Binary Decision Trees

• Definition: A binary decision tree is a tree whose non-terminal nodes are labelled with boolean variables x, y, z, … and whose terminal nodes are labelled with either 0 or 1. Each non- terminal node has two edges, one dashed line and one solid line.

y y

1 0 0 0 1 Binary Decision Trees (continued)

• Definition: Let T be a finite binary decision tree. Then T determines a unique Boolean function of the variables in non-terminal nodes, in the following way. Given an assignment of 0s and 1s to the Boolean variables occurring in T, we start at the root of T and take the dashed line whenever the value at the current node is 0; otherwise, we travel along the solid line. The function value is the value of the terminal node we reach.

2 An example

• f(x, y) = ¬(x + y) x

y y

1 0 0 0

3 Some optimization techniques Binary Decision Diagrams (BDDs)

y y

1 0

4 Some optimization techniques (continued)

1 0

5 A BDD with duplicated sub-BDDs

x x

y y y y

0 1

6 After removal of one of the duplicate y-nodes

x x

y y y

0 1

7 After removal of another duplicate y-nodes and then a redundant x-decision point

y y

0 1

8 Three ways of reducing a BDD

• Removal of a duplicate terminals. • Removal of a redundant tests. • Removal of duplicate non-terminals.

9 Directed Acyclic Graphs (DAGs)

• A directed acyclic graph(dag) is a directed graph that does not have any cycles. A node of a dag is initial if there are no edges pointing to that node. A node is terminal if there are no edges out of that node.

10 Binary Decision Diagram (BDD)

• Definition: A binary decision diagram(BDD) is a finite dag with a unique initial node, where all terminal nodes are labeled with 0 or 1 and all non-terminal nodes are labeled with a Boolean variable. Each non-terminal node has exactly two edges from that node to others: one labeled 0 and one labeled 1 ( we represent them as a dashed line and a solid line, respectively). A BDD is said to be reduced if none of the earlier three optimization techniques can be applied (i.e., no more reductions are possible).

11 Ordered Binary Decision Diagrams (OBDDs)

• Definition: Let [x 1, …, x n] be an ordered list of variables without duplications and let B be a BDD all of whose variables occur somewhere in

the list. We say B has the ordering [x 1, …, x n] if all variables labels of B occur in that list and, for

every occurrence of x j followed by x j along any path in B, we have i < j. • An orderedBDD (OBDD) is a BDD which has an ordering for some list of variables.

12 Examples of a BDD (left) and an OBDD (right)

x x

y z y y

x y x z

1 0 1 0

13 Reduced OBDDs as Canonical forms

• Theorem: The reduced OBDD representing a given function f is unique.

That is to say, let B 1 and B 2 be two reduced OBDDs with a compatible

variable orderings. If B 1 and B 2 represent the same boolean function, then they have identical structure.

14 An OBDD for the even parity function for four bits

y y

z z

w w

1 0

15 The OBDD for (x + y) . (z + w) . (u + v) with variable ordering [x, y, z, w, u, v]

y z

u v

0 1

16 The OBDD for (x + y) . (z + w) . (u + v) with variable ordering [x, z, u, y, w, v]

x z z

u u u u

y y y y

w w

0 1

17 The importance of canonical representation • Absence of redundant variables. • Test for semantic equivalence. • Test for validity. • Test for implication. • Test for satisfiability.

18 The algorithm reduce in general terms

• If the ordering of B is [x 1, x 2, …, x l], then B has at most l +1 layers. The algorithm reduce now traverses B layers in a bottom-up fashion (beginning with the terminal nodes). In traversing B, it assigns an integer label id(n) to each node n of B, in such a way that subOBDDs with root nodes n and m denote the same boolean function if, and only if, id(n) equals id(m).

19 The algorithm reduce

• Definition: Given a non-terminal node n in a BDD, we define lo(n) to be the node pointed to via the dashed line from n. Dually, hi(n) is the node pointed to via the solid line from n. • Let us assume that reduce has already assigned integer labels to all nodes of a layer > i (i.e., all terminal nodes

and x j-nodes with j > i). We describe how nodes of layer i (i.e., x i-nodes) are being handled as follows.

20 The algorithm reduce (continued)

Given an x i-node n, there are three ways in which it may get its label: • If the label id(lo(n)) is the same as id(hi(n)), then we set id(n) to be that label • If there is another node m such that n and m have the

same variable x i, and id(lo(n)) = id(lo(m)) and id(hi(n)) = id(hi(m)), then we set id(n) to be id(m). • Otherwise, we set id(n) to the next unused integer label.

21 An example execution of the algorithm reduce

#4 #4 x1 x1 #3 #2 x2 #3 x2 x2

#2 #2 #2 x3 x3 x3

0 1 0 1 0 1 #0 #1 #0 #1 #0 #1

22 restrictions of f

• Definition: Let f be a Boolean formula and x a variable. We denote by f[0/x] the Boolean formula obtained by replacing all occurrences of x in f by 0. The formula f[1/x] is defined similarly. The f[0/x] and f[1/x] are called restrictionsof f.

23 The algorithm restrict in general terms

• Given an OBDD B f representing a boolean formula f, the call restrict (0, x, B f) computes the reduced OBDD representing f[ 0/x] using the same variable ordering as

Bf.

24 The algorithm restrict

• The algorithm restrict(0, x, B f) works as follows. For each node n labelled with x, incoming edges are redirected to lo(n) and n is removed. Then we call reduce on the

resulting OBDD. The call restrict(1, x, B f) proceeds similarly, only we now redirect incoming edges to hi(n) .

25 Examples of restrictions of f

• OBDD for f • OBDD for f[0 | x 3] OBDD for f[0 | x 3]

26 The algorithm apply in general terms

• Given OBDDs B f and B g for Boolean formulas f and g, respectively, the call

apply (op, B f, B g) computes the reduced OBDD of the Boolean formula f op g , where op denotes any function from {0, 1} X {0, 1} to {0, 1}.

27 The algorithm apply

The algorithm operates recursively on the structure of the two OBDDs: • Let v be the variable highest in the ordering (=

leftmost in the list) which occurs in B f or B g. • Split the problem into sub-problems for v being 0 and v being 1 and solve recursively; • At the leaves, apply the Boolean operation op directly.

28 Shannon expansion

• Lemma: For all Boolean formulas f and all Boolean variables x (even those not occurring in f) we have f ≡ ¬x . f[0/x] + x . f[1/x]. • The function apply is based on the above lemma for f op g :

f op g ≡ ¬x i . (f[0/x i] op g[0/x i]) + x i . (f[1/x i] op g[1/x i]).

29 Time complexity

Algorithm Input OBDD(s) Output OBDD Time complexity

reduce B reduced B O(|B| . Log |B|)

apply Bf op g (reduced) O(|B f| . |B g|) Bf, B g (reduced )

Bf[0/x] or B f[1/x] restrict Bf (reduced) (reduced) O(|B f| . Log |B f|)

30 Symbolic Model Checking

• Model checking using OBDDs is called symbolic model checking. • The term emphasizes that individual states are not represented; rather, sets of states are represented symbolically, namely, those which satisfy the formula being checked.

31 Representing subsets of the set of states

• Let S be a finite set of states. The task is to represent the various subsets of S as OBDDs. • The way to do this in general is to assign to each element s ε S a ε unique vector of boolean values (v 1, v 2, …, v n), each v i {0, 1}. Then, we represent a subset T by the boolean function f T which ε maps (v 1, v 2, …, v n) onto 1 if s T and maps it onto 0 otherwise. • Note that 2n-1 < |S| ≤ 2n . n → • The function f T :{0, 1} {0, 1} which tells us, for each s, represented by (v 1, v 2, …, v n), whether it is in the set T or not, is called the characteristic functionof T.

32 An example of a Kripke structure (S, R, L)

• S = {s0, s1,s2} • R = {(s0, s1), (s1, s2), (s2, s0), (s2, s2)} • L(s0) = {x1} • L(s 1) = {x 2} • L(s2) = Ø s0 x1 x2 s1

33 Representation of subsets of states of the previous example

set of states representation by boolean representation by boolean values function

Ø 0

{s0} (1,0) x1 .¬x2

{s1} (0,1) ¬x1 .x2

{s2} (0,0) ¬x1 .¬x2

{s0, s1} (1,0), (0,1) x1 .¬x2 + ¬x1 .x2

{s0, s2} (1,0), (0,0) x1 .¬x2 + ¬x1 .¬x2

{s1, s2} (0,1), (00) ¬x1 .x2 + ¬x1 .¬x2

{s0, s1, s2} (1,0), (0,1), (00) x1 .¬x2 + ¬x1 .x2 + ¬x1 .¬x2 34 Two OBDDs for the set {s0, s1} of previous example

x1 x1

x2 x2 x2

0 1 0 1

x1 .¬x2 + ¬x1 .x2 x1 + x2

35 Representing transition relations

• The transition relation R of a Kripke structure (S, R, L) is a subset of S × S. So, as a subset of a given set, it may be represented as an OBDD.

• Thus, a transition (s, s’) ε R can be represented by a pair

of Boolean vectors (v 1, …, v n), (v’ 1, …, v’ n), where (v 1, …, vn) represents s and (v’ 1, …, v’ n) represents s’.

36 The Boolean function representation for the transition relation of the previous example

• fR = x1 .¬x2 . ¬x’1 .x’2 + ¬x1 .x2 .¬x’1 .¬x’2 + ¬x 1 .¬x 2 . ¬x’ 1 .x’ 2 + ¬x 1 .¬x 2 .¬x’ 1 .¬x’ 2

37 The truth table for the transition relation of the previous example with ordering [x1, x2, x’1, x’2]

x1 x2 x’1 x’2 R

0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 1 _ 0 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 1 _ 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 _ 1 1 0 0 _ 1 1 0 1 _ 1 1 1 0 _ 1 1 1 1 _

38 The truth table for the transition relation of the previous example with ordering [x1, x’1, x2, x’2]

x1 x’1 x2 x’2 R 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 _ 0 1 1 0 0 0 1 1 1 _ 1 0 0 0 0 1 0 0 1 1 1 0 1 0 _ _ 1 0 1 1 1 1 0 0 0 1 1 0 1 _ 1 1 1 0 _ 1 1 1 1 _

39 An OBDD for the transition relation of the previous example.

X’1 X’1

X’2 X’2

0 1

40 Computation of the Satisfaction Sets