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NASA NTRS2020

Formal Verification of a Solution to the n-Queens Problem

This report describes a formal verification of a concise algorithm that computes a solution to the n-Queens problem for all natural numbers n, such that n > 3. The formal proof of the algorithm is completed in the Prototype Verification System (PVS) theorem prover. This verification effort serves two purposes. First, it is presented as a pedagogical example for learning a theorem prover, such as PVS, and second, as a candidate benchmark for comparing other formal methods tools to PVS.

Mahyar R Malekpour↗
NASA NTRSDate not supplied

Formal Verification of the Interaction Between Semi-Algebraic Sets and Real Analytic Functions

Semi-algebraic sets and real analytic functions are fundamental concepts in Real Algebraic Geometry and Real Analysis, respectively. These concepts interact 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 predicable 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 favorable way that a real analytic function enters and exits a semi-algebraic set is proven. It is further shown that if the function is assumed to be smooth, a slightly weaker assumption than real analytic, these favorable interactions with semi-algebraic sets may fail.

Real analytic functions↗
NASA NTRSDate not supplied

Formal Verification of Termination Criteria for​ First-Order Recursive Functions

This talk presents a formalization of several termination criteria for first-order recursive functions. The formalization, which is developed in the Prototype Verification System (PVS), includes the specification and proof of equivalence of semantic termination, Turing termination, size change principle, calling context graphs, and matrix-weighted graphs. These termination criteria are defined on a computational model that consists of a basic functional language called PVS0, which is an embedding of recursive first-order functions. Through this embedding, the native mechanism for checking termination of recursive functions in PVS could be soundly extended with semi-automatic termination criteria such as calling contexts graphs.

Termination↗
NASA NTRSDate not supplied

Interpretation and Formalization of the Right-of-Way Rules

This paper presents an interpretation and mathematical definition of the right-of-way rules as stated in USA, Title 14 of the Code of Federal Regulations, Part 91, Section 91.113 (14 CFR 91.113). In an encounter between two aircraft, the right-of-way rules in 14 CFR 91.113 defines which aircraft, if any, has the right-of-way and which aircraft must maneuver to stay well clear of the other aircraft. The objective of the work presented in this paper is to give an unambiguous interpretation of the rules. From the interpretation, a precise mathematical formulation is created that can be used for analysis and proof of properties. The mathematical formulation has been defined in the Prototype Verification System (PVS) and properties of well formedness and core properties of the formalization have been mechanically proved. Additional objectives are to use the mathematical formulation to code a rule compliant virtual pilot that can be used in simulation experiments and to possibly use the rule compliant virtual pilot in autonomous operations.

right-of-way↗
NASA NTRSDate not supplied

Interpretation and Formalization of the Right-of-Way Rules

This paper presents an interpretation and mathematical definition of the right-of-way rules as stated in USA, Title 14 of the Code of Federal Regulations, Part 91, Section 91.113 (14 CFR 91.113). In an encounter between two aircraft, the right-of-way rules defines which aircraft, if any, has the right-of-way and which AQ2 aircraft must maneuver to stay well clear of the other aircraft. The objective of the work presented in this paper is to give an unambiguous interpretation of the rules. From the interpretation, a precise mathematical formulation is created that can be used for analysis and proof of properties. The mathematical formulation has been defined in the Prototype Verification System (PVS) and properties of well formedness and core properties of the formalization have been mechanically proved. This mathematical formulation can be implemented digitally, so that right-of-way rules can be used in simulation or in future autonomous operations.

Right-of-Way↗
NASA NTRSDate not supplied

A Temporal Differential Dynamic Logic Formal Embedding

Differential dynamic logic is a formal framework to specify and reason about hybrid programs (HPs). The core of dL is a proof calculus that contains a collection of axioms and rules for the rigorous verification of properties of HPs. Recently, dL has been embedded within the theorem prover Prototype Verification System (PVS) resulting in the tool Plaidypvs2. The integration of dL into PVS expands its expressive power; user defined functions, such as trigonometric and other transcendental functions, can be used inside the dL framework, and meta-reasoning about HPs can be performed, including reasoning about entire classes of HPs, specified using dependent types in PVS. The differential temporal dynamic logic (dTL2) extends dL with temporal logic operators to reason about all the states reachable during the execution of an HP. This paper presents a work in progress focusing on embedding dTL2 in PVS as an extension of Plaidypvs. Plaidypvs is expanded with the formalization of a trace semantics for HPs, the definition of the LTL temporal operators eventually and globally, and the implementation of the proof calculus for dTL2. This new embedding has the same capabilties as Plaidypvs, which allows user defined functions and meta-reasoning of properties of HPs. To the best of the authors’ knowledge this is the first implementation of dTL2.

differential dynamic logic↗
NASA NTRSDate not supplied

A Formal Verification Framework for Runtime Assurance

The simplex architecture is an instance of Runtime Assurance (RTA) where a trusted component takes control of a safety-critical system when an untrusted component violates a safety property. This paper presents a formalization of the simplex RTA framework in the language of hybrid programs. A feature of this formal verification framework is that, for a given system, a specific instantiation can be created and its safety properties are guaranteed by construction. Instantiations may be kept at varying levels of generality, allowing for black box components, such as ML/AI-based controllers, to be modeled. The framework is written in the Prototype Verification System (PVS) using Plaidypvs, an embedding of differential dynamic logic in PVS. As a proof of concept, the framework is illustrated on an automatic vehicle braking system.

Runtime assurance↗
NASA NTRSDate not supplied

A Formal Verification Framework for Runtime Assurance

The simplex architecture is an instance of Runtime Assurance (RTA) where a trusted component takes control of a safety-critical system when an untrusted component violates a safety property. This paper presents a formalization of the simplex RTA framework in the language of hybrid programs. A feature of this formal verification framework is that, for a given system, a specific instantiation can be created and its safety properties are guaranteed by construction. Instantiations may be kept at varying levels of generality, allowing for black box components, such as ML/AI-based controllers, to be modeled. The framework is written in the Prototype Verification System (PVS) using Plaidypvs, an embedding of differential dynamic logic in PVS. As a proof of concept, the framework is illustrated on an automatic vehicle braking system.

Runtime assurance↗
NASA NTRSDate not supplied

Formalization of the Bellman-Ford Algorithm for Airspace Applications

This paper describes the formal verification of one of the most well-known algorithms for finding the shortest path between all vertices in a directed graph, namely the Bellman-Ford algorithm. This formal verification, performed in the Prototype Verification System (PVS), is motivated by two applications in the aerospace domain which use the algorithm for path planning. The first is a pre-flight calculation that uses an adapted version of Bellman-Ford to find a route intended to maximize GNSS availability throughout the flight. The second is a more traditional application intended to find the shortest path between an autonomous aircraft's current position and a goal waypoint, while avoiding regions of space specified by geofences. A novel aspect of this formal verification effort is the inclusion of two distinct models of computation for the algorithm, one being a traditional serial computation, and the other being an explicitly parallel computation. The ability to use parallel computation in the Bellman-Ford algorithm is in fact why it was chosen over other traditionally more performant algorithms, especially for the GNSS application, where the size of the graph makes a purely serial computation infeasible.

formal verification↗
NASA NTRSDate not supplied

A Verification Framework for Runtime Assurance of Autonomous UAS

Runtime Assurance (RTA) is a design-time architecture for safety-critical systems where an internal monitor acts upon detecting a violation of a property. The simplex architecture is an instance of RTA, where the action taken is to hand control of the overall system to a trusted controller when an untrusted one violates a safety property. Simplex RTA is emerging as a method for allowing AI/ML and other unverified software to be integrated into safety-critical applications like aircraft. To this end, the American Society for Testing and Materials (ASTM) and NASA have each published guidelines on the use of RTA in such systems. In the simplex RTA framework, a system has an advanced controller (AC) and a reversionary controller (RC). The system is allowed to operate with the AC until a runtime monitor detects that some property has been violated and then the RC takes over. Assuming that the sample rate of the monitor will detect improper functioning with enough time for the RC to correct the impending problem, and that the RC is trusted, the system will operate as intended. This use of the simplex RTA framework can allow for the integration of untrusted, but possibly more performant, controllers in a safe way. This paper presents a formalization of a simplex RTA framework in the Prototype Verification System (PVS) theorem prover using an embedding of differential dynamic logic (DDL) called Plaidypvs. A novel feature of this framework is that it can be instantiated at different levels of abstraction. This feature allows for the formal verification of a system with an untrusted black box component, such as an AI/ML controller. This paper does not address the many difficulties in deploying RTA in an industrial-level system. Instead, the focus is on the formal verification of the simplex RTA framework in the language of hybrid programs. Hybrid programs are programs that include both discrete and continuous dynamics and can be used to model complex cyber-physical systems. Plaidypvs is a tool that enables formalization of hybrid programs in the PVS theorem prover. Plaidypvs enables the verification of the general simplex RTA framework and then, by specializing some components of the hybrid program, verifying instances of the framework while treating the untrusted component as a black box. A selection of Unmanned Aircraft Systems (UAS) operations are shown as instances of the general RTA framework in PVS. This offers the benefit of design time verification of relevant safety properties to the system, and it also gives requirements on the sample rate of sensors that determine the time interval in which the ‘switch’ property of the RTA framework is checked.

PVS↗
NASA NTRS2010

TPSAS-NF1676L-9990-DND

PVS (Prototype Veri cation System)1 is an interactive environment for the specification and verification of systems. PVS provides a strongly typed specification language, which is based on Higher-Order Logic. The type system of PVS supports: sub-typing, dependent-types, abstract data types, parametric types, records, unions, and tuples. The PVS theorem prover includes decision procedures for a variety of theories such as linear arithmetic, propositional logic, and temporal logic. This seminar will provide a gentle introduction to the basic and advanced features of PVS, including: theory interpretations, real number proving, batch proving, rapid prototyping, and strategy development. All these features are illustrated with simple examples and exercises.

César Muñoz↗
NASA NTRS2010

Proceedings of the Second NASA Formal Methods Symposium

This publication contains the proceedings of the Second NASA Formal Methods Symposium sponsored by the National Aeronautics and Space Administration and held in Washington D.C. April 13-15, 2010. Topics covered include: Decision Engines for Software Analysis using Satisfiability Modulo Theories Solvers; Verification and Validation of Flight-Critical Systems; Formal Methods at Intel -- An Overview; Automatic Review of Abstract State Machines by Meta Property Verification; Hardware-independent Proofs of Numerical Programs; Slice-based Formal Specification Measures -- Mapping Coupling and Cohesion Measures to Formal Z; How Formal Methods Impels Discovery: A Short History of an Air Traffic Management Project; A Machine-Checked Proof of A State-Space Construction Algorithm; Automated Assume-Guarantee Reasoning for Omega-Regular Systems and Specifications; Modeling Regular Replacement for String Constraint Solving; Using Integer Clocks to Verify the Timing-Sync Sensor Network Protocol; Can Regulatory Bodies Expect Efficient Help from Formal Methods?; Synthesis of Greedy Algorithms Using Dominance Relations; A New Method for Incremental Testing of Finite State Machines; Verification of Faulty Message Passing Systems with Continuous State Space in PVS; Phase Two Feasibility Study for Software Safety Requirements Analysis Using Model Checking; A Prototype Embedding of Bluespec System Verilog in the PVS Theorem Prover; SimCheck: An Expressive Type System for Simulink; Coverage Metrics for Requirements-Based Testing: Evaluation of Effectiveness; Software Model Checking of ARINC-653 Flight Code with MCP; Evaluation of a Guideline by Formal Modelling of Cruise Control System in Event-B; Formal Verification of Large Software Systems; Symbolic Computation of Strongly Connected Components Using Saturation; Towards the Formal Verification of a Distributed Real-Time Automotive System; Slicing AADL Specifications for Model Checking; Model Checking with Edge-valued Decision Diagrams; and Data-flow based Model Analysis.

Munoz, Cesar↗
NASA NTRS1997

Analyzing Tabular and State-Transition Requirements Specifications in PVS

We describe PVS's capabilities for representing tabular specifications of the kind advocated by Parnas and others, and show how PVS's Type Correctness Conditions (TCCs) are used to ensure certain well-formedness properties. We then show how these and other capabilities of PVS can be used to represent the AND/OR tables of Leveson and the Decision Tables of Sherry, and we demonstrate how PVS's TCCs can expose and help isolate errors in the latter. We extend this approach to represent the mode transition tables of the Software Cost Reduction (SCR) method in an attractive manner. We show how PVS can check these tables for well-formedness, and how PVS's model checking capabilities can be used to verify invariants and reachability properties of SCR requirements specifications, and inclusion relations between the behaviors of different specifications. These examples demonstrate how several capabilities of the PVS language and verification system can be used in combination to provide customized support for specific methodologies for documenting and analyzing requirements. Because they use only the standard capabilities of PVS, users can adapt and extend these customizations to suit their own needs. Those developing dedicated tools for individual methodologies may find these constructions in PVS helpful for prototyping purposes, or as a useful adjunct to a dedicated tool when the capabilities of a full theorem prover are required. The examples also illustrate the power and utility of an integrated general-purpose system such as PVS. For example, there was no need to adapt or extend the PVS model checker to make it work with SCR specifications described using the PVS TABLE construct: the model checker is applicable to any transition relation, independently of the PVS language constructs used in its definition.

Owre, Sam↗