1. 1 Regression-Verification Benny Godlin Ofer Strichman Technion

2. 2 Software Formal Verification Tony Hoare’s grand challenge: "verifying compiler" Suppose we do not require completeness. Still, there are two major bottlenecks when applying functional verification:  Needs formal specification assertions, or higher level temporal properties  Complexity.

3. 3 A more modest challenge: Regression Verification “For every problem that you cannot solve, there is an easier problem that you cannot solve” Develop a method for formally verifying the equivalence of two closely-related programs. Does not require formal specification. Computationally easier than functional verification On the other hand: defines a weaker notion of correctness.

4. 4 Advantages of Regression Verification Specification by reference to an earlier version. Can be applied from early development stages. Equivalence proofs are potentially easier than functional verification.  Ideally, the complexity should depend on the semantic difference between the programs, and not on their size.

5. 5 Advantages of Regression Verification Potential usage:  Checking for backward compatibility  Checking for correctness after refactoring  Checking for correctness after performance optimization  …

6. 6 Regression-based technologies In Testing: Regression testing is the most popular automatic testing method In hardware verification: Equivalence checking is the most used verification method in the industry

7. 7 Hoare’s 1969 paper has it all.... …and 6 pages later:

8. 8 Previous work In the theorem-proving world:  Not dealing with realistic programs / realistic programming languages  Not utilizing the equivalence of most of the code for simplifying the computational challenge Industrial / realistic programs:  Loop-free/Recursion-free code: (microcode @ Intel, embedded code @ Hu)

9. 9 Goals Develop notions of equivalence Develop corresponding proof rules Develop a prototype for verifying equivalence of C programs, that will  incorporate the equivalence rules  and be sensitive to the magnitude of change rather than the magnitude of the programs

10. 10 Notions of equivalence (1 / 6) Partial equivalence  For any 2 terminating executions of P1 and P2 on equal inputs  the results of the executions are equal. Undecidable

11. 11 Notions of equivalence (2 / 6) Mutual termination  Given equal inputs, P1 terminates, P2 terminates Undecidable

12. 12 Notions of equivalence (3 / 6) Reactive equivalence  Let P1 and P2 be reactive programs.  Given equal inputs, every 2 executions of P1 and P2 which also receive the same input sequence, emit the same output sequence. The sequences may be finite or infinite Undecidable

13. 13 Notions of equivalence (4 / 6) k-equivalence  For any 2 executions of P1 and P2 on equal inputs where each loop and recursion is restricted to iterate up to k times, results in equal output. Decidable

14. 14 Notions of equivalence (5 / 6) Full equivalence  P1 and P2 are partially equivalent and mutually terminate Undecidable

15. 15 Notions of equivalence (6 / 6) Total equivalence  P1 and P2 are partially equivalent and both terminate Undecidable

16. 16 Notions of equivalence: summary 1. Partial equivalence 2. Mutual termination 3. Reactive equivalence 4. k-equivalence 5. Full equivalence* = Partial equivalence + mutual termination 6. Total equivalence* = partial equivalence + both terminate * Appeared in earlier publications.

17. 17 Outline Inference rules for proving equivalence  Hoare’s rule of recursion  Rule 1: partial equivalence  Rule 2: mutual termination  Rule 3: reactive equivalence Regression verification for C programs  CBMC – the back engine  Architecture  Decomposition  Handling dynamic data structures

18. 18 Hoare ’ s Rule of Recursion Where symbolizes a sound proof relation. (explained on the next slide) Let A be a recursive procedure.

19. 19 A recursive run: Diagram level 1: A(...) {... call A(...);... } level 2: A(...) {... call A(...);... }... last level: A(...) {... call A(...);... }

20. 20 Rule of Recursion: Diagram level 1: {p} A(...) {... {p} call A(...); {q}... } {q} level d-1: {p} A(...) {... {p} call A(...); {q}... } {q}... last level (d): {p} A(...) {... call A(...);... } {q} {p} {q} {p} {q}

21. 21 Rule 1: partial equivalence of recursive procedures. //in[A] A(... ) {... //in[call A] call A(...); //out[call A]... } //out[A] //in[B] B(... ) {... // in[call B] call B(...); //out[call B]... } //out[B] A B

22. 22 Rule 1: encoding the premise The only thing we know about the called function is that it satisfies the congruence axiom: (congruence) Calls to U with the same input values return the same output values. Making the called function Uninterpreted is enough. We call such a program isolated.

23. 23 Rule 1. Isolation example unsigned gcd1(unsigned a, unsigned b) { unsigned g; if (b == 0) g = a; else { a = a % b; g = gcd1(b, a); } return g; } unsigned gcd1(unsigned a, unsigned b) { unsigned g; if (b == 0) g = a; else { a = a % b; g = U(b, a); } return g; } U is an uninterpreted function

24. 24 Rule 1. Definitions f : a function UF ( f ): an uninterpreted function such that f, UF ( f ) have the same prototype. F UF : the isolated version of a function F :

25. 25 Rule 1: partial equivalence made simple Earlier we had: Now we have:

26. 26 Rule 1: partial equivalence made simple //in[A UF ] A UF (... ) {... call U(...);... } //out[A UF ] //in[B UF ] B UF (... ) {... call U(...);... } //out[B UF ] A UF B UF  U is an uninterpreted function

27. 27 Rule 1: partial equivalence made simple I has to reason about a combination of  programs without loops or functions calls, and  uninterpreted functions. For a programming language such as C, there exists decision Procedures (and tools) with these features.

28. 28 unsigned gcd1(unsigned a, unsigned b) { unsigned g; if (b == 0) g = a; else { a = a % b; g = gcd1(b, a); } return g; } unsigned gcd2(unsigned x, unsigned y) { unsigned z; z = x; if (y > 0) z = gcd2(y, z % y); } return z; } Rule 1: example ?=?=

29. 29 Rule 1: example (side 1) First, compute the transition relation using Static Single Assignment (SSA) unsigned gcd1(unsigned a, unsigned b) { unsigned g; if (b == 0) g = a; else { a = a % b; g = H(b, a); } return g; }

30. 30 Rule 1: example (side 2) unsigned gcd2(unsigned x, unsigned y) { unsigned z; z = x; if (y > 0) z = H(y, z % y); } return z; }

31. 31 Rule 1: example unsigned gcd1(unsigned a, unsigned b) {... return g; } unsigned gcd2(unsigned x, unsigned y) {... return z; }

32. 32 Rule 2: Proving Mutual termination The rule:  If all paired procedures satisfy : 1. Partial equivalence, 2. The conditions under which they call each pair of mapped procedures are equal, and 3. The read arguments of the called procedures are the same when they are called, then  all paired procedures mutually terminate.

33. 33 Rule 3: Proving reactive equivalence The rule:  If all paired procedures satisfy 1. given the same arguments and the same input sequences, they return the same values, 2. they consume the same number of inputs, and 3. the interleaved sequence of procedure calls and output statements inside the mapped procedures is the same (and the procedure calls are made with the same arguments) then  all paired procedures are reactive equivalent.

34. 34 Outline Inference rules for proving equivalence  Hoare’s rule of recursion  Rule 1: partial equivalence  Rule 2: mutual termination  Rule 3: reactive equivalence Regression verification for C programs  CBMC – the back engine  Architecture  Decomposition  Handling dynamic data structures

35. 35 CBMC A C verification tool by D. Kroening Normal use – bounded model checking of C programs. We use CBMC as follows:  Given a loop-free, recursion-free C program and an assertion, check whether the assertion can fail. Supports “assume(…)” construct to enforce constraints.

36. 36 The Regression Verification Tool (RVT) Version AVersion B CBMC  rename identical globals  enforce equality of inputs.  insert check points code  Supports:  Decomposition  Abstraction  some static analysis  … feedback  result  counterexample C program RVT Preprocessor

37. 37 How to compare programs ? Deciding program equivalence based only on output  misleading – output may not reflect well complex computation  impractical – program run may be too long before output  impossible - no defined output in some stages of development User may want to compare values which are not output Therefore we use check points. Check points are places (labels) at both programs where some expression value should be equal on both sides.

38. 38 Checkpoints Define (in a separate file): checkpoints h label, exp, condition i for both programs. Update an array with the value of exp each time it reaches label and condition holds. In all rules, rather than checking simple I/O relation:  when executed on the same input  the sequences of values collected at checkpoints are equal. exp 1 exp 2... P1: exp ’ 1 exp ’ 2... P2: = ===

39. 39 The main challenge: complexity The goal: complexity depends on the semantic difference The key: decomposition. Preprocessing: all loops are rewritten as recursive functions. Map the functions in the two programs.  Mapping heuristics: Name, signature, ‘role’ played in the code. Mapped functions that have already been proven to be equal are abstracted with uninterpreted functions.

40. 40 Call Graph Algorithm A: B: f1() f2() f5() f3()f4() f6() f1’() f2’() f3’()f4’() f5’() Equivalent pair Syntactically equivalent pair Equivalence undecided yet Non-equivalent pair Legend: CBMC Unpaired function f7’() U UUU U U

41. 41 Mutual Recursion Mutually Recursive Functions:  Recognized by MSCCs in the call graph  We can treat this case by an extension of the solution to the simple recursion case.

42. 42 Call Graph Algorithm (with SCCs) A: B: f1() f2() f5() f3()f4() f6() f1’() f3’()f4’() f5’() f6’() Equivalent pair Syntactically equivalent pair Equivalence undecided yet Non-equivalent pair Legend: Equivalent if MSCC U UUU U U CBMC U U U U f2’()

43. 43 The Regression Verification Tool (RVT) We tested RVT on several programs:  Small programs that calculate GCD, power, sum of digits...  Simple interactive (reactive) calculator program.  Automatically generated sizable programs up-to thousands of code lines but without arrays (unsupported yet) Limited-size industrial programs:  Parts of TCAS - Traffic Alert and Collision Avoidance System.  Core of MicroC/OS - real-time kernel for embedded systems.  Matlab examples: parts of engine-fuel-injection simulation.

44. 44 Testing RVT on programs: Conclusions For equivalent programs, partial-equivalence checks were very fast:  proving equivalence in minutes. For non-equivalent programs:  RVT attempts to prove partial-equivalence but fails  then RVT tries to prove k-equivalence  k-equivalence sometimes succeeds to show counter examples: executions on equal inputs with non-equal values at check-points  k-equivalence may run for several hours or even run out of memory.

45. 45 Long list of future work Finish implementation of:  Rules 2/3, arrays (especially in UFs), slicing,… Future work:  How to strengthen the rules with invariants.  Test on real changes (CVS repository). Use other program verification tools for C to prove equivalence.

46. 46 More Challenges Identifying the connection between functional properties and equivalence properties:  Assume that a high level property P is satisfied by some version of the program,  Q: Which values should be followed by regression verification to guarantee that this property still holds ? Q: What is the ideal gap between two versions of the same program, that makes Regression Verification most effective ?