BDD MODEL CHECKING B I N A R Y D E C I S I O N D I A G R A M S - - PowerPoint PPT Presentation

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BDD MODEL CHECKING B I N A R Y D E C I S I O N D I A G R A M S Loc Massin University of Oslo INF5140 / Spring 2017 1 B ASIC M ODEL C HECKING P ROBLEM System describe by states. Basic approach : represent each state individually.

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BDD MODEL CHECKING

B I N A R Y D E C I S I O N D I A G R A M S

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Loïc Massin University of Oslo INF5140 / Spring 2017

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BASIC MODEL CHECKING PROBLEM

System describe by states. Basic approach : represent each state individually. Problem, size of the state space increases exponentially. State Space Explosion.

  • Need too much memory;
  • Need too much time.

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ONE SOLUTION

Symbolic model checking :

  • Idea: represent set of states by Boolean formula over

Boolean variables. f : Booln → Bool

  • Need efficient representation and manipulation for

state sets and transition relation.  Use Binary Decision Diagrams

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BINARY DECISION TREES

  • Directed acyclic graphs.
  • One or two Terminal nodes / Leaves: labelled with

0 or 1;

  • Set of variables nodes u of out-degree two:
  • Non-Terminal nodes: each are labelled with a

variable var(u);

  • Branches / Children: low(v) / high(v), correspond to

assignment of 0 or 1 for the variable in the node

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EXAMPLE OF BDT

a b b c c c c 1 1 1

a ∧ (¬b ∨ c)

Dashed lines denote low-branches, solid lines high-branches

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PROBLEMS

  • Still exponential;
  • Several BDT can verify the same formula.

a b b c c c c 1 1 1 a c b c c b b 1 1 1

a ∧ (¬b ∨ c)

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BASICS BDD PROPERTIES

To move from BDT to BDD: Merge terminal nodes; Ordered BDD (OBDD): Define a variable ordering: on all paths from root to leafs, variables appear in same order, without repetitions (there exists a global ordering of variables).

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EXAMPLE OF OBDD

a b b c c c c 1 1 1

a ∧ (¬b ∨ c) with ordering a < b < c

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REDUCED ORDERED BDD (1)

Uniqueness: no two distinct nodes v and w have the same variable name and low- and high- children.  Merge isomorphic subgraphs; Non-redundant tests: No variable node v has identical low- and high- children.  Remove redundancy.

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MERGE ISOMORPHIC SUBGRAPHS

a b b c c c c 1 1 1

a ∧ (¬b ∨ c)

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REMOVE REDUNDANCY

a b c 1 1 1

a ∧ (¬b ∨ c)

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ROBDD (2)

Canonical (unique) representation of a Boolean formula for a particular variable order: For every function f : Booln → Bool and variable ordering x1 < x2 < · · · < xn, there exists exactly one ROBDD representing this function. Equivalence checking in linear time, and satisfiability checking in constant time.

Most of time, we will refer to ROBDD simply as BDDs.

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SENSITIVITY TO VARIABLE ORDERING (1)

a1 b1 a2 b2 a3

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b3 a1 a2 a2 a3 a3 a3 a3 b1 b1 b1 b1 b2 b2 b3

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(a1 ∧ b1) ∨ (a2 ∧ b2) ∨ (a3 ∧ b3)

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  • Two different variable ordering lead to tow different

ROBDD.

  • Crucial importance in practice, determine the

efficiency of ROBDD-based model checking.

  • Finding the best variable ordering is NP-hard. It

exists several heuristics to approach the problem.

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SENSITIVITY TO VARIABLE ORDERING (2)

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THE ALGORITHM APPLY (1)

  • If Bf and By are two OBDDs, the call apply(op, Bf,

By) computes the OBBD of the formula f op y.

  • Operates recursively on the structure of the two

OBDDs:

  • We start at the root and follow parallel paths on the

two OBDDs to the leaves;

  • Once we arrive at the leaves, we apply the given

boolean operation to the boolean constants 0 and 1 to form the result for that particular path.

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THE ALGORITHM APPLY (2)

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a a b b b 1 1

f(a,b) = a with a>b f(a,b) = b with a>b

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THE ALGORITHM APPLY (2)

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a b 1

f(a,b) = a ∨ b with a>b

a b b 1 a b

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THE ALGORITHM APPLY (2)

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a b 1

f(a,b) = a ∨ b with a>b

a b b 1 a b 1

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THE ALGORITHM APPLY (2)

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a b 1

f(a,b) = a ∨ b with a>b

a b b 1 a b 1 b

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THE ALGORITHM APPLY (2)

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a b 1

f(a,b) = a ∨ b with a>b

a b b 1 a b 1 b

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THE ALGORITHM RESTRICT (1)

  • If Bf is a OBDD, the call restrict(0, x, Bf)

(respectively restrict(1, x, Bf)) the OBDD for f[0/x] (respectively f[1/x]).

  • restrict(0, x, Bf)
  • For each node v labeled with x:

Incoming edges are redirected to low(v); Node v is removed.

  • restrict(1, x, Bf)
  • As above but redirected to high(v).

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THE ALGORITHM RESTRICT (1)

a1 b1 a2 b2 a3

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b3

Bf

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a1 b1 a2 b2

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restrict(0, a3, Bf)

a1 b1 a2 b2

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b3

restrict(1, a3, Bf)

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REFERENCES

  • Henrik Reif Andersen, An Introduction to Binary

Decision Diagrams. The IT University of Copenhagen, Fall 1999

  • Alessandro Artale, Formal Methods Lecture VI,

Binary Decision Diagrams. http://www.inf.unibz.it/~artale/FM/slide7.pdf (visited on 05.17.2017)

  • A. Pnueli, Symbolic Model Checking.

http://www.cs.nyu.edu/courses/spring07/G22.3033- 002/lecture6_h4.pdf (visited on 05.17.2017)

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