Binary Decision Diagrams. Graphs. Boolean Functions

Transcription

1 Binary Decision Diagrams Graphs Binary Decision Diagrams (BDDs) are a class of graphs that can be used as data structure for compactly representing boolean functions. BDDs were introduced by R. Bryant in 986. BDDs are used to solve equivalence problems between formulas of propositional logic. Recall some basic graph-theoretical concepts: directed graph node edge predecessor successor path cycle acyclic graph tree forest Very important in the areas of hardware design and hardware optimization. Boolean Functions A boolean function of arity n is a function {, } n {, }. Examples: { if x = or x 2 = or(x, x 2 ) = if x = or x 2 = { if x = if then else(x, x 2, x ) = x if x = Z.B.: if then else(,, ) =, if then else(,, ) = { if x + x 2 = x x 4 sum(x, x 2, x, x 4 ) = otherwise Z.B.: sum(,,, ) = (because + = ), sum(,,, ) = (because + = ).

2 Formulas and boolean functions if the majority of majority n (x,..., x n ) = x,..., x n has value otherwise Z.B.: majority 4 (,,, ) =, majority (,, ) = if the number of inputs x,..., x n parity n (x,..., x n ) = equal to is even otherwise Z.B.: parity (,, ) =, parity 2 (, ) = Let F be a formula, and let n be a number such that all atomic formulas occurring in F belong to {A,..., A n }. Example: F = A A 2, n = 2, but also n =! We define the boolean function f n F : {, }n {, }: f n F (x,..., x n ) = truth value of F under the assignment that sets A,..., A n to x,..., x n 5 Example: For F = A A 2 : ff 2 (, ) = value of = ff (,, ) = value of = Remark: If all of {A,..., A n } occur in F, then ff n is essentially the truth table of F. Convention: We write e.g. f(x, x 2, x ) = x (x 2 x ), meaning f = f F for the formula F = A (A 2 A ). Fact: Let F and F 2 be two formulas, and let n be a number such that all atomic formulas occurring in F or F 2 belong to {A,..., A n }. Then f n F = f n F 2 iff F F 2. Example: F = A, F 2 = A (A 2 A 2 ). f 2 F (, ) = = f 2 F 2 (, ) f 2 F (, ) = = f 2 F 2 (, ) f 2 F (, ) = = f 2 F 2 (, ) f 2 F (, ) = = f 2 F 2 (, ) Convention: The constants and represent the only two boolean functions of arity. 7

3 sum as binary decision tree Variable order A boolean function can be represented by a decision tree x A decision tree can use a variable order different from the order used in the function. x4 x4 x4 x4 x4 x4 x4 x4 x x4 x4 x4 x4 x 9 Binary decision trees A variable order is a bijection b: {,..., n} {x,..., x n } We say that b(), b(2), b(),..., b(n) are the first, second, third,..., n-th variable w.r.t. the order b. We denote the bijection b() = x i,..., b(n) = x in x i < x i2 <... < x in. by A decision tree for the variable order x i <... < x in is a tree satisfying the following conditions: () All leaves are labelled by or by. (2) All other nodes are labelled by a variable and have exactly two children, the -child and the -child. The edges leading to these children are labelled by resp. by. () If the root is not a leave, then it is labelled by x i. (4) If a node is labelled by x in then its two children are leaves. (5) If a node is labelled by x ij and j < n, then its two children are labelled by x ij+.

4 Binary Decision Diagrams (informally) Every path of a decision tree determines an assignment of the variables x i,... x in and vice versa. The boolean function f T represented by a decision tree T is defined as follows: f T (x,..., x n ) = label of the leaf reached by the path corresponding A BDD (multibdd) is a compact representation of a binary decision tree (decision forest). A BDD (multibdd) is obtained from a decision tree (forest) through repeated application of two compression rules (see example in the next slide): Rule : Sharing of identical subtrees. to the assignment x i x i2... x in A binary decision forest ist a forest of decision trees with the same variable order. A decision forest represents the set of functions represented by its elements. Rule 2: Elimination of nodes for which the -child and the -child coincide (redundant nodes). The rules are applied until all subtrees are different and there are no redundant nodes. Example: sharing of subtrees Example: sharing of subtrees x x x4 x4 x4 x4 x4 x4 x4 x4 x4 x4 x4 x4 x4 x4 x4 x4 All - und -leaves are merged. All - und -leaves are merged. 5

5 Example: sharing of subtrees Example: sharing of subtrees x x x4 x4 x4 x4 x4 x4 Identical x 4 -nodes are merged. Identical x -nodes are merged. 7 Example: removing redundant nodes Example: removing redundant nodes x x x4 x4 x4 x4 x4 Redundant x 4 -node is removed Redundant x 4 -node is removed 9 2

6 Formal Definition of BDDs MultiBDDs A BDD for a given variable order is an acyclic directed graph satisfying the following properties: () There is exactly one node without predecessors (the root) (2) There is one or two nodes without succesors, labelled by or (if there are two then they carry different labels). () All other nodes are labelled by a variable and have exactly two distinct children, the -child and the -child. The edges leading to these children are labelled by resp. by. (4) A child of a node is labelled by, by, or by a variable larger than the label of its parent w.r.t. the variable order. (5) All descendant-closed subgraphs of the graph are non-isomorphic. A multibdd is an acyclic graph satisfying (2)-(5) together a distinguished nonempty subset of nodes called the roots. Every node without predecessors is a root, but other nodes may also be roots. A multibdd represents a set of boolean functions, one for each root. 2 2 Remarks Relevance of variable orders Remark: A closed subgraph of a BDD is again a BDD. Remark: The function true n (x,..., x n ) given by true n (x,..., x n ) = for every x, x n {, } n is represented, for every n and for every variable order, by the BDD consisting of one single node labelled by. Similarly for false n (x,..., x n ) The variable order can have large impact on the size of the BDD. Example: f(x,..., x 2n ) = (x x n+ ) (x 2 x n+2 ) (x n x 2n ) Size grows exponentially in n for x < < x n < x n+ < < x 2n. Size grows linearly in n for x < x n+ < x 2 < x n+2 <... < x n < x 2n. Problem in practice: finding a good order. 2 2

7 Canonicity of BDDs The functions f[] und f[] Lemma I: Let f be a boolean function of arity n. There are exactly two boolean functions f[] und f[] of arity (n ) satisfying f(x,..., x n ) = ( x f[](x 2,..., x n )) (x f[](x 2,..., x n )) ( ) We show that for a given boolean function and a given variable order there is a unique BDD representing the function. More generally (but simpler to prove!), we show that for every set of boolean functions of the same arity and for every variable order there is a unique multibdd representing the set. Proof: The functions f[] and f[] defined by f[](x 2,..., x n ) = f(, x 2,..., x n ) and f[](x 2,..., x n ) = f(, x 2,..., x n ) satisfy (*). Let f and f be arbitrary functions satisfying (*). Then f(x,..., x n ) = ( x f (x 2,..., x n )) (x f (x 2,..., x n )) By the properties of and we have f(, x 2,..., x n ) = f (x 2,..., x n ) and together with f(, x 2,..., x n ) = f[](x 2,..., x n ) we get f = f[]. The proof that f = f[] holds is analogous Let f: {, } n {, } be a boolean function, let B be a BDD with variable order x < x 2 <... < x n, and let v be the root of B. Define the nodes v[] and v[] as follows: () If v is labelled by x, then v[] and v[] are the -child and the -child of v. (2) Otherwise, v[] = v = v[]. Lemma II: B represents the function f iff v[] and v[] represent the functions f[] and f[], respectively. Proof: Easy. Theorem: Let F be a nonempty set of boolean functions of arity n and let x i <... < x in be a variable order. There is exactly one multibdd that follows this order and represents F. Proof: We consider the order x < x 2 <... < x n, for other orders the proof is similar. Proof by induction on the arity n. Basis: n =. There are exactly two boolean functions with n =, namely the constants and, and two BDDs K, K consisting of one single node labelled by or by. The set {} is represented by K, the set {} by K, and the set {, } by the multibdd consisting of K and K. 27 2

8 Step: n >. Let F = {f,..., f k }. Define F = {f [], f [],..., f k [], f k []}, where f i [] and f i [] are as in Lemma I. By induction hypothesis there is exactly one multibdd B with roots v, v,..., v k, v k representing F. I.e., for every function f i [j] the root v ij represents f i [j]. Let B be the multibdd with roots v,... v n obtained from B after executing the following steps for i =, 2,..., k: If v i = v i then set v i := v i. (In this case v i represents f i.) If v i v i and B has a node v such with v i as -child and v i as -child then set v i := v. If v i v i and B contains no such node then add a new node v i having v i as -Kind and v i as -Kind. (So v i represents f i, see Lemma II.) Clearly, B represents F. We now show that B is the only multibdd representing F. 29 Computing BDDs from Formulas Let B be an arbitrary multibdd with roots ṽ,... ṽ n representing F. By Lemma II, B contains nodes ṽ [], ṽ [],..., ṽ k [], ṽ k [] representing the functions of F. By induction hypothesis, the multibdd containing these nodes and all its descendants is the multibdd B. In particular, we have v ij = ṽ i [j] for every i {,..., k} and j {, }. Let v i and ṽ i be the roots of B and B, representing f i. By Lemmas I und II, v i and ṽ i [] represent f[], and v i and ṽ i [] represent f[]. Since v i [] = ṽ i [] and v i [] = ṽ i [] we get v i = ṽ i. So B and B are equal. Goal: Given a formula F over the atomic formulas A,..., A n and a variable order for {x,..., x n }, compute a BDD representing f F (x,..., x n ). Naive procedure: Compute the decision tree of f F and reduce it using the compression rules. Problem: The decision tree is too large! Better procedure (idea): Compute recursively the multibdd representing {f F [Ai /], f F [Ai /]} for a suitable A i, and derive from it the BDD for f F, where F [A i /] bzw. F [A i /] are the formulas obtained by replacing every occurrence of A i by resp. by. In the next slides we formalize this idea.

9 The function multibdd(s) Let S = {F,..., F n } be a nonempty set of formulas. We define a procedure multibdd(s) that returns the roots of a multibdd representing the set {f F,..., f Fn }. K denotes the BDD with only one node labelled by. K denotes the BDD with only one node labelled by. A proper formula is a formula containing at least one occurrence of a variable (i.e., not only and ). An atomic formula A i is smaller than A j if x i appears before x j in the variable order. if S contains no proper formulas then if all formulas of S are equivalent to then return {K } else if all formulas in S are equivalent to then return {K } else return {K, K } else choose a proper formula F S. Let A i be the smallest atomic formula occurring in F. Let B = multibdd( (S \ {F }) {F [A i /], F [A i /]} ). Let v, v be the roots of B representing F [A i /], F [A i /]. if v = v then return B else add a new node v with v, v as - and -child (if such a node does not exist yet); return (B \ {v, v }) {v} Equivalence problems Operations on BDDs Given two formulas F, F 2, the following algorithm decides whether F F 2 holds: Choose a suitable variable order x <... < x n. Compute a multibdd for {F, F 2 }. Check whether the roots v F, v F2 are equal. For digital circuits: the BDDs are not derived from formulas, but directly from the circuits. Given: two formulas F, G over the atomic formulas A,..., A n, a variable order for {x,..., x n }, a multibdd with two roots v F, v G representing the functions f F (x,..., x n ) and f F (x,..., x n ), and a binary boolean operation (e.g.,,, ) Goal: compute a BDD for the function f F G (x,..., x n ). With our convention we have f F G = f F f G 5

10 Idea The function Or(v F, v G ) Lemma: (f F f G )[] = f F [] f G [] and (f F f G )[] = f F [] f G []. Proof: Exercise. Algorithm: (for the order x < x 2 <... < x n, similar for others) Compute a multibdd for {f F [] f G [], f F [] f G []}. (Recursively.) Use the Lemma to build a BDD for f F G (x,..., x n ). if v F = K or v F = K then return K else if v F = v G = K then return K else let v F, v G be the nodes for F [], G[] and let v F, v G be the nodes for F [], G[] v := Or(v F, v G ); v := Or(v F, v G ) if v = v then return v else add a new node v with v, v as - and -child (if such a node does not exist yet); return v Implementing BDDs 7 Data structures BDD-nodes coded as numbers,, 2,... with, for the end nodes. BDD-nodes are stored in a table T : u (i, l, h) Source: An introduction to Binary Decision Diagrams Prof. H.R. Andersen where i, l, h are the label, the -child and the -child of u. (Here l stands for low and h for high.) We maintain a second table so that following invarinat holds: H: (i, l, h) u T (u) = (i, l, h) iff H(i, l, h) = u 9 4

11 The function Make(i, l, h) Basic operations on T : init(t ): Initializes T with and add(t, i, l, h): Adds node with attributes (i, l, h) to T and returns it var(u), low(u), high(u): Returns the variable, -child, -child of u Basic operations on H: init(h): Initializes H as the empty table member(h, i, l, h): Checks whether (i, l, h) belongs to H lookup(h, i, l, h): Returns the node H(i, l, h) insert(h, i, l, h, u): Adds (i, l, h) u to H (if not yet there) Look in H for a node with atributes (i, l, h). If found, then return it. Otherwise create a new node and return it. Make(i, l, h) : if l = h then return l 2: else if member(h, i, l, h) then : return lookup(h, i, h, l) 4: else u := add(t, i, l, h) 5: insert(h, i, l, h, u) 6: return u 4 4 Implementing Or Problem: the function can be called many times with the same arguments. Solution: dynamic programming. The results of all calls are stored. Each call checks first if the result has already been computed earlier. Or(u, u 2 ) : init G 2: return Or (u, u 2 ) Or (u, u 2 ) : if G(u, u 2 ) empty then return G(u, u 2 ) 2: else if u = or u 2 = then return : else if u = and u 2 = then return 4: else if var(u ) = var(u 2 ) then 5: u := Make(var(u ), Or (low(u ), low(u 2 )), Or (high(u ), high(u 2 ))) 6: else if var(u ) < var(u 2 ) then 7: u := Make(var(u ), Or (low(u ), u 2 ), Or (high(u ), u 2 )) 8: else u := Make(var(u 2 ), Or (u, low(u 2 )), Or (u, high(u 2 ))) 9: G(u, u 2 ) = u : return u 4 4