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. x 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) x y y 1 0 4 Some optimization techniques (continued) x y 1 0 5 A BDD with duplicated sub-BDDs z x x y y y y 0 1 6 After removal of one of the duplicate y-nodes z x x y y y 0 1 7 After removal of another duplicate y-nodes and then a redundant x-decision point z x 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 ordered BDD (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 x 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] x y z w 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 v 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 restrictions of 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 function of 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 s2 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.