Toward the Formalization of Macroscopic Models of Traffic Flow Using Higher-Order-Logic Theorem Proving

Next-generation transportation will be integrated, interconnected and highly autonomous. One key challenge in traffic management is ensuring safety while maintaining the required level of service quality. In such future mobility systems, rigorous formalization and validation will thus become critical to ensure that the transportation network operates as intended, traffic properties are reliably maintained, and services resume in a timely manner after potential disruptions. Formal methods have recently gained considerable attention as a modeling and verification paradigm capable of addressing many challenges associated with next-generation transportation systems. In this regard, we propose to use higher-order-logic theorem proving for formally analyzing transportation systems. As a first step towards this direction, we present a logical framework for the formal analysis of macroscopic models of traffic flow. Leveraging upon the high expressiveness of the underlying logic, we formally model the continuous components of macroscopic models while capturing their real behavior. In particular, we present a formalization of some foundation concepts of macroscopic models, namely density, flow rate, mean speed, relative occupancy, and shockwave using the higher-order-logic theorem prover
HOL Light
. This choice is primarily motivated by the fact that the macroscopic models deal with the traffic flow dynamics and thus play a vital role in planning strategies in allocating resources for implementing optimized and balanced transportation systems. For illustration, we perform the formal input-output and shockwave analysis of a German freeway. The case study demonstrated the practicability of this formal approach due to the high expressiveness of the underlying logic. The proposed research is first step towards formalizing the foundational mathematical theories and core concepts of traffic flow theory. This accomplishment will open new ways to plan and model various components of the transportation systems such as highway links, ramp metering, merging behavior and eventually address the problem of routing vehicles in a network of automated vehicles.