Enumerating Prime Implicants of Propositional Formulae in Conjunctive Normal Form

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1 Enumerating Prime Implicants of Propositional Formulae in Conjunctive Normal Form Said Jabbour 1, Joao Marques-Silva 2, Lakhdar Sais 1, and Yakoub Salhi 1 1 CRIL, Université d Artois & CNRS, Lens, France {jabbour,sais,salhi}@cril.fr 2 CASL, University College Dublin, Ireland jpms@ucd.ie Abstract. In this paper, a new approach for enumerating the set prime implicants (PI) of a Boolean formula in conjunctive normal form (CNF) is proposed. It is based on an encoding of the input formula as a new one whose models correspond to the set of prime implicants of the original theory. This first PI enumeration approach is then enhanced by an original use of the boolean functions or gates usually involved in many CNF instances encoding real-world problems. Experimental evaluation on several classes of CNF instances shows the feasibility of our proposed framework. 1 Introduction The problem of enumerating prime implicants (PIs) of Boolean functions is an important research topic from the early days of computer science. It was used in the context of boolean function minimization by Quine [29, 28] and McCluskey [23]. This first application of the prime implicant canonical form is important as it allows to reduce digital circuit size and cost while improving the computing speed (e.g. [34]). In addition to digital circuit analysis and optimisation, PIs have found several other application domains including fault tree analysis [6, 11], bioinformatics [1], databases [10], model based diagnosis [8], knowledge representation and reasoning [4]. The computation of prime implicants is also important in many subfields of artificial intelligence such us knowledge compilation [3, 7], automated and non-monotonic reasoning [14], multi-agent systems [33]. Unfortunately, the problem of generating all prime implicants of a given propositional theory is a highly complex task. First, the number of prime implicants of a given theory can be exponential in the size of the theory, while finding just one prime implicant is an NP-hard task. Consequently, enumerating all PIs cannot be done in polynomial total time unless P=NP [16]. Despite this computational bottleneck, several techniques have been proposed in the literature. Many of these PI enumeration techniques are based on some adaptation of the well-known search paradigm, namely branch and bound/backtrack search procedures. Additionally, most of these techniques consider propositional formulae in conjunctive normal form (CNF), the standard representation of propositional

2 knowledge bases. One can cite two related adaptations of the well-known DPLL procedure for prime implicants generation [5, 32, 30] or the modification of modern SAT solvers for computing one prime implicant [9]. Concurrently, almost simultaneously, a 0-1 integer linear programming (ILP) formulation [21, 19] was proposed for computing minimum-size prime implicant improving the formulation given in [27]. In addition, a new algorithm for solving ILP was developed, built on top of a CDCL SAT solver. In [26], Palopoli et al. formulated two algorithms for PIs computation. The first one (called Enumerative Prime Implicants/Feasible Solutions - EPI/FS) search for a feasible solution of the linear program and then extract a prime implicant from it. The second variant, called EPI/OS, is obtained by simply adding a suitable minimization function to the ILP formulation. In contrast to PI enumeration based on an adaptation of DPLL-like procedure (e.g. [5, 32]) and ILP based formulations [21, 27], in this paper, we first propose an original approach that rewrite the CNF formula Φ as a new CNF formula Φ such that the prime implicants of Φ correspond the models of the new CNF Φ. In this way, prime implicant enumeration is reduced to the problem of finding all models of a CNF formula. Such correspondence, allows us to benefit from the recent and continuous advances in practical SAT solving at least for finding one prime implicant. From the ILP formulation mentioned above [21, 27], our PI enumeration encoding borrows only the idea of literal renaming. As such renaming substantially increases the number of variables and clauses, in our second contribution, an enhanced encoding is derived thanks to the structural knowledge recovered from the CNF formula. Indeed, by exploiting Boolean functions encoded in the formula, our new encoding allows significant reductions in the number of variables and clauses. Surprisingly, despite the numerous studies on this issue, to the best of our knowledge, there is no available PI enumeration tool. To compare our proposed approach, an additional PI enumeration algorithm is implemented, which is based on adapting previous work (see Section 5). More precisely, the CNF formula Φ is encoded as a partial MaxSAT formula Φ P using a reformulation of the previous mentioned 0-1 ILP model. Thanks to the correspondence between minimal correction subsets (MCSs) of Φ P and the prime implicants of Φ, we exploit the MCSs enumeration tool proposed recently in [22]. The paper is organized as follows. After some preliminary definitions and necessary notations, our SAT-based encoding of PI enumeration problem is provided in Section 3. Then, we describe in Section 4 our structure-based enhancement of PI enumeration. Then, an alternative approach for PIs generation is discussed in Section 5. An extensive experimental evaluation of our proposed approaches is provided (Section 6) before concluding. 2 Preliminary Definitions and Notations We first introduce the satisfiability problem (SAT) and some necessary notations. SAT corresponds to the problem of deciding if a formula of propositional classical

3 logic is consistent or not. It is one of the most studied NP-complete decision problem. We consider the conjunctive normal form (CNF) representation for propositional formulas. A CNF formula Φ is a conjunction of clauses, where a clause is a disjunction of literals. A literal is a positive (p) or negated ( p) propositional variable. The two literals p and p are called complementary. We denote by l the complementary literal of l, i.e., if l = p then l = p and if l = p then l = p. For a set of literals L, L is defined as { l l L}. A CNF formula can also be seen as a set of clauses, and a clause as a set of literals. Let us recall that any propositional formula can be translated to CNF using linear Tseitin s encoding [35]. We denote by V ar(φ) (respectively Lit(Φ)) the set of propositional variables (respectively literals) occurring in Φ. Φ x denotes the formula obtained from Φ by assigning x the truth-value true. Formally Φ x = {c c Φ, {x, x} c = } {c\{ x} c Φ, x c} (that is: the clauses containing x are therefore satisfied and removed; and those containing x are simplified). Φ denotes the formula Φ closed under unit propagation (UP closure), defined recursively as follows: (1) Φ = Φ if Φ does not contain any unit clause, (2) Φ = if Φ contains two unit-clauses {x} and { x}, (3) otherwise, Φ = (Φ x ) where x is the literal appearing in a unit clause of Φ. The set of unit propagated literals by applying UP closure on Φ is denoted UP (Φ). An interpretation B of a propositional formula Φ is a function which associates a value B(p) {0, 1} (0 corresponds to false and 1 to true) to the variables p V ar(φ). An interpretation B of Φ is alternatively represented by a set of literals, i.e., B = x V ar(φ) f(x), where f(x) = x (respectively f(x) = x), if B(x) = true (respectively B(x) = false). A model or an implicant of a formula Φ is an interpretation B that satisfies the formula, noted B Φ. The SAT problem consists in deciding if a given formula admits a model or not. An implicant B of Φ is called a prime implicant (in short PI), iff for all literals l B, B\{l} Φ. We define P I(Φ) as the disjunction of all prime implicant of Φ. Obviously, P I(Φ) is logically equivalent to Φ, while its size might be exponential in the worst case. 3 SAT-based Encoding of PI Enumeration Problem In this section, we describe our SAT-based encoding of the prime implicant enumeration problem. The idea consists in reformulating the PI enumeration problem of a given CNF Φ as the model enumeration problem of a CNF Φ. Our encoding borrows the idea of literals renaming used in the ILP formulations proposed in [27, 19, 20]. Let Φ be a CNF formula. We associate to each element l of Lit(Φ) a propositional variable x l. We define the CNF formula Φ R

4 as the formula obtained from Φ by renaming each literal l in Lit(Φ) by its corresponding propositional variable x l, and by adding the following binary clauses: x p x p (1) p V ar(φ) One can easily see that Φ and Φ R are equisatisfiable. Example 1. Let us consider the following CNF formula: Φ = (p q r) ( p r) (q r). Then, we have Φ R = (x p x q x r ) (x p x r ) (x q x r ) ( x p x p ) ( x q x q ) ( x r x r ). As we can see, the formula Φ R is a conjunction of two monotone CNF formulae. We now propose a new constraint in order to establish a bijection between prime implicants of a CNF formula and the models of the resulting CNF formula. This additional formula M(Φ R ) is defined as follows: M(Φ R ) = x l Cl(Φ R, x l ) (2) l Lit(Φ) where Cl(Φ R, x l ) corresponds to the restriction of Φ R to the clauses containing x l without the latter, i.e., Cl(Φ R, x l ) c Φ R,x l c c \ {x l}. For instance, if we consider again the formula Φ given in Example 1, M(Φ R ) corresponds to the the following formula: (x p (x q x r )) (x p x r ) (x q x r ) (x q (x p x r )) (x r (x p x q )) (x r (x q x p )). Theorem 1. If B is a model of the formula Φ R M(Φ R ), then the set of literals I B = {l Lit(Φ) B(x l ) = 1} is a prime implicant of Φ. Proof. Using the fact that Φ R is nothing else than a renaming of the literals of Φ, we have I B is an implicant of Φ. Assume now that I B is not a prime implicant of Φ. Then, there exists l 0 I such that I B \ {l 0 } is an implicant of Φ. Let B be a Boolean interpretation of Φ R M(Φ R ) defined as follows: { B B(xl ) if l l (x l ) = 0 0 otherwise Clearly, we have B (x l0 ) = B (x l0 ) = 0. Indeed, B (x l0 ) = 0 (see the definition of B ). Let us show that B (x l0 ) = 0. By definition, we have B (x l0 ) = B(x l0 ) because l 0 l 0. As l 0 I B, we have B(x l0 ) = 1. Also, as I B is an implicant of Φ, then it satisfies the binary clause ( x l0 x l0 ) (see formula (1). Consequently, we deduce that B( x l0 ) = 1 i.e. B(x l0 ) = 0. Finally, B (x l0 ) = 0. Using the formula M(Φ R ), B(x l0 Cl(Φ R, x l0 )) = 1 holds. Hence, we have B(Cl(Φ R, x l0 )) = 0, since B(x l0 ) = 1 because l 0 I B. Since B is obtained from B by setting the truth value of x l0 to 0 (B (x l0 ) = 0), then B(Cl(Φ R, x l0 )) = B (Cl(Φ R, x l0 )) = 0. We then obtain that B is not a model of Φ R and we get a contradiction with I B \ {l 0 } is an implicant of Φ. Therefore, I is a prime implicant of Φ.

5 Theorem 2. Let I be a prime implicant of Φ and B a Boolean interpretation of Φ R M(Φ R ) defined as follows: { 1 if l I B(x l ) = 0 otherwise Then, B is a model of Φ R M(Φ R ). Proof. Clearly, B is a model of Φ R because I is a prime implicant of Φ. We now show that B is also a model of M(Φ R ). Let l be a literal in Lit(Φ). If B(x l ) = 0 then B(x l Cl(Φ R, x l )) = 1 holds. Otherwise, we have B(x l ) = 1 and l I. If B(Cl(Φ R, x l )) = 1 then I \ {l} is an implicant of Φ and we get a contradiction. Therefore, B(Cl(Φ R, x l )) = 0 holds. Consequently, B is a model of M(Φ R ) and then of Φ R M(Φ R ). Corollary 1. The number of prime implicants of Φ is equal to the number of models of Φ R M(Φ R ). Proof. For all B 1 and B 2 two different models of Φ R M(Φ R ), I B1 I B2 holds. Thus, using Theorem 1, we obtain that the number of prime implicants of Φ is greater than or equal to the number of models of Φ R M(Φ R ). Moreover, using Theorem 2, the number of prime implicants of Φ is smaller than or equal to the number of models of Φ R M(Φ R ). Therefore, the number of prime implicants of Φ is equal to the number of models of Φ R M(Φ R ). 4 Structure-Based Enhancement of PI Enumeration CNF formulae encoding real words problems usually involve a large fraction of clauses encoding different kind of boolean functions or gates. These Boolean functions result from the problem specification itself or introduced during the CNF transformation using the well-known Tseitin extension principle [35]. Tseitin s encoding consists in introducing fresh variables to represent sub-formulae in order to represent their truth values. For example, given a Boolean formula, containing the variables a and b, and v a fresh variable, one can add the definition v a b (called extension) to the formula while preserving satisfiability. Tseitin s extension principle is at the basis of the linear transformation of general Boolean formulae into CNF. Boolean functions express strong relationships between variables, the goal of this section is to show how such variable dependencies can be exploited in the context of prime impliquant generation. Let us first introduce some formal definitions and notations about Boolean functions or gates. A Boolean function or Gate is an expression of the form l = f(l 1,..., l n ), where l, l 1,..., l n are literals and f is a logical connective among {,, }. It allows us to express that the truth value of l is determined by f(l 1,..., l n ). According to f, a Boolean gate can be defined as a conjunction of clauses as follows:

6 1. l = (l 1,..., l n ) represents the set of clauses { l 1 l n l, l l 1,..., l l n }; 2. l = (l 1,..., l n ) represents the set of clauses { l l 1 l n, l 1 l,..., l n l }; 3. l = (l 1,..., l n ) represents the following equivalence chain (also called biconditional formula) (l l 1,..., l n ). Let us note that the Boolean functions using the connective ( -gates) and those using ( -gates) are dual. Indeed, any Boolean gate l = (l 1,..., l n ) (resp. l = (l 1,..., l n )) is equivalent to l = ( l 1,..., l n ) (resp. l = ( l 1,..., l n )). It is also important to note that for the third Boolean function (equivalence chain), its equivalent representation in CNF leads to a huge number of clauses (2 n clauses). In general, we say that an expression l = f(l 1,..., l n ) is a gate of a CNF formula Φ if it is a logical consequence of Φ. Unfortunately, this deduction problem is Co-NP Complete. However, several recent contributions addressed the issue of recovering Boolean functions from CNF formulae (e.g. [25, 15, 31, 13, 2]). In [25], a polynomial and syntactical approach, which recovers only Boolean functions implicitly present in the CNF formula. Another detection technique is proposed by Roy et al. in [31]. It operates by translating the gate matching problem into one of recognizing sub hypergraph isomorphism. This approach is clearly intractable. In [15], a new technique based on deduction restricted to unit propagation process is proposed. It extends the syntactical approach [25] and allows the detection of some hidden Boolean functions. Given a formula CNF Φ, we can use two different methods for detecting Boolean functions encoded by the clauses of Φ. The first detection method, called syntactical method, is a pattern matching approach that allows us to detect the Boolean functions that appear directly in the structure of the CNF formula [25]. Example 2. Let Φ {(y x 1 x 2 x 3 ), ( y x 1 ), ( y x 2 ), ( y x 3 )}. In this example, we can detect syntactically the function y = (x 1, x 2, x 3 ). The second method is a semantic detection approach where the functions are detected using Unit Propagation (UP) [15]. Indeed, this method allows us to detect hidden Boolean functions linearly in the size of the CNF. Example 3. Let Φ {(y x 1 x 2 x 3 ), ( y x 1 ), ( x 1 x 4 ), ( x 4 x 2 ), ( x 2 x 5 ), ( x 4 x 5 x 3 )}. In this example, UP (Φ y) = {x 1, x 4, x 2, x 5, x 3 } the set of unit propagated (UP) literals and we have the clause c = (y x 1 x 2 x 3 ) Φ which is such that c\{y} UP (Φ y). So, we can discover the Boolean function y = (x 1, x 2, x 3 ), that the above syntactical method does not help us to discover.

7 In our implementation, we exploit the semantic or UP-based approach proposed in [15]. Consequently, the boolean function of the form l = (l 1,..., l n ) is not considered in our experiments. Let us now describe how these Boolean functions can be used to improve the PI enumeration CNF encoding. Proposition 1. Let Φ be a CNF formula and l = f(l 1,..., l n ) a gate of Φ. then, for every prime implicant I, we have either l I or l I. Proof. Assume that there exists a prime implicant I such that both l / I and l / I hold. Consider I an extension of I which is obtained by assigning truth values to all the literals l 1,..., l n and without assigning any truth value to l. Thus, we have either I (f(l 1,..., l n )) = 0 or I (f(l 1,..., l n )) = 1. If I (f(l 1,..., l n )) = 0 then I {l } is a counter-model of Φ and we get a contradiction. Otherwise, I { l } is a counter-model of Φ and we also get contradiction. Let Φ be a CNF formula and {l 1 = e 1,..., l n = e n } a set of its gates. Using Proposition 1, we know that it is not necessary to associate in our encoding fresh propositional variables (of the form x l ) to the literals in S = {l 1,..., l n, l 1,..., l n }. In this case, Φ R is obtained by renaming only the literals in Lit(Φ) \ S and by redefining the formula (1) as follows: We also redefine (2) as follows: M(Φ R ) = p V ar(φ)\s l Lit(Φ)\S x p x p (3) x l Cl(Φ R, x l ) (4) Proposition 2. Let Φ be a CNF formula and l = (l 1,..., l n ) a gate of Φ. then, for every prime implicant I, we have either {l, l 1,..., l n } I or l I. Proof. In the same way as the proof of Proposition 1. Using Proposition 2, one can reduce the formula M(Φ R ). Indeed, given a Boolean gate l = (l 1,..., l n ) of Φ, every formula of the form x l Cl(Φ R, x l ) where l {l 1,..., l n } can be reduced as follows: (l (x l Cl(Φ R, x l, l ))) (5) where Cl(Φ R, x l, l ) corresponds to the formula c Φ R, l / c,x l c c\{x l, l }. This redefinition comes from the fact that if l is in a prime implicant I then l 1,..., l n are also in I. Thus, it is not necessary to reduce a model by removing literals in {l 1, l 1,..., l n, l n } when l is true. In the same way, we have also the following proposition:

8 Proposition 3. Let Φ be a CNF formula and l = (l 1,..., l n ) a gate of Φ. then, for every prime implicant I and every l {l, l 1,..., l n }, we have either l I or l I. Using Proposition 3, we know that it is not necessary to associate in our encoding fresh propositional variables (of the form x l ) to the literals appearing in the equivalence chains. Other related well known XOR constraints of the form (l 1,..., l n ) can be exploited in the same way. Their detection can be done using a pattern matching approach [25]. In this paper, the integration to our encoding of these specific Boolean functions is left as an interesting perspective. 5 Prime Implicant Enumeration: Alternative Approaches As mentioned in the introduction, to the best of our knowledge, there is no available PI enumeration tool. To compare our proposed approach, we discuss an alternative that will be used in our comparative experimental evaluation (Section 6). mcsls One can envision a number of alternative approaches for enumerating prime implicants, by exploiting the 0-1 ILP model for computing a minimum size prime implicant [19]. Given a CNF formula Φ, let Φ i denote the 0-1 ILP model associated with Φ: minimize subject to l Lit(Φ) x l x l 1 for c Φ (6) l c x p + x p 1 for p V ar(φ) (7) This model can be re-formulated as a partial MaxSAT formula [17] Φ P. A simple observation is that any minimal correction subset (MCS) of Φ P is a prime implicant of Φ. Therefore, a tool capable of enumerating the MCSes Φ P can be used for enumerating the prime implicants of Φ. A number of approaches have been proposed in recent years for enumerating MCSes [18, 24, 22]. These either use MaxSAT [18, 24] or dedicated algorithms [22], with recent results indicating that the dedicated algorithms outperform MaxSAT-based solutions. 6 Experiments In this section, we present an experimental evaluation of our approach which consists in enumerating prime implicants of a CNF formula. As described above, our transformation allows to translate the problem of enumerating prime implicants of a formula Φ to that of enumerating models of a new formula Φ. In this context, we use a modified version of MiniSAT [12] solver to enumerate the

9 models of Φ. Each time a model is found a prime implicant is extracted and the clause representing the negation of the model is added to seek for the next model until Φ becomes unsatisfiable. In order to evaluate the performances of our approach, we consider a comparison with the prime implicants enumerator (mcsls described in Section 5. All the experimental results presented in this section have been obtained with a Quad-core Intel Xeon X5550 (2.66GHz, 32 GB RAM) cluster. Our experiments are conducted on benchmarks coming from 2012 MaxSAT Evaluation 3. More precisely, we consider the enumeration of prime implicants of the hard parts of partial MaxSAT instances. 1e+07 1e+08 1e+06 1e e+06 #Vars (Trans.) #Cls (Trans.) e+06 #Vars (Orig.) e+06 1e+07 #Cls (Orig.) Fig. 1. Size of the Encoding With Boolean Gates Let us first illustrate the size of our encoding enhanced with Boolean functions (or gates). In the scatter plot of Figure 1 (left hand side), each dot (x, y) represents the number of variables x of the original (Orig.) formula and the number of variables y of new formula (Trans.) respectively. As we can see, except for some few instances, the resulting encoding is, in general, of reasonable size (in many cases, the number of variables does not exceed ten times the original one). The same observation can be made on the number of clauses (see Figure 1 - right hand side). Figure 2 shows the size of the encoding with and without using Boolean gates. As we can see, exploiting Boolean functions allows us to improve the encoding both in the number of variables and clauses. All the dots are below the diagonal. On a representative sample of instances, Table 6 highlights some characteristics of the original formula and of those obtained using our encodings. In column 1, 2 and 3, we report respectively the name of the instance, its number of variables (#vars) and clauses (#cls). Columns 4 and 5 provide the increasing factor of the number of variables ( #vars) and clauses ( #cls) of the encoding without using Boolean Gates (P I-Encoding). In column 6, we provide the number of Boolean functions or Gates (#bg) detected from the original instance. The two 3 MaxSAT Evaluations:

10 1e+07 1e+08 1e+06 1e+07 #Vars with BG (Trans.) #Cls with BG (Trans.) 1e e+06 1e+07 #Vars without BG (Trans.) e+06 1e+07 1e+08 #Cls without BG (Trans.) Fig. 2. Size of the Encoding With and Without Boolean Gates last columns show the increasing factor of the number of variables and clauses obtained using our enhanced encoding (with Boolean gates - P I-Encoding + BG). We ran the two prime implicant enumerators, primp (our encoding without Boolean gates), mcsls (see Section 5) on the set of the 497 instances taken from the 2012 MaxSAT evaluation. In the first experiment, we limit the size of the output (number of prime implicants) to PIs and we compare the time needed for each approach to output this number of PIs or all PIs of the formula whenever it does not exceed PIs (with a time out of 1 hour). Figure 3, provides a comparison between mcsls and our primp approach. One can see that the majority of the dots are under the diagonal which illustrates the ability of primp to efficiently generate prime implicants compared to mcsls primp mcsls Fig. 3. mcls vs primp: CPU Time Needed to Compute PIs For a deeper analysis, we ran primp and mcsls with a time limit fixed to 1 hour. We compare the number of prime implicants found by each PIs enumerator. In Figure 4, each dot (x, y) represents the number of prime implicants found

11 Original Instance PI-Encoding PI-Encoding+BG instances #vars #cls (#vars) (#cls) #bg (#vars) (#cls) normalized-f20c10b 020 area delay ,89 6, ,86 1,90 normalized-f20c10b 014 area delay ,12 6, ,86 1,97 normalized-fir09 area partials ,11 22, ,33 2,03 normalized-f20c10b 025 area delay ,81 11, ,20 6,52 normalized-fir04 area delay ,07 9, ,81 2,86 normalized-fir01 area opers ,62 7, ,07 3,45 normalized-f20c10b 003 area delay ,88 6, ,80 1,80 normalized-fir05 area partials ,91 28, ,96 11,59 normalized-fir10 area partials ,67 18, ,21 4,63 normalized-m r ,00 903, ,00 903,00 normalized-m r , , , ,00 normalized-max1024.pi ,97 49,08 0 5,97 49,08 normalized-m r , , , ,00 15tree801posib ,46 6, ,46 1,44 15tree201posib ,64 6, ,47 1,44 10tree515p ,63 6, ,42 1,40 15tree901p ,92 7, ,32 1,28 10tree315p ,70 6, ,44 1,42 normalized-s pb ,66 9, ,91 7,01 normalized-s pb ,71 9, ,94 7,04 normalized-hanoi ,94 4, ,81 2,28 normalized-ssa ,51 3, ,14 1,15 normalized-ii32e ,98 74, ,98 74,61 normalized-ii16b ,29 11,30 0 7,29 11,30 normalized-par32-2-c ,56 8, ,92 8,12 normalized-ii32e ,61 89, ,61 89,88 normalized-par32-4-c ,57 8, ,95 8,14 normalized-ii32b ,29 85, ,29 85,54 normalized-ii16b ,72 10,79 0 6,72 10,79 splitedreads 158.matrix ,35 11,22 0 9,35 11,22 splitedreads 0.matrix ,53 10, ,53 10,94 SU3 simp-genos.haps ,75 6, ,75 6,14 simp-ibd ,33 6, ,33 6,58 simp-ibd ,10 6, ,10 6,69 simp-ibd ,38 6, ,32 6,89 simp-test chr10 JPT ,43 6, ,43 6,85 1knt.5pti.g.wcnf.t ,00 1,00 0 2,00 1,00 2knt.5pti.g.wcnf.t ,00 1,00 0 2,00 1,00 6ebx.1era.g.wcnf.t ,00 1,00 0 2,00 1,00 3ebx.1era.g.wcnf.t ,00 1,00 0 2,00 1,00 Table 1. Size of the Encoding: Highlighting Results

12 by mcsls (x) and primp (y). This experiment confirms the efficiency of our approach. Indeed, the enumerator primp outperforms mcsls. Note that primp is able to generate 10 times more prime implicants than mcsls on several instances. 1e+08 1e+07 1e+06 primp e+06 mcsls Fig. 4. mcsls vs primp : Number of Generated PIs in less than 1 hour In our last experiment, we evaluate our encoding enhanced with Boolean gates. Figure 5, compares primp encoding (with and without Boolean gates) in terms of CPU time (in seconds) needed to generate the first PIs, under a time out of 1 hour. Note that the CPU time (seconds) includes the time needed for recovering Boolean gates from the CNF formula. The results demonstrate that exploiting Boolean functions makes significant improvements. Indeed, most of the dots are above the diagonal. Dots that are near the diagonal correspond to instances that involve a marginal number of Boolean functions primp without BG primp with BG Fig. 5. primp with/without Boolean gates: CPU Time Needed to Compute PIs

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