Qualitätssicherung von Software (SWQS)

1. Qualitätssicherung von Software (SWQS) Prof. Dr. Holger Schlingloff Humboldt-Universität zu Berlin und Fraunhofer FOKUS 28.5.2013: Modellprüfung II - BDDs

2. Existenzgründer gesucht!

3. Fragen zur Wiederholung • Unterschied Verifikation – Validierung? • Wie kann man Sudoku aussagenlogisch beschreiben? • Wie ist die Komplexität des Erfüllbarkeitsproblems? • Was versteht man unter Modellprüfung? • Unterschied Sudoku – Schiebepuzzle?

4. 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 byvar 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 224=108 states • e.g. buffer of length 10 with 10-bit values  1030 states • Representation of large sets of states?

5. Representation of Sets

6. Truth table and tree form formula Reduction: Replace Ite (v,ψ,ψ) by ψ

7. Abbreviations • Introduce abbreviations • maximally abbreviated • for any given order of variables the maximal abbreviated form is uniquely determined!

8. Binary Decision Trees (BDTs) • Binary decision tree • Elimination ofisomorphic subtrees(abbreviations)

9. Binary Decision Diagrams (BDDs) • Elimination ofredundant nodes(redundant subformulas) Ite (v,ψ,ψ) by ψ formula: ((V1  V2)  V4)

10. Calculation of BDDs

11. Boolean operations on BDDs

12. Satisfiability • This procedure can be applied for arbitrary boolean connectives (or, and, not) • BDD() is the constant node  • p = (p), (pq) = (pq) etc. • direct algorithms for ,  possible • this amounts to set union, intersection, and complement with respect to the base set • Formula φ is satisfiable iff BDD(φ)   • any path through the BDD to T defines a model

13. Binary Encoding of Relations • A relation is a subset of the product of two sets • Thus, a relation is nothing but a set • Example: var v: {0..3}, w:{0..7}; var v0, v1, w0, w1, w2: boolean; “divides”-Relation: v divides w iff v=1, or v=2 and w even,or v=3 and w in {0,3,6} boolean formula:

14. The Influence of Variable Ordering

15. Boolean Quantification • Substitution by constants is trivial • Boolean quantification: • ! This works for arbitrary finite domains !

16. Bounded Model Checking • State s is reachable from s0 iff • it is reachable in 0 steps: s=s0, or • it is reachable in 1 step: R(s0,s), or • it is reachable in 2 steps: s1(R(s0,s1)  R(s1,s)), or • it is reachable in 3 steps: s1s2(R(s0,s1)  R(s1,s2)  R(s2,s)), or • ..., or • it is reachable in n steps, where n is the diameter of the model • Idea: Check each of these formulas sequentially

17. Transitive Closure • Each finite (transition) relation can be represented as a BDD • The transitive closure of a relation R is defined recursively by • Thus, transitive closure be calculated by an iteration on BDDs

18. Reachability • State s is reachable iff s0R*s, where s0S0 is an initial state and R is the transition relation • Reachability is one of the most important properties in verification • most safety properties can be reduced to it • in a search algorithm, is the goal reachable? • Can be arbitrarily hard • for infinite state systems undecidable • Can be efficiently calculated with BDDs