An obvious way to implement retraction would be to mark propositions that have been asserted by the user; to implement Retract, start over with an empty database and Assert all marked propositions except the one that was retracted. However, this would be inefficient, since the retraction is likely to have only limited effects.
A Justification-based Truth Maintenance System or JTMS keeps for each derived proposition a list of sets of propositions from which it was derived, and for each proposition, a list of propositions it supports. When a proposition is retracted, the system checks each proposition that depends on it, and if the proposition no longer has support, retracts it.
In some systems, it may be likely that a retracted proposition may be re-asserted in the future. Rather than removing a proposition from the database, the proposition can be marked as IN if it is currently believed, or OUT if it is currently not believed.
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