#over_approximate#abstract_interpretation #precision #fixpoint Widening is a technique used in [[Abstract Interpretation|abstract interpretation]] to accelerate the convergence of the analysis process. It is applied when the analysis reaches a [[Fixpoint|fixpoint]] too slowly or fails to terminate due to infinite ascending chains. In abstract interpretation, a fixpoint represents a stable approximation of the program's behavior, where no further refinements or changes occur. The analysis iteratively refines the abstract domain until it reaches a fixpoint. However, in some cases, the analysis may get stuck in an infinite loop or converge very slowly to the fixpoint due to the widening gap between successive iterations. The purpose of widening is to make the abstraction more conservative by introducing an over-approximation that "widens" the gap between successive iterations. This helps to accelerate the convergence by making larger steps towards the fixpoint and potentially avoiding infinite ascending chains. The widening operator takes two abstract values as input, typically the current abstract value and the value from the previous iteration. It computes a new abstract value that is an over-approximation of the behavior observed in both inputs. The widening operator allows the analysis to "jump" over parts of the state space, leading to a faster convergence. The choice of the widening operator depends on the abstract domain used in the analysis. The operator should strike a balance between [[Precision|precision]] and efficiency. A common form of widening is the widening operator based on the widening operator used in the domain of numerical intervals. It widens the abstract value by introducing special markers or widening points that represent imprecise or unknown information. It's important to note that widening introduces imprecision and may cause a loss of precision in the analysis results. To mitigate this, narrowing can be applied after widening to refine the abstraction and regain precision. The widening and narrowing operators work together iteratively to improve the analysis process and converge to a more precise fixpoint. In summary, widening is a technique used in abstract interpretation to accelerate the convergence of the analysis by introducing a more conservative over-approximation. It helps to avoid infinite ascending chains and improve the efficiency of the analysis process, although at the cost of some precision. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Widening") or contains(subject, "widening") ```