Boosting MUC extraction in unsatisfiable constraint networks
Grégoire, É., Lagniez, JM. & Mazure, B. Boosting MUC extraction in unsatisfiable constraint networks. Appl Intell 41, 1012–1023 (2014). https://doi.org/10.1007/s10489-014-0549-6
https://doi.org/10.1007/s10489-014-0549-6
各种提取一个MUC的方法：
本文方法和参考文献 [2, 10, 11, 13–17, 34].
Abstract
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One very fertile domain of applied Artificial Intelligence is constraint solving technologies. Especially, constraint networks that concern problems that can be represented using discrete variables, together with constraints on allowed instantiation values for these variables. Every solution to a constraint network must satisfy every constraint. 译文：一个非常丰富的应用人工智能领域是约束解决技术。特别是约束网络，它涉及的问题可以用离散变量表示，以及这些变量允许的实例化值的约束。约束网络的每个解必须满足每个约束 |
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When no solution exists, the user might want to know the actual reasons leading to the absence of global solution. In this respect, extracting MUCs (Minimal Unsatisfiable Cores) from an unsatisfiable constraint network is a useful process when causes of unsatisfiability must be understood so that the network can be re-engineered and relaxed to become satisfiable. 译文：当不存在解决方案时，用户可能想知道导致没有全局解决方案的实际原因。在这方面，当不能满足的原因必须被理解时，从一个不能满足的约束网络中提取MUCs(最小不可满足的核心)是一个有用的过程，这样网络就可以被重新设计和放松，从而变得可以满足。 |
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Despite bad worst-case computational complexity results, various MUC-finding approaches that appear tractable for many real-life instances have been proposed. Many of them are based on the successive identification of so-called transition constraints. 译文：尽管最坏情况下的计算复杂度结果很糟糕，但已经提出了各种似乎可以处理许多现实生活实例的muc -find方法。其中许多是基于所谓的过渡约束的连续识别。 |
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In this respect, we show how local search can be used to possibly extract additional transition constraints at each main iteration step. 译文：在这方面，我们展示了如何使用局部搜索在每个主要迭代步骤提取额外的转换约束。 In the general constraint networks setting, the approach is shown to outperform a technique based on a form of model rotation imported from the SAT-related technology and that also exhibits additional transition constraints. 译文：在一般约束网络设置中，该方法优于一种基于从sat相关技术导入的模型旋转形式的技术，该技术也显示了额外的过渡约束。 Our extensive computational experimentations show that this enhancement also boosts the performance of state-of-the-art DC(WCORE)-like MUC extractors. 译文：我们大量的计算实验表明，这种增强也提高了最先进的DC(WCORE)类MUC提取器的性能。 |
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Introduction
| One very fertile domain of applied Artificial Intelligence is constraint solving technologies. Especially, constraint networks concern problems that can be represented using discrete variables, together with constraints linking allowed instantiation values for these variables (see for example [19] and [30] for an introduction to the fields of constraint networks and constraint-solving problems, in short CSP). Every solution to a constraint network is an assignment of values to all variables such that every constraint is satisfied. In this paper, the focus is on unsatisfiable constraint networks, i.e., constraint networks that do not have any solution. When no solution exists, the user might want to know the actual reasons for this. From a technical point of view, these reasons are related to the existence of unsatisfiable subsets of constraints that are minimal in the sense that dropping any constraint in each of these subsets would allow the resulting network to become satisfiable. When causes of unsatisfiability must be understood and the network must be re-engineered and relaxed to become satisfiable, extracting such minimal sets of incompatible constraints can be a cornerstone issue since each such set provides one of the causes for unsatisfiability. | |
| Technically, this paper introduces a new approach for extracting minimal cores (or, MUCs for Minimally Unsatisfiable Cores) of constraint networks. A MUC is a minimal (w.r.t. ⊆) set of constraints that cannot be satisfied all together. Despite bad worst-case computational complexity results, various approaches for extracting one MUC have been proposed that appear tractable for many instances [2, 10, 11, 13–17, 34]. | |
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A MUC of a network can also be defined as an unsatisfiable sub-network formed of transition constraints, which are constraints that would allow this sub-network to become satisfiable if any of them was removed. Powerful approaches to MUC extraction are founded on transition constraints, both in the CSP [10, 13] and the Boolean satisfiability (i.e., SAT) [5, 8, 9, 20, 20, 32, 35] domains. 译文：一个网络的MUC也可以被定义为一个由转换约束构成的不可满足的子网络，这些约束允许这个子网络在任何一个转换约束被移除后变得可满足。在CSP[10, 13]和布尔可满足性(即SAT)[5, 8, 9, 20, 20, 32, 35]域中，基于过渡约束建立了强大的MUC提取方法。
In the latter area, a recent approach [4, 26] focuses on the following intuition. An assignment of values to the variables that satisfies all constraints except one is called a transition assignment and the unsatisfied constraint is a transition constraint: additional transition constraints might be discovered by so-called model rotation, i.e., by examining other assignments differing from the transition assignment on the value of one variable, only. |
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| In the paper, an approach both extending and enhancing this latter technique is proposed in the general constraint network framework. The main idea is to use local search for exploring the neighborhood of transition assignments in an attempt to find out other transition constraints. The technique is put to work in a so-called dichotomy destructive strategy à la DC(WCORE) [13] to extract one MUC. Extensive computational experimentations show that this approach outperforms both the model rotation technique from [4] in the general constraint networks setting and the performance of state-of-the-art MUC extractors. | |
| The paper is organized as follows. In the next section, basic concepts, definitions and notations are provided. Section 3 focuses on existing techniques for MUC extraction, including DC(WCORE)-like ones, and then on model rotation. In section 4, a local search procedure for exhibiting additional transition constraints is presented and motivated, whereas section 5 describes the full algorithm for MUC extraction. Section 6 describes our experimental investigations and results before some promising paths for further research are presented in the conclusion. | |
Conclusion and perspectives
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Clearly, the technique proposed in this paper to extract one MUC in a constraint network outperforms previous approaches. 译文：显然，本文提出的在约束网络中提取一个MUC的技术优于以往的方法。 |
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| Although dichotomy strategies, as explored in this paper, are known to be the most efficient ones, it could be interesting to graft this local search scheme to constructive or QuickXplain-like methods. Also, note that we have not tried to fine-tune the various parameters of this local search scheme. In this respect, it would be interesting to devise forms of dynamical settings for these parameters that better take the recorded information about the previous search steps into account, as explored in [10]. In the future, we plan to explore more advanced concepts that are related to transition constraints in the goal of better guiding the local search towards promising parts of the search space. Especially, so-called critical clauses [8] in the Boolean framework could be generalized in various ways in the full constraint networks setting. Exploring the possible ways according to which LSTC could benefit from this is a promising path for further research. | |
Notes
The d o m/w d e g scores collected during the WCORE step are used to rank-order constraints.
The benchmarks are available at http://www.cril.univ-artois.fr/~lecoutre
Third international CSP solver competition. http://cpai.ucc.ie/08/ 2008
Fourth international constraint solver competition. http://cpai.ucc.ie/09/ 2009
The executable is available at http://www.cril.univ-artois.fr/~lagniez
References
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