Proceedings of the Twenty-Seventh AAAI Conference on Artificial Intelligence Partial MUS Enumeration Alessandro Previti Joao Marques-Silva Complex and Adaptive Systems Laboratory CASL, University College Dublin, Ireland University College Dublin, Ireland IST/INESC-ID, Portugal Email: [email protected] Email: [email protected] Abstract Gregoire,´ Mazure, and Piette 2007; Stern et al. 2012), that build on earlier work (Reiter 1987; Han and Lee 1999). Minimal explanations of infeasibility find a wide range Despite the important improvements observed in MUS of uses. In the Boolean domain, these are referred to as Minimal Unsatisfiable Subsets (MUSes). In some set- enumeration, a number of drawbacks can be identified. First, tings, one needs to enumerate MUSes of a Boolean for- earlier work can require a very large number of calls to mula. Most often the goal is to enumerate all MUSes. a SAT solver (de la Banda, Stuckey, and Wazny 2003; In cases where this is computationally infeasible, an Bailey and Stuckey 2005). This issue is addressed in more alternative is to enumerate some MUSes. This paper recent work (Liffiton and Sakallah 2005; 2008). However, develops a novel approach for partial enumeration of a drawback of (Liffiton and Sakallah 2008) is the enumer- MUSes, that complements existing alternatives. If the ation of Minimal Correction Subformulas (MCSes) before enumeration of all MUSes is viable, then existing alter- hitting set dualization is applied for enumerating MUSes. natives represent the best option. However, for formu- The worst-case number of MCSes (and of MUSes) is expo- las where the enumeration of all MUSes is unrealistic, nential on the size of the formula. Thus, when the worst- our approach provides a solution for enumerating some MUSes within a given time bound. The experimental re- case number of MCSes is exercised, resources are exceeded sults focus on formulas for which existing solutions are before enumeration of MUSes takes place. In (Liffiton and unable to enumerate MUSes, and shows that the new Sakallah 2008) a solution to this problem is to use Partial approach can in most cases enumerate a non-negligible Correction Subsets (PCSes), subsets of MCSes such that number of MUSes within a given time bound. some MUSes are preserved. PCSes are smaller in number than MCSes, but enumeration of MUSes using PCSes may not be complete. However, as will be shown in this paper, Introduction even the enumeration of PCSes as proposed in (Liffiton and Minimal Unsatisfiable Subformulas (MUSes) of Boolean Sakallah 2008) may be unrealistic for some formulas. To our formulas in conjunctive normal form (CNF) are used in best knowledge, for formulas having a very large number of different settings. Examples of applications include incon- MCSes or PCSes, there exists no solution that can enumer- sistency measurement (Hunter and Konieczny 2006; 2010; ate a non-negligible number of MUSes. In this paper, the Xiao and Ma 2012), type error debugging (de la Banda, problem of enumerating a non-trivial subset (i.e. with more Stuckey, and Wazny 2003; Bailey and Stuckey 2005), de- than 1 element) of the MUSes of an unsatisfiable formula bugging of relational specifications (Torlak, Chang, and is referred to as partial MUS enumeration. Despite focusing Jackson 2008; Torlak, Vaziri, and Dolby 2010), analysis of on the enumeration of MUSes (and MCSes) of CNF formu- over-constrained temporal problems (Liffiton et al. 2005), las, the work in this paper can be related with model-based axiom pinpointing in description logics (Schlobach et al. diagnosis (e.g. (Reiter 1987)), where MUSes correspond to 2007), software and hardware model checking (Andraus, minimal conflict sets and MCSes correspond to minimal di- Liffiton, and Sakallah 2008), among many others. Enumer- agnoses. ation of MUSes is also tightly related with model-based This paper proposes a novel algorithm for MUS enumer- diagnosis (Reiter 1987). In most applications, it is im- ation, that represents an effective approach for partial MUS portant to provide different explanations of infeasibility. enumeration. The algorithm works by generating candidate For Boolean formulas, this means that one needs to enu- sets of clauses with a key property: each set either contains merate some (or preferably all) MUSes. A number of al- an MUS or represents the complement of an MCS. As a re- gorithms for enumeration of MUSes have been proposed sult, the partial enumeration of MUSes does not necessarily in recent years (de la Banda, Stuckey, and Wazny 2003; require the complete enumeration of MCSes or PCSes. As Bailey and Stuckey 2005; Liffiton and Sakallah 2005; 2008; the experimental results show, for formulas for which ex- isting solutions fail to enumerate MUSes, our approach is Copyright © 2013, Association for the Advancement of Artificial able to partially enumerate MUSes. (As indicated by the ex- Intelligence (www.aaai.org). All rights reserved. perimental results, the number of MUSes that can be enu- 818 merated depends on the target formula.) It should be noted Proposition 1. Let MUSes(F) and MCSes(F) be the set of that the new approach does not replace the state of the art in all MUSes and MCSes of F respectively. Then the following MUS enumeration (Liffiton and Sakallah 2008). For formu- hold: las where enumeration is viable, existing approaches are the 1. A subset M of F is an MUS if and only if M is a minimal preferred option. However, for formulas where enumeration hitting set of MCSes(F). or partial enumeration is impossible with existing solutions, the approach proposed in this paper is able to compute a 2. A subset N of F is an MCS if and only if N is a minimal number of MUSes that is bounded by available (time and hitting set of MUSes(F). memory) resources. Enumeration of MUSes Preliminaries In recent years, a number of different algorithms have been This section introduces the basic definitions used throughout proposed for MUS enumeration (Han and Lee 1999; de la the paper. Although the new algorithm for MUS enumera- Banda, Stuckey, and Wazny 2003; Bailey and Stuckey 2005; tion can applied to any constraint programming problem, the Liffiton and Sakallah 2008; Gregoire,´ Mazure, and Piette paper addresses the more specific problem of Boolean for- 2007). These approaches can be divided into two main cate- mula satisfiability. In this paper Boolean formulas are repre- gories: sented in Conjunctive Normal Form (CNF), where a formula is a conjunction of clauses, and each clause is disjunction of • MUSes obtained by direct computation; literals. A literal is either a Boolean variable or its comple- • MUSes obtained by hitting set dualization. ment. Formulas can also be viewed as sets of clauses, and clauses as sets of literals. Boolean variables can be assigned The direct computation of MUSes (Han and Lee 1999; a value in set f0; 1g. An interpretation is a mapping from the de la Banda, Stuckey, and Wazny 2003) is based on explicit set of variables V to f0; 1g, m : X ! f0; 1g. A formula is enumeration of the subsets of a formula. The enumeration satisfiable if it admits a model, that is an interpretation that of subsets is organized in a tree (the CS-tree) to avoid repe- satisfies it. Otherwise, F is said to be unsatisfiable. Given tition of subsets. An order of the clauses is used to specify a CNF formula F, the Boolean Satisfiability (SAT) prob- how the subsets are considered. Each node of the CS-tree is lem consists in deciding whether F is satisfiable. This pa- labeled with a subset S and each of its children is labeled per studies unsatisfiable formulas F, and the definitions be- with a subset S0 ⊂ S. The nodes of the tree are visited in low will be used throughout (e.g. see (Buning¨ and Kullmann a depth first manner and at each node a satisfiability test is 2009; Liffiton and Sakallah 2008)). Minimal Unsatisfiable performed on that subset. A subset S of a node n is marked Subsets (MUSes) are defined as follows: as an MUS if S is unsatisfiable and all the children of n are Definition 1. A subset M ⊆ F is an MUS if M is unsatis- labeled with satisfiable subsets. (de la Banda, Stuckey, and fiable and 8C ⊂ M with C 6= ;; M n C is satisfiable. Wazny 2003) develop a number of improvements over the original approach (Han and Lee 1999), but the explicit enu- Every unsatisfiable formula contains at least one MUS. meration of subsets coupled with the iterative SAT tests is A related concept is the one of Minimal Correction Subset often a performance bottleneck (Bailey and Stuckey 2005). (MCS): A different alternative can be obtained from Reiter’s algo- Definition 2. A subset N of F is an MCS if F n N is satis- rithm for computing diagnoses (Reiter 1987). Although the fiable and 8G ⊆ N ^ G= 6 ;; (F n N ) [G is unsatisfiable. original purpose was to enumerate diagnoses, the algorithm An MCS can be defined as the complement of a Maximal can be easily adapted to enumerate all MUSes. The algo- Satisfiable Subset (MSS): rithm also makes use of a tree (the HS-tree), whose goal is to enumerate all conflict sets without repetition. Each time Definition 3. A satisfiable subset S ⊆ F is an MSS if 8C ⊆ a new node is expanded, a new conflict set is computed by F n S ^ C 6= ;; S[C is unsatisfiable. means of a theorem prover or a new diagnosis is extracted For partial MUS enumeration (Liffiton and Sakallah from the tree. The algorithm can be modified to enumer- 2008) use Partial Correction Subsets (PCSes): ate MUSes, if in place of generic conflict sets, the theorem Definition 4.