1 / 30

Recap

Summary of worklist algorithm for reaching definitions & using lattices in fixed point computation. Explore monotonicity, termination guarantees, and examples.

rorcutt
Download Presentation

Recap

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Recap • Let’s do a recap of what we’ve seen so far • Started with worklist algorithm for reaching definitions

  2. Worklist algorithm for reaching defns let m: map from edge to computed value at edge let worklist: work list of nodes for each edge e in CFG do m(e) := ; for each node n do worklist.add(n) while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0 .. info_out.length do let new_info := m(n.outgoing_edges[i]) [ info_out[i]; if (m(n.outgoing_edges[i])  new_info]) m(n.outgoing_edges[i]) := new_info; worklist.add(n.outgoing_edges[i].dst);

  3. Generalized algorithm using lattices let m: map from edge to computed value at edge let worklist: work list of nodes for each edge e in CFG do m(e) := ? for each node n do worklist.add(n) while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0 .. info_out.length do let new_info := m(n.outgoing_edges[i]) t info_out[i]; if (m(n.outgoing_edges[i])  new_info]) m(n.outgoing_edges[i]) := new_info; worklist.add(n.outgoing_edges[i].dst);

  4. Next step: removed outer join • Wanted to remove the outer join, while still providing termination guarantee • To do this, we re-expressed our algorithm more formally • We first defined a “global” flow function F, and then expressed our algorithm as a fixed point computation

  5. Fixedpoint view • We want to find a fixed point of F, that is to say a map m such that m = F(m) • Define ?, which is ? lifted to be a map: ? =  e. ? • Compute F(?), then F(F(?)), then F(F(F(?))), ... until the result doesn’t change anymore

  6. Guarantees • If F is monotonic and height of lattice is finite: iterative algorithm terminates • If F is monotonic, the fixed point we find is the least fixed point. • Any questions so far?

  7. What about if we start at top? • What if we start with >: F(>), F(F(>)), F(F(F(>))) • We get the greatest fixed point • Why do we prefer the least fixed point? • More precise

  8. Graphically y 10 10 x

  9. Graphically y 10 10 x

  10. Graphically y 10 10 x

  11. Graphically, another way

  12. Another example: constant prop • Set D = in x := N Fx := n(in) = out in x := y op z Fx := y op z(in) = out

  13. Another example: constant prop • Set D = 2 { x ! N | x 2 Vars Æ N 2 Z } in x := N Fx := n(in) = in – { x ! * } [ { x ! N } out in x := y op z Fx := y op z(in) = in – { x ! * } [ { x ! N | ( y ! N1 ) 2 in Æ ( z ! N2 ) 2 in Æ N = N1 op N2 } out

  14. Another example: constant prop in Fx := *y(in) = x := *y out in F*x := y(in) = *x := y out

  15. Another example: constant prop in Fx := *y(in) = in – { x ! * } [ { x ! N | 8 z 2 may-point-to(x) . (z ! N) 2 in } x := *y out in F*x := y(in) = in – { z ! * | z 2 may-point(x) } [ { z ! N | z 2 must-point-to(x) Æ y ! N 2 in } [ { z ! N | (y ! N) 2 in Æ (z ! N) 2 in } *x := y out

  16. Another example: constant prop in *x := *y + *z F*x := *y + *z(in) = out in x := f(...) Fx := f(...)(in) = out

  17. Another example: constant prop in *x := *y + *z F*x := *y + *z(in) = Fa := *y;b := *z;c := a + b; *x := c(in) out in x := f(...) Fx := f(...)(in) = ; out

  18. Another example: constant prop in s: if (...) out[0] out[1] in[0] in[1] merge out

  19. Lattice • (D, v, ?, >, t, u) =

  20. Lattice • (D, v, ?, >, t, u) = (2 { x ! N | x 2 Vars Æ N 2 Z } , ¶, D, ;, Å, [)

  21. Example x := 5 v := 2 w := 3 y := x * 2 z := y + 5 x := x + 1 w := v + 1 w := w * v

  22. Back to lattice • (D, v, ?, >, t, u) = (2 { x ! N | x 2 Vars Æ N 2 Z } , ¶, D, ;, Å, [) • What’s the problem with this lattice?

  23. Back to lattice • (D, v, ?, >, t, u) = (2 { x ! N | x 2 Vars Æ N 2 Z } , ¶, D, ;, Å, [) • What’s the problem with this lattice? • Lattice is infinitely high, which means we can’t guarantee termination

  24. Better lattice • Suppose we only had one variable

  25. Better lattice • Suppose we only had one variable • D = {?, > } [ Z • 8 i 2 Z . ?v i Æ i v> • height = 3

  26. For all variables • Two possibilities • Option 1: Tuple of lattices • Given lattices (D1, v1, ?1, >1, t1, u1) ... (Dn, vn, ?n, >n, tn, un) create: tuple lattice Dn =

  27. For all variables • Two possibilities • Option 1: Tuple of lattices • Given lattices (D1, v1, ?1, >1, t1, u1) ... (Dn, vn, ?n, >n, tn, un) create: tuple lattice Dn = ((D1£ ... £ Dn), v, ?, >, t, u) where ? = (?1, ..., ?n) > = (>1, ..., >n) (a1, ..., an) t (b1, ..., bn) = (a1t1 b1, ..., antn bn) (a1, ..., an) u (b1, ..., bn) = (a1u1 b1, ..., anun bn) height = height(D1) + ... + height(Dn)

  28. For all variables • Option 2: Map from variables to single lattice • Given lattice (D, v, ?, >, t, u) and a set V, create: map lattice V ! D = (V ! D, v, ?, >, t, u)

  29. Back to example in Fx := y op z(in) = x := y op z out

  30. Back to example in Fx := y op z(in) = in [ x ! in(y) op in(z) ] where a op b = x := y op z out

More Related