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Lecture 8: Binary Decision Diagrams

Lecture 8: Binary Decision Diagrams 1. Classical Boolean expression representations 2. If-then-else Normal Form (INF) 3. Binary Decision Diagrams. Today’s Program. [13:00-13:55 ] Computation based representations Boolean expressions and Boolean functions

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Lecture 8: Binary Decision Diagrams

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  1. Lecture 8: Binary Decision Diagrams 1. Classical Boolean expression representations2. If-then-else Normal Form (INF)3. Binary Decision Diagrams

  2. Today’s Program • [13:00-13:55] Computation based representations • Boolean expressions and Boolean functions • Classical representations of Boolean expressions • Truth tables • Two-level normal forms: CNF, DNF • Multi-level representations: circuits and formulas • [14:05-15:00] Evaluation based representations • If-then-else normal form • Decision trees • Ordered Binary Decision Diagrams (OBDDs) • Reduced Ordered Binary Decision Diagrams (ROBDDs or just BDDs) 2007 Rune M. Jensen

  3. Classical Representations of Boolean Expressions 2007 Rune M. Jensen

  4. Boolean Expressions • BooleanExpressions t ::= x | 0 | 1 | t | t  t | t t | t t | t  t • Literals l ::= x | x • Precedence , , ,  , (obs. two last swapped in HRA notes) • Terminology • Boolean expression = Boolean formula, Propositional formula, Sentence in propositional logic • Boolean variable = Propositional symbol/letter/variable 2007 Rune M. Jensen

  5. Boolean functions • DefinitionAn n-variable function f : Bn Bis called a Boolean function if it is induced by a Boolean expression E. • B = {0,1} • E represents the Boolean function 2007 Rune M. Jensen

  6. Properties • Order of arguments matterf(x,y) = x y  f(y,x) = x y • Several expressions may represent a functionf(x,y) = x y = x y = (x y)  (x x) = … • Equalityf = giff x . f(x) = g(x) • Number of Boolean functions f : Bn B 2007 Rune M. Jensen

  7. Desirable properties of a representation • Compact • Equality check easy (true for canonical representations) • Easy to evaluate the truth-value of an assignment • Boolean operationsefficient • SAT checkefficient • Tautology checkefficient • Easy to implement 2007 Rune M. Jensen

  8. The curse of Boolean function representation • Boolean functions in nvariables… • How do we find a single compact representation for all of them? Ex. Boolean circuits : how many Boolean circuits of size gwith n inputs and 1 output? • Choice of gates: 16g • Choice of connections: (2g+1)(n+g) • Total: 16g (2g+1)(n+g) In1 In2 Inn … 1 2 16 Out1 2007 Rune M. Jensen

  9. The curse of Boolean function representation • Assume g = O(nk) • Thus the fraction of Boolean functions of n variables with a polynomial circuit size in n is Curse of Boolean function representation: This problem exists for all representations we know! 2007 Rune M. Jensen

  10. Truth tables • Compact table size2n • Equality check easy canonical • Easy to evaluate the truth-value of an assignmentlinear • Boolean operations efficient linear • SAT check efficient linear • Tautology check efficient linear • Easy to implement e.g., DBs 2007 Rune M. Jensen

  11. Two Level Normal Forms OP1 • One level normal forms • Two level normal Forms … Lit Lit Lit OP1 … OP2 OP2 OP2 … … … Lit Lit Lit Lit Lit Lit Lit Lit Lit 2007 Rune M. Jensen

  12. Two-level normal forms: DNF CNF • -monomials : x1  x2  x3  x4 • -monomials : x1  x2  x3  x4 • ( ,)-polynomials: -product of -monomials disjunctive normal form (DNF): • Ex: x  y  x   y • (, )-polynomials: -product of -monomials conjunctive normal form (CNF): • Ex: (x   y)  (x  y) 2007 Rune M. Jensen

  13. Two-level normal forms: DNF CNF • Is there a DNF and CNF of every expression? • Given a truth table representation of a Boolean formula, can we easily define a DNF and CNF of the formula? 2007 Rune M. Jensen

  14. Two-level normal forms: DNF CNF • Example DNF of xyz x  y  z  x y  z  x   y  z  x y  z  x y  z 2007 Rune M. Jensen

  15. Two-level normal forms: DNF CNF • Example CNF of xyz  ( x  y  z)  (x y  z)  (x   y  z) 2007 Rune M. Jensen

  16. Two-level normal forms: DNF CNF • Example CNF of xyz (x y  z) (x   y  z)  (x y  z) 2007 Rune M. Jensen

  17. Two-level normal forms: DNF CNF • The special version DNF and CNF representations produced from on and off-tuples are canonical and called cDNF and cCNF • Are cDNF and cCNF minimum size DNF and CNF representations? Every Boolean formula has a DNF and CNF representation 2007 Rune M. Jensen

  18. Two-level normal forms: DNF CNF • Properties of DNF and CNF • Problem: conversion between CNFs and DNFs is exponential 2007 Rune M. Jensen

  19. Two-level normal forms: DNF CNF • Example • CNF • Corresponding DNF 2007 Rune M. Jensen

  20. Two-level normal forms: DNF CNF • Compact Translating a formula to either CNF or DNF may cause an exponential blow-up in expression size • Equality check easy ”compact” CNF and DNF formulas are not canonical • Easy to evaluate the truth-value of an assignment linear • Boolean operations efficient disjunction e.g. efficient for DNF • SAT check efficient Not for CNF • Tautology check efficient Not for DNF • Easy to implement programmable logic arrays (PLAs) 2007 Rune M. Jensen

  21. Circuits • Circuits are multi-level representations • CNF and DNF are special circuits with depth 3 • Thus, circuits are not canonical x y z    Out 2007 Rune M. Jensen

  22. Circuits • Compact But still suffers from the curse of Booelan function representation. • Equality check easy Not canonical • Easy to evaluate the truth-value of an assignment linear • Boolean operations efficient constant • SAT check efficient (NP-Hard) • Tautology check efficient Not canonical • Easy to implement Integrated circuits 2007 Rune M. Jensen

  23. Formulas • Formulas are circuits where each node has out-degree 1 • Multiple use of intermediate functions is not allowed ((x y)  (z  y)) ((x y)  (z  x)) 2007 Rune M. Jensen

  24. Formulas • Compact At least as large as a circuit • Equality check easy Not canonical • Easy to evaluate the truth-value of an assignment linear • Boolean operations efficient constant • SAT check efficient (Circuit-SAT ≤p Formula-SAT) • Tautology check efficient Not canonical • Easy to implement Simpler algorithmic handling than circuits 2007 Rune M. Jensen

  25. Classical Representations • Observations • There is a trade off between the size of a representation and the efficiency of operations and checks • Truthtables: very large, but all checks and Boolean operations can be carried out efficiently • Circuits: quite compact, but SAT and equality checks are very hard • Next: Binary decision diagrams, probably the best known balance between size and efficiency! 2007 Rune M. Jensen

  26. Binary Decision Diagrams 2007 Rune M. Jensen

  27. Computation and Evaluation Based Representation • Computation Process Representation • Representation gives a way to compute the expression value • Truth-tables, DNF, CNF, circuits and formulas • Evaluation Process Representation • Representation gives a way to classify the expression value by means of tests • Decision trees 2007 Rune M. Jensen

  28. If-then-else operator • The if-then-else Boolean operator is defined by xy1 , y0 = ( x  y1 )  ( x  y0 ) • We have (xy1 , y0) [1/x] = ( 1  y1 )  ( 0  y0 ) = y1 (xy1 , y0) [0/x] = ( 0  y1 )  ( 1  y0 ) = y0 2007 Rune M. Jensen

  29. If-then-else operator • All operators can be expressed using • only  operators and the constants 0 and 1 • tests on un-negated variables • What are if-then-else expressions for • x, x • x  y, x y • x y 2007 Rune M. Jensen

  30. INF normal form • Proposition: any Boolean expression t is equivalent to an expression in INF Proof:t = x t[1/x], t[0/x](Shannon expansion of t)Apply the Shannon expansion recursively on t. The recursion must terminate in 0 or 1, since the number of variables is finite An if-then-else Normal Form (INF) is a Boolean expression build entirely from the if-then-else operator and the constants 0 and 1 such that all test are performed only on variables 2007 Rune M. Jensen

  31. Example • Example: t = (x1 y1)  (x2 y2) • Shannon expansion of t in order x1,y1,x2,y2 t = x1  t1, t0 2007 Rune M. Jensen

  32. Example • Example: t = (x1 y1)  (x2 y2) • Shannon expansion of t in order x1,y1,x2,y2 t = x1  t1, t0 t0 = y1  t01, t00 t1 = y1  1, t10 2007 Rune M. Jensen

  33. Example • Example: t = (x1 y1)  (x2 y2) • Shannon expansion of t in order x1,y1,x2,y2 t = x1  t1, t0 t0 = y1  t01, t00 t1 = y1  1, t10 t01 = x2  t011, 0 t00 = x2  t001, 0 t10 = x2  t101, 0 2007 Rune M. Jensen

  34. Example • Example: t = (x1 y1)  (x2 y2) • Shannon expansion of t in order x1,y1,x2,y2 t = x1  t1, t0 t0 = y1  t01, t00 t1 = y1  1, t10 t01 = x2  t011, 0 t00 = x2  t001, 0 t10 = x2  t101, 0 t011 = y2  1,0 t001 = y2  1,0 t101 = y2  1,0 2007 Rune M. Jensen

  35. 0 1 0 0 0 0 1 0 1 1 Decision tree • Example: t = (x1 y1)  (x2 y2) • Shannon expansion of t in order x1,y1,x2,y2 t = x1  t1, t0 t0 = y1  t01, t00 t1 = y1  1, t10 t01 = x2  t011, 0 t00 = x2  t001, 0 t10 = x2  t101, 0 t011 = y2  1,0 t001 = y2  1,0 t101 = y2  1,0 t x1 t0 t1 y1 y1 t01 t10 t00 x2 x2 x2 t001 t011 t101 y2 y2 y2 2007 Rune M. Jensen

  36. 0 1 Reduction I: substitute identical subtrees • Example: t = (x1 y1)  (x2 y2) • Shannon expansion of t in order x1,y1,x2,y2 t = x1  t1, t0 t0 = y1  t01, t01 t1 = y1  1, t01 t01 = x2  t011, 0 t011 = y2  1,0 t x1 t0 t1 y1 y1 t01 x2 t011 y2 Result: an Ordered Binary Decision Diagram (OBDD) 2007 Rune M. Jensen

  37. 0 1 Reduction II: remove redundant tests • Example: t = (x1 y1)  (x2 y2) • Shannon expansion of t in order x1,y1,x2,y2 t = x1  t1, t01 t1 = y1  1, t01 t01 = x2  t011, 0 t011 = y2  1,0 t x1 t1 y1 t01 x2 Result: a Reduced Ordered Binary Decision Diagram (ROBDD) [often called a BDD] t011 y2 2007 Rune M. Jensen

  38. Reductions x x x Uniqueness requirement Non-redundant tests requirement 2007 Rune M. Jensen

  39. x y y z z z 0 1 Another reduction example 2007 Rune M. Jensen

  40. x y y z z z 0 1 Another reduction example Redundant test Equal subtrees 2007 Rune M. Jensen

  41. x y y z z 0 1 Another reduction example Equal subtrees 2007 Rune M. Jensen

  42. x y y z 0 1 Another reduction example 2007 Rune M. Jensen

  43. x y y z 0 1 Another reduction example Equal subtrees 2007 Rune M. Jensen

  44. x y z 0 1 Another reduction example Redundant test 2007 Rune M. Jensen

  45. y z 0 1 Another reduction example 2007 Rune M. Jensen

  46. z 0 1 Another reduction example y  z y 2007 Rune M. Jensen

  47. Definition of ROBDDs 2007 Rune M. Jensen

  48. Canonicity of ROBDDs • Canonicity Lemma: for any function f : Bn B there is exactly one ROBDD u with a variable ordering x1 < x2 < … < xn such that fu = f(x1,…,xn) Proof (by induction on n) 2007 Rune M. Jensen

  49. Canonicity of ROBDDs • The canonicity of ROBDDs simplifies • Equality check • Tautology check • Satisfiability check Why? 2007 Rune M. Jensen

  50. Practice • What are the ROBDDs of • 1 • 0 • xyorder x, y • (x y)  zorder x, y, z 2007 Rune M. Jensen

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