Binary Decision Diagrams

Presenter: Haroon Rashid

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Contents

Motivation

Binary Decision Diagrams (BDDs)

Ordered BDDs

Reduced OBDD algorithms

reduce

apply

restrict

exists

Applications of OBDDs

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Motivation: Boolean Function Representation

Boolean functions are mostly represented by truth tables and propositional formulas

Representation with truth table:

Space inefficient. 100 variables results in 2100 lines.

Checking satisfiability and equivalence is also inefficient.

Representation with propositional formula:

Better than truth table in terms of compact representation.

Deciding whether two prop. formulas f and g denote same Boolean function is expensive.

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Binary Decision Diagrams (BDDs)

Another way of representing Boolean functions. BDDs are trees with terminal nodes labeled with either 0 or 1 and non-

terminal nodes labeled with Boolean variables x, y, z…

Each non-terminal node has two edges One dashed line and one solid line

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x=0 x=1

Solid line: Corresponds when variable value = 1 Dashed line: Corresponds when variable value = 0

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BDD functions

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f(x, y)=?

x y f(x, y)

0 0 1

0 1 0

1 0 0

1 1 1

f(x, y)= (x+y)

Cons: f depending on n variables will have at least 2n+1 -1 nodes. This shows that this representation is not compact. But, we can exploit its redundancy in order to improve it.

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BDD Contraction

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• [C1]: Removal of duplicate terminals

• [C2]: Removal of redundant tests

• [C3]: Removal of duplicate non-terminals

Duplicate Terminals Duplicate Non-terminals Terminals 11/21/2014

BDD Contraction

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(a) Duplicated BDDs (b) Removal of duplicate y (c) Removal of redundant decision point x

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Satisfiability, Validity and operations (.,+)

Satisfiability: A BDD represents satisfiable function if a 1-terminal node is reachable from the root along a consistent path*.

Validity: Same as satisfiability but here we check no 0-terminal is reachable by consistent path.

. () operation on BDDs Bf and Bg : Resulting BDD obtained by replacing all 1-terminals with by Bg.

+ () operation on BDDs Bf and Bg : Resulting BDD obtained by replacing all 0-terminals with by Bg.

8 * A consistent path is one which, for every variable, has only dashed lines or only solid lines leaving nodes labeled by that variable.(OR we can’t assign a variable the values 0 & 1 simultaneously)

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BDD Cons

• BDDs with multiple occurrences of a Boolean variable along a path make

it inefficient.

Checking equivalence of two BDDs is still difficult. See figure (a) and (b)

9 (a) (b)

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Ordered BDDs

Ordered BDDs ensure the variables appear in the same order along all paths from the root to the leaves.

There are not multiple occurrences of any variable along a path.

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Unordered BDD Ordered BDD 11/21/2014

Ordered BDDS cont..

With OBDDs we cannot have more than one BDD representing the same function, i.e., one function -> one OBDD.

Follows that checking equivalence with OBDDs is immediate.

OBDDS for f = (x1+x2 ). (x3+x4 ). (x5+x6 )

11 Ordering: [x1, x2, x3, x4, x5, x6 ] Ordering: [x1, x3, x5, x2, x4, x6 ]

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Reduced OBDDs Algorithm: reduce

Notation: Given a non-terminal node, n, then lo(n) denotes the node pointed via the dashed line, while hi(n) denotes the node pointed by the solid line.

Given an OBDD, reduce proceeds bottom-up assigning an integer label, id(n), to each node:

1. Assign label 0 to all 0-terminals and label 1 to all 1-terminals. Given now a non-terminal node for xi, say n, then:

2. If id(lo(n)) = id(hi(n)), then id(n) = id(lo(n));

3. If there is another node for xi, say m, such that id(lo(n)) = id(lo(m)) and id(hi(n)) = id(hi(m)), then, id(n) = id(m);

4. Otherwise we set id(n) to the next unused integer.

Final Step: Collapsing nodes with the same label and redirecting edges accordingly with the node collapsing. Example on next slide

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reduce

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lo(x1) hi(x1)

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Algorithm apply

apply is used to implement operations like ., + on Boolean functions

Definition: Let β be a formula and x a variable. We denote by β[0/x] (β[1/x]) the formula obtained by replacing all occurrences of x in β by 0 (1).

Lemma [Shannon Expansion]: Let β be a formula and x a variable, then:

β = x . β[0/x] + x. β[1/x]

The function apply is based on Shannon Expansion for α op β :

α op β = x. (α .[0/x] op β[0/x] ) + x. (α .[1/x] op β[1/x] )

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Algorithm apply

apply (op, Bj, By) proceeds from the roots downward.

Let rj, ry be the roots of Bj, By respectively:

If both rj, ry are terminal nodes, then,

apply(op, Bj, By) = B(rj op ry)

If both roots are xi-nodes, then create an xi-node with a dashed line to apply(op, Blo(rj), Blo(ry)) and a solid line to apply(op, Bhi(rj), Bhi(ry))

If rj is an xi-node, but ry is a terminal node or an xj-node with j > i (i.e., y[0/xi] ≡ y[1/xi] ≡ y), then create an xi-node with dashed line to

apply(op, Blo(rj), By) and solid line to apply(op, Bhi(rj), By)

If ry is an xi-node, but rj is a terminal node or an xj-node with j > i, is handled as above.

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Algorithm apply cases

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1

2

3

4

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Algorithm apply cont.

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a) apply(+, Bj, By) b) Recursive call structure of apply

c)Result of apply(+, Bj, By) 11/21/2014

Algorithm restrict

Given an OBDD Bf representing Boolean formula f, algorithm restrict(0, x, Bf) computes reduced OBDD representing f [0/x] as:

Bf[0/x] = restrict(0, x, Bf).

For each node n labeled with x, then:

1. Incoming edges are redirected to lo(n)

2. n is removed.

Bf[1/x] = restrict(1, x, Bf). Same as above, but redirecting incoming edges to hi(n)

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Algorithm exists [apply+restrict]

exists is used to perform operation on Boolean function while relaxing constraint on some variable.

We write (∃x. ƒ) for the Boolean function ƒ with the constraint on x relaxed and it can be expressed as:

∃x. ƒ = ƒ [0/x]+ ƒ[1/x]

i.e., there exists x on which the constraint is relaxed

The exists algorithm can be implemented in terms of algorithms apply and restrict as

∃x. ƒ = apply(+, restrict(0, x, Bf), restrict(1, x, Bf))

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Algorithm exists example

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Bf

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OBDDs Application

Symbolic Model Checking

Model checking using OBDDs is called symbolic model checking

OBDDs allow systems with a large space to be verified.

Normally, Labeling algorithm takes a CTL formula and returns a set of states manipulating intermediate set of states.

The algorithm is changed by storing set of states as OBDD's and then manipulating them.

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Symbolic Representation

Three states X1, X2, X3:

{000,001,010,011} represented as first bit false: ┐X1

With five state variables X1, X2, X3, X4, X5:

{00000,00001,00010,00011,00100,00101,00110, 00111, ,…, 01111} still represented as first bit false: ┐X1

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Conclusion

BDDs and OBDDs

Application of various algorithms

Application of OBDDs in model checking

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References

Logics in Computer Science, Modelling and Reasoning about Systems, Chapter 6, M. Huth and M. Ryan.

Slides of Alessandro Artale, http://www.inf.unibz.it/~artale/FM/slide7.pdf

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Questions ?

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