1. Software Verification 2 Automated Verification Prof. Dr. Holger Schlingloff Institut für Informatik der Humboldt Universität and Fraunhofer Institut für Rechnerarchitektur und Softwaretechnik

2. Slide 2 H. Schlingloff, SS2012: SWV 2 LTL Model Checking Algorithm 12.4.2012

3. Slide 3 H. Schlingloff, SS2012: SWV 2 Improvements bitstate hashing incomplete hashing partial order techniques 12.4.2012

4. Slide 4 H. Schlingloff, SS2012: SWV 2 CTL model checking For each LTS/model there is exactly one computation tree  CTL model checking works directly on the model (no need to extract computation sequences) For all subformulas of a formula and all states of a given model, mark whether the state satisfies the subformula  iteration on formulas according to their inductive definition  if p is an atomic proposition, then p M = I(p)   M ={}  (φ  ψ) M = (M-φ M +ψ M )  (EXφ) M = {w |  w‘ (wRw‘  w‘  φ M )}  E(φU + ψ) M = {w | there is a path α from w and a w‘ on α such that (w<w‘  w‘  ψ M )  w‘‘ (w<w‘‘<w‘  w‘‘  φ M )}  A(φU + ψ) M = {w | for all paths α from w there is a w‘ on α such that (w<w‘  w‘  ψ M )  w‘‘ (w<w‘‘<w‘  w‘‘  φ M )} 12.4.2012

5. Slide 5 H. Schlingloff, SS2012: SWV 2 Actual Calculation How to calculate (EX ψ) M from ψ M ?  Inverse image construction How to calculate E(φU + ψ) M or A(φU + ψ) M from φ M and ψ M ? 12.4.2012

6. Slide 6 H. Schlingloff, SS2012: SWV 2 12.4.2012

7. Slide 7 H. Schlingloff, SS2012: SWV 2 Inverse reachability calculation 12.4.2012

8. Slide 8 H. Schlingloff, SS2012: SWV 2 Symbolic Representation Modelchecking algorithm deals with sets of states and with relations (sets of pairs of states) Need an efficient representation 12.4.2012

9. Slide 9 H. Schlingloff, SS2012: SWV 2 6.5.2008 Binary Encoding of Domains Any variable on a finite domain D can be replaced by log(D) binary variables  similar to encoding of data types by compilers  e.g. var v: {0..15} can be replaced by var v1,v2,v3,v4: boolean (0=0000, 1= 0001, 2=0010, 3=0011,..., 15=1111) State space  still in the order of original domain!  e.g. three int8-variables can have 2 24 =10 8 states  e.g. array of length 10 with 10-bit values  10 30 states Representation of large sets of states?

10. Slide 10 H. Schlingloff, SS2012: SWV 2 6.5.2008 Representation of Sets

11. Slide 11 H. Schlingloff, SS2012: SWV 2 6.5.2008 Ordered Tree Form Normal form for propositional formulas Uses only the connective Ite Linear ordering on the set of propositions  e.g., most significant bit first Shannon expansion

12. Slide 12 H. Schlingloff, SS2012: SWV 2 6.5.2008 Truth table and tree form formula Reduction: Replace Ite (v,ψ,ψ) by ψ

13. Slide 13 H. Schlingloff, SS2012: SWV 2 6.5.2008 Abbreviations Introduce abbreviations maximally abbreviated

14. Slide 14 H. Schlingloff, SS2012: SWV 2 6.5.2008 Binary Decision Trees (BDTs) Binary decision tree Elimination of isomorphic subtrees (abbreviations)

15. Slide 15 H. Schlingloff, SS2012: SWV 2 6.5.2008 Binary Decision Diagrams (BDDs) Elimination of redundant nodes (redundant subformulas) Ite (v,ψ,ψ) by ψ

16. Slide 16 H. Schlingloff, SS2012: SWV 2 6.5.2008 Calculation of BDDs