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From last lecture

Learn how to compute fixed points in lattices using iterative algorithms. Understand the concepts of least and greatest fixed points and why least fixed points are preferred. Explore examples of constant propagation and live variables analysis.

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From last lecture

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  1. From last lecture • 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

  2. From last lecture • 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.

  3. 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

  4. Graphically y 10 10 x

  5. Graphically y 10 10 x

  6. Graphically y 10 10 x

  7. Graphically, another way

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

  9. 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

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

  11. 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

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

  13. 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

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

  15. Lattice • (D, v, ?, >, t, u) =

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

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

  18. Better lattice • Suppose we only had one variable

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

  20. 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 =

  21. 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)

  22. 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)

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

  24. 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

  25. General approach to domain design • Simple lattices: • boolean logic lattice • powerset lattice • incomparable set: set of incomparable values, plus top and bottom (eg const prop lattice) • two point lattice: just top and bottom • Use combinators to create more complicated lattices • tuple lattice constructor • map lattice constructor

  26. May vs Must • Has to do with definition of computed info • Set of x ! y must-point-to pairs • if we compute x ! y, then, then during program execution, x must point to y • Set of x! y may-point-to pairs • if during program execution, it is possible for x to point to y, then we must compute x ! y

  27. May vs must

  28. May vs must

  29. Direction of analysis • Although constraints are not directional, flow functions are • All flow functions we have seen so far are in the forward direction • In some cases, the constraints are of the form in = F(out) • These are called backward problems. • Example: live variables • compute the of variables that may be live

  30. Example: live variables • Set D = • Lattice: (D, v, ?, >, t, u) =

  31. Example: live variables • Set D = 2 Vars • Lattice: (D, v, ?, >, t, u) = (2Vars, µ, ; ,Vars, [, Å) in Fx := y op z(out) = x := y op z out

  32. Example: live variables • Set D = 2 Vars • Lattice: (D, v, ?, >, t, u) = (2Vars, µ, ; ,Vars, [, Å) in Fx := y op z(out) = out – { x } [ { y, z} x := y op z out

  33. Example: live variables x := 5 y := x + 2 y := x + 10 x := x + 1 ... y ...

  34. Example: live variables x := 5 y := x + 2 y := x + 10 x := x + 1 ... y ...

  35. Theory of backward analyses • Can formalize backward analyses in two ways • Option 1: reverse flow graph, and then run forward problem • Option 2: re-develop the theory, but in the backward direction

  36. Precision • Going back to constant prop, in what cases would we lose precision?

  37. Precision • Going back to constant prop, in what cases would we lose precision? if (...) { x := -1; } else x := 1; } y := x * x; ... y ... if (p) { x := 5; } else x := 4; } ... if (p) { y := x + 1 } else { y := x + 2 } ... y ... x := 5 if (<expr>) { x := 6 } ... x ... where <expr> is equiv to false

  38. Precision • The first problem: Unreachable code • solution: run unreachable code removal before • the unreachable code removal analysis will do its best, but may not remove all unreachable code • The other two problems are path-sensitivity issues • Branch correlations: some paths are infeasible • Path merging: can lead to loss of precision

  39. MOP: meet over all paths • Information computed at a given point is the meet of the information computed by each path to the program point if (...) { x := -1; } else x := 1; } y := x * x; ... y ...

  40. MOP • For a path p, which is a sequence of statements [s1, ..., sn] , define: Fp(in) = Fsn( ...Fs1(in) ... ) • In other words: Fp = • Given an edge e, let paths-to(e) be the (possibly infinite) set of paths that lead to e • Given an edge e, MOP(e) = • For us, should be called JOP...

  41. MOP vs. dataflow • As we saw in our example, in general,MOP  dataflow • In what cases is MOP the same as dataflow? x := -1; y := x * x; ... y ... x := 1; y := x * x; ... y ...

  42. MOP vs. dataflow • As we saw in our example, in general,MOP  dataflow • In what cases is MOP the same as dataflow? • Distributive problems. A problem is distributive if: 8 a, b . F(a t b) = F(a) t F(b)

  43. Summary of precision • Dataflow is the basic algorithm • To basic dataflow, we can add path-separation • Get MOP, which is same as dataflow for distributive problems • Variety of research efforts to get closer to MOP for non-distributive problems • To basic dataflow, we can add path-pruning • Get branch correlation • Will see example of this later in the course • To basic dataflow, can add both: • meet over all feasible paths

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