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

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Require: n, E, L function normalize_positive(n, E, L) nnew = n nold = NULL while nold = nnew do (nold, E, L) = i_normalize1(nnew, E, L) nnew = rename(nold, L) return (nnew, E, L) function i_normalize1(n, E, L) if n = 1 then E = E ∪ y ∈ Vars ∃x ∈ E . (x, y) ∈ L L = L \ (x, y) x ∈ E or y ∈ E return (n, E, L) else nnew = n nold = NULL Lnew = L Lold = NULL while nold = nnew or Lold = Lnew do nold = nnew Lold = Lnew E = E ∪ entailed_vars(n) (nnew, E) = elim_search_true_vars(nnew, E) E = E ∪ y ∈ Vars ∃x ∈ E . (x, y) ∈ Lnew Lnew = Lnew \ (x, y) x ∈ E or y ∈ E T = equivalent_vars(nnew) if T = ∅ then nnew = squeeze_equivalent_vars(nnew, T ) Lnew = Lnew ∧ T return (nnew, E, Lnew) Algorithm 23: The normalize_positive function 4.2 Algorithms with composite representations 40
more. Formally, given m composite representations 〈n1, E1, D1, L1〉, . . . , 〈nm, Em, Dm, Lm〉, their conjunction can be computed, for each i ∈ N : 1 ≤ i ≤ m, by: 〈n ′ i, E ′ i, D ′ i, L ′ i〉 = η ni, m m m Ej, Dj, j=1 j=1 j=1 Lj , 〈n1, E1, D1, L1〉 ∧ . . . ∧ 〈nm, Em, Dm, Lm〉 = η m n ′ m j, E ′ m j, D ′ m j, j=1 j=1 j=1 j=1 L ′ j . The actual implementation, in fact, does not utilize this exact expression: whenever we perform normalization (except, of course, the final normalization) we actually avoid looking for all entailed, disentailed and equivalent variables in the ROBDDs; instead we just reduce the current ROBDD using the information on entailed, disentailed and equivalent variables that we have collected so far by using algorithms elim_search_true_vars, elim_search_false_vars and rename. If elim_search_true_vars or elim_search_false_vars find new entailed or disentailed variables then they are added to the sets of collected variables. We also conjunct each ROBDD immediately after this partial reduction instead of reducing the remaining ones before. 4.2.4 Disjunction Given the same arguments as conjunction, disjunction can be performed as: where: 〈n1, E1, D1, L1〉 ∨ 〈n2, E2, D2, L2〉 = 〈n ′ 1 ∨ n ′ 2, E1 ∩ E2, D1 ∩ D2, L ′ 1 ∨ L ′ 2〉 L ′ 1 = L1 ∧ L ′ 2 = L2 ∧ (x,y)∈E1\E2 x≺y (x,y)∈E2\E1 x≺y (x, y) ∧ (x, y) ∧ (x,y)∈D1\D2 x≺y (x,y)∈D2\D1 x≺y (x, y) , (x, y) , E ′ 1 = λL ′(x) ′ x ∈ E1 \ E2 , E 2 = λL ′(x) x ∈ E2 \ E1 , D ′ 1 = λL ′(x) ′ x ∈ D1 \ D2 , D 2 = λL ′(x) x ∈ D2 \ D1 , L ′′ 1 = (L ′ 1 \ L ′ 2) ∨ L1, L ′′ 2 = (L ′ 2 \ L ′ 1) ∨ L2, n ′ 1 = n1 ∧ x∈E ′ 1 n ′ 2 = n2 ∧ x∈E ′ 2 x ∧ x∈D ′ 1 x ∧ x∈D ′ 2 x ∧ (x,y)∈L ′′ 1 x ∧ (x,y)∈L ′′ 2 x ⇔ y, x ⇔ y. (recall that the ∨ operator on equivalence relations represents their intersection). If the two arguments were already normalized, then it is proven that the result of the application of this algorithm will be already normalized (otherwise a further call to the normalization operator is required). 4.2 Algorithms with composite representations 41
Page 1 and 2: UNIVERSITÀ DEGLI STUDI DI PARMA Fa
Page 3 and 4: Abstract The purpose of this work i
Page 5 and 6: 3.2.9 Combined search for entailed,
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: Require: n, E, D, L function norm
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 57 and 58: 5.2.3 ROBDD plus entailed and equiv
Page 61: Bibliography [Bry86] R. E. Bryant.

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

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