Binary Decision Diagrams

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1 Binary Decision Diagrams Hao Zheng Department of Computer Science and Engineering University of South Florida Tampa, FL Phone: (813) Fax: (813) Hao Zheng (CSE, USF) Comp Sys Verification 1 / 44

2 1 Binary Decision Tree 2 Ordered Binary Decision Diagram 3 From BDT to BDDs 4 Variable Ordering 5 Logic Operations by BDDs Hao Zheng (CSE, USF) Comp Sys Verification 2 / 44

3 Binary Decision Diagrams Binary decision diagrams (BDDs) are graphs representing Boolean functions. They can be made canonical. They can be very compact for many applications. They are important since many applications can be converted to sequences of Boolean operations. References: R. Bryant, Graph-Based Algorithms for Boolean Function Manipulation, IEEE Transactions on Computers, R. Bryant Symbolic Boolean Manipulation with Ordered Binary Decision Diagrams, ACM Computing Surveys, Textbook, Hao Zheng (CSE, USF) Comp Sys Verification 3 / 44

4 Representing Switching Functions Truth Tables Satisfiability and equivalence check: easy; boolean operations also easy. Very space inefficient: 2 n entries for n variables. Disjunctive Normal Form (DNF) Satisfiability is easy: find a disjunct without complementary literals. Negation and conjunction complicated. Conjunctive Normal Form (CNF) Satisfiability problem is NP-complete (Cook s theorem). Negation and disjunction complicated. Hao Zheng (CSE, USF) Comp Sys Verification 4 / 44

5 Representing Switching Functions representation compact? sat equ truth table never hard hard hard hard hard DNF sometimes easy hard hard easy hard CNF sometimes hard hard easy hard hard propositional formula often hard hard easy easy easy reduced ordered binary decision diagram often easy easy medium medium easy Provided appropriate implementation techniques are used. Hao Zheng (CSE, USF) Comp Sys Verification 5 / 44

6 Symbolic xample Symbolic Encoding: Representation An Exampleof a Transition System /0 s 0 s 1 a s 3 s 2 {a,b } Switching function: (x 1, x itching function: (x 1,x {z } 2,x1,, x 0 }{{} 1, x 2) = 1 if and only if s 2)=1 if and only if s! s 0 s }{{} s {z } s s s (x 0 1,, x 1, x 2 ) = ( x 1 x 1 x 2 ) (x 1,,,x0 2 )= ( ( x x 1 ^ x 2 ^ x x 0 ^ x0 _ ( x ^ ^ 0 1 x 2 ) ( x 1 x 1 ^ x0 2 ) _ ( x 1 ^ ^ x 2 )... 1 ^ x0 2 ) _ (x... 1 x _ (x 1 ^ ^ x 2 ) 1 ^ x0 2 ) Hao Zheng (CSE, USF) Comp Sys Verification 6 / 44 b

7 Contents 1 Binary Decision Tree 2 Ordered Binary Decision Diagram 3 From BDT to BDDs 4 Variable Ordering 5 Logic Operations by BDDs Hao Zheng (CSE, USF) Comp Sys Verification 7 / 44

8 Transition Relation as a BDT Binary Decision Tree: Example x The BDT for function f on x = { x 1,..., x n } has depth n. Every A BDT node representing is labeled with for our a variable. example using ordering x 1 < < < x0 2. Every node labeled with x i has two outgoing edges. 0-edge: x i = 0 (dashed) and, 1-edge: x i = 1 (solid). Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 110 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 8 / 44

9 Binary Decision Transition Tree: Example Relation as a BDT x Every non-terminal node n has two successor nodes. low(n): A BDT representing the node at the forend our of example the 0-edge using of ordering node n. x 1 < < < x0 2. high(n): the node at the end of the 1-edge of node n. Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 110 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 8 / 44

10 Binary Decision Transition Tree: Example Relation as a BDT x The edge labelings of a path from the root to a terminal is an evaluation A BDT representing s = [x 1 = b 1 for,.. our., xexample m = b m using ], where ordering b i x 1 {0, < x1}. 2 < < x0 2. The labeling of the terminal node is the output of f(s). Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 110 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 8 / 44

11 Transition Relation as a BDT Binary Decision Tree: Example x The subtree of node v at level i for variable ordering x 1 <... < x n represents: A BDT representing for our example using ordering x 1 < < < x0 2. f v = f x1 =b 1,...,x i 1 =b i 1 which is a switching function over { x i,..., x n } and where x 1 = b 1,..., x i 1 = b i 1 are the decisions made along the path from root to node v. Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 110 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 8 / 44

12 Binary Decision Tree The BDT for function f on Var = {z 1,..., z m } has depth m with outgoing edges for node at level i stand for z i = 0 (dashed) and z i = 1 (solid). For evaluation s = [z 1 = b 1,..., z m = b m ], f(s) is the value of the leaf reached by traversing the BDT from the root using branch z i = b i. The subtree of node v at level i for variable ordering z 1 <... < z m represents: f v = f z1 =b 1,...,z i 1 =b i 1 which is a switching function over {z i,..., z m } and where z 1 = b 1,..., z i 1 = b i 1 are the decisions made along the path from root to node v. Hao Zheng (CSE, USF) Comp Sys Verification 9 / 44

13 Considerations on BDTs BDTs are a different form of truth tables. BDTs are not compact: A BDT for switching function f on n variables has 2 n leafs. The size of a BDT does not change if the variable order changes. They are as space inefficient as truth tables! BDTs contain a lot of redundancy: All leafs with value one (zero) could be collapsed into a single leaf. A similar scheme could be adopted for isomorphic subtrees. Two graphs rooted at nodes u and v are isomorphic, denoted as u v when both following conditions hold. value(u) = value(v) if u and v are terminals. low(u) low(v) high(u) high(v), otherwise. Hao Zheng (CSE, USF) Comp Sys Verification 10 / 44

14 Contents 1 Binary Decision Tree 2 Ordered Binary Decision Diagram 3 From BDT to BDDs 4 Variable Ordering 5 Logic Operations by BDDs Hao Zheng (CSE, USF) Comp Sys Verification 11 / 44

15 Ordered Binary Decision Diagram (OBDD) OBDDs rely on compactions of BDT representations. Idea: skip redundant fragments of BDT representations. Collapse subtrees with all terminals having same value. Identify nodes with isomorphic subtrees. This yields directed acyclic graphs with outdegree two. Inner nodes are labeled with variables. Leafs are labeled with function values (zero and one). A unique variable ordering is followed by every path. Each variable is assigned an unique index. Each BDD node v has an index index(v) which is the index of the variable labeled in v. index(v) < index(low(v)) if low(v) is a non-terminal, index(v) < index(high(v)) if high(v) is a non-terminal. Hao Zheng (CSE, USF) Comp Sys Verification 12 / 44

16 Transition Relation as a BDT Transition Relation as a BDT x A BDT A BDT representing for forour example using orderingxx 1 < < x < < x 1 < x 2. Chris Hao J. Zheng Myers (Lecture (CSE, USF) 6: CTL) Verification Comp of Cyber-Physical Sys Verification Systems / 155 / 44

17 Transition Relation as a BDD Transition Relation as an ROBDD x An example ROBDD representing for our example using x 1 < < < x0 2. A BDT representing for our example using ordering x 1 < < x 1 < x 2. Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 120 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 14 / 44

18 OBBDs and Boolean Functions Let be a variable ordering for Var where = (z 1,..., z m ). Every OBDD is defined wrt a given variable ordering. The nodes in every path are labeled with variables in the order as in. A terminal node represents a constant Boolean function either 1 or 0. For a non-terminal node n labeled with z representing a Boolean function f n, its two successor nodes represent Boolean functions: Node at the end of the 0-edge (low(n)): f n z=0. Node at the end of the 1-edge (high(n)): f n z=1. Therefore, f n = z f n z=0 z f n z=1 A OBDD is reduced (i.e. ROBDD) if for every pair (v, w) of nodes, v w implies f v f w. Hao Zheng (CSE, USF) Comp Sys Verification 15 / 44

19 Universality and Canonicity Theorem [Fortune, Hopcroft & Schmidt, 1978] Let x be a finite set of Boolean variables and a variable ordering for x. (a) For each switching function f for x there exists a -ROBDD OBDD with f OBDD = f. (b) For any -ROBDDs G and H with f G = f H, G and H are isomorphic, i.e., agree up to renaming of the nodes. ROBDDs are canonical for a fixed variable ordering. Hao Zheng (CSE, USF) Comp Sys Verification 16 / 44

20 The Importance of Canonicity Absence of redundant vertices: If f B does not depend on x i, ROBDD B does not contain an x i node. Test for equivalence: f(x 1,..., x n ) g(x 1,..., x n )? Generate ROBDDs B f and B g, and check isomorphism. Test for validity: f(x 1,..., x n ) = 1? Generate ROBDD B f and check whether it only consists of a 1-leaf. Test for implication: f(x 1,..., x n ) g(x 1,..., x n )? Generate ROBDD B f g and check if it just consists of a 0-leaf. Test for satisfiability: f is satisfiable if and only if B f has a reachable 1-leaf. Hao Zheng (CSE, USF) Comp Sys Verification 17 / 44

21 Contents 1 Binary Decision Tree 2 Ordered Binary Decision Diagram 3 From BDT to BDDs 4 Variable Ordering 5 Logic Operations by BDDs Hao Zheng (CSE, USF) Comp Sys Verification 18 / 44

22 Reducing OBDDs Generate an OBDD (or BDT) for a boolean expression, then reduce by means of a recursive descent over the OBDD. Elimination rule: If low(v) = high(v) = w, eliminate v and redirect all incoming edges to v to node w. Isomorphism rule: If v w are roots of isomorphic subtrees, remove v, and redirect all incoming edges to v to node w. (Special case) Combine all 0/1-leaves, redirect all incoming edges. Hao Zheng (CSE, USF) Comp Sys Verification 19 / 44

23 From BDT to Transition ROBDD Relation as a BDT x An OBDD representing for our example using ordering x 1 < < x A BDT representing for our example using ordering x 1 < < 1 < < x 2. x0 2. Hao Zheng (CSE, USF) Comp Sys Verification 20 / 44

24 Transition Relation as a OBDD From BDT to ROBDD x After isomorphism rule Next, Isomorphism elimination rule. rule Hao Zheng (CSE, USF) Comp Sys Verification 21 / 44

25 From BDT Transition to ROBDDRelation as a OBDD x Next, isomorphism rule Elimination rule. Hao Zheng (CSE, USF) Comp Sys Verification 22 / 44

26 From BDT Transition to ROBDD Relation as a OBDD x Next, elimination rule Isomorphism rule. Hao Zheng (CSE, USF) Comp Sys Verification 23 / 44

27 Transition Relation as a OBDD From BDT to ROBDD x Next, isomorphism rule Elimination rule. Hao Zheng (CSE, USF) Comp Sys Verification 24 / 44

28 Transition Relation as a OBDD From BDT to ROBDD x Next, isomorphism rule Isomorphism rule. Hao Zheng (CSE, USF) Comp Sys Verification 25 / 44

29 From BDT to ROBDD Transition Relation as an ROBDD x Isomorphism rule. Final reduced BDD Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 132 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 26 / 44

30 Contents 1 Binary Decision Tree 2 Ordered Binary Decision Diagram 3 From BDT to BDDs 4 Variable Ordering 5 Logic Operations by BDDs Hao Zheng (CSE, USF) Comp Sys Verification 27 / 44

31 Variable Ordering Variable Ordering x 1 x (a) ordering x 1 < < < (b) ordering x 1 < 0 < 0 < 0 Hao Zheng (CSE, USF) Comp Sys Verification 28 / 44

32 Variable Ordering and Size of BDDs The size of the ROBDD crucially depends on the variable ordering. # nodes in ROBDD OBDD = # of -consistent co-factors of f. Some switching functions have linear and exponential ROBDDs. e.g., the addition function, or the stable function. Some switching functions only have polynomial ROBDDs. This holds, e.g., for symmetric functions. Examples f(...) = x 1... x n, or f(...) = 1 iff k variables x i are true. Some switching functions only have exponential ROBDDs. This holds, e.g., for the middle bit of the multiplication function. Hao Zheng (CSE, USF) Comp Sys Verification 29 / 44

33 The Function Stable with Exponential ROBDD The Function Stable with Exponential ROBDD x 1 x 3 x 3 x 3 x 3 y 1 y 1 y 1 y 1 y 1 y 1 y 1 y 1 y 2 y 2 y 2 y 2 y 3 y 3 1 The ROBDD of f stab (x,y)=(x 1 $ y 1 ) ^... ^ (x n $ y n ) The has ROBDD 3 2 n 1of vertices f stab (x, under y) ordering = (x 1 x 1 <...<x y 1 ) n. <.. y 1 <...<y (x n n. y n ) has 3 2 n 1 vertices under ordering x 1 <... < x n < y 1 <... < y n. Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 135 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 30 / 44

34 The Function Stable with Linear ROBDD The Function Stable with Linear ROBDD x 1 y 1 y 1 y 2 y 2 x 3 y 3 y 3 1 The ROBDD of f stab (x,y)=(x 1 $ y 1 ) ^... ^ (x n $ y n ) The ROBDD has 3 n + of2fvertices stab (x, under y) = ordering (x 1 x 1 y< 1 ) y 1 <...<x... n < (x y n. y n ) has 3 n + 2 vertices under ordering x 1 < y 1 <... < x n < y n. Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 136 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 31 / 44

35 Another Function with an Exponential ROBDD Another Function with an Exponential ROBDD z1 z2 z2 z3 z3 z3 z3 y1 y1 y1 y1 y2 y2 y3 0 1 ROBDD for f 3 (z,y) =(z 1 ^ y 1 ) _ (z 2 ^ y 2 ) _ (z 3 ^ y 3 ) for the variable ordering z 1 < z 2 < z 3 < y 1 < y 2 < y 3. ROBDD for f 3 ( z, y) = (z 1 y 1 ) (z 2 y 2 ) (z 3 y 3 ) Chrisfor J. Myers the (Lecture variable 6: CTL) ordering Verification ofzcyber-physical 1 < z 2 Systems < z 3 < y 1 < y 2 < y / 155 Hao Zheng (CSE, USF) Comp Sys Verification 32 / 44

36 An Optimal Linear ROBDD An Optimal Linear ROBDD z 1 y 1 z 2 y 2 z 3 ROBDD for f 3 ( ) = (z 1 y 1 ) (z 2 y 2 ) (z 3 y 3 ). ROBDD for f 3 ( ) =(z 1 ^ y 1 ) _ (z 2 ^ y 2 ) _ (z 3 ^ y 3 ). For ordering z 1 < y 1 < z 2 < y 2 < z 3 < y 3. For ordering z 1 < y 1 < z 2 < y 2 < z 3 < y 3. Asall all variables are are essential, essential, this ROBDD this ROBDD is optimal. is optimal. For no variable ordering, a smaller ROBDD exists. For no variable ordering, a smaller ROBDD exists. y Chris J. Myers (Lecture 6: CTL) Verification of Cyber-Physical Systems 138 / 155 Hao Zheng (CSE, USF) Comp Sys Verification 33 / 44

37 The Multiplication Function Consider two n-bit integers: Let b n 1 b n 2... b 0 and c n 1 c n 2... c 0 where b n 1 is the most significant bit, and b 0 the least significant bit. Multiplication yields a 2n-bit integer: The ROBDD OBDD fn 1 has at least 1.09 n vertices where f n 1 denotes the (n 1)-st output bit of the multiplication. Hao Zheng (CSE, USF) Comp Sys Verification 34 / 44

38 Optimal Variable Ordering The size of ROBDDs is dependent on the variable ordering. Is it possible to determine such that the ROBDD has minimal size? To check whether a variable ordering is optimal is NP-hard. Polynomial reduction from the 3SAT problem. [Bollig & Wegener, 1996] There are many switching functions with large ROBDDs: For almost all switching functions the minimal size is in Ω( 2n n ). How to deal with this problem in practice? Guess a variable ordering in advance. Rearrange the variable ordering during the ROBDD manipulations. Not necessary to test all n! orderings, best known algorithm in O(3 n n 2 ). Hao Zheng (CSE, USF) Comp Sys Verification 35 / 44

39 Dynamic Re-ordering Finding an optimal ordering is NP-hard. Static ordering does not work well across different applications, or for BDDs that need to be transformed during different stages of an application. Automated dynamic re-ordering rearranges the variable orders periodically to reduce the size of BDDs. Rudell s sifting is widely used. Try moving a variable to all other positions, leaving the others fixed. Then place variable in the position that minimizes BDD size. Do this for all variables. Hao Zheng (CSE, USF) Comp Sys Verification 36 / 44

40 Dynamic Re-ordering Greatly improved effectiveness of BDDs. It is usually performed in the background. It may slow down the performance. BDD operations stop when re-ordering is activated. It makes a difference between success and failure in complete an application. Some functions are inherently hard, e.g. outputs of integer multiplier. Hao Zheng (CSE, USF) Comp Sys Verification 37 / 44

41 Contents 1 Binary Decision Tree 2 Ordered Binary Decision Diagram 3 From BDT to BDDs 4 Variable Ordering 5 Logic Operations by BDDs Hao Zheng (CSE, USF) Comp Sys Verification 38 / 44

42 Logic Operations on BDDs Restriction f[b/x]: replacing variable x with a value 0 or 1. if b = 0, direct all incoming edges of node labeled with x to low(v), or if b = 1, direct all incoming edges of node labeled with x to high(v), remove node x and its outgoing edges. The result of restriction is a cofactor of f. a a b c c 0 1 f 0 1 f[1/b] Hao Zheng (CSE, USF) Comp Sys Verification 39 / 44

43 Logical Operations on BDDs (cont d) Negation is a constant time operation with OBDDs. Swap terminal nodes. The binary operations are based on the Shannon expansion. f g = x (f[0/x] g[0/x]) x (f[1/x] g[1/x]). where is some binary logic operator. Both BDDs must have the same variable ordering. The new BDD for f g is constructed as follows. The root of f g is the root of f or g with the smaller index. For any node in f g, low(v) = f[0/x] g[0/x], and high(v) = f[1/x] g[1/x] Repeat the above step for low(v) and high(v) until either of them becomes terminal. Reduce the constructed BDD to make it canonical. Hao Zheng (CSE, USF) Comp Sys Verification 40 / 44

44 Logical Operations on BDDs Example f = (a ^ c) f _ g a g = b ^ c a f a=1 _ g b b f a=1 c c f a=1 _ g b=0 f a=1 _ g b=1 c c Hao Zheng (CSE, USF) Comp Sys Verification 41 / 44

45 Logical Operations on BDDs Example f _ g a f a=1 _ g b f _ g a f a=1 _ g b f a=1 _ g b=0 f a=1 _ g b=1 c c f a=1 _ g b=0 c Hao Zheng (CSE, USF) Comp Sys Verification 42 / 44

46 Variants of BDDs Various kinds of BDDs for compactness or different applications. Compactness Multi-rooted BDDs, free BDDs, partitioned OBDDs, etc Arithmetic operations Multi-terminal BDDs (ADDs), edge-valued BDDs, binary moment diagrams (BMDs), etc. Hao Zheng (CSE, USF) Comp Sys Verification 43 / 44

47 Implementation: Shared OBDDs A shared -OBDD is an OBDD with multiple roots. 0 1 Shared OBDD representing z 1 z }{{} 2, z 2, z }{{} 1 z 2 and z }{{} 1 z }{{} 2 f 1 f 2 f 3 f 4. Main underlying idea: combine several OBDDs with same variable ordering such that common -consistent co-factors are shared. Hao Zheng (CSE, USF) Comp Sys Verification 44 / 44

Binary Decision Diagrams

Binary Decision Diagrams Hao Zheng Department of Computer Science and Engineering University of South Florida Tampa, FL 33620 Email: zheng@cse.usf.edu Phone: (813)974-4757 Fax: (813)974-5456 Hao Zheng