Flow Analysis, Linearity, and PTIME

Abstract

Abstract. 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.

Control Flow Analysis for SF Combinator Calculus

“…The best known algorithm for 0CFA uses an efficient representation of the sets in Γ to achieve O(n 3 / log n) complexity [3]. Van Horn and Mairson showed that, for linear programs (in which each bound variable occurs exactly once), 0CFA gives the same result as actually evaluating the program; hence it is PTIME-complete [10].…”

“…A term t l can be analysed by finding a Γ and ϕ such that Γ, ϕ |= t l . This is done by solving the 6 2 , @ (7,2) } Γ(11) = {F 10 1 , @ (10,9) , @ (17,8) , @ (7,2) } ϕ(13) = true ϕ(15) = true Γ, ϕ |= (S 15 @ 16 (F 13 @ 14 F 12 )@ 17 (F 10 @ 11 F 9 ))@ 18 (S 6 @ 7 (F 4 @ 5 F 3 )@ 8 (F 1 @ 2 F 0 )) Figure 8: Solution of the analysis for application of identity to itself in SF-calculus.…”