1. SS 2017 Software Verification CTL model checking, BDDs
Prof. Dr. Holger Schlingloff 1,2
Dr. Esteban Pavese 1
(1) Institut für Informatik der Humboldt Universität
(2) Fraunhofer Institut für offene Kommunikationssysteme FOKUS

2. Recap What is a tableau? How is it interpreted?
Propositional closing rules?
LTL tableaus?
Closing rules?
Model generation from an LTL tableau?
Connection to LTL model checking?

3. 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 pM= I(p)
M={}
(φψ)M = (M-φM +ψ M)
(EXφ)M = {w | w‘ (wRw‘ w‘φM )}
E(φU+ψ)M = {w | there is a path α from w and a w‘ on α such that (w<w‘ w‘ ψM ) w‘‘ (w<w‘‘<w‘ w‘‘ φM )}
A(φU+ψ)M = {w | for all paths α from w there is a w‘ on α such that (w<w‘ w‘ ψM ) w‘‘ (w<w‘‘<w‘ w‘‘ φM )}
(AXφ)M = {w | w‘ (wRw‘ w‘ (wRw‘ w‘φM ))}

4. Actual Calculation How to calculate (EX ψ)M from ψM?
Inverse image construction
How to calculate (AX ψ)M from ψM?
How to calculate E(φU+ψ)M or A(φU+ψ)M from φM and ψM?

5. Inverse reachability calculation

7. Symbolic Representation
Modelchecking algorithm deals with sets of states and with relations (sets of pairs of states)
Need an efficient representation

8. 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 224=108 states
e.g. array of length 10 with 10-bit values  1030 states
Representation of large sets of states?

9. Representation of Sets

10. 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

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

12. Abbreviations
Introduce abbreviations
maximally abbreviated

13. Binary Decision Trees (BDTs)
Elimination of isomorphic subtrees (abbreviations)

14. Binary Decision Diagrams (BDDs)
Elimination of redundant nodes (redundant subformulas) Ite (v,ψ,ψ) by ψ

15. Calculation of BDDs

16. Operations on BDDs Negation: easy (exchange T and F) Falsum: trivial
and, or: Shannon expansion
(φ OP ψ) = x  (φ{x:=T} OP ψ{x:=T})  ¬ x  (φ{x:=} OP ψ{x:=})
(φψ) = (x  (φ{x:=T}  ψ{x:=T}))  (¬ x  (φ{x:=}  ψ{x:=}))
BDD realization?

17. BDD-implies

18. The Influence of Variable Ordering
Heuristics: keep dependent variables close together!