Abstract

Abductive reasoning is a non-monotonic formalism stemming from the work of Peirce. It describes the process of deriving the most plausible explanations of known facts. Considering the positive version, asking for sets of variables as explanations, we study, besides the problem of wether there exists a set of explanations, two explanation size limited variants of this reasoning problem (less than or equal to, and equal to a given size bound). In this paper, we present a thorough two-dimensional classification of these problems: the first dimension is regarding the parameterized complexity under a wealth of different parameterizations, and the second dimension spans through all possible Boolean fragments of these problems in Schaefer’s constraint satisfaction framework with co-clones (T. J. Schaefer. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1–3, 1978, San Diego, California, USA, R.J. Lipton, W.A. Burkhard, W.J. Savitch, E.P. Friedman, A.V. Aho eds, pp. 216–226. ACM, 1978). Thereby, we almost complete the parameterized complexity classification program initiated by Fellows et al. (The parameterized complexity of abduction. In Proceedings of the Twenty-Sixth AAAI Conference on Articial Intelligence, July 22–26, 2012, Toronto, Ontario, Canada, J. Homann, B. Selman eds. AAAI Press, 2012), partially building on the results by Nordh and Zanuttini (What makes propositional abduction tractable. Artificial Intelligence, 172, 1245–1284, 2008). In this process, we outline a fine-grained analysis of the inherent parameterized intractability of these problems and pinpoint their FPT parts. As the standard algebraic approach is not applicable to our problems, we develop an alternative method that makes the algebraic tools partially available again.

中文翻译：

---

Schaefer框架中绑架的参数化复杂性

摘要

归纳推理是源自皮尔斯的工作的非单调形式主义。它描述了对已知事实进行最合理解释的过程。考虑正向版本，要求变量集作为解释，我们研究，除了存在一组解释的问题以外，该推理问题的两个解释大小受限变体（小于或等于并等于给定大小界）。在本文中，我们对这些问题进行了全面的二维分类：第一维涉及在许多不同参数化情况下的参数化复杂性，第二维跨越了Schaefer约束满足框架中这些问题的所有可能的布尔片段。共克隆（TJ Schaefer。可满足性问题的复杂性。第十届ACM计算理论年度研讨会论文集，1978年5月1日至3日，美国加利福尼亚州圣地亚哥，RJ Lipton，WA Burkhard，WJ Savitch，EP Friedman，AV Aho eds，第216-226页。ACM，1978年）。因此，我们几乎完成了由Fellows等人发起的参数化复杂度分类程序。（绑架的参数化复杂性。在第二十六届AAAI人工智能会议论文集，2012年7月22日至26日，加拿大安大略省多伦多市， J。Homann，B。Selman编辑，AAAI出版社，2012年），部分构建基于Nordh和Zanuttini的研究结果（是什么使命题绑架变得易于处理。人工智能，172，1245–1284，2008）。在此过程中，我们概述了对这些问题的固有参数化难处理性的细粒度分析，并指出了它们的FPT部分。由于标准代数方法不适用于我们的问题，因此我们开发了另一种方法，使代数工具再次部分可用。