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

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Step 1 If the false successor is 0 add the variable by which this node is labeled to the set and proceed to the true successor. Step 2 If the true successor is 0 proceed to the false successor. Step 3 If both successors are non-terminal nodes, start a new instance of the algorithm on both successors in order to obtain two new sets of variables: the resulting set is the union of the set that has already been computed by this instance with the intersection between these two new sets. Step 4 Otherwise, stop. The proof that this algorithm finds all and only the entailed variables can be found in [BS98]. Notice that we dont’t consider the case when the ROBDD is 0, as all variables in Vars are entailed and disentailed for ⊥ so we would have to return an infinite set. Pseudo-code for both algorithms is given in Algorithm 5 and Algorithm 6. Require: a ROBDD node n function entailed_vars(n) if nfalse = 0 then return nvar ∪entailed_vars(ntrue) else if ntrue = 0 then return entailed_vars(nfalse) else if is_not_leaf(ntrue) and is_not_leaf(nfalse) then return entailed_vars(ntrue) ∩ entailed_vars(nfalse) else return ∅ Algorithm 5: The entailed_vars function Require: a ROBDD node n function disentailed_vars(n) if ntrue = 0 then return nvar ∪disentailed_vars(nfalse) else if nfalse = 0 then return disentailed_vars(ntrue) else if is_not_leaf(ntrue) and is_not_leaf(nfalse) then return disentailed_vars(ntrue) ∩ disentailed_vars(nfalse) else return ∅ Algorithm 6: The disentailed_vars function 3.2 ROBDD algorithms 18
3.2.9 Combined search for entailed, disentailed and equivalent variables The algorithm equivalent_entailed_disetailed_vars_no_cache is used to find all equivalent, entailed and disentailed vars in a ROBDD by performing a single visit on it. The logic used for detecting entailed and disentailed variables is the same as the one of the previous algorithm, while for equivalent variables we say that the equivalent variables of a ROBDD having its root labeled with variable x are the ones that are equivalent for both its true and false successors, plus the equivalences given by the fact that x is equivalent to the variables that are both entailed in the true successor and disentailed in the false successor. This statement has been proven in [BS98]. We must also consider the fact that all entailed and disentailed variables are equivalent to each other: you can see how we handle this in Algorithm 7. Note that when whenever we compare the current variable with the maximum variable in a set, we are exploiting the ordering property of ROBDDs. 3.2.10 Build a ROBDD representing some variable equivalences Given an equivalence relation L over Vars, the algorithm make_equivalent_vars can be used to build the ROBDD representing the Boolean function that is true if and only if ∀(x, y) ∈ L : x ⇔ y. We define the current variable xc and the set of true leaders T (it contains the leaders for L such that the branch of the ROBDD that we are currently building is reached only when all of these leaders are true); the first is initialized to the smallest variable in dom(L), while the second is initialized to ∅. The algorithm proceeds as follows: Step 1 If xc is undefined (which means that we have already processed all variables in dom(L)) return 1. Step 2 Let xn be the smallest variable in dom(L) that is greater than xc and xl = λL(xc). If xl = xc (xc is a leader for L) return the ROBDD having its root labeled with xc, its false successor being the result of the recursive application of this algorithm with xc = xn and its true successor being the result of the recursive application of this algorithm with xn as the current variable and T = T ∪ {xc}. Step 3 Otherwise, if xl ∈ T , return the ROBDD having its root labeled with xc, its true successor being the result of the recursive application of this algorithm with xc = xn and its false successor being 0. Step 4 Otherwise, return the ROBDD having its root labeled with xc, its false successor being the result of the recursive application of this algorithm with xc = xn and its true successor being 0. 3.2.11 Projection onto a set The algorithm project_set is used to compute f | X given a ROBDD representing f and a set of variables X. We start from the root and follow these steps: 3.2 ROBDD algorithms 19
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: Case 4 If fvar = hvar < gvar, retur
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 57 and 58: 5.2.3 ROBDD plus entailed and equiv
Page 61: Bibliography [Bry86] R. E. Bryant.

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