1. 1 Systematic Domain Design Some Remarks

2. 2 Best (Conservative) interpretation abstract representation Set of states concretization Abstract semantics statement s abstract representation abstraction Operational semantics statement s Set of states

3. 3 Galois Connections u For –A complete lattice (L 1,  1 ) = (L 1, ,  1,  1,  1,  1 ) –A complete lattice (L 2,  2 ) = (, ,  2,  2,  2,  2 ) –  :L 1  L 2 –  : L 2  L 1 u We say that (L 1, , , L 2 ) is a Galois connection –  and  are monotone –For all c  L 1 :  (  (c))  c –For all a  L 2 :  (  (a))  a

4. 4 Best (Induced) Abstract Transformer u For –A Galois connection (L 1, , , L 2 ) –A function f 1 : L 1  L 1 u Define f 2 = f 1  ( ,  ): L 2  L 2 –f 2 (l 2 ) =  (f 1 (  (l 2 ) ) u Theorem: –f 2 is monotone if f 1 is –f 2 is a sound approximation of f 1 »  l 1  L 1  (f 1 (l 1 ))  f 2 (  (l 1 )) »  l 2  L 2 f 1 (  ( l 2 ))   (f 2 (l 2 )) »For every sound approximation f’ 2 is a sound approximation of f 1 u f 2  f’ 2 u Sometimes we can actually implement the best transformer

5. 5 Best (Induced) Abstract Transformer u But… u It may be difficult to implement u It may be very imprecise –  (f 1 (l 1 ))  f 2 (  (l 1 )) u Indicates inappropriate abstract domain u Sometimes it is better to pick a more expressive domain and work with suboptimal solutions