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Unformatted text preview: 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 doesnt change anymore 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. 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 Graphically x y 10 10 Graphically x y 10 10 Graphically x y 10 10 2 Graphically, another way Another example: constant prop Set D = x := N in out F x := n (in) = x := y op z in out F x := y op z (in) = Another example: constant prop Set D = 2 { x & N  x Vars N Z } x := N in out F x := n (in) = in { x & * } { x & N } x := y op z in out F x := y op z (in) = in { x & * } { x & N  ( y & N 1 ) in ( z & N 2 ) in N = N 1 op N 2 } Another example: constant prop *x := y in out F *x := y (in) = x := *y in out F x := *y (in) = Another example: constant prop *x := y in out F *x := y (in) = in { z & *  z maypoint(x) } { z & N  z mustpointto(x) y & N in } { z & N  (y & N) in (z & N) in } x := *y in out F x := *y (in) = in { x & * } { x & N  z maypointto(x) . (z & N) in } Another example: constant prop x := f(...) in out F x := f(...) (in) = *x := *y + *z in out F *x := *y + *z (in) = 3 Another example: constant prop x := f(...) in out F x := f(...) (in) = & *x := *y + *z in out F *x := *y + *z (in) = F a := *y;b := *z;c := a + b; *x := c (in) Another example: constant prop...
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This note was uploaded on 02/19/2008 for the course CSE 231 taught by Professor Lerner during the Fall '06 term at UCSD.
 Fall '06
 Lerner

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