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Another example: constant prop

Another example: constant prop. Set D = 2 { x ! N | x 2 Vars Æ N 2 Z }. in. x := N. F x := N (in) = in – { x ! * } [ { x ! N }. out. in. x := y op z. F x := y op z (in) = in – { x ! * } [ { x ! N | ( y ! N 1 ) 2 in Æ ( z ! N 2 ) 2 in Æ

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Another example: constant prop

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  1. 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

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

  3. 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

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

  5. 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

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

  7. Lattice • (D, ⊑, ⊥, ⊤, ⊔, ⊓) =

  8. Lattice • (D, ⊑, ⊥, ⊤, ⊔, ⊓) = (2 A , ¶, A, ;, Å, [) where A = { x ! N | x ∊ Vars Æ N ∊ Z }

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

  10. Back to lattice • (D, ⊑, ⊥, ⊤, ⊔, ⊓) = (2 A , ¶, A, ;, Å, [) where A = { x ! N | x ∊ Vars Æ N ∊ Z } • What’s the problem with this lattice?

  11. Back to lattice • (D, ⊑, ⊥, ⊤, ⊔, ⊓) = (2 A , ¶, A, ;, Å, [) where A = { x ! N | x ∊ Vars Æ N ∊ Z } • What’s the problem with this lattice? • Lattice is infinitely high, which means we can’t guarantee termination

  12. Better lattice • Suppose we only had one variable

  13. Better lattice • Suppose we only had one variable • D = {⊥, ⊤ } [ Z • 8 i ∊ Z . ⊥⊑ i Æ i ⊑⊤ • height = 3

  14. 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 =

  15. 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)

  16. For all variables • Option 2: Map from variables to single lattice • Given lattice (D, v1, ?1, >1, t1, u1) and a set V, create: map lattice V ! D = (V ! D, v, ?, >, t, u)

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

  18. 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

  19. 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

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