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Termination Proofs from Tests

Aditya Nori Rahul Sharma MSR India Stanford University. Termination Proofs from Tests . Goal. Prove termination of a program Program terminates if all loops terminate H ard problem, undecidable in general Need to exploit all available information.

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Termination Proofs from Tests

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  1. AdityaNori Rahul Sharma MSR India Stanford University Termination Proofs from Tests

  2. Goal • Prove termination of a program • Program terminates if all loops terminate • Hard problem, undecidable in general • Need to exploit all available information

  3. Tests • Previous techniques are static • Tests are a neglected source of information • Tests have previously been used • Safety properties, empirical complexity, … • This work, use tests for termination proofs

  4. Example: GCD gcd(intx,int y) assume(x>0 && y>0); while( x!=y ) do if( y > x ) y = y–x; if( x > y) x = x-y; od return x; x=1, y=1 x=2, y=1

  5. Infer-and-Validate Approach (1,1) (2,1) … while … … … while … print x print y x=1, y=3 Data … while … … assert … ML

  6. Infer-and-Validate Approach (1,1) (2,1) … while … … … while … print x print y x=1, y=3 Data … while … … assert … ML

  7. Instrument the Program gcd(int x, int y) assume(x>0 && y>0); a := x; b := y; c := 0; while( x!=y ) do c := c + 1; if( y > x ) y := y–x; if( x > y) x := x-y; od print ( a, b, c ); • New variables to capture initial values • Introduce a loop counter • Print values of input variables and counter

  8. Infer-and-Validate Approach (1,1) (2,1) … while … … … while … print x print y x=1, y=3 Data … while … … assert … ML

  9. Generating Data gcd(int x, int y) assume(x>0 && y>0); a := x; b := y; c := 0; while( x!=y ) do c := c + 1; if( y > x ) y := y–x; if( x > y) x := x-y; od print( a, b, c) For on inputs , the loop iterates times Infer a bound using and

  10. Infer-and-Validate Approach (1,1) (2,1) … while … … … while … print x print y x=1, y=3 Data … while … … assert … ML

  11. Regression • Predict number of iterations (final value ofc) • As a linear expression in a and b • Find • Find • But we want • Addas a constraint • Solvable by quadratic programming

  12. Quadratic Program (QP) • The quadratic program is: • Solved in MATLAB • quadprog(A’*A,-A’*C,-A,-C) • For gcd example, • Bound

  13. Naïve Regression

  14. Quadratic Program

  15. Infer-and-Validate Approach (1,1) (2,1) … while … … … while … print x print y x=1, y=3 Data … while … … assert … ML

  16. Verification Burden assume(x>0 && y>0); a := x; b := y; c := 0; while( x!=y ) do c := c + 1; if( y > x ) y := y–x; if( x > y) x := x-y; assert(c <= a+b-2); od • Bound: • Difficult to validate • Infer invariants from tests

  17. Regression for Invariant assume(x>0 && y>0); a := x; b := y; c := 0; while( x!=y ) do print(c, a, b, x, y); c := c + 1; if( y > x ) y := y–x; if( x > y) x := x-y; assert(c <= a+b-2); od • Predict a bound onc • Same tests, more data • Solve same QP • has five columns • [1,a,b,x,y] • hascat every iteration

  18. Free Invariant assume(x>0 && y>0); a:=x; b:=y; c := 0; free_inv(c<=a+b-x-y); while( x!=y ) do c := c + 1; if( y > x ) y := y – x; if( x > y) x := x-y; assert(c <= a+b-2 ); od • Obtain • Add as a free invariant • Use if checker can prove • Otherwise discard

  19. Validate • Give program to assertion checker • Inductive invariant for gcd example: • If check fails then return a cex as a new test

  20. Non-linear Example u := x;v := y;w := z; while ( x >= y ) do if ( z > 0) z := z-1; x := x+z; else y := y+1; od • Given degree 2, • Bound: • After rounding:

  21. Assertion Checker • Requirements from assertion checker: • Handle non-linear arithmetic • Consume free invariants • Produce tests as counter-examples • Micro-benchmarks: Use SGHAN’13 • Handles non-linear arithmetic, no counter-examples • Windows Device Drivers: Use Yogi (FSE’ 06) • Cannot handle non-linear, produce counter-examples

  22. Micro-benchmarks

  23. Experiments with WDK

  24. Related Work • Regression: Goldsmith et al. ‘07 , Huang et al. ’10, … • Mining specifications from tests: Dallmeier et al. `12,… • Termination: Cousot `05, ResAna, Lee et al. ’12, … • Bounds analysis: SPEED, WCET, Gulavani et al. `08, … • Invariant inference: Daikon, InvGen, Nguyen et al.`12, …

  25. Conclusion • Use tests for termination proofs • Infer bounds and invariants using QP • Use off-the-shelf assertion checkers to validate • Future work: disjunctions, non-termination

  26. Disjunctions Example a = i ; b = j ; while(i<M || j<N) i= i+1; j = j+1; • Partition using predicates • Control flow refinement • Sharma et al. ’11

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