The Truth Maintenance System (TMS) is a problem-solving subsystem that allows reasoning programs to make assumptions and revise their beliefs when new information contradicts these assumptions. It records and maintains the reasons for program beliefs, which are useful for constructing explanations of program actions and guiding problem-solving. This paper describes the TMS's representation and structure, mechanisms for revising beliefs, dependency-directed backtracking, techniques for summarizing explanations, dialectical arguments, models of others' beliefs, and methods for embedding control structures in assumptions. The paper emphasizes the need for problem solvers to choose between alternative belief systems and outlines a mechanism for selecting what to believe, want, and do. The TMS is used to determine the current set of beliefs from the current set of reasons and to update them in an incremental fashion. It traces the reasons for beliefs to find the consequences of changes in assumptions. The paper discusses the conventional view of reasoning, which is based on monotonicity, and proposes a different view that allows for non-monotonic changes in beliefs. This view incorporates new concepts that address the problems of commonsense reasoning, the frame problem, and control. The TMS uses support-list (SL) and conditional-proof (CP) justifications to represent reasons for beliefs. SL-justifications are valid if each node in the inlist is in and each node in the outlist is out. CP-justifications are valid if the consequent node is in whenever the inhypotheses are in and the outhypotheses are out. The TMS handles CP-justifications by converting them into equivalent SL-justifications when necessary. The TMS also deals with circular arguments, which can lead to inconsistencies. It ensures that nodes are not believed for circular reasons and uses a constraint-relaxation process to resolve circularities. The TMS processes justifications to determine the support-status of nodes and updates beliefs accordingly. It also handles dependency-directed backtracking to revise assumptions when inconsistencies are discovered. The TMS is designed to efficiently manage the revision of beliefs and support the development of rational programs.The Truth Maintenance System (TMS) is a problem-solving subsystem that allows reasoning programs to make assumptions and revise their beliefs when new information contradicts these assumptions. It records and maintains the reasons for program beliefs, which are useful for constructing explanations of program actions and guiding problem-solving. This paper describes the TMS's representation and structure, mechanisms for revising beliefs, dependency-directed backtracking, techniques for summarizing explanations, dialectical arguments, models of others' beliefs, and methods for embedding control structures in assumptions. The paper emphasizes the need for problem solvers to choose between alternative belief systems and outlines a mechanism for selecting what to believe, want, and do. The TMS is used to determine the current set of beliefs from the current set of reasons and to update them in an incremental fashion. It traces the reasons for beliefs to find the consequences of changes in assumptions. The paper discusses the conventional view of reasoning, which is based on monotonicity, and proposes a different view that allows for non-monotonic changes in beliefs. This view incorporates new concepts that address the problems of commonsense reasoning, the frame problem, and control. The TMS uses support-list (SL) and conditional-proof (CP) justifications to represent reasons for beliefs. SL-justifications are valid if each node in the inlist is in and each node in the outlist is out. CP-justifications are valid if the consequent node is in whenever the inhypotheses are in and the outhypotheses are out. The TMS handles CP-justifications by converting them into equivalent SL-justifications when necessary. The TMS also deals with circular arguments, which can lead to inconsistencies. It ensures that nodes are not believed for circular reasons and uses a constraint-relaxation process to resolve circularities. The TMS processes justifications to determine the support-status of nodes and updates beliefs accordingly. It also handles dependency-directed backtracking to revise assumptions when inconsistencies are discovered. The TMS is designed to efficiently manage the revision of beliefs and support the development of rational programs.