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In this paper, a formal language called the aircraft intent description language (AIDL) is proposed as a standard, interoperable means of describing and exchanging predicted aircraft trajectories in trajectory-based operations (TBO). The AIDL provides the necessary elements to unambiguously formulate aircraft intent, which, in the context of trajectory prediction, refers to the information that describes how the aircraft is to be operated within a certain time interval. By expressing aircraft intent according to the AIDL, it is ensured that each instance of aircraft intent defines a unique trajectory. It is anticipated that sharing aircraft intent information expressed in a structured and formal manner, e.g. according to the AIDL, can facilitate the synchronization of the predicted aircraft trajectories held by different automation systems in the context of TBO. The AIDL is characterized by an alphabet and a grammar. The definition of the alphabet and the grammar rules are based on a rigorous mathematical analysis of the trajectory computation process at the core of a TP. This analysis relies on a novel approach to modeling the trajectory computation process based on the theory of differential algebraic equations (DAEs). The paper presents the alphabet of the AIDL, which contains a set of instructions that capture the individual commands and guidance modes available to direct the motion of an aircraft in the ATM context. Then, the AIDL grammar rules, which define the possible combinations of the instructions in the alphabet, are defined and mathematically justified. The AIDL seeks to exploit the physical and mathematical foundations underlying the trajectory computation process to allow describing a priori (i.e. before a trajectory is actually computed) any possible motion behavior that can reasonably be elicited from an aircraft in the ATM context. The objective is that the AIDL encompasses all other methods and formats that may be used to describe aircraft intent,- i.e. they would be subsets of the AIDL. The AIDL could thus be seen as a metalanguage for aircraft intent description, containing any other language that may be used to describe aircraft intent in the context of TBO.