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Basic Concepts of Abstract Interpretation Patrick Cousot cole normale suprieure 45 rue dUlm 75230 Paris cedex 05, France Patrick.Cousot@ens.fr www.di.ens.fr/~cousot IFIP WCC Topical day on Abstract Interpretation P.

  1. Undecidability – The concrete mathematical semantics of a program is an “tinfinite” mathematical object, not computable ; – All non trivial questions on the concrete program se- mantics are undecidable . Example: termination – Assume termination(P) would always terminates and returns true iff P always terminates on all input data; – The following program yields a contradiction P ” while termination(P) do skip od . ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 22 —

  2. Graphic example: Safety properties The safety properties of a program express that no possi- ble execution in any possible execution environment can reach an erroneous state. ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 23 —

  3. Graphic example: Safety property x ( t ) �������������� ��������� ������������ t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 24 —

  4. Safety proofs – A safety proof consists in proving that the intersection of the program concrete semantics and the forbidden zone is empty; – Undecidable problem (the concrete semantics is not computable); – Impossible to provide completely automatic answers with finite computer resources and neither human in- teraction nor uncertainty on the answer 2 . 2 e.g. probabilistic answer. ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 25 —

  5. Test/debugging – consists in considering a subset of the possible execu- tions; – not a correctness proof; – absence of coverage is the main problem. ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 26 —

  6. Graphic example: Property test/simulation x ( t ) �������������� ��������� ��������� ������������ �������������������������� t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 27 —

  7. Abstract interpretation – consists in considering an abstract semantics , that is to say a superset of the concrete semantics of the pro- gram; – hence the abstract semantics covers all possible con- crete cases; – correct: if the abstract semantics is safe (does not in- tersect the forbidden zone) then so is the concrete se- mantics ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 28 —

  8. Graphic example: Abstract interpretation x ( t ) �������������� ��������� ������������ ������������������������������� t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 29 —

  9. Formal methods Formal methods are abstract interpretations, which dif- fer in the way to obtain the abstract semantics: – “ model checking ”: - the abstract semantics is given manually by the user; - in the form of a finitary model of the program exe- cution; - can be computed automatically, by techniques rele- vant to static analysis. ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 30 —

  10. – “ deductive methods ”: - the abstract semantics is specified by verification con- ditions; - the user must provide the abstract semantics in the form of inductive arguments (e.g. invariants); - can be computed automatically by methods relevant to static analysis. – “ static analysis ”: the abstract semantics is computed automatically from the program text according to pre- defined abstractions (that can sometimes be tailored automatically/manually by the user). ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 31 —

  11. Required properties of the abstract semantics – sound so that no possible error can be forgotten; – precise enough (to avoid false alarms); – as simple/abstract as possible (to avoid combinatorial explosion phenomena). ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 32 —

  12. Graphic example: The most abstract correct and precise semantics x ( t ) �������������� ��������� ������������ t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 33 —

  13. Graphic example: Erroneous abstraction — I x ( t ) �������������� ��������� ��������� ������������ �������������������������������� t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 34 —

  14. Graphic example: Erroneous abstraction — II x ( t ) �������������� ��������� ��������� ������������ �������������������������������� t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 35 —

  15. Graphic example: Imprecision ) false alarms x ( t ) �������������� ����������� ��������� ������������ �������������������������������� t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 36 —

  16. Abstract domains Standard abstractions – that serve as a basis for the design of static analyzers: - abstract program data, - abstract program basic operations; - abstract program control (iteration, procedure, con- currency, . . . ); – can be parametrized to allow for manual adaptation to the application domains. ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 37 —

  17. Graphic example: Standard abstraction by intervals x ( t ) �������������� ������������ ��������� ������������ ��������������������������������������������� t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 38 —

  18. Graphic example: A more refined abstraction x ( t ) �������������� ��������� ������������ ����������������������� t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 39 —

  19. A very informal introduction to static analysis algorithms ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 40 —

  20. Standard operational semantics ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 41 —

  21. Standard semantics – Start from a standard operational semantics that de- scribes formally: - states that is data values of program variables, - transitions that is elementary computation steps; – Consider traces that is successions of states correspond- ing to executions described by transitions (possibly in- finite). ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 42 —

  22. Graphic example: Small-steps transition semantics x ( t ) ��������� ��������� ������������ t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 43 —

  23. Example: Small-steps transition semantics of an assignment int x; ... l: x := x + 1; l’: f l : x = v ! l 0 : x = v + 1 j v 2 [ min _ int ; max _ int ` 1] g [ f l : x = max _ int ! l 0 : x = ˙ g (runtime error) ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 44 —

  24. Example: Small-steps transition semantics of a loop 3 l1 : : : : 7 7 7 7 l1 : x = ` 1 & 7 7 7 7 7 7 l1: l1 : x = 0 ! l2 : x = 1 7 7 7 7 x := 1; 7 7 % l1 : x = 1 7 7 l2: 7 7 7 7 l1 : : : : 7 while x < 10 do 7 5 l3: l2 : x = 1 ! l3 : x = 1 x := x + 1 l3 : x = 1 ! l4 : x = 2 l4: od l4 : x = 2 ! l3 : x = 2 l5: l3 : x = 2 ! l4 : x = 3 : : : l4 : x = 10 ! l5 : x = 10 ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 45 —

  25. Example: Trace semantics of loop l1: x := 1; l2: while x < 10 do l3: x := x + 1 l4: od 3 l5: l1 : : : : 7 7 7 7 l1 : x = ` 1 & 7 7 7 7 7 7 ! l2 : x = 1 ! l3 : x = 1 ! l4 : x = 2 ! l1 : x = 0 7 7 7 7 7 7 l1 : x = 1 % 7 7 7 7 7 7 l1 : : : : 7 7 5 l3 : x = 2 ! l4 : x = 3 : : : ! l4 : x = 10 ! l5 : x = 10 ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 46 —

  26. Transition systems – h S; t !i where: - S is a set of states/vertices/. . . t - ! 2 } ( S ˆ S ) is a transition relation/set of arcs/. . . t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 47 —

  27. Collecting semantics in fixpoint form ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 48 —

  28. Collecting semantics – consider all traces simultaneously; – collecting semantics: - sets of states that describe data values of program variables on all possible trajectories; - set of states transitions that is simultaneous elemen- tary computation steps on all possible trajectories; ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 49 —

  29. Graphic example: sets of states x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 50 —

  30. Graphic example: set of states transitions x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 51 —

  31. � Example: Reachable states of a transition system I ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 52 —

  32. Reachable states in fixpoint form F ( X ) = I [ f s 0 j 9 s 2 X : s t ! s 0 g „ R = lfp ; F = + 1 n =0 F n ( ; ) f 0 ( x ) = x where [ f n +1 ( x ) = f ( f n ( x )) ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 53 —

  33. Example of fixpoint iteration „ ; –X . I [ f s 0 j 9 s 2 X : s t for reachable states lfp ! s 0 g I � � � � � � � � � � � � � � � �� � � ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 54 —

  34. Example of fixpoint iteration „ ; –X . I [ f s 0 j 9 s 2 X : s t for reachable states lfp ! s 0 g � � � � � F � � � � � � � � � � �� � � ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 54 —

  35. Example of fixpoint iteration „ ; –X . I [ f s 0 j 9 s 2 X : s t for reachable states lfp ! s 0 g � � � � � F � � F � � � � � � � �� � � ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 54 —

  36. Example of fixpoint iteration „ ; –X . I [ f s 0 j 9 s 2 X : s t for reachable states lfp ! s 0 g � � � � � F � � F � � F � � � � �� � � ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 54 —

  37. Example of fixpoint iteration „ ; –X . I [ f s 0 j 9 s 2 X : s t for reachable states lfp ! s 0 g � � � � � F � � F � � F � � F � �� � � ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 54 —

  38. Abstraction by Galois connections ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 55 —

  39. Abstracting sets (i.e. properties) – Choose an abstract domain, replacing sets of objects (states, traces, . . . ) S by their abstraction ¸ ( S ) – The abstraction function ¸ maps a set of concrete ob- jects to its abstract interpretation; – The inverse concretization function ‚ maps an abstract set of objects to concrete ones; – Forget no concrete objects: (abstraction from above) S „ ‚ ( ¸ ( S )) . ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 56 —

  40. Interval abstraction ¸ � �� f x : [1 ; 99] ; y : [2 ; 77] g � � � �� ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 57 —

  41. Interval concretization ‚ � �� f x : [1 ; 99] ; y : [2 ; 77] g � � � �� ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 58 —

  42. The abstraction ¸ is monotone � �� �� f x : [33 ; 89] ; y : [48 ; 61] g v �� f x : [1 ; 99] ; y : [2 ; 90] g � � � �� �� �� X „ Y ) ¸ ( X ) v ¸ ( Y ) ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 59 —

  43. The concretization ‚ is monotone f x : [33 ; 89] ; y : [48 ; 61] g v f x : [1 ; 99] ; y : [2 ; 90] g X v Y ) ‚ ( X ) „ ‚ ( Y ) ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 60 —

  44. The ‚ ‹ ¸ composition is extensive � �� f x : [1 ; 99] ; y : [2 ; 77] g � � �� � X „ ‚ ‹ ¸ ( X ) ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 61 —

  45. The ¸ ‹ ‚ composition is reductive � �� f x : [1 ; 99] ; y : [2 ; 77] g = = v f x : [1 ; 99] ; y : [2 ; 77] g � � � �� ¸ ‹ ‚ ( Y ) = = v Y ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 62 —

  46. Correspondance between concrete and abstract properties – The pair h ¸; ‚ i is a Galois connection: ‚ ` ` ` h } ( S ) ; „i ` hD ; vi ` ` ! ¸ ‚ ` ` ` ` – h } ( S ) ; „i ` hD ; vi when ¸ is onto (equivalently ` `! ` ! ¸ ¸ ‹ ‚ = 1 or ‚ is one-to-one). ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 63 —

  47. Galois connection ‚ ` ` ` hD ; „i ` hD ; vi ` ` ! ¸ iff 8 x; y 2 D : x „ y = ) ¸ ( x ) v ¸ ( y ) ^ 8 x; y 2 D : x v y = ) ‚ ( x ) „ ‚ ( y ) ^ 8 x 2 D : x „ ‚ ( ¸ ( x )) ^ 8 y 2 D : ¸ ( ‚ ( y )) v x iff 8 x 2 D ; y 2 D : ¸ ( x ) v y ( ) x „ ‚ ( y ) ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 64 —

  48. Graphic example: Interval abstraction x ( t ) ����������������������������� t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 65 —

  49. Graphic example: Abstract transitions x ( t ) ������������������� t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 66 —

  50. Example: Interval transition semantics of assignments int x; ... l: x := x + 1; l’: f l : x 2 [ ‘; h ] ! l 0 : x 2 [ l + 1 ; min( h + 1 ; max _ int )] [ f ˙ j h = max _ int g j ‘ » h g where [ ‘; h ] = ; when h < ‘ . ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 67 —

  51. �������� ������ Function abstraction � � F ] = ¸ ‹ F ‹ ‚ i : e : F ] =  ‹ F � �������� ������ ‚ ` ` ` h P; „i ` h Q; vi ) ` ` ! ¸ –F ] . ‚ ‹ F ] ‹ ¸ ! P; _ ` ` ` ` ` ` ` ` ` ` ! Q; _ mon h Q mon h P 7` „i ` 7` vi ` ` ` ` ` ` ` ` ` ! –F . ¸ ‹ F ‹ ‚ ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 68 —

  52. Example: Set of traces to trace of intervals abstraction Set of traces: ¸ 1 # Trace of sets: ¸ 2 # Trace of intervals ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 69 —

  53. Example: Set of traces to reachable states abstraction Set of traces: ¸ 1 # Trace of sets: ¸ 3 # Reachable states ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 70 —

  54. Composition of Galois Connections The composition of Galois connections: ‚ 1 ` ` ` h L; »i ` h M; vi ` ` ! ¸ 1 and: ‚ 2 ` ` ` h M; vi ` h N; —i ` ` ! ¸ 2 is a Galois connection: ‚ 1 ‹ ‚ 2 ` ` ` ` ` ` h L; »i ` h N; —i ` ` ` ` ` ! ¸ 2 ‹ ¸ 1 ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 71 —

  55. Abstract semantics in fixpoint form ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 72 —

  56. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  57. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  58. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  59. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  60. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  61. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  62. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  63. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  64. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  65. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  66. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  67. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  68. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  69. Graphic example: traces of sets of states in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 73 —

  70. Graphic example: traces of intervals in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 74 —

  71. Graphic example: traces of intervals in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 74 —

  72. Graphic example: traces of intervals in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 74 —

  73. Graphic example: traces of intervals in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 74 —

  74. Graphic example: traces of intervals in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 74 —

  75. Graphic example: traces of intervals in fixpoint form x ( t ) t ľ P. Cousot IFIP WCC — Topical day on Abstract Interpretation, Toulouse, 24 August 2004 — 74 —

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