A note on the complexity of the causal ordering problem

详细信息    查看全文

• 作者：Bernardo Gonç ; alves^a ; ^{begoncalves@acm.org" class="auth_mail" title="E-mail the corresponding author} ; Fabio Porto^b ; ^{fporto@lncc.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Causal ordering ; Causal reasoning ; Structural equations ; Hypothesis management
• 刊名：Artificial Intelligence
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：238
• 期：Complete
• 页码：154-165
• 全文大小：516 K

文摘

In this note we provide a concise report on the complexity of the causal ordering problem, originally introduced by Simon to reason about causal dependencies implicit in systems of mathematical equations. We show that Simon's classical algorithm to infer causal ordering is NP-Hard—an intractability previously guessed but never proven. We present then a detailed account based on Nayak's suggested algorithmic solution (the best available), which is dominated by computing transitive closure—bounded in time by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0004370216300674&_mathId=si1.gif&_user=111111111&_pii=S0004370216300674&_rdoc=1&_issn=00043702&md5=d5a19d3caed7c92a45cc65196febf9b2" title="Click to view the MathML source">O(|V|⋅|S|)class="mathContainer hidden">class="mathCode">$O(|\mathcal{V}|\cdot |\mathcal{S}|)$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0004370216300674&_mathId=si173.gif&_user=111111111&_pii=S0004370216300674&_rdoc=1&_issn=00043702&md5=8627cabbf8b2e3f9a18cae7db921e964" title="Click to view the MathML source">S(E,V)class="mathContainer hidden">class="mathCode">$\mathcal{S}(\mathcal{E},\mathcal{V})$ is the input system structure composed of a set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0004370216300674&_mathId=si3.gif&_user=111111111&_pii=S0004370216300674&_rdoc=1&_issn=00043702&md5=3421d89761e3f480d33ab00ea9ad8bc6" title="Click to view the MathML source">Eclass="mathContainer hidden">class="mathCode">$\mathcal{E}$ of equations over a set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0004370216300674&_mathId=si20.gif&_user=111111111&_pii=S0004370216300674&_rdoc=1&_issn=00043702&md5=ec586d7ed89f56f131a5ec74115f1ff7" title="Click to view the MathML source">Vclass="mathContainer hidden">class="mathCode">$\mathcal{V}$ of variables with number of variable appearances (density) class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0004370216300674&_mathId=si5.gif&_user=111111111&_pii=S0004370216300674&_rdoc=1&_issn=00043702&md5=237cc1eacad1f8409b5a334139a7be1d" title="Click to view the MathML source">|S|class="mathContainer hidden">class="mathCode">$|\mathcal{S}|$. We also comment on the potential of causal ordering for emerging applications in large-scale hypothesis management and analytics.