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A Kripke Logical Relation Between ML and Assembly

A Kripke Logical Relation Between ML and Assembly. Chung- Kil Hur and Derek Dreyer MPI-SWS POPL 2011. Compiler Correctness by Simulation Relation. execution. compiler. Read. Read. Write. Write. Send. Send. Limitation of Simulation Relation: Lack of Compositionality. compiler 1.

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A Kripke Logical Relation Between ML and Assembly

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  1. A Kripke Logical Relation Between ML and Assembly Chung-KilHur and Derek Dreyer MPI-SWS POPL 2011

  2. Compiler Correctness by Simulation Relation execution compiler Read Read Write Write Send Send

  3. Limitation of Simulation Relation: Lack of Compositionality compiler1 compiler2 ? ?

  4. Essential Question When is a high-level program “equivalent” to a low-level program? Want a notion of “program equivalence” that is • preserved under linking • as large as possible

  5. What is Contextual Equivalence? • Canonical notion of program equivalence for high-level programs: C[] and C[] return the same observable results for every program context C

  6. What is good about Contextual Equivalence? • Compositionality: • Extensionality:

  7. What is wrong with Contextual Equivalence? But: • Does not work for low-level languages • Contexts have too much distinguishing power • Does not work for relating different languages • Can’t use same context for two different languages

  8. Logical Relations to the Rescue! [Benton & Hur, ICFP’09] • Typically used as technique for proving contextual equivalence in a higher-order language • Take logical relation as “definition of program equivalence” between high-level and low-level languages

  9. Why is Logical Relation a good definition? apply( • Compositionality and extensionality are built in! • Unlike , logical relations can be adapted to relate different languages. • ICFP’09 paper related pure -calculus and SECD

  10. Contributions of This Work • Scale ideas to ML-like high-level language and assembly-like low-level language • Builds on recent work on step-indexed Kripke logical relations (by Dreyer et al. POPL’09, ICFP’10) • Handle interaction with a garbage collector • Distinguish between logical and physical memories (idea from McCreight et al. PLDI’07, ICFP’10)

  11. Kripke Logical Relations • Relating ML to assembly is challenging • Need a way to “tame” state, at both high and low levels • Kripke logical relations arelogical relations indexed by “possible worlds” • Worlds encode invariants about state • Dreyer et al. [ICFP’10] generalize worlds to express state transition systems • Supports reasoning about how state evolves over time

  12. Irreversible State Change[Dreyer et al.] (no successor) (can be in either state)

  13. Self-Modifying Code bg move wk4 bg+3 move wk5 1 jmpalloc move [wk5+0]h bg+5 jmp wk0 bg+5 bg+13 move wk3 bg+10 jmpbg+12 bg+5 isr wk4 wk3 minus wk4wk4 666 bg+13 bg+5 isw wk3 wk4 (jneq bg+6 wk3 bg+21)+666 plus wk3wk3 1 (isw bg+5 (jmp bg+12) )+666 01100110010101001110110001101011 jneq bg+6 wk3 bg+21 bg+12 (move [wk1+0]h bg+13 )+666 01101010101101110101000100100101 isw bg+5 (jmpbg+12) bg+13 (plus spsp 1 )+666 bg+12 10110101001111010101010111000100 bg+12move [wk1+0]h bg+13 (move [sp-1]s wk0)+666 bg+13 01010110101001010101110110001010 bg+13plus spsp 1 (move wk1 wk2 )+666 11011001010111000101010101001010 move [sp-1]s wk0 (move wk0 bg+18 )+666 11010101000101011010010101010101 move wk1 wk2 (jmp [wk1+0]h)+666 01110110100101011100101010100101 move wk0 bg+18 (move wk5 1 )+666 bg+5 10101010100100110110001000111010 jmp [wk1+0]h (minus spsp1 )+666 10010001000011101111001111001010 move wk5 1 (jmp [sp-0]s)+666 01110100010010101010010101010010 minus spsp 1 01100010001000111100111100111011 jmp [sp-0]s

  14. Reasoning about Self-Modifying Code decrypted (CASE 2) (CASE 3) (CASE 1) decrypted encrypted decrypted encrypted bg move wk4 bg+3 3 move wk5 1 jmpalloc move [wk5+0]h bg+5 jmp wk0 encrypted move wk3 bg+10 jmpbg+12 bg+5 3 isr wk4 wk3 minus wk4wk4 666 isw wk3 wk4 plus wk3wk3 1 01100110010101001110110001101011 jneq bg+6 wk3 bg+21 01101010101101110101000100100101 isw bg+5 (jmpbg+12) bg+12move [wk1+0]h bg+13 bg+12 10110101001111010101010111000100 bg+13plus spsp 1 bg+13 01010110101001010101110110001010 bg+5 bg+13 11011001010111000101010101001010 move [sp-1]s wk0 11010101000101011010010101010101 move wk1 wk2 01110110100101011100101010100101 move wk0 bg+18 10101010100100110110001000111010 jmp [wk1+0]h 10010001000011101111001111001010 move wk5 1 minus spsp 1 01110100010010101010010101010010 01100010001000111100111100111011 jmp [sp-0]s

  15. What Else is in the Paper? • Public vs. private transitions • Idea from Dreyer et al., useful for modeling “well-bracketed” state change, e.g. assumptions about stack & callee-save registers • Logical vs. physical memories • Idea from McCreight et al., critical for defining logical relations that enjoy monotonicity property in the presence of GC • Symmetric, “language-generic” presentation of logical relation • Compiler correctness for simple compiler • Full proofs in online appendix

  16. Conclusion • Progress toward a more general notion of equivalence between high- and low-level programs • We’ve generalized previous work to a much more realistic setting • Applying cool new ideas from recent work on both logical relations and certified compilers

  17. Future Work • Compositional Compiler Correctness for C(Linking for CompCert) • Realistic Compilation • Multiphase Compilation

  18. Future Work • Linking between different languages ML ML+C C

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