, computing with words is a necessity when the available information is too imprecise to justify the use of numbers, and second, when there is a tolerance for imprecision which can be exploited to achieve tractability, robustness, low solution cost, and better rapport with reality. Exploitation of the tolerance for imprecision is an issue of central importance in CW.

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The Gaseous Content of the Universe at Zeta less than 1.6

Together with graduate student Hsiao-Wen Chen, I have measured and analyzed structural and morphological parameters of 38 galaxies in eight fields for which sensitive measurements of corresponding Ly(alpha) absorption toward background QSOs are available. These measurements are based on Wide Field Planetary Camera 2 (WFPC2) observations obtained with the Hubble Space Telescope (HST) and provide a first look at how the incidence and extent of tenuous gas around galaxies depends on galaxy luminosity, size, and morphological type and on geometry of the impact. The primary result of the analysis is that the amount of gas encountered along the line of sight depends on the galaxy impact parameter and B-band luminosity but does not depend strongly on the galaxy average surface brightness, disk-to-bulge ratio, or redshift. This result confirms and improves upon an anti-correlation between Ly(alpha) absorption equivalent width and galaxy impact parameter found previously. More importantly, this result provides the first quantitative means of relating statistics of faint galaxies to statistics of Ly(alpha) absorption systems. which we plan to exploit to constrain the luminosity function of galaxies beyond the realm of current surveys. Results have been submitted for publication and will greatly improve our statistical conclusions . Together with graduate student Noriaki Yahata. I have measured and classified spectral properties of over 1000 faint galaxies and stars obtained in our low-resolution spectroscopic survey. The goal of this project is two-fold: (1) to exhaustively characterize the spectral properties of all faint galaxies that comprise our current survey, and (2) to gain experience with our measurement and classification code. which ultimately will be used on a data base of 20,000 galaxies to be obtained with the Two-Degree Field (2df) spectrograph at the Anglo-Australian Telescope (AAT). The results will ultimately be used for many goals, but so far we have concentrated on using the results to make a binary classification of the galaxies (i.e. early type versus late type) and to then exploit the density-morphology relationship to obtain a crude density indicator. The primary result of the analysis is that the incidence and extent of tenuous gas around galaxies shows no strong preference for local galaxy environment, at least over the range of densities spanned by the current observations. Along similar lines, two instances of Ly(alpha) absorption lines that arise in groups or clusters were examined. Analysis demonstrates that some can produce corresponding absorption lines and that LY(alpha) absorption lines do not avoid a high-density environment. A new measure of the galaxy-absorber cross-correlation function defines the statistical criterion by which galaxies and absorber pairs are to be matched. I have identified a damped Ly(alpha) absorption system at redshift z equals approximately 0.16, the lowest redshift confirmed to date. The most important results of the analysis are learning that the metal abundances of the absorption system are less than 10 percent of the solar metal abundance and that the absorbing gas is not rotating with the galaxy disk.

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Effect of Marangoni Convection Generated by Voids on Segregation During Low-G and 1-G Solidification

Solidification experiments, especially microgravity solidification experiments are often hampered by the evolution of unwanted voids or bubbles in the melt. Although these voids and/or bubbles are highly undesirable, there are currently no effective means of preventing their formation or eliminating their adverse effects, particularly, during low-g experiments. Marangoni Convection caused by these voids can drastically change the transport processes in the melt and, therefore, introduce enormous difficulties in interpreting the results of the space investigations. Recent microgravity experiments by Matthiesen, Andrews, and Fripp are all good examples of how the presence of voids and bubbles affect the outcome of costly space experiments and significantly increase the level of difficulty in interpreting their results. In this work we examine mixing caused by Marangoni convection generated by voids and bubbles in the melt during both 1-g and low-g solidification experiments. The objective of the research is to perform a detailed and comprehensive combined numerical-experimental study of Marangoni convection caused by voids during the solidification process and to show how it can affect segregation and growth conditions by modifying the flow, temperature, and species concentration fields in the melt. While Marangoni convection generated by bubbles and voids in the melt can lead to rapid mixing that would negate the benefits of microgravity processing, it could be exploited in some terrestrial processing to ensure effective communication between a melt/solid interface and a gas phase stoichiometry control zone. Thus we hope that this study will not only aid us in interpreting the results of microgravity solidification experiments hampered by voids and bubbles but to guide us in devising possible means of minimizing the adverse effects of Marangoni convection in future space experiments or of exploiting its beneficial mixing features in ground-based solidification.

Kassemi, M.↗

Novel Highly Parallel and Systolic Architectures Using Quantum Dot-Based Hardware

VLSI technology has made possible the integration of massive number of components (processors, memory, etc.) into a single chip. In VLSI design, memory and processing power are relatively cheap and the main emphasis of the design is on reducing the overall interconnection complexity since data routing costs dominate the power, time, and area required to implement a computation. Communication is costly because wires occupy the most space on a circuit and it can also degrade clock time. In fact, much of the complexity (and hence the cost) of VLSI design results from minimization of data routing. The main difficulty in VLSI routing is due to the fact that crossing of the lines carrying data, instruction, control, etc. is not possible in a plane. Thus, in order to meet this constraint, the VLSI design aims at keeping the architecture highly regular with local and short interconnection. As a result, while the high level of integration has opened the way for massively parallel computation, practical and full exploitation of such a capability in many applications of interest has been hindered by the constraints on interconnection pattern. More precisely. the use of only localized communication significantly simplifies the design of interconnection architecture but at the expense of somewhat restricted class of applications. For example, there are currently commercially available products integrating; hundreds of simple processor elements within a single chip. However, the lack of adequate interconnection pattern among these processing elements make them inefficient for exploiting a large degree of parallelism in many applications.

Fijany, Amir↗

Evaluation of Methods for Multidisciplinary Design Optimization (MDO)

A new MDO method, BLISS, and two different variants of the method, BLISS/RS and BLISS/S, have been implemented using iSIGHT's scripting language and evaluated in this report on multidisciplinary problems. All of these methods are based on decomposing a modular system optimization system into several subtasks optimization, that may be executed concurrently, and the system optimization that coordinates the subtasks optimization. The BLISS method and its variants are well suited for exploiting the concurrent processing capabilities in a multiprocessor machine. Several steps, including the local sensitivity analysis, local optimization, response surfaces construction and updates are all ideally suited for concurrent processing. Needless to mention, such algorithms that can effectively exploit the concurrent processing capabilities of the compute servers will be a key requirement for solving large-scale industrial design problems, such as the automotive vehicle problem detailed in Section 3.4.

Kodiyalam, Srinivas↗

SHIVA-(Spaceflight Holography in a Virtual Apparatus)

SHIVA (Spaceflight Holography Investigation in a Virtual Apparatus) will expand our understanding of the fundamental physics of particle movement in fluids by exploiting the power of holography in a spaceflight experiment'. In addition, the study will exploit the movement of particles in fluids to observe and quantify microgravity phenomena that are extremely important in materials sciences with applications both in space and on earth. The regime under scrutiny is the low Reynolds number, Stokes regime or creeping flow, which covers particles and bubbles moving at very low velocity. The equations describing this important regime have been under development and investigation for over 100 years and yet a complete analytical solution of the general equation had remained elusive yielding only approximations and numerical solutions. In the course of the ongoing NASA NRA, the first analytical solution of the general equation was produced by members of the investigator team using the mathematics of fractional derivatives. This opened the way to an even more insightful and important investigation of the phenomena in microgravity.

Trolinger, James D.↗

Some Problems and Solutions in Transferring Ecosystem Simulation Codes to Supercomputers

Many computer codes for the simulation of ecological systems have been developed in the last twenty-five years. This development took place initially on main-frame computers, then mini-computers, and more recently, on micro-computers and workstations. Recent recognition of ecosystem science as a High Performance Computing and Communications Program Grand Challenge area emphasizes supercomputers (both parallel and distributed systems) as the next set of tools for ecological simulation. Transferring ecosystem simulation codes to such systems is not a matter of simply compiling and executing existing code on the supercomputer since there are significant differences in the system architectures of sequential, scalar computers and parallel and/or vector supercomputers. To more appropriately match the application to the architecture (necessary to achieve reasonable performance), the parallelism (if it exists) of the original application must be exploited. We discuss our work in transferring a general grassland simulation model (developed on a VAX in the FORTRAN computer programming language) to a Cray Y-MP. We show the Cray shared-memory vector-architecture, and discuss our rationale for selecting the Cray. We describe porting the model to the Cray and executing and verifying a baseline version, and we discuss the changes we made to exploit the parallelism in the application and to improve code execution. As a result, the Cray executed the model 30 times faster than the VAX 11/785 and 10 times faster than a Sun 4 workstation. We achieved an additional speed-up of approximately 30 percent over the original Cray run by using the compiler's vectorizing capabilities and the machine's ability to put subroutines and functions "in-line" in the code. With the modifications, the code still runs at only about 5% of the Cray's peak speed because it makes ineffective use of the vector processing capabilities of the Cray. We conclude with a discussion and future plans.

Skiles, J. W.↗

Designing a Unique Single Point Cross Over Method

The idea behind genetic algorithms is to extract optimization strategies nature uses successfully - known as Darwinian Evolution - and transform them for application in mathematical optimization theory to find the global optimum in a defined phase space. One could imagine a population of individual 'explorers' sent into the optimization phase-space. Each explorer is defined by its genes, what means, its position inside the phase-space is coded in his genes. Every explorer has the duty to find a value of the quality of his position in the phase space. (Consider the phase-space being a number of variables in some technological process, the value of quality of any position in the phase space - in other words: any set of the variables - can be expressed by the yield of the desired chemical product.) Then the struggle of 'life' begins. The three fundamental principles are selection, mating/crossover, and mutation. Only explorers (= genes) sitting on the best places will reproduce and create a new population. This is performed in the second step (mating/crossover). The 'hope' behind this part of the algorithm is, that 'good' sections of two parents will be recombined to yet better fitting children. In fact, many of the created children will not be successful (as in biological evolution), but a few children will indeed fulfill this hope. These good sections are named in some publications as building blocks. Now there appears a problem. Repeating these steps, no new area would be explored. The two former steps would only exploit the already known regions in the phase space, which could lead to premature convergence of the algorithm with the consequence of missing the global optimum by exploiting some local optimum. The third step, mutation, ensures the necessary accidental effects. One can imagine the new population being mixed up a little bit to bring some new information into this set of genes. Whereas in biology a gene is described as a macro-molecule with four different bases to code the genetic information, a gene in genetic algorithms is usually defined as a bitstring (a sequence of b 1's and 0's).

Wilson, Richard Phillip↗

Application-Controlled Demand Paging for Out-of-Core Visualization

In the area of scientific visualization, input data sets are often very large. In visualization of Computational Fluid Dynamics (CFD) in particular, input data sets today can surpass 100 Gbytes, and are expected to scale with the ability of supercomputers to generate them. Some visualization tools already partition large data sets into segments, and load appropriate segments as they are needed. However, this does not remove the problem for two reasons: 1) there are data sets for which even the individual segments are too large for the largest graphics workstations, 2) many practitioners do not have access to workstations with the memory capacity required to load even a segment, especially since the state-of-the-art visualization tools tend to be developed by researchers with much more powerful machines. When the size of the data that must be accessed is larger than the size of memory, some form of virtual memory is simply required. This may be by segmentation, paging, or by paged segments. In this paper we demonstrate that complete reliance on operating system virtual memory for out-of-core visualization leads to poor performance. We then describe a paged segment system that we have implemented, and explore the principles of memory management that can be employed by the application for out-of-core visualization. We show that application control over some of these can significantly improve performance. We show that sparse traversal can be exploited by loading only those data actually required. We show also that application control over data loading can be exploited by 1) loading data from alternative storage format (in particular 3-dimensional data stored in sub-cubes), 2) controlling the page size. Both of these techniques effectively reduce the total memory required by visualization at run-time. We also describe experiments we have done on remote out-of-core visualization (when pages are read by demand from remote disk) whose results are promising.

Cox, Michael↗

By Hand or Not By-Hand: A Case Study of Alternative Approaches to Parallelize CFD Applications

While parallel processing promises to speed up applications by several orders of magnitude, the performance achieved still depends upon several factors, including the multiprocessor architecture, system software, data distribution and alignment, as well as the methods used for partitioning the application and mapping its components onto the architecture. The existence of the Gorden Bell Prize given out at Supercomputing every year suggests that while good performance can be attained for real applications on general purpose multiprocessors, the large investment in man-power and time still has to be repeated for each application-machine combination. As applications and machine architectures become more complex, the cost and time-delays for obtaining performance by hand will become prohibitive. Computer users today can turn to three possible avenues for help: parallel libraries, parallel languages and compilers, interactive parallelization tools. The success of these methodologies, in turn, depends on proper application of data dependency analysis, program structure recognition and transformation, performance prediction as well as exploitation of user supplied knowledge. NASA has been developing multidisciplinary applications on highly parallel architectures under the High Performance Computing and Communications Program. Over the past six years, the transition of underlying hardware and system software have forced the scientists to spend a large effort to migrate and recede their applications. Various attempts to exploit software tools to automate the parallelization process have not produced favorable results. In this paper, we report our most recent experience with CAPTOOL, a package developed at Greenwich University. We have chosen CAPTOOL for three reasons: 1. CAPTOOL accepts a FORTRAN 77 program as input. This suggests its potential applicability to a large collection of legacy codes currently in use. 2. CAPTOOL employs domain decomposition to obtain parallelism. Although the fact that not all kinds of parallelism are handled may seem unappealing, many NASA applications in computational aerosciences as well as earth and space sciences are amenable to domain decomposition. 3. CAPTOOL generates code for a large variety of environments employed across NASA centers: MPI/PVM on network of workstations to the IBS/SP2 and CRAY/T3D.

Yan, Jerry C.↗

Parallelization of NAS Benchmarks for Shared Memory Multiprocessors

This paper presents our experiences of parallelizing the sequential implementation of NAS benchmarks using compiler directives on SGI Origin2000 distributed shared memory (DSM) system. Porting existing applications to new high performance parallel and distributed computing platforms is a challenging task. Ideally, a user develops a sequential version of the application, leaving the task of porting to new generations of high performance computing systems to parallelization tools and compilers. Due to the simplicity of programming shared-memory multiprocessors, compiler developers have provided various facilities to allow the users to exploit parallelism. Native compilers on SGI Origin2000 support multiprocessing directives to allow users to exploit loop-level parallelism in their programs. Additionally, supporting tools can accomplish this process automatically and present the results of parallelization to the users. We experimented with these compiler directives and supporting tools by parallelizing sequential implementation of NAS benchmarks. Results reported in this paper indicate that with minimal effort, the performance gain is comparable with the hand-parallelized, carefully optimized, message-passing implementations of the same benchmarks.

Waheed, Abdul↗

The Dynamics of Cascaded Monod System Models Through Five Levels

In the context of this paper, a Monod system model is a set of ordinary differential equations in which the terms resemble those which Motion presented in his 1949 paper. Attention is directed to the multiple trophic level case in which each trophic level exploits only one of the trophic levels for its perpetuation, and no two trophic entities exploit the same trophic level (cascaded). The treatment expands from a primary producer progressively through five trophic levels. Types of stability are identified and are related to persistence, and the consequences of some intuitive scaling structures are developed. These considerations are relevant to some theoretical questions in ecology and to applications such as bioreactor operation.

Blackwell, Charles↗

Practical Computer Security through Cryptography

The core protocols upon which the Internet was built are insecure. Weak authentication and the lack of low level encryption services introduce vulnerabilities that propagate upwards in the network stack. Using statistics based on CERT/CC Internet security incident reports, the relative likelihood of attacks via these vulnerabilities is analyzed. The primary conclusion is that the standard UNIX BSD-based authentication system is by far the most commonly exploited weakness. Encryption of Sensitive password data and the adoption of cryptographically-based authentication protocols can greatly reduce these vulnerabilities. Basic cryptographic terminology and techniques are presented, with attention focused on the ways in which technology such as encryption and digital signatures can be used to protect against the most commonly exploited vulnerabilities. A survey of contemporary security software demonstrates that tools based on cryptographic techniques, such as Kerberos, ssh, and PGP, are readily available and effectively close many of the most serious security holes. Nine practical recommendations for improving security are described.

McNab, David↗

Pressure Measurement Systems

System 8400 is an advanced system for measurement of gas and liquid pressure, along with a variety of other parameters, including voltage, frequency and digital inputs. System 8400 offers exceptionally high speed data acquisition through parallel processing, and its modular design allows expansion from a relatively inexpensive entry level system by the addition of modular Input Units that can be installed or removed in minutes. Douglas Juanarena was on the team of engineers that developed a new technology known as ESP (electronically scanned pressure). The Langley ESP measurement system was based on miniature integrated circuit pressure-sensing transducers that communicated pressure information to a minicomputer. In 1977, Juanarena formed PSI to exploit the NASA technology. In 1978 he left Langley, obtained a NASA license for the technology, introduced the first commercial product, the 780B pressure measurement system. PSI developed a pressure scanner for automation of industrial processes. Now in its second design generation, the DPT-6400 is capable of making 2,000 measurements a second and has 64 channels by addition of slave units. New system 8400 represents PSI's bid to further exploit the $600 million U.S. industrial pressure measurement market. It is geared to provide a turnkey solution to physical measurement.

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Optimization of Ocean Color Algorithms: Application to Satellite Data Merging

The objective of our program is to develop and validate a procedure for ocean color data merging which is one of the major goals of the SIMBIOS project. The need for a merging capability is dictated by the fact that since the launch of MODIS on the Terra platform and over the next decade, several global ocean color missions from various space agencies are or will be operational simultaneously. The apparent redundancy in simultaneous ocean color missions can actually be exploited to various benefits. The most obvious benefit is improved coverage. The patchy and uneven daily coverage from any single sensor can be improved by using a combination of sensors. Beside improved coverage of the global Ocean the merging of Ocean color data should also result in new, improved, more diverse and better data products with lower uncertainties. Ultimately, ocean color data merging should result in the development of a unified, scientific quality, ocean color time series, from SeaWiFS to NPOESS and beyond. Various approaches can be used for ocean color data merging and several have been tested within the frame of the SIMBIOS program. As part of the SIMBIOS Program, we have developed a merging method for ocean color data. Conversely to other methods our approach does not combine end-products like the subsurface chlorophyll concentration (chl) from different sensors to generate a unified product. Instead, our procedure uses the normalized water-leaving radiances (L(sub WN)(lambda)) from single or multiple sensors and uses them in the inversion of a semi-analytical ocean color model that allows the retrieval of several ocean color variables simultaneously. Beside ensuring simultaneity and consistency of the retrievals (all products are derived from a single algorithm), this model-based approach has various benefits over techniques that blend end-products (e.g. chlorophyll): 1) it works with single or multiple data sources regardless of their specific bands, 2) it exploits band redundancies and band differences, 3) it accounts for uncertainties in the (L(sub WN)(lambda)) data and, 4) it provides uncertainty estimates for the retrieved variables.

Maritorena, Stephane↗

Coherent Effects in Tiny Optics: Tunneling Through the Looking Glass

I will discuss two types of one-dimensional photonic bandgap (PBG) effects that can arise in systems of coupled spherical resonators: (1) nearly-free-photon Fabry-Perot photonic bands that arise in quarter-wave concentrically stratified spheres and, (2) tight- binding photonic bands that arise in weakly-coupled mutually-resonant spheres as a result of whispering-gallery mode splitting. These effects can be derived directly from Mie theory, in a more straightforward manner, by exploiting an analogy with stratified planar systems. For odd numbers of mutually-resonant lossless coupled ring resonators, the circulating intensity can increase exponentially with the number of resonators, which can potentially be exploited for the development of advanced sensors. For even numbers of resonators, mode splitting and classical destructive interference lead to a cancellation of absorption and slow light on-resonance, reminiscent of electromagnetic induced transparency. The analogy between these coherent photon trapping effects and population trapping in an atomic system will be explored.

Smith, David D.↗

Optimization Of Ocean Color Algorithms: Application To Satellite And In Situ Data Merging

The objective of our program is to develop and validate a procedure for ocean color data merging which is one of the major goals of the SIMBIOS project (McClain et al., 1995). The need for a merging capability is dictated by the fact that since the launch of MODIS on the Terra platform and over the next decade, several global ocean color missions from various space agencies are or will be operational simultaneously. The apparent redundancy in simultaneous ocean color missions can actually be exploited to various benefits. The most obvious benefit is improved coverage (Gregg et al., 1998; Gregg & Woodward, 1998). The patchy and uneven daily coverage from any single sensor can be improved by using a combination of sensors. Beside improved coverage of the global ocean the merging of ocean color data should also result in new, improved, more diverse and better data products with lower uncertainties. Ultimately, ocean color data merging should result in the development of a unified, scientific quality, ocean color time series, from SeaWiFS to NPOESS and beyond. Various approaches can be used for ocean color data merging and several have been tested within the frame of the SIMBIOS program (see e.g. Kwiatkowska & Fargion, 2003, Franz et al., 2003). As part of the SIMBIOS Program, we have developed a merging method for ocean color data. Conversely to other methods our approach does not combine end-products like the subsurface chlorophyll concentration (chl) from different sensors to generate a unified product. Instead, our procedure uses the normalized waterleaving radiances (LwN( )) from single or multiple sensors and uses them in the inversion of a semianalytical ocean color model that allows the retrieval of several ocean color variables simultaneously. Beside ensuring simultaneity and consistency of the retrievals (all products are derived from a single algorithm), this model-based approach has various benefits over techniques that blend end-products (e.g. chlorophyll): 1) it works with single or multiple data sources regardless of their specific bands, 2) it exploits band redundancies and band differences, 3) it accounts for uncertainties in the LwN( ) data and, 4) it provides uncertainty estimates for the retrieved variables.

Maritorena, Stephane↗

Toward Synthesis, Analysis, and Certification of Security Protocols

Implemented security protocols are basically pieces of software which are used to (a) authenticate the other communication partners, (b) establish a secure communication channel between them (using insecure communication media), and (c) transfer data between the communication partners in such a way that these data only available to the desired receiver, but not to anyone else. Such an implementation usually consists of the following components: the protocol-engine, which controls in which sequence the messages of the protocol are sent over the network, and which controls the assembly/disassembly and processing (e.g., decryption) of the data. the cryptographic routines to actually encrypt or decrypt the data (using given keys), and t,he interface to the operating system and to the application. For a correct working of such a security protocol, all of these components must work flawlessly. Many formal-methods based techniques for the analysis of a security protocols have been developed. They range from using specific logics (e.g.: BAN-logic [4], or higher order logics [12] to model checking [2] approaches. In each approach, the analysis tries to prove that no (or at least not a modeled intruder) can get access to secret data. Otherwise, a scenario illustrating the &tack may be produced. Despite the seeming simplicity of security protocols ("only" a few messages are sent between the protocol partners in order to ensure a secure communication), many flaws have been detected. Unfortunately, even a perfect protocol engine does not guarantee flawless working of a security protocol, as incidents show. Many break-ins and security vulnerabilities are caused by exploiting errors in the implementation of the protocol engine or the underlying operating system. Attacks using buffer-overflows are a very common class of such attacks. Errors in the implementation of exception or error handling can open up additional vulnerabilities. For example, on a website with a log-in screen: multiple tries with invalid passwords caused the expected error message (too many retries). but let the user nevertheless pass. Finally, security can be compromised by silly implementation bugs or design decisions. In a commercial VPN software, all calls to the encryption routines were incidentally replaced by stubs, probably during factory testing. The product worked nicely. and the error (an open VPN) would have gone undetected, if a team member had not inspected the low-level traffic out of curiosity. Also, the use secret proprietary encryption routines can backfire, because such algorithms often exhibit weaknesses which can be exploited easily (see e.g., DVD encoding). Summarizing, there is large number of possibilities to make errors which can compromise the security of a protocol. In today s world with short time-to-market and the use of security protocols in open and hostile networks for safety-critical applications (e.g., power or air-traffic control), such slips could lead to catastrophic situations. Thus, formal methods and automatic reasoning techniques should not be used just for the formal proof of absence of an attack, but they ought to be used to provide an end-to-end tool-supported framework for security software. With such an approach all required artifacts (code, documentation, test cases) , formal analyses, and reliable certification will be generated automatically, given a single, high level specification. By a combination of program synthesis, formal protocol analysis, certification; and proof-carrying code, this goal is within practical reach, since all the important technologies for such an approach actually exist and only need to be assembled in the right way.

Schumann, Johann↗