a modern C++ library for the manipulation of Boolean functions

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Contents 1 Boolean functions 1 1.1 Fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Definition of Boolean functions . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Denoting specific Boolean functions . . . . . . . . . . . . . . . . . . 1 1.1.3 Notable sets of variables . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.4 Equivalence relations over sets of variables . . . . . . . . . . . . . . 2 1.1.5 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.6 Positive Boolean functions . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.7 Shannon expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Reduced Ordered Binary Decision Diagrams 5 2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 ROBDD definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 ROBDD semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 ROBDD properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Canonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 Influence of variable ordering . . . . . . . . . . . . . . . . . . . . . 7 2.3 Operations on ROBDDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Implementation of algebraic operators . . . . . . . . . . . . . . . . 8 2.3.2 Implementation of the restriction operation . . . . . . . . . . . . . 10 3 Algorithms in CORAL 11 3.1 Basic data structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.1 Sets of Boolean variables . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.2 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.3 Hash table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.4 Caches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 ROBDD algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2.1 Memory management . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2.2 Algorithms with upward approximation . . . . . . . . . . . . . . . 12 3.2.3 Complexity limitation . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.4 Conjunction and disjunction . . . . . . . . . . . . . . . . . . . . . . 13 3.2.5 if/then/else operator . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.6 Specializations of if/then/else . . . . . . . . . . . . . . . . . . . . . 15 3.2.7 Variables on which a Boolean function depends . . . . . . . . . . . 17 3.2.8 Entailed and disentailed variables . . . . . . . . . . . . . . . . . . . 17 i
3.2.9 Combined search for entailed, disentailed and equivalent variables . 19 3.2.10 Build a ROBDD representing some variable equivalences . . . . . . 19 3.2.11 Projection onto a set . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.12 Shift renaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.13 Renaming with respect to an equivalence relation . . . . . . . . . . 22 3.2.14 Elimination of entailed and disentailed variables . . . . . . . . . . . 23 3.2.15 Elimination of equivalent variables . . . . . . . . . . . . . . . . . . 24 3.2.16 Combined search and elimination of entailed and disentailed variables 25 3.2.17 Forcing a set of variables to true or false . . . . . . . . . . . . . . . 25 3.2.18 Forcing some variables to be equivalent in a positive function . . . 29 3.2.19 Exotic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 Policy-based composite representations 34 4.1 Composite representations of Boolean functions . . . . . . . . . . . . . . . 34 4.1.1 Beyond ROBDDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.1.2 Implementation policies . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Algorithms with composite representations . . . . . . . . . . . . . . . . . . 37 4.2.1 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2.2 Implementing the normalization algorithm . . . . . . . . . . . . . . 37 4.2.3 Conjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.4 Disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2.5 Entailed, disentailed and equivalent variables . . . . . . . . . . . . 42 4.2.6 Projections and variable renaming . . . . . . . . . . . . . . . . . . 42 5 Experimental evaluation 43 5.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2.1 Plain ROBDD representation . . . . . . . . . . . . . . . . . . . . . 45 5.2.2 ROBDD plus entailed variables representation . . . . . . . . . . . . 48 5.2.3 ROBDD plus entailed and equivalent variables representation . . . 51 6 Conclusions 54 6.1 Future developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 ii
Page 1 and 2: UNIVERSITÀ DEGLI STUDI DI PARMA Fa
Page 3: Abstract The purpose of this work i
Page 7 and 8: 1 Boolean functions 1.1 Fundamental
Page 9 and 10: L is denoted as leaders(L) and its
Page 11 and 12: 2 Reduced Ordered Binary Decision D
Page 13 and 14: onto nfalse if a(nvar) = 0, otherwi
Page 15 and 16: operator has a dominant value (for
Page 17 and 18: 3 Algorithms in CORAL 3.1 Basic dat
Page 19 and 20: 1 inline const ROBDD* 2 ROBDD::and_
Page 21 and 22: We have four terminal cases (note t
Page 23 and 24: Case 4 If fvar = hvar < gvar, retur
Page 25 and 26: 3.2.9 Combined search for entailed,
Page 27 and 28: Step 1 If the node is 1 or 0 return
Page 29 and 30: Step 5 If λR(xi) ∈ T return the
Page 31 and 32: The implementation is quite simple:
Page 33 and 34: Step 1 If the node is 0 return this
Page 35 and 36: 3.2.18 Forcing some variables to be
Page 37 and 38: Require: a variable v and a set of
Page 39 and 40: Require: a set of variables V funct
Page 41 and 42: 4.1.2 Implementation policies The N
Page 43 and 44: 4.2 Algorithms with composite repre
Page 45 and 46: Require: n, E, D, L function norm
Page 47 and 48: more. Formally, given m composite r
Page 49 and 50: 5 Experimental evaluation 5.1 Metho
Page 51 and 52: 5.2 Results 5.2.1 Plain ROBDD repre
Page 53 and 54: Program China China + CORAL Differs
Page 55 and 56:
Program China China + CORAL Differs
Page 57 and 58:
5.2.3 ROBDD plus entailed and equiv
Page 59 and 60:
Program China China + CORAL Differs
Page 61:
Bibliography [Bry86] R. E. Bryant.
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a modern C++ library for the manipulation of Boolean functions

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