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NASA NTRS1978

Radiative transfer calculated from a Markov chain formalism

The theory of Markov chains is used to formulate the radiative transport problem in a general way by modeling the successive interactions of a photon as a stochastic process. Under the minimal requirement that the stochastic process is a Markov chain, the determination of the diffuse reflection or transmission from a scattering atmosphere is equivalent to the solution of a system of linear equations. This treatment is mathematically equivalent to, and thus has many of the advantages of, Monte Carlo methods, but can be considerably more rapid than Monte Carlo algorithms for numerical calculations in particular applications. We have verified the speed and accuracy of this formalism for the standard problem of finding the intensity of scattered light from a homogeneous plane-parallel atmosphere with an arbitrary phase function for scattering. Accurate results over a wide range of parameters were obtained with computation times comparable to those of a standard 'doubling' routine. The generality of this formalism thus allows fast, direct solutions to problems that were previously soluble only by Monte Carlo methods. Some comparisons are made with respect to integral equation methods.

Esposito, L. W.↗

NASA NTRS1981

Adaptation of the TH Epsilon Mu formalism for the analysis of the equivalence principle in the presence of the weak and electroweak interaction

The TH epsilon mu formalism, used in analyzing equivalence principle experiments of metric and nonmetric gravity theories, is adapted to the description of the electroweak interaction using the Weinberg-Salam unified SU(2) x U(1) model. The use of the TH epsilon mu formalism is thereby extended to the weak interactions, showing how the gravitational field affects W sub mu (+ or -1) and Z sub mu (0) boson propagation and the rates of interactions mediated by them. The possibility of a similar extension to the strong interactions via SU(5) grand unified theories is briefly discussed. Also, using the effects of the potentials on the baryon and lepton wave functions, the effects of gravity on transition mediated in high-A atoms which are electromagnetically forbidden. Three possible experiments to test the equivalence principle in the presence of the weak interactions, which are technologically feasible, are then briefly outline: (1) K-capture by the FE nucleus (counting the emitted X-ray); (2) forbidden absorption transitions in high-A atoms' vapor; and (3) counting the relative Beta-decay rates in a suitable alpha-beta decay chain, assuming the strong interactions obey the equivalence principle.

Fennelly, A. J.↗

NASA NTRS1983

The use of a formal simulator to verify a simple real time control program

The authors present an initial and elementary investigation of the formal specification and mechanical verification of programs that interact with environments. They describe a mechanical proof that a simple, real time control program keeps a vehicle on a straightline course in a variable crosswind. To formalize the specification they define a mathematical function which models the interaction of the program and its environment. They then state and proved two theorems about this function: the simulated vehicle never gets farther than three units away from the intended course, and it comes to the course if the wind ever remains steady for at least four sampling units.

Boyer, R. S.↗

NASA NTRS1984

Formal verification of mathematical software

Methods are investigated for formally specifying and verifying the correctness of mathematical software (software which uses floating point numbers and arithmetic). Previous work in the field was reviewed. A new model of floating point arithmetic called the asymptotic paradigm was developed and formalized. Two different conceptual approaches to program verification, the classical Verification Condition approach and the more recently developed Programming Logic approach, were adapted to use the asymptotic paradigm. These approaches were then used to verify several programs; the programs chosen were simplified versions of actual mathematical software.

Sutherland, D.↗

NASA NTRS1985

New formalism for two-photon quantum optics. I - Quadrature phases and squeezed states. II - Mathematical foundation and compact notation

A new formalism for analyzing two-photon devices, such as parametric amplifiers and phase-conjugate mirrors, is proposed in part I, focusing on the properties and the significance of the quadrature-phase amplitudes and two-mode squeezed states. Time-stationary quasi-probability noise is also detailed for the case of Gaussian noise, and uncertainty principles for the quadrature-phase amplitudes are outlined, as well as some important properties of the two-mode states. Part II establishes a mathematical foundation for the formalism, with introduction of a vector notation for compact representation of two-mode properties. Fundamental unitary operators and special quantum states are also examined with an emphasis on the two-mode squeezed states. The results are applied to a previously studied degenerate limit (epsilon = 0).

Caves, C. M.↗

NASA NTRS1986

Statistical foundation of the fluid analogue of the soliton formalism

A fully nonlinear analysis is used to develop a general soliton formalism for the description of the nonlinear evolution of soliton fluctuations in both plasmas and classical fluids. From the Navier-Stokes equations for plasmas and compressible fluids of two scales, two equations for the propagation of density waves are derived. A fast soliton field is spontaneously created by rarefaction, and a slow density wave modulates the field intensity as a ponderomotive force. Constitutive properties are demonstrated using a Lagrangian-kinetic formalism of the fluctuation-dissipation theory.

Tchen, C. M.↗

NASA NTRS1988

Toward a mathematical formalism of performance, task difficulty, and activation

The rudiments of a mathematical formalism for handling operational, physiological, and psychological concepts are developed for use by the man-machine system design engineer. The formalism provides a framework for developing a structured, systematic approach to the interface design problem, using existing mathematical tools, and simplifying the problem of telling a machine how to measure and use performance.

Samaras, George M.↗

NASA NTRS1986

Structuring the formal definition of Ada

The structure of the formal definition of Ada are described. At present, a difficult subset of Ada has been defined and the experience gained so far by this work is reported. Currently, the work continues towards the formal definition of the Ada language.

Hansen, Kurt W.↗

NASA NTRS1989

Formalisms for user interface specification and design

The application of formal methods to the specification and design of human-computer interfaces is described. A broad outline of human-computer interface problems, a description of the field of cognitive engineering and two relevant research results, the appropriateness of formal specification techniques, and potential NASA application areas are described.

Auernheimer, Brent J.↗

NASA NTRS1990

Improving system reliability through formal analysis and use of checks in software

Software is playing increasingly important roles in avionics systems. It is widely used in navigation and, in some cases, in control loops that maintain aircraft stability. To guarantee the safety of flight systems, the FAA requires that critical components have a probability of failure no greater than 10(exp -9) per hour of flight. Software is being used to diagnose system components for failure. SIFT (Software Implemented Fault Tolerance) was a computer system developed to study the use of software to check for failure and manage processor reconfiguration. To guarantee that software satisfies its specifications, formal verification can be used. With this a program and its specification are viewed as mathematical objects, and a mathematical proof is used to show that the program and its specification are equivalent. In previous research, a theory of checking was developed to offer assistance in analyzing specifications and designing run-time checks. In the theory, checking is considered abstractly in terms of n-ary relations much like those of relational database theory. Within the theory check are categorized, checks on input and checks on results are considered, and formal attention is given to the minimization and logical combination of checks. The focus is upon input checks and the obstacles in checking input to critical systems. A central concern is with a property referred to as independence. The concern is with circumstances under which it is possible to apply isolated, independent checks to separate sensor inputs and be assure that all illegal input will be properly detected. Presently, independence is being investigated and checked in the context of the GCS (Guidance and Control System). The GCS simulator is intended for testing software that implements control laws for landing spacecraft. The large number of inputs and their complex interrelationships provide an exciting context in which to investigate independence and the difficulties of supplying input checks.

Staknis, Mark E.↗

NASA NTRS1990

Design for validation, based on formal methods

Validation of ultra-reliable systems decomposes into two subproblems: (1) quantification of probability of system failure due to physical failure; (2) establishing that Design Errors are not present. Methods of design, testing, and analysis of ultra-reliable software are discussed. It is concluded that a design-for-validation based on formal methods is needed for the digital flight control systems problem, and also that formal methods will play a major role in the development of future high reliability digital systems.

Butler, Ricky W.↗

NASA NTRS1990

What can formal methods offer to digital flight control systems design

Formal methods research begins to produce methods which will enable mathematic modeling of the physical behavior of digital hardware and software systems. The development of these methods directly supports the NASA mission of increasing the scope and effectiveness of flight system modeling capabilities. The conventional, continuous mathematics that is used extensively in modeling flight systems is not adequate for accurate modeling of digital systems. Therefore, the current practice of digital flight control system design has not had the benefits of extensive mathematical modeling which are common in other parts of flight system engineering. Formal methods research shows that by using discrete mathematics, very accurate modeling of digital systems is possible. These discrete modeling methods will bring the traditional benefits of modeling to digital hardware and hardware design. Sound reasoning about accurate mathematical models of flight control systems can be an important part of reducing risk of unsafe flight control.

Good, Donald I.↗

NASA NTRS1990

Applications of formal simulation languages in the control and monitoring subsystems of Space Station Freedom

The notions, benefits, and drawbacks of numeric simulation are introduced. Two formal simulation languages, Simpscript and Modsim are introduced. The capabilities of each are discussed briefly, and then the two programs are compared. The use of simulation in the process of design engineering for the Control and Monitoring System (CMS) for Space Station Freedom is discussed. The application of the formal simulation language to the CMS design is presented, and recommendations are made as to their use.

Lacovara, R. C.↗

NASA NTRS1991

Report on the formal specification and partial verification of the VIPER microprocessor

The formal specification and partial verification of the VIPER microprocessor is reviewed. The VIPER microprocessor was designed by RSRE, Malvern, England, for safety critical computing applications (e.g., aircraft, reactor control, medical instruments, armaments). The VIPER was carefully specified and partially verified in an attempt to provide a microprocessor with completely predictable operating characteristics. The specification of VIPER is divided into several levels of abstraction, from a gate-level description up to an instruction execution model. Although the consistency between certain levels was demonstrated with mechanically-assisted mathematical proof, the formal verification of VIPER was never completed.

Brock, Bishop↗

NASA NTRS1991

Verification of the FtCayuga fault-tolerant microprocessor system. Volume 2: Formal specification and correctness theorems

Presented here is a formal specification and verification of a property of a quadruplicately redundant fault tolerant microprocessor system design. A complete listing of the formal specification of the system and the correctness theorems that are proved are given. The system performs the task of obtaining interactive consistency among the processors using a special instruction on the processors. The design is based on an algorithm proposed by Pease, Shostak, and Lamport. The property verified insures that an execution of the special instruction by the processors correctly accomplishes interactive consistency, providing certain preconditions hold, using a computer aided design verification tool, Spectool, and the theorem prover, Clio. A major contribution of the work is the demonstration of a significant fault tolerant hardware design that is mechanically verified by a theorem prover.

Bickford, Mark↗

NASA NTRS1990

Improving software quality - The use of formal inspections at the Jet Propulsion Laboratory

The introduction of software formal inspections (Fagan Inspections) at JPL for finding and fixing defects early in the software development life cycle are reviewed. It is estimated that, by the year 2000, some software efforts will rise to as much as 80 percent of the total. Software problems are especially important at NASA as critical flight software must be error-free. It is shown that formal inspections are particularly effective at finding and removing defects having to do with clarity, correctness, consistency, and completeness. A very significant discovery was that code audits were not as effective at finding defects as code inspections.

Bush, Marilyn↗

NASA NTRS1990

Effect of formal specifications on program complexity and reliability: An experimental study

The results are presented of an experimental study undertaken to assess the improvement in program quality by using formal specifications. Specifications in the Z notation were developed for a simple but realistic antimissile system. These specifications were then used to develop 2 versions in C by 2 programmers. Another set of 3 versions in Ada were independently developed from informal specifications in English. A comparison of the reliability and complexity of the resulting programs suggests the advantages of using formal specifications in terms of number of errors detected and fault avoidance.

Goel, Amrit L.↗

NASA NTRS1992

Formal design and verification of a reliable computing platform for real-time control. Phase 2: Results

The design and formal verification of the Reliable Computing Platform (RCP), a fault tolerant computing system for digital flight control applications is presented. The RCP uses N-Multiply Redundant (NMR) style redundancy to mask faults and internal majority voting to flush the effects of transient faults. The system is formally specified and verified using the Ehdm verification system. A major goal of this work is to provide the system with significant capability to withstand the effects of High Intensity Radiated Fields (HIRF).

Butler, Ricky W.↗