Structured natural languages provide a trade space between ambiguous natural languages that make up most written requirements, and mathematical formal specifications such as Linear Temporal Logic. FRETish is a structured natural language for the elicitation of system requirements developed at NASA. The related open-source tool Fret provides support for translating FRETish requirements into temporal logic formulas that can be input to several verification and analysis tools. In the context of safety-critical systems, it is crucial to ensure that a generated formula captures the semantics of the corresponding FRETish requirement precisely. This paper presents a rigorous formalization of the FRETish language including a new denotational semantics and a proof of semantic equivalence between FRETish specifications and their temporal logic counterparts computed by Fret. The complete formalization and the proof have been developed in the Prototype Verification System (PVS) theorem prover.

Semi-algebraic sets and real analytic functions are fundamental concepts in Real Algebraic Geometry and Real Analysis, respectively. These concepts appear in the study of Differential Equations, where the real analytic solution to a differential equation is known to enter or exit a semi-algebraic set in a predictable way. Motivated to enhance the capability to reason about differential equations in the Prototype Verification System (PVS), a formalization of multivariate polynomials, semi-algebraic sets, and real analytic functions is developed. The way that a real analytic function behaves in a neighborhood around a point where the function meets the boundary of a semi-algebraic set is described and verified. It is further shown that if the function is assumed to be smooth, a slightly weaker assumption than real analytic, the behavior around the boundary of a semi-algebraic set can be very different.

Abstract The Automatic Dependent Surveillance-Broadcast (ADS-B) system allows aircraft to communicate current state information, including position and velocity messages, to other aircraft in their vicinity and to ground stations. The Compact Position Reporting (CPR) algorithm is the ADS-B protocol responsible for the encoding and decoding of aircraft positions. CPR is sensitive to computer arithmetic since it relies on functions that are intrinsically unstable such as floor and modulus. In this paper, a formal verification of the CPR algorithm is presented. In contrast to previous work, the algorithm presented here encompasses the entire range of message types supported by ADS-B. The paper also presents two implementations of the CPR algorithm, one in double-precision floating-point and one in 32-bit unsigned integers, which are both formally verified against the real-number algorithm. The verification proceeds in three steps. For each implementation, a version of CPR, which is simplified and manipulated to reduce numerical instability and leverage features of the datatypes, is proposed. Then, the Prototype Verification System (PVS) is used to formally prove real conformance properties, which assert that the ideal real-number counterpart of the improved algorithm is mathematically equivalent to the standard CPR definition. Finally, the static analyzer Frama-C is used to verify software conformance properties, which say that the software implementation of the improved algorithm is correct with respect to its idealized real-number counterpart. In concert, the two properties guarantee that the implementation meets the original specification. The two implementations will be included in the revised version of the ADS-B standards document as the reference implementation of the CPR algorithm.