((LecLec 3 3)) BinaryBinary DecisionDecision Diagrams:Diagrams: RepresentationRepresentation ^ What you know X Lots of useful, advanced techniques from Boolean algebra X Lots of cofactor-related manipulations X A little bit of computational strategy X Cubelists, positional cube notation X Unate recursive paradigm ^ What you don’t know X The “right” data structure for dealing with Boolean functions: BDDs X Properties of BDDs X Graph representation of a Boolean function X Canonical representation X Efficient algorithms for creating, manipulating BDDs X Again based on recursive divide&conquer strategy (Thanks to Randy Bryant for nice BDD pics+slides) © R. Rutenbar 2001, CMU 18-760, Fall 2001 1 CopyrightCopyright NoticeNotice © Rob A. Rutenbar, 2001 All rights reserved. You may not make copies of this material in any form without my express permission. © R. Rutenbar 2001, CMU 18-760, Fall 2001 2 Page 1 HandoutsHandouts ^ Physical X Lecture 03 -- BDDs: Representation X Paper: Symbolic Boolean Manipulation with Ordered Binary Decision Diagrams, ACM Computing Surveys, Sept 1992. ^ Electronic X Nothing today ^ Reminder X HW1 is due Thu in class © R. Rutenbar 2001, CMU 18-760, Fall 2001 3 WhereWhere AreAre We?We? ^ Still doing Boolean background, now focussed on data structs M T W Th F Aug 27 28 29 30 31 1 Introduction Sep 3 4 5 6 7 2 Advanced Boolean algebra 10 11 12 13 14 3 JAVA Review 17 18 19 20 21 4 Formal verification 24 25 26 27 28 5 2-Level logic synthesis Oct 1 2 3 4 5 6 Multi-level logic synthesis 8 9 10 11 12 7 Technology mapping 15 16 17 18 19 8 Placement 22 23 24 25 26 9 Routing 29 30 31 1 2 10 Static timing analysis Nov 5 6 7 8 9 11 12 13 14 15 16 12 Electrical timing analysis Thnxgive 19 20 21 22 23 13 Geometric data structs & apps 26 27 28 29 30 14 Dec 3 4 5 6 7 15 10 11 12 13 14 16 © R. Rutenbar 2001, CMU 18-760, Fall 2001 4 Page 2 ReadingsReadings ^ In De Micheli book X pp 75-85 does BDDs, but not in as much depth as the notes ^ Randy Bryant paper X Symbolic Boolean Manipulation with Ordered Binary Decision Diagrams, ACM Computing Surveys, Sept 1992. X Lots more detail (some of it you don’t need just yet) but very complete, if a bit terse. © R. Rutenbar 2001, CMU 18-760, Fall 2001 5 BDDBDD HistoryHistory ^ A little history... X Original idea for Binary Decision Diagrams due to Lee (1959) and Akers (1978) X Critical refinement–Ordered BDDs–due to Bryant (1986) X Refinement imposes some restrictions on structure X Restrictions needed to make result canonical representation ^ A little terminology X A BDD is a directed acyclic graph X Graph: vertices connected by edges X Directed: edges have direction (draw them with an arrow) X Acyclic: no cycles possible by following arrows in graph X Often see this shortened to “DAG” © R. Rutenbar 2001, CMU 18-760, Fall 2001 6 Page 3 GraphsGraphs ^ DAGs -- a reminder of some technicalities... vertex directed edge edge A graph A directed graph A directed acyclic graph vertices + edges ...but not acyclic ...note that a “loop” is not a directed cycle, you are only allowed to follow edges along direction that the arrow points © R. Rutenbar 2001, CMU 18-760, Fall 2001 7 BinaryBinary DecisionDecision DiagramsDiagrams ^ Big Idea #1: Binary Decision Diagram X Turn a truth table for the Boolean function into a Decision Diagram Vertices = Edges = Leaf nodes = X In simplest case, resulting graph is just a tree ^ Aside X Convention is that we don’t actually draw arrows on the edges in the DAG representing a decision diagram X Everybody knows which way they point, implicitly X Point from parent to child in the decision tree ^ Look at a simple example... © R. Rutenbar 2001, CMU 18-760, Fall 2001 8 Page 4 BinaryBinary DecisionDecision DiagramsDiagrams Truth Table Decision Tree x1 x2 x3 f x1 0 0 0 0 0 0 1 0 0 1 0 0 x2 x2 0 1 1 1 1 0 0 0 1 0 1 1 x3 x3 x3 x3 1 1 0 0 1 1 1 1 • Vertex represents a decision • Follow green (dashed) line for value 0 • Follow red (solid) line for value 1 • Function value determined by leaf value. © R. Rutenbar 2001, CMU 18-760, Fall 2001 9 BinaryBinary DecisionDecision DiagramsDiagrams ^ Some terminology A ‘variable’ vertex The ‘lo’ pointer to ‘lo’ son or x1 The ‘hi’ pointer child of the to ‘hi’ son or vertex child of the x2 x2 vertex A ‘constant’ vertex x x x x 3 3 3 3 at the bottom leaves of the tree 0 0 0 1 0 1 0 1 The ‘variable ordering’, which is the order in which decisions about vars are made. Here, it’s X1 X2 X3 © R. Rutenbar 2001, CMU 18-760, Fall 2001 10 Page 5 OrderingOrdering ^ Note: Different variable orders are possible x1 x2 x2 Order for this subtree is X2 then X3 x3 x3 x3 x3 0 0 0 1 0 1 0 1 Here, it’s x1 X3 then X2 x x 2 2x3 x3 x3 xx32 xx32 0 0 0 1 00 10 01 11 © R. Rutenbar 2001, CMU 18-760, Fall 2001 11 BinaryBinary DecisionDecision DiagramsDiagrams ^ Observations X Each path from root to leaf traverses variables in a some order X Each such path constitutes a row of the truth table, ie, a decision about what output is when vars take particular values X But we have not yet specified anything about the order of decisions X This decision diagram is not canonical for this function ^ Reminder: canonical forms X Representation that does not depend on the logic gate implementation of a Boolean function X Same function (ie, truth table) of same vars always produces this exact same representation X Example: a truth table is canonical a minterm list, for our function f = Σ m(3,5,7), is canonical © R. Rutenbar 2001, CMU 18-760, Fall 2001 12 Page 6 BinaryBinary DecisionDecision DiagramsDiagrams ^ What’s wrong with this representation? X It’s not canonical, X Way too big to be useful X ...in fact it’s every bit as big as a truth table: 1 leaf per row ^ Big idea #2: Ordering X Restrict global ordering of variables Means: X Note X It’s OK to omit a variable if you don’t need to check it to decide which leaf node to reach for final value of function © R. Rutenbar 2001, CMU 18-760, Fall 2001 13 TotalTotal OrderingOrdering ^ Assign arbitrary total ordering to variables X x1 < x2 < x3 X Variables must appear in this specific order along all paths OK Not OK x1 x 1 x3 x 1 x2 x2 x3 x3 x1 x1 ^ Properties X No conflicting variable assignments along path (see each var at most once walking down the path). X Simplifies manipulation © R. Rutenbar 2001, CMU 18-760, Fall 2001 14 Page 7 BinaryBinary DecisionDecision DiagramsDiagrams ^ OK, now what’s wrong with it? X Variable ordering simplifies things... X ...but representation still too big X ...and still not necessarily canonical x1 Original decision diagram x2 x2 Equivalent, but x3 x3 x3 x3 different diagram 0 0 0 1 0 1 0 1 x1 x2 x2 x3 x3 x3 x3 0 0 0 1 0 1 0 1 1 © R. Rutenbar 2001, CMU 18-760, Fall 2001 15 BinaryBinary DecisionDecision DiagramsDiagrams ^ Big Idea #3: Reduction X Identify redundancies in the DAG that can remove unnecessary nodes and edges X Removal of X2 node and its children, replacement with X3 node is an example of this sort of reduction ^ Why are we doing this? X To combat size problem: want DAGs as small as possible X To achieve canonical form: for same function, given total variable order, want there to be exactly one graph that represents this function © R. Rutenbar 2001, CMU 18-760, Fall 2001 16 Page 8 ReductionReduction RulesRules ^ Reduction Rule 1: Merge equivalent leaves a a a X ‘a’ is either a constant 1 or constant 0 here X Just keep one copy of the leaf node X Redirect all edges that went into the redundant leaves into this one kept node © R. Rutenbar 2001, CMU 18-760, Fall 2001 17 ReductionReduction RulesRules ^ Apply Rule 1 to our example... x1 x1 x2 x2 x2 x2 x3 x3 x3 x3 x3 x3 x3 x3 0 0 0 1 0 1 0 1 0 1 © R. Rutenbar 2001, CMU 18-760, Fall 2001 18 Page 9 ReductionReduction RulesRules ^ Reduction Rule 2: Merge isomorphic nodes x x x y z y z X Isomorphic: Means 2 nodes with same var and identical children X You cannot tell these nodes apart from how they contribute to decisions as you decend thru DAG X Note: means exact same physical child nodes, not just children with same labels X Remove redundant node (extra ‘x’ node here) X Redirect all edges that went into the redundant node into the one copy that you kept (edges into right ‘x’ node now into left as well) © R. Rutenbar 2001, CMU 18-760, Fall 2001 19 ReductionReduction RulesRules ^ Apply Rule 2 to our example x1 x2 x2 x3 x3 x3 x3 0 1 © R. Rutenbar 2001, CMU 18-760, Fall 2001 20 Page 10 ReductionReduction RulesRules ^ Reduction Rule #3: Eliminate Redundant Tests x y y X Test: means a variable node here... X It’s redundant since both of its children go to same node... X ...so we don’t care what value x node takes in this diagram X Remove redundant node X Redirect all edges into the redundant node (x) into the one child node (y) of the removed node © R.