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DOE OSTI2020

Derivation of K-matrix reaction theory in a discrete basis formalism

The usual derivations of the S and K matrices for two-particle reactions proceed through the Lippmann–Schwinger equation with formal definitions of the incoming and outgoing scattering states. Here we present a simpler alternative derivation that is carried out completely in the Hamiltonian representation, using a discrete basis of configurations for the scattering channels as well as the quasi-bound configurations of the combined fragments. We use matrix algebra to derive an explicit expression for the K matrix in terms of the Hamiltonian of the internal states of the compound system and the coupling between the channels and the internal states. The formula for the K matrix includes explicitly a real dispersive shift matrix to the internal Hamiltonian that is easily computed in the formalism. That expression is applied to derive the usual form of the S matrix as a sum over poles in the complex energy plane. Some extensions and limitations of the discrete-basis Hamiltonian formalism are discussed in the concluding remarks and in the Appendix.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2024

Thermodynamically consistent Cahn–Hilliard–Navier–Stokes equations using the metriplectic dynamics formalism

Cahn–Hilliard–Navier–Stokes (CHNS) systems describe flows with two-phases, e.g., a liquid with bubbles. Obtaining constitutive relations for general dissipative processes for such systems, which are thermodynamically consistent, can be a challenge. We show how the metriplectic 4-bracket formalism (Morrison and Updike, 2024) achieves this in a straightforward, in fact algorithmic, manner. First, from the noncanonical Hamiltonian formulation for the ideal part of a CHNS system we obtain an appropriate Casimir to serve as the entropy in the metriplectic formalism that describes the dissipation (e.g. viscosity, heat conductivity and diffusion effects). General thermodynamics with the concentration variable and its thermodynamics conjugate, the chemical potential, are included. Having expressions for the Hamiltonian (energy), entropy, and Poisson bracket, we describe a procedure for obtaining a metriplectic 4-bracket that describes thermodynamically consistent dissipative effects. The 4-bracket formalism leads naturally to a general CHNS system that allows for anisotropic surface energy effects. Furthermore, this general CHNS system reduces to cases in the literature, to which we can compare.

Cahn–Hilliard↗

DOE OSTI2022

Variational approach to quantum state tomography based on maximal entropy formalism

Quantum state tomography is an integral part of quantum computation and offers the starting point for the validation of various quantum devices. One of the central tasks in the field of state tomography is to reconstruct, with high fidelity, the quantum states of a quantum system. From an experiment on a real quantum device, one can obtain the mean measurement values of different operators. With such data as input, in this report we employ the maximal entropy formalism to construct the least biased mixed quantum state that is consistent with the given set of expectation values. Even though, in principle, the reported formalism is quite general and should work for an arbitrary set of observables, in practice we shall demonstrate the efficacy of the algorithm on an informationally complete (IC) set of Hermitian operators. Such a set possesses the advantage of uniquely specifying a single quantum state from which the experimental measurements have been sampled and hence renders the rare opportunity not only to construct a least-biased quantum state but even replicate the exact state prepared experimentally within a preset tolerance. Here, the primary workhorse of the algorithm is reconstructing an energy function which we designate as the effective Hamiltonian of the system, and parameterizing it with Lagrange multipliers, according to the formalism of maximal entropy. These parameters are thereafter optimized variationally so that the reconstructed quantum state of the system converges to the true quantum state within an error threshold. To this end, we employ a parameterized quantum circuit and a hybrid quantum-classical variational algorithm to obtain such a target state, making our recipe easily implementable on a near-term quantum device.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Scrutinizing formally Ni IV centers through the lenses of core spectroscopy, molecular orbital theory, and valence bond theory

Nickel K- and L 2,3 -edge X-ray absorption spectra (XAS) are discussed for 16 complexes and complex ions with nickel centers spanning a range of formal oxidation states from II to IV. K-edge XAS alone is shown to be an ambiguous metric of physical oxidation state for these Ni complexes. Meanwhile, L 2,3 -edge XAS reveals that the physical d-counts of the formally Ni IV compounds measured lie well above the d 6 count implied by the oxidation state formalism. The generality of this phenomenon is explored computationally by scrutinizing 8 additional complexes. The extreme case of NiF 6 2- is considered using high-level molecular orbital approaches as well as advanced valence bond methods. The emergent electronic structure picture reveals that even highly electronegative F-donors are incapable of supporting a physical d 6 Ni IV center. The reactivity of Ni IV complexes is then discussed, highlighting the dominant role of the ligands in this chemistry over that of the metal centers.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

Efficient real space formalism for hybrid density functionals

We present an efficient real space formalism for hybrid exchange-correlation functionals in generalized Kohn–Sham density functional theory (DFT). In particular, we develop an efficient representation for any function of the real space finite-difference Laplacian matrix by leveraging its Kronecker product structure, thereby enabling the time to solution of associated linear systems to be highly competitive with the fast Fourier transform scheme while not imposing any restrictions on the boundary conditions. We implement this formalism for both the unscreened and range-separated variants of hybrid functionals. We verify its accuracy and efficiency through comparisons with established planewave codes for isolated as well as bulk systems. In particular, we demonstrate up to an order-of-magnitude speedup in time to solution for the real space method. We also apply the framework to study the structure of liquid water using ab initio molecular dynamics, where we find good agreement with the literature. Overall, the current formalism provides an avenue for efficient real-space DFT calculations with hybrid density functionals.

Chemistry↗

DOE OSTI2022

Comparison of the canonical transformation and energy functional formalisms for ab initio calculations of self-localized polarons

In materials with strong electron-phonon (e-ph) interactions, charge carriers can distort the surrounding lattice and become trapped, forming self-localized (small) polarons. We recently developed an ab initio approach based on canonical transformations to efficiently compute the formation and energetics of small polarons A different approach based on a Landau-Pekar energy functional has been proposed in the recent literature. Here in this work, we analyze and compare these two methods in detail. We show that the small polaron energy is identical in the two formalisms when using the same polaron wave function. We also show that our canonical transformation formalism can predict polaron band structures and can properly treat zero- and finite-temperature lattice vibration effects, although at present using a fixed polaron wave function. Conversely, the energy functional approach can compute the polaron wave function, but as we show here, it neglects lattice vibrations and cannot address polaron self-localization and thermal band narrowing. Taken together, this work relates two different methods developed recently to study polarons from first-principles, highlighting their merits and shortcomings and discussing them both in a unified formalism.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Correspondence between Color Glass Condensate and High-Twist Formalism

The color glass condensate (CGC) effective theory and the collinear factorization at high twist (HT) are two well-known frameworks describing perturbative QCD multiple scatterings in nuclear media. It has long been recognized that these two formalisms have their own domain of validity in different kinematic regions. Taking direct photon production in proton-nucleus collisions as an example, we clarify for the first time the relation between CGC and HT at the level of a physical observable. We show that the CGC formalism beyond shock-wave approximation, and with the Landau-Pomeranchuk-Migdal interference effect is consistent with the HT formalism in the transition region where they overlap. Such a unified picture paves the way for mapping out the phase diagram of parton density in nuclear medium from dilute to dense region.

Parton distribution functions↗

DOE OSTI2024

Multi-Rigor Agile Verification and Rapid Prototyping for Formally Verified Software

We propose a novel approach to developing formally verified systems through Multi-rigor Agile Verification. Multi-rigor Agile Verification is rooted in the hypothesis of Rigor Independence, that a system’s specification and verification architecture depend primarily on the system requirements to be verified, and they depend very little on the rigor level of the methods used to verify those requirements. Due to its iterative nature, Multi-rigor Agile Verification promises to mitigate many of the high upfront design costs experienced by formally verified systems and to deliver a better-architected, and thus better-trusted, system in the end. We then discuss the tooling needed to perform Multi-rigor Agile Verification and go in depth to build one of those tools, which directly generates executable prototype code from declarative formal specifications using the Maude rewrite-logic framework.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1979

Formal methods for achieving reliable software

Requirements for reliable avionic systems are discussed in terms of the effectiveness of programming methodology. The need for methods to cope with the complexity of critical real-time systems is emphasized. Some general concepts about formal methods are presented and an example is given of the SRI hierarchical development methodology taken from the executive system of the SIFT fault tolerant computer. Formal methods with alternatives are compared and the prospects for introducing formal methods into practice are considered.

Goldberg, J.↗

NASA NTRS1982

Formal convergence characteristics of elliptically constrained incremental Newton-Raphson algorithms

Various aspects of the convergence, uniqueness, and existence properties associated with solutions generated via the elliptically constrained incremental Newton-Raphson (ECINR) algorithm are analyzed. Several theorems are developed, and the formal behavior of the elliptically constrained scheme developed by Padovan (1981) is discussed in detail. Consideration is given to global and local rates of convergence, to the determination of the occurrence of safety zones wherein the algorithm yields inherently convergent results, to formal limitations on the class of functions which the scheme can be applied to solve, and to single and multidimensional formalisms on existence uniqueness and convergence. Special attention is given to functions whose Jacobian matrix exhibit positive, negative, semi and indefinite properties. Several significant advantages of ECINR over the classical INR are mentioned.

Padovan, J.↗

NASA NTRS1989

Formal verification of a fault tolerant clock synchronization algorithm

A formal specification and mechanically assisted verification of the interactive convergence clock synchronization algorithm of Lamport and Melliar-Smith is described. Several technical flaws in the analysis given by Lamport and Melliar-Smith were discovered, even though their presentation is unusally precise and detailed. It seems that these flaws were not detected by informal peer scrutiny. The flaws are discussed and a revised presentation of the analysis is given that not only corrects the flaws but is also more precise and easier to follow. Some of the corrections to the flaws require slight modifications to the original assumptions underlying the algorithm and to the constraints on its parameters, and thus change the external specifications of the algorithm. The formal analysis of the interactive convergence clock synchronization algorithm was performed using the Enhanced Hierarchical Development Methodology (EHDM) formal specification and verification environment. This application of EHDM provides a demonstration of some of the capabilities of the system.

Rushby, John↗

NASA NTRS1989

Formal verification of AI software

The application of formal verification techniques to Artificial Intelligence (AI) software, particularly expert systems, is investigated. Constraint satisfaction and model inversion are identified as two formal specification paradigms for different classes of expert systems. A formal definition of consistency is developed, and the notion of approximate semantics is introduced. Examples are given of how these ideas can be applied in both declarative and imperative forms.

Rushby, John↗

NASA NTRS1991

Beyond formalism

The ongoing debate over the role of formalism and formal specifications in software features many speakers with diverse positions. Yet, in the end, they share the conviction that the requirements of a software system can be unambiguously specified, that acceptable software is a product demonstrably meeting the specifications, and that the design process can be carried out with little interaction between designers and users once the specification has been agreed to. This conviction is part of a larger paradigm prevalent in American management thinking, which holds that organizations are systems that can be precisely specified and optimized. This paradigm, which traces historically to the works of Frederick Taylor in the early 1900s, is no longer sufficient for organizations and software systems today. In the domain of software, a new paradigm, called user-centered design, overcomes the limitations of pure formalism. Pioneered in Scandinavia, user-centered design is spreading through Europe and is beginning to make its way into the U.S.

Denning, Peter J.↗

NASA NTRS1992

Moving formal methods into practice. Verifying the FTPP Scoreboard: Results, phase 1

This report documents the Phase 1 results of an effort aimed at formally verifying a key hardware component, called Scoreboard, of a Fault-Tolerant Parallel Processor (FTPP) being built at Charles Stark Draper Laboratory (CSDL). The Scoreboard is part of the FTPP virtual bus that guarantees reliable communication between processors in the presence of Byzantine faults in the system. The Scoreboard implements a piece of control logic that approves and validates a message before it can be transmitted. The goal of Phase 1 was to lay the foundation of the Scoreboard verification. A formal specification of the functional requirements and a high-level hardware design for the Scoreboard were developed. The hardware design was based on a preliminary Scoreboard design developed at CSDL. A main correctness theorem, from which the functional requirements can be established as corollaries, was proved for the Scoreboard design. The goal of Phase 2 is to verify the final detailed design of Scoreboard. This task is being conducted as part of a NASA-sponsored effort to explore integration of formal methods in the development cycle of current fault-tolerant architectures being built in the aerospace industry.

Srivas, Mandayam↗

NASA NTRS1992

Formal mechanization of device interactions with a process algebra

The principle emphasis is to develop a methodology to formally verify correct synchronization communication of devices in a composed hardware system. Previous system integration efforts have focused on vertical integration of one layer on top of another. This task examines 'horizontal' integration of peer devices. To formally reason about communication, we mechanize a process algebra in the Higher Order Logic (HOL) theorem proving system. Using this formalization we show how four types of device interactions can be represented and verified to behave as specified. The report also describes the specification of a system consisting of an AVM-1 microprocessor and a memory management unit which were verified in previous work. A proof of correct communication is presented, and the extensions to the system specification to add a direct memory device are discussed.

Schubert, E. Thomas↗

NASA NTRS1993

A formally verified algorithm for interactive consistency under a hybrid fault model

Consistent distribution of single-source data to replicated computing channels is a fundamental problem in fault-tolerant system design. The 'Oral Messages' (OM) algorithm solves this problem of Interactive Consistency (Byzantine Agreement) assuming that all faults are worst-cass. Thambidurai and Park introduced a 'hybrid' fault model that distinguished three fault modes: asymmetric (Byzantine), symmetric, and benign; they also exhibited, along with an informal 'proof of correctness', a modified version of OM. Unfortunately, their algorithm is flawed. The discipline of mechanically checked formal verification eventually enabled us to develop a correct algorithm for Interactive Consistency under the hybrid fault model. This algorithm withstands $a$ asymmetric, $s$ symmetric, and $b$ benign faults simultaneously, using $m+1$ rounds, provided $n is greater than 2a + 2s + b + m$, and $m\geg a$. We present this algorithm, discuss its subtle points, and describe its formal specification and verification in PVS. We argue that formal verification systems such as PVS are now sufficiently effective that their application to fault-tolerance algorithms should be considered routine.

Lincoln, Patrick↗

NASA NTRS1993

Formal methods for dependable real-time systems

The motivation for using formal methods to specify and reason about real time properties is outlined and approaches that were proposed and used are sketched. The formal verifications of clock synchronization algorithms are concluded as showing that mechanically supported reasoning about complex real time behavior is feasible. However, there was significant increase in the effectiveness of verification systems since those verifications were performed, at it is to be expected that verifications of comparable difficulty will become fairly routine. The current challenge lies in developing perspicuous and economical approaches to the formalization and specification of real time properties.

Rushby, John↗

NASA NTRS1995

Formal methods and their role in digital systems validation for airborne systems

This report is based on one prepared as a chapter for the FAA Digital Systems Validation Handbook (a guide to assist FAA certification specialists with advanced technology issues). Its purpose is to explain the use of formal methods in the specification and verification of software and hardware requirements, designs, and implementations; to identify the benefits, weaknesses, and difficulties in applying these methods to digital systems used in critical applications; and to suggest factors for consideration when formal methods are offered in support of certification. The presentation concentrates on the rationale for formal methods and on their contribution to assurance for critical applications within a context such as that provided by DO-178B (the guidelines for software used on board civil aircraft); it is intended as an introduction for those to whom these topics are new.

Rushby, John↗