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CHAPTER 12 WORKING WITH SY MBOLIC REPRES ENTATIONS
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Representing Propositional Formulae Efficiently Using BDDs
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In practice, propositional formulae to describe hardware can be enormous, involving hundreds of thousands of nodes. As a result, hardware companies have an interest in smart algorithms to process these formulae and check them for correctness. The circuits in the computers you use from day to day have almost certainly been verified using advanced propositional logic techniques, often using a functional language as the means to drive and control the analysis of the circuits. A major advance in the application of symbolic techniques to hardware design occurred in the late 1980s with the discovery of binary decision diagrams, a representation for propositional logic formulae that is compact for many common circuit designs. BDDs represent a propositional formula via the use of if ... then ... else conditionals alone, which you write as (variable => true-branch | false-branch). Special nodes are used for true and false at the leaves: we ll write these as T and F. Every BDD is constructed with respect to a global variable ordering, so x AND NOT y can be represented as (x => (y => F | T) | F) if x comes before y in this ordering and as (y => F | (x => T | F)) if y comes before x. The variable ordering can be critical for performance of the representation. BDDs are efficient because they use some of the language representation techniques you saw in 9. In particular, they work by uniquely memoizing all BDD nodes that are identical, which works by representing a BDD as an integer index into a lookup table that stores the real information about the node. Furthermore, negative indexes are used to represent the negation of a particular BDD node without creating a separate entry for the negated node. Listing 12-15 shows our implementation of BDDs. Fully polished BDD packages are often implemented in C. It is easy to access those packages from F# using the techniques described in 19. Here we are content with a clear and simple implementation entirely in F# code. Listing 12-15. Implementing Binary Decision Diagrams open System.Collections.Generic let memoize f = let tab = new Dictionary<_,_>() fun x -> if tab.ContainsKey(x) then tab.[x] else let res = f x in tab.[x] <- res; res type type type type BddIndex = int Bdd = Bdd of BddIndex BddNode = Node of Var * BddIndex * BddIndex BddBuilder(order : Var -> Var -> int) = // The core data structures that preserve uniqueness let uniqueTab = new Dictionary<BddNode,BddIndex>() let nodeTab = new Dictionary<BddIndex,BddNode>() // Keep track of the next index let mutable nextIdx = 2 let trueIdx = 1 let falseIdx = -1
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CHAPTER 12 WORKING WITH S YMBOLIC REPRES ENTATIONS
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let trueNode = Node("",trueIdx,trueIdx) let falseNode = Node("",falseIdx,falseIdx) // Map indexes to nodes. Negative indexes go to their negation. The special // indexes -1 and 1 go to special true/false nodes. let idxToNode(idx) = if idx = trueIdx then trueNode elif idx = falseIdx then falseNode elif idx > 0 then nodeTab.[idx] else let (Node(v,l,r)) = nodeTab.[-idx] Node(v,-l,-r) // Map nodes to indexes. Add an entry to the table if needed. let nodeToUniqueIdx(node) = if uniqueTab.ContainsKey(node) then uniqueTab.[node] else let idx = nextIdx uniqueTab.[node] <- idx nodeTab.[idx] <- node nextIdx <- nextIdx + 1 idx // Get the canonical index for a node. Preserve the invariant that the // left-hand node of a conditional is always a positive node let mkNode(v:Var,l:BddIndex,r:BddIndex) = if l = r then l elif l >= 0 then nodeToUniqueIdx(Node(v,l,r) ) else -nodeToUniqueIdx(Node(v,-l,-r)) // Construct the BDD for a conjunction "m1 AND m2" let rec mkAnd(m1,m2) = if m1 = falseIdx or m2 = falseIdx then falseIdx elif m1 = trueIdx then m2 elif m2 = trueIdx then m1 else let Node(x,l1,r1) = idxToNode(m1) let Node(y,l2,r2) = idxToNode(m2) let v,(la,lb),(ra,rb) = match order x y with | c when c = 0 -> x,(l1,l2),(r1,r2) | c when c < 0 -> x,(l1,m2),(r1,m2) | c -> y,(m1,l2),(m1,r2) mkNode(v,mkAnd(la,lb), mkAnd(ra,rb)) // Memoize this function let mkAnd = memoize mkAnd
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