arXiv subjects
Publications and source records attributed to Harry G. Mairson.
We give an exact characterization of the computational complexity of the $k$CFA hierarchy. For any $k > 0$, we prove that the control flow decision problem is complete for deterministic exponential time. This theorem validates empirical observations that such control flow analysis is intractable. It also provides more general insight into the complexity of abstract interpretation.
Flow analysis is a ubiquitous and much-studied component of compiler technology---and its variations abound. Amongst the most well known is Shivers' 0CFA; however, the best known algorithm for 0CFA requires time cubic in the size of the analyzed program and is unlikely to be improved. Consequently, several analyses have been designed to approximate 0CFA by trading precision for faster computation. Henglein's simple closure analysis, for example, forfeits the notion of directionality in flows and enjoys an "almost linear" time algorithm. But in making trade-offs between precision and complexity, what has been given up and what has been gained? Where do these analyses differ and where do they coincide? We identify a core language---the linear $λ$-calculus---where 0CFA, simple closure analysis, and many other known approximations or restrictions to 0CFA are rendered identical. Moreover, for this core language, analysis corresponds with (instrumented) evaluation. Because analysis faithfully captures evaluation, and because the linear $λ$-calculus is complete for PTIME, we derive PTIME-completeness results for all of these analyses.