Citation Context

...orithm 20 return νR 21 return νR computing precisely that by enabling passing the literals L as assumptions. Those assumptions that are part of the core are then returned in the final conflict clause =-=[8]-=-. 5 Backbone Filtering With the exception of Algorithm 2, all the algorithms utilize implicants to prune the backbone estimate. Indeed, according to Proposition 1, if ν is an implicant then any litera...

Citation Context

...empty intersection with each clause of φ. Since the set cover problem is NP-hard, approximate solutions are often used. One example is a greedy approximation algorithm for the set cover problem (e.g. =-=[5]-=-). In the following we refer to these techniques (lifting, set cover) as implicant reduction, i.e. we say that an implicant ν was reduced to an implicant ν′ and thus filtering out the literals ν r ν′ ...

Citation Context

... backdoors [10] and in probabilistic message-passing algorithms [12]. Backbones appear in a number of practical applications of SAT. One concrete example is SATbased interactive product configuration =-=[2]-=-, where the identification of a backbone was utilized in the recent past [19, 16, 14, 15]. Identification of a backbone during the configuration process prevents the user from choosing values that can...

Citation Context

...emonstrates that one of the novel algorithms significantly outperforms the existing ones. 1 Introduction A backbone of a propositional formula φ is formed by literals that are true in all models of φ =-=[27, 3, 17]-=-. Alternatively, one can view a backbone as the set of necessary assignments: if a literal l is in the backbone of φ, any assignment satisfying φ must set l to true. ∗This paper is based on, but signi...

Citation Context

...concept of a backbone is closely related to the concept of failed literals. A literal l is failed in a formula φ in conjunctive normal form if unit prop1 agation derives falsity for the formula φ ∧ l =-=[9]-=-. If a literal l is a failed literal, then the complementary literal l̄ is part of the backbone of φ (but not the other way around). Besides uses in practical applications, backbones provide relevant ...

Citation Context

...algorithm enumerates implicants one by one and updates the backbone estimate in each iteration. In order to avoid finding the same implicant again, the algorithm adds to the formula a blocking clause =-=[29, 25]-=-. A blocking clause for an implicate ν is defined as the clause ∨ l∈ν l̄, as shown in line 7. Adding the blocking clauses to the 4 Algorithm 1: Enumeration-based backbone computation Input : Satisfiab...

Citation Context

...meaningful to increase K for formulas that are “easy” for the SAT solver and conversely decrease K for hard SAT formulas. One could search inspiration in frameworks like SATzilla [33, 32] or ParamILS =-=[13]-=- to heuristically determine the formula’s hardness and determine an appropriate strategy for the value of K. Portfolios. A natural way how to avail of the different algorithms and their parametrizatio...

Citation Context

... of the calls, it is meaningful to increase K for formulas that are “easy” for the SAT solver and conversely decrease K for hard SAT formulas. One could search inspiration in frameworks like SATzilla =-=[33, 32]-=- or ParamILS [13] to heuristically determine the formula’s hardness and determine an appropriate strategy for the value of K. Portfolios. A natural way how to avail of the different algorithms and the...

Citation Context

...ed in research on the phase transitions in Boolean Satisfiability (SAT), where the size of a backbone was related with search complexity. In addition, backbones have also been studied in random 3-SAT =-=[7]-=- and in optimization problems [6, 31, 18, 34], including Maximum Satisfiability (MaxSAT) [35, 26]. Finally, backbones have been the subject of recent interest, in the analysis of backdoors [10] and in...

Citation Context

...emonstrates that one of the novel algorithms significantly outperforms the existing ones. 1 Introduction A backbone of a propositional formula φ is formed by literals that are true in all models of φ =-=[27, 3, 17]-=-. Alternatively, one can view a backbone as the set of necessary assignments: if a literal l is in the backbone of φ, any assignment satisfying φ must set l to true. ∗This paper is based on, but signi...

Citation Context

...ing heuristically the best solver for the given instance, as for example in SATzilla [33], or executing the algorithms in parallel and terminate once the winner terminates, as for instance in ManySAT =-=[11]-=-. Table 1: Legend for Configurations of Algorithms element meaning number size of chunk u chunk size var(φ) cb core-based algorithm l approximate set covering reduction r rotating variables VBS virtua...

Citation Context

...itions in Boolean Satisfiability (SAT), where the size of a backbone was related with search complexity. In addition, backbones have also been studied in random 3-SAT [7] and in optimization problems =-=[6, 31, 18, 34]-=-, including Maximum Satisfiability (MaxSAT) [35, 26]. Finally, backbones have been the subject of recent interest, in the analysis of backdoors [10] and in probabilistic message-passing algorithms [12...

Citation Context

...emonstrates that one of the novel algorithms significantly outperforms the existing ones. 1 Introduction A backbone of a propositional formula φ is formed by literals that are true in all models of φ =-=[27, 3, 17]-=-. Alternatively, one can view a backbone as the set of necessary assignments: if a literal l is in the backbone of φ, any assignment satisfying φ must set l to true. ∗This paper is based on, but signi...

Citation Context

...he variable in the literal, i.e. var(x) = x, var(x̄) = x. Analogously, for a clause ω, var(ω) = {var(l) | l ∈ ω} and for a formula φ, var(φ) = ⋃ ω∈φ var(ω). The following definitions are adopted from =-=[24]-=-. An assignment ν is a mapping from X to {0, u, 1}, ν : X → {0, u, 1}. The constant u has the meaning of an unspecified value, and we define 0 < u < 1 and 1−u = u. The assignment ν is a complete assig...

Citation Context

...algorithm enumerates implicants one by one and updates the backbone estimate in each iteration. In order to avoid finding the same implicant again, the algorithm adds to the formula a blocking clause =-=[29, 25]-=-. A blocking clause for an implicate ν is defined as the clause ∨ l∈ν l̄, as shown in line 7. Adding the blocking clauses to the 4 Algorithm 1: Enumeration-based backbone computation Input : Satisfiab...

Citation Context

...itions in Boolean Satisfiability (SAT), where the size of a backbone was related with search complexity. In addition, backbones have also been studied in random 3-SAT [7] and in optimization problems =-=[6, 31, 18, 34]-=-, including Maximum Satisfiability (MaxSAT) [35, 26]. Finally, backbones have been the subject of recent interest, in the analysis of backdoors [10] and in probabilistic message-passing algorithms [12...

Citation Context

...nes appear in a number of practical applications of SAT. One concrete example is SATbased interactive product configuration [2], where the identification of a backbone was utilized in the recent past =-=[19, 16, 14, 15]-=-. Identification of a backbone during the configuration process prevents the user from choosing values that cannot be extended to a model (or configuration). Another recent application of backbones is...

Citation Context

... backbone was related with search complexity. In addition, backbones have also been studied in random 3-SAT [7] and in optimization problems [6, 31, 18, 34], including Maximum Satisfiability (MaxSAT) =-=[35, 26]-=-. Finally, backbones have been the subject of recent interest, in the analysis of backdoors [10] and in probabilistic message-passing algorithms [12]. Backbones appear in a number of practical applica...

Citation Context

... of the calls, it is meaningful to increase K for formulas that are “easy” for the SAT solver and conversely decrease K for hard SAT formulas. One could search inspiration in frameworks like SATzilla =-=[33, 32]-=- or ParamILS [13] to heuristically determine the formula’s hardness and determine an appropriate strategy for the value of K. Portfolios. A natural way how to avail of the different algorithms and the...

Citation Context

...itions in Boolean Satisfiability (SAT), where the size of a backbone was related with search complexity. In addition, backbones have also been studied in random 3-SAT [7] and in optimization problems =-=[6, 31, 18, 34]-=-, including Maximum Satisfiability (MaxSAT) [35, 26]. Finally, backbones have been the subject of recent interest, in the analysis of backdoors [10] and in probabilistic message-passing algorithms [12...

Citation Context

...itions in Boolean Satisfiability (SAT), where the size of a backbone was related with search complexity. In addition, backbones have also been studied in random 3-SAT [7] and in optimization problems =-=[6, 31, 18, 34]-=-, including Maximum Satisfiability (MaxSAT) [35, 26]. Finally, backbones have been the subject of recent interest, in the analysis of backdoors [10] and in probabilistic message-passing algorithms [12...

Citation Context

...rms of the subset ordering). 4.2 Iterative SAT Testing Enumerating implicants has its clear limitations since the number of implicants is in the worst-case exponential in the number of variables (cf. =-=[4]-=-). An alternative to enumerating implicants is to focus at each literal separately and test whether it is a backbone literal or not. Proposition 3 shows that a literal is in the backbone iff SAT(φ ∪ {...

Citation Context

...nes appear in a number of practical applications of SAT. One concrete example is SATbased interactive product configuration [2], where the identification of a backbone was utilized in the recent past =-=[19, 16, 14, 15]-=-. Identification of a backbone during the configuration process prevents the user from choosing values that cannot be extended to a model (or configuration). Another recent application of backbones is...

Citation Context

...nes appear in a number of practical applications of SAT. One concrete example is SATbased interactive product configuration [2], where the identification of a backbone was utilized in the recent past =-=[19, 16, 14, 15]-=-. Identification of a backbone during the configuration process prevents the user from choosing values that cannot be extended to a model (or configuration). Another recent application of backbones is...

Citation Context

...process prevents the user from choosing values that cannot be extended to a model (or configuration). Another recent application of backbones is post-silicon fault localization in integrated circuits =-=[37, 36]-=-. Manolios and Papavasileiou show that backbones enable improving the solving of pseudo-Boolean constraints via compilation to SAT [21]. Lonsing and Biere use backbones to preprocess quantified Boolea...

Citation Context

... literature, the concept of backbone appears under various terms. For instance, inadmissible and necessary variables [16], bound literals [15], fixed assignments [37], units [21], or frozen variables =-=[1]-=-. Also, we should note that the concept of a backbone is closely related to the concept of failed literals. A literal l is failed in a formula φ in conjunctive normal form if unit prop1 agation derive...

Citation Context

... a literal l is in the backbone of φ, any assignment satisfying φ must set l to true. ∗This paper is based on, but significantly extends, papers presented at ECAI 2010 and RCRA 12 on the same subject =-=[22, 23]-=-. This is a preprint of a paper accepted to RCRA 2012 AICom special issue. The term backbone was coined in research on the phase transitions in Boolean Satisfiability (SAT), where the size of a backbo...

Citation Context

...34], including Maximum Satisfiability (MaxSAT) [35, 26]. Finally, backbones have been the subject of recent interest, in the analysis of backdoors [10] and in probabilistic message-passing algorithms =-=[12]-=-. Backbones appear in a number of practical applications of SAT. One concrete example is SATbased interactive product configuration [2], where the identification of a backbone was utilized in the rece...

Citation Context

...nes appear in a number of practical applications of SAT. One concrete example is SATbased interactive product configuration [2], where the identification of a backbone was utilized in the recent past =-=[19, 16, 14, 15]-=-. Identification of a backbone during the configuration process prevents the user from choosing values that cannot be extended to a model (or configuration). Another recent application of backbones is...

Citation Context

...es is post-silicon fault localization in integrated circuits [37, 36]. Manolios and Papavasileiou show that backbones enable improving the solving of pseudo-Boolean constraints via compilation to SAT =-=[21]-=-. Lonsing and Biere use backbones to preprocess quantified Boolean formulas [20]. It is worth mentioning that throughout the literature, the concept of backbone appears under various terms. For instan...

Citation Context

...m 3-SAT [7] and in optimization problems [6, 31, 18, 34], including Maximum Satisfiability (MaxSAT) [35, 26]. Finally, backbones have been the subject of recent interest, in the analysis of backdoors =-=[10]-=- and in probabilistic message-passing algorithms [12]. Backbones appear in a number of practical applications of SAT. One concrete example is SATbased interactive product configuration [2], where the ...

Citation Context

...and Papavasileiou show that backbones enable improving the solving of pseudo-Boolean constraints via compilation to SAT [21]. Lonsing and Biere use backbones to preprocess quantified Boolean formulas =-=[20]-=-. It is worth mentioning that throughout the literature, the concept of backbone appears under various terms. For instance, inadmissible and necessary variables [16], bound literals [15], fixed assign...

Citation Context

... backbone was related with search complexity. In addition, backbones have also been studied in random 3-SAT [7] and in optimization problems [6, 31, 18, 34], including Maximum Satisfiability (MaxSAT) =-=[35, 26]-=-. Finally, backbones have been the subject of recent interest, in the analysis of backdoors [10] and in probabilistic message-passing algorithms [12]. Backbones appear in a number of practical applica...

Citation Context

...e beneficial for these implicants to be as diverse as possible in order to reduce the backbone estimate more rapidly. Existing techniques for SAT solution diversification could be applied to this end =-=[28]-=-. Heuristic parametrization. The article introduces two novel chunking algorithms. Both of them are parametrized by the size of the chunk K. Hence, these algorithms can be fine-tuned by setting this p...

Citation Context

...process prevents the user from choosing values that cannot be extended to a model (or configuration). Another recent application of backbones is post-silicon fault localization in integrated circuits =-=[37, 36]-=-. Manolios and Papavasileiou show that backbones enable improving the solving of pseudo-Boolean constraints via compilation to SAT [21]. Lonsing and Biere use backbones to preprocess quantified Boolea...