1. Non-Termination of Affine Loops Kevin Durant, Corina Pasareanu, Willem Visser Stellenbosch University and NASA/CMU

2. public String preserveTags(String body) {…}

3. Infinite loops are the worst kind of error, since it is input driven and therefore can reappear frequently, in fact infinitely often!

4. Symbolic String Analysis (Almost) All Java String operations covered Mixed Integer and String constraints Automata and SMT (bitvector) back-ends Part of Symbolic PathFinder M.Sc. by Gideon Redelinghuys Collaborators – Jaco Geldenhuys (Stellenbosch)

5. Infinite Loop? while (x > 0) (x,y) = (x+y+2,-x); Try (2,-3)

6. We only consider affine transformations on loop variables and simple loop conditions such as x>0 and x>=0

7. x,y are inputs while (x >= 0) { x := x – y; } Infinite Loop?

8. x,y are inputs while (x >= 0) { assert(‘x > x); x := x – y; } Ranking functions

9. Use ranking functions for non-termination!

10. x,y are inputs while (x >= 0) { assert(‘x > x); x := x – y; } Ranking functions ‘x <= x … {c /\ wp(s,‘x <= x)} s {c /\ wp(s,‘x <= x)}

11. x,y are inputs while (x >= 0) { assert(‘x > x); x := x – y; } Inductive? {x >= 0 /\ wp(x:=x-y,‘x <= x)} x := x - y {x >= 0/\ wp(x:=x-y,‘x <= x)} wp(x:=x-y,’x<=x) = {x <= x-y} {x >= 0 /\ y <= 0} x := x - y {x >= 0 /\ y <= 0}

12. So how about just… while (c) { s; } {c /\ wp(s,!rr)} s {c /\ wp(s,!rr)}

13. x,y are inputs while (x >= 0) { assert(‘x > x); x := x + y; y := 1 – y; }

14. x,y are inputs while (x >= 0) { assert(‘x > x); x := x + y; y := 1 – y; } {x >= 0 /\ wp(x:=x+y;y:=1-y,‘x <= x)} x := x – y; y := 1 – y; {x >= 0/\ wp(x:=x+y;y:=1-y,‘x <= x)} wp(x:=x+y;y:=1-y,’x<=x) = {x <= x+(1-y)} {x >= 0 /\ y <= 1} x:=x+y;y:=1-y; {x >= 0 /\ y <= 1}

15. ‘x <= x … ‘x <= x … N while (c) { s; } {c /\ wp(s n,!rr)} s n {c /\ wp(s n,!rr)}

16. while (x 0 > 0) { f(x) = Ax+b; } We conjecture that if there is an infinite loop then there exist n such that for all x for which the following is true you will loop infinitely x 0 > 0 /\ f 1 (x) > 0 /\ … /\ f 2n-1 (x) > 0 /\ x 0 ≤ f n (x) => f n (x) ≤ f 2n (x) Can we derive n from the number of variables in x? For 1 variable n = 2For 2 variables n >= 6 For 3 variables there is no n

17. Infinite Loop and no “n” while (x > 0) { x = 5x+y+z, y = 4y+3z, z = -3y+4z; }

18. Implementation JavaPathFinder Symbolic PathFinder SPF AffineLoopListener Model Checker for Java Open Source http://babelfish.arc.nasa.gov/trac/jpf Symbolic Execution extension for JPF called jpf-symbc Custom Listener on SPF Tries n = 0..6

19. To Do Study how many loops can be handled? Combine with abstraction Nested Loops?