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**7.17** A propositional 2-CNF expression is a conjunction of clauses, each containing *exactly* 2 literals, e.g.,

$$(A \lor B) \land (\lnot A \lor C) \land (\lnot B \lor D) \land (\lnot C \lor G) \land (\lnot D \lor G)$$

**a**. Prove using resolution that the above sentence entails G.**b**. Two clauses are *semantically distinct* if they are not logically equivalent. How many semantically distinct 2-CNF clauses can be constructed from n proposition symbols?**c**. Using your answer to (b), prove that propositional resolution always terminates in time polynomial in n given a 2-CNF sentence containing no more than n distinct symbols.**d**. Explain why your argument in (c) does not apply to 3-CNF.

详细的证明请见 pdf.

(Thanks to Lv Feng.)

本次作业，利用课本讲授的包含11条推演规则的命题逻辑系统，以形式推演的方式证明了课本中的

- 定理2.6.4
- 定理2.6.9

详细的证明请见 pdf.

(Thanks to Rongqing Wang.)

向陆钟万先生致敬！

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We also find that we can also perform the scout step even for the first child of a node. It means that the constraint on the “first action” (see Line 6 of the algorithm in the pdf) is **NOT** necessary. It is only necessary when $\alpha = -\infty $.

Please check the following pdf.

(Thanks to Rongqing Wang.)

- If a heuristic is consistent, it must be admissible. We also give an example heuristic which is admissible, but not consistent.
`A*`

of graph search is**NOT**optimal with**admissible heuristic**.

For the second property, the course book only says that “`A*`

of graph search is optimal with **consistent heuristic**”. Some students are wondering whether the conclusion that

** A* of graph search is optimal with admissible heuristic **

is true or not (see the page). Here, we show this conclusion is wrong. The key point is that the frontier set in graph search include both the visited and unvisited nodes.

Please check the following pdf.

(Thanks to Xingcheng Ruan.)

- The graph separation property in graph search
- The optimality on uniform-cost search

Please check the following pdf.

(Thanks to Qinlin Zhu.)

Thank you for your interest in our materials developed for AI Fundamental course (2017) at University of Chinese Academy of Sciences. At this web site, you will find the following:

- Some lecture slides (mainly for the third part of this course)
- Detailed analysis on some key questions, which are not well addressed in the course book. These questions are usually used as the homework for the students in the course.

The whole course includes the following three parts:

Part 1: Search

Part 2: Knowledge

Part 3: Learning

The materials for the first two parts are mainly based on AIMA (CS 188 at Berkeley) and the AI course at Taiwan University. The materials for the last part are mainly developed by the instructor himself.

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