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Static program analysis is in general more precise if it is sensitive to execution contexts (execution paths). But then it is also more expensive in terms of memory consumption. For languages with conditions and iterations, the number of contexts grows exponentially with the program size. This problem is not just a theoretical issue. Several papers evaluating inter-procedural context-sensitive data-flow analysis report severe memory problems, and the path-explosion problem is a major issue in program verification and model checking. In this paper we propose χ-terms as a means to capture and manipulate context-sensitive program information in a data-flow analysis. χ-terms are implemented as directed acyclic graphs without any redundant subgraphs. We introduce the k-approximation and the l-loop-approximation that limit the size of the context-sensitive information at the cost of analysis precision. We prove that every context-insensitive data-flow analysis has a corresponding k,l-approximated context-sensitive analysis, and that these analyses are sound and guaranteed to reach a fixed point. We also present detailed algorithms outlining a compact, redundancy-free, and DAG-based implementation of χ-terms.

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