AI 中文总结
本文设计作业让完成CS1/CS2的本科生小组自行证明Fortune定理，帮助学生理解复杂性理论相关知识，提升问题解决与研究技能。
AI 中文摘要
本文提供了一项作业设计，旨在让完成本科CS1/CS2系列课程的本科生以小组形式自行证明Fortune定理。（Fortune定理指出：若布尔可满足性问题的补集可多项式时间归约到一个稀疏集，则布尔可满足性问题是多项式时间可计算的。该作业不要求学生预先了解布尔可满足性问题、多项式时间归约或稀疏集，而是在作业中教授这些内容。注：用复杂性理论的技术术语重述后，Fortune定理表明：除非P=NP，否则不存在任何稀疏集是coNP-难的。Fortune定理是对难度与密度之间关系理解的重大进展。）我们提供了作业讲义（作为本报告主体内容加上附录A）和作业解答（作为附录B，当然在学生提交作业后才会提供给他们）。作业讲义的安排是（教师可调整）：学生以小组形式在课堂上用一整节课开始作业，然后以同样的小组形式作为带回家的作业完成，并在下节课前提交。我们发现学生小组经常能在这项挑战中取得部分或全部成功。这对学生意义重大：他们会意识到自己取得了一项进展，而该进展首次提出时，发表在当时可说是复杂性理论研究领域顶级期刊的刊物上。这能让他们获得信心，相信当真正投入到给定挑战中时，自己具备扎实的问题解决技能（基本等同于研究技能）。
英文摘要
This article provides an assignment designed to let undergraduate students who have completed an undergraduate CS1/CS2 sequence try to themselves, in groups, prove Fortune's Theorem. (Fortune's Theorem states that if the complement of the Boolean satisfiability problem polynomial-time reduces to a sparse set, then the Boolean satisfiability problem is polynomial-time computable. The assignment does not assume that students have previously seen the Boolean satisfiability problem, polynomial-time reductions, or sparse sets. Rather, it teaches those within the assignment. Note: Reworded into the technical vocabulary of complexity theory, Fortune's Theorem states that no sparse set is coNP-hard unless P=NP. Fortune's Theorem was a major advance in the understanding of the relationship between hardness and density.) We provide both the assignment handout (as the main body of this report plus Appendix A) and a solution to the assignment (as Appendix B, which would of course not be made available to the students until after they had handed in the assignment). The assignment handout, though the instructor can change this, is framed as having the students starting the assignment in teams in class for a whole class session, and then finishing it in those same teams as a take-home assignment, and handing it in before the next class session. We have found that student groups often succeed, partially or completely, in this challenge. This can mean a lot to the students: they see that they were able to make an advance that, when it was first obtained, appeared in what was arguably at the time the top journal venue for complexity theory research. This can give them confidence that they have substantial problem-solving skills (which basically means research skills) when they truly apply themselves to a given challenge.
Comments22 pages; 1 figure
