Approximate Solutions for the Minimal Revision Problem of Specification Automata Kangjin Kim and Georgios E. Fainekos Abstract— As robots are being integrated into our daily lives, it becomes necessary to provide guarantees of safe and provably correct operation. Such guarantees can be provided using automata theoretic task and mission planning where the requirements are expressed as temporal logic specifications. However, in real-life scenarios, it is to be expected that not all user task requirements can be realized by the robot. In such cases, the robot must provide feedback to the user on why it cannot accomplish a given task. Moreover, the robot should indicate what tasks it can accomplish which are as “close” as possible to the initial user intent. Unfortunately, the latter problem, which is referred to as minimal specification revision problem, is NP complete. This paper presents an approximation algorithm that can compute good approximations to the min- imal revision problem in polynomial time. The experimental study of the algorithm demonstrates that in most problem instances the heuristic algorithm actually returns the optimal solution. Finally, some cases where the algorithm does not return the optimal solution are presented. I. I NTRODUCTION As robots become mechanically more capable, they are going to be more and more integrated into our daily lives. Non-expert users will have to communicate with the robots in a natural language setting and request a robot or a team of robots to accomplish complicated tasks. Therefore, we need methods that can capture the high-level user requirements, solve the planning problem and map the solution to low level continuous control actions. In addition, such frameworks must come with mathematical guarantees of safe and correct operation for the whole system and not just the high level planning or the low level continuous control. Linear Temporal Logic (LTL) [1] can provide the mathe- matical framework that can bridge the gap between (i) natural language and high-level planning algorithms [2], [3], and (ii) high-level planning algorithms and control [4]–[8]. LTL has been utilized as a specification language in a range of robotics applications (see [4]–[14]). All the previous methods are based on the assumption that the LTL planning problem has a feasible solution. However, in real-life scenarios, it is to be expected that not all complex task requirements can be realized by a robot or a team of robots. In such failure cases, the robot needs to provide feedback to the non-expert user on why the specification failed. Furthermore, it would be desirable that the robot proposes a number of plans that can be realized by the robot and which are as “close” as possible to the initial user intent. This work has been partially supported by award NSF CNS 1116136. K. Kim and G. Fainekos are with the School of Computing, Informatics and Decision Systems Engineering, Arizona State University, Tempe, AZ 85281, USA {Kangjin.Kim,fainekos}@asu.edu Then, the user would be able to understand what are the limitations of the robot and, also, he/she would be able to choose among a number of possible feasible plans. In [15], we made the first steps towards solving the debugging (i.e., why the planning failed) and revision (i.e., what the robot can actually do) problems for automata theoretic LTL planning [16]. We remark that many robotic applications, e.g., [4], [6], [10]–[12], [14], are utilizing this particular method. In the follow-up paper [17], we studied the theoretical foundations of the specification revision problem when both the system and the specification can be repre- sented by ω-automata [18]. We focused on the Minimal Re- vision Problem (MRP), i.e., finding the “closest” satisfiable specification to the initial specification, and we proved that the problem is NP-complete even when severely restricting the search space. Furthermore, we presented an encoding of MRP as a satisfiability problem and we demonstrated experimentally that we can quickly get the exact solution to MRP for small problem instances. In this paper, we revisit the MRP problem that we in- troduced in [17]. We present a heuristic algorithm that can approximately solve the MRP problem in polynomial time. We experimentally establish that in most cases the heuristic algorithm returns the optimal solution on random problem instances and on LTL planning scenarios from our previous work. Furthermore, we demonstrate that now we can quickly return a solution to the MRP problem on large problem instances. Finally, we provide examples where the algorithm is guaranteed not to return the optimal solution. Related Research: The automatic specification revision problem for automata based planning techniques is a rela- tively new problem. Finding out why a specification is not satisfiable on a model is a problem that is very related to the problems of vacuity and coverage in model checking [19]. In the context of general planners, the problem of finding good excuses on why the planning failed has been studied in [20]. Another related problem is the detection of the causes of unrealizability in LTL games. In this case, a number of heuristics have been developed in order to localize the error and provide meaningful information to the user for debugging [21], [22]. Along these lines, LTLMop [23] was developed to debug unrealizable LTL specifications in reactive planning for robotic applications. II. PROBLEM FORMULATION In this paper, we work with discrete abstractions (Finite State Machines) of the continuous robotic control system [4]. Each state of the Finite State Machine (FSM) T is labeled