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Proving Exception Freedom within High Integrity Software Systems

Proving Exception Freedom within High Integrity Software Systems. Bill J. Ellis (bill@macs.hw.ac.uk) Dependable Systems Group Heriot-Watt University (Project page: http://www.macs.hw.ac.uk/~air/clamspark/). Overview. High integrity software Proving exception freedom in SPARK

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Proving Exception Freedom within High Integrity Software Systems

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  1. Proving Exception Freedom within High Integrity Software Systems Bill J. Ellis (bill@macs.hw.ac.uk) Dependable Systems Group Heriot-Watt University (Project page: http://www.macs.hw.ac.uk/~air/clamspark/)

  2. Overview • High integrity software • Proving exception freedom in SPARK • Property discovery • Abstract interpretation • Recurrence relations • Interval arithmetic • Proofs • Conclusions

  3. High Integrity Software • Software standards encourage or enforce proof for high integrity software: • MOD 00-55: requirements for the procurement of safety critical software in defence equipment. • Formal methods and proof mandatory • ITSEC: Information technology security evaluation criteria • Formal methods mandatory

  4. Praxis and SPARK • SPARK is developed by Praxis Critical Systems for building high integrity software: • Formally defined safe subset of Ada • Information and data flow static analysis • Supports proofs of: • Partial correctness • Exception freedom (No run time errors) • SPARK is used in industry: • BAE prove exception freedom (Unnamed project) • Praxis completed SHOLLIS and MULTOS CA • Many more...

  5. SPARK Proof In Industry • Partial correctness (Rare): • User supplied specification • Proofs usually deep • Very limited automation • Exception freedom (Increasingly common): • Automatic specification • Proofs usually shallow • Good (90%) automation via Praxis Simplifier • Remaining 10% may number in the thousands...

  6. Exception Freedom in SPARK • Storage_Error(Static memory requirement) • Program_Error • Tasking_Error • Constraint_Error(Some can occur in SPARK) • Access_Check  • Discriminant_Check  • Tag_Check  • Division_Check  • Index_Check  • Range_Check  • Overflow_Check  Proving exception freedom in SPARK is proving variables stay within legal bounds

  7. Example Code subtype Index is Integer range 0 .. 9; type D is array (Index) of Integer; R:=0; For I in Index loop if D(I) >= 0 and D(I) <= 100 then R := R + D(I); end if; end loop;

  8. Example Code(Exception Freedom Checks) subtype Index is Integer range 0 .. 9; type D is array (Index) of Integer; --#check 0>=-32768 and 0<=32767; R:=0; For I in Index loop --#invariant I>=0 and I<=9 and --#forall J in 0..9 D(J)>=-32768 and D(J)<=32767; --#check I>=0 and I<=9; if D(I) >= 0 and D(I) <= 100 then --#check I>=0 and I<=9; --#check R+D(I)>=-32768 and R+D(I)<=32767; R := R + D(I); end if; end loop;

  9. Pre-condition Prove properties hold between between cut points Check ... Check Check Invariant Prove exception freedom VCs using properties ... Check Check ... Post-condition Check Exception Freedom VCs

  10. Proof Strategy Success! Prove Exception Freedom VCs Try proof again Fail Discover properties (Typically invariants) Prove properties

  11. Abstract Interpretation (AI) • Evaluate a program, replacing concrete variables with abstract values. • Concrete integer variable: -32768 to +32767 • An abstract integer variable: {-,0,+}. • Abstract interpretation provides a framework to reason about programs in the abstract.

  12. Example (AI Flowchart) Pre-condition R:=0 I:=0 Loop junction node Invariant N Simple junction node D(I)0 and D(I) 100 I:=I+1 R:=R+D(I) Normalised form of a SPARK for loop Post-condition I=9

  13. Recurrence Relations • Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.. • Recurrence relation: A0=0, A1=1, An=A(n-1) +A(n-2) • Liner recurrence relations with constant coefficients (LRRCs): • An=c1*A(n-1)+…+ck*A(n-k)+f(n) • nk • Only use first powers of previous terms (A(n-1)^1) • Coefficients ck are constants • f(n) is a function • Can automatically solve LRRCs (and a few other special cases) using: • Mathematica • The Parma University's Recurrence Relation Solver (PURRS)

  14. Interval Arithmetic/Algebra • A theory for reasoning about bounds. • Bounds on x: x1 and x 5 • Interval: X=[1, 5], lower: X=1, upper: X=5 • Operations are defined on intervals: • X+Y = [X+Y, X+Y] • X-Y = [X-Y, X-Y] • Example: • [0, 2] + [-4, 6] = [0+(-4), 2+6] = [-4, 8] • [0, 2] - [-4, 6] = [0-6, 2-(-4)] = [-6, 6]

  15. Property Discovery • Abstract interpretation: • Focus on bounds of variables and arrays. • Discover and propagate bounds: • Using code semantics. (Of assignment, test, …) • Exploit semantics of exception freedom. • Use interval algebra and proof planning. • Loop junction nodes (Invariant discovery): • Exploit recurrence relation solvers. • Replace ‘n’ with program variables.

  16. R :[0,0] R:[min(R), max(R)] D[0,0]:[0, 100] R:[(R (n-1))+(D(I)), ((R (n-1)) +(D(I))] R:merge([(R (n-1))+0, (R (n-1)) +100], [0, 0]) Example (Variable R) Pre-condition R:=0 I:=0 Invariant N D(I)0 and D(I) 100 I:=I+1 R:=R+D(I) R:[(R (n-1))+(D(0)), ((R (n-1)) +(D(0))] R:[(R (n-1))+0, ((R (n-1)) +100] I=9 Post-condition

  17. Example (Variable R) Solve lower: Rn = R (n-1)+0 Rn=R0 0 R :[0, 0] Arriving at the loop Solve upper: Rn = R (n-1)+100 Rn=R0 + 100*n 0 + 100*n  100*n R:merge( [(R (n-1))+0, (R (n-1)) +100], [0, 0]) R:merge([0, 100*n], [0, 0]) R:[0, 100*n] Returning from first iteration Starting second iteration...

  18. Invariant Discovery (Eliminate n) Properties for I Invariant N Some details... R:[0, 100*n] I  0 and I  9 I:[n,n] | {exit([min(I), 9]) or exit([10, max(I)])} Properties for R Express n in terms of I: I=n  n=I Substitute n for I in R: R:[0, 100*I] Stabilised abstract values for R and I at the invariant R  0 and R  100*I

  19. Example (Discovered invariant) subtype Index is Integer range 0 .. 9; type D is array (Index) of Integer; --#check 0>=-32768 and 0<=32767; R:=0; For I in Index loop --#invariant I>=0 and I<=9 and R>=0 and R<=100*I and --#forall J in 0..9 D(J)>=-32768 and D(J)<=32767; --#check I>=0 and I<=9; if D(I) >= 0 and D(I) <= 100 then --#check R+D(I)>=-32768 and R+D(I)<=32767; --#I>=0 and I<=9; R := R + D(I); end if; end loop;

  20. And the proofs? • Invariant property VCs: • Rippling reduces VC to a residue • Prove residue using proof planning • Exception freedom VCs: • Middle-out reasoning? • ‘Look ahead’ using interval algebra?

  21. Implementation (Underway…) SPARK code Add new properties to code Praxis Examiner Light weight SPARK Parser VCs Rule files SPARK structure Method: Abstracting to bounds NuSPADE Subprogram Spider Proof Planner Subprogram Details Method: Rippling Proof scripts CLAM

  22. Related Work • RUNCHECK (Steven M. German) (1981) • Proves exception freedom VCs for Pascal • Uses a few rewrite rules (7) to solve recurrence relations as a final stage • Does not exploit program context • Limited treatment of arrays (Considered array initialisation) • Abstract Interpretation (Patrick Cousot, Radhia Cousot) (1976) • Is algorithmic and always generates correct results • Good automatic linear property generation for programs with linear assignments and no arrays • Used for compiler optimisation, exception detection, program documentation, program visualisation...

  23. Conclusions Abstract Interpretation (Framework) + Interval Algebra (Theory) + Recurrence Relations (Off the shelf tools) + Proof Planning (Light reasoning) = Property generation for exception freedom Properties + Rippling (For invariants) + Proof Planning (Suitable methods) = Automated exception freedom proof Note: Is not complete -- Can fail!

  24. EOF

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