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+startxref +112031 +%%EOF \ No newline at end of file diff --git a/logic/main.typ b/logic/main.typ new file mode 100644 index 0000000..e562b1c --- /dev/null +++ b/logic/main.typ @@ -0,0 +1,96 @@ +#set page(paper: "a4", margin: 2cm) +#set text(font: ("Noto Sans CJK SC", "SimSun"), size: 10.5pt) +#set heading(numbering: "1.") + +#align(center)[ + #block(text(weight: "bold", size: 18pt)[人工智能课程ch07-09逻辑作业报告]) +] + += 命题逻辑合取范式转换 +*题目:* 求 $(P and (Q -> R)) -> S$ 的合取范式。 + +*解答:* +1. 消去蕴含符号: + $ not (P and (not Q or R)) or S $ +2. 应用德·摩根定律: + $ (not P or not (not Q or R)) or S $ + $ (not P or (Q and not R)) or S $ +3. 应用分配律: + $ ((not P or Q) and (not P or not R)) or S $ +4. 再次应用分配律(将 $S$ 结合进去): + $ (not P or Q or S) and (not P or not R or S) $ + +*最终结果:* $(not P or Q or S) and (not P or not R or S)$ + + += 复合语句分析 +*题目:* 考虑语句 $[((F or D) => P) or (M => P)] => [(F and M) => P]$ (简写变量名)。 + +*(a) 通过枚举模型判断有效性:* +设 $F, D, M, P$ 为布尔变量。考虑一种情况:使结论 $(F and M) => P$ 为假。 +只有当 $F=T, M=T, P=F$ 时,结论为假。 +代入左边:$[((T or D) => F) or (T => F)] equiv [F or F] equiv F$。 +由于“假 $=>$ 假”为真,该语句在所有使结论为假的解释下依然为真。经穷举,该语句在所有解释下均为真。 +*结论:* 该语句是 *有效的 (Valid)*。 + +*(b) 转换为合取范式 (CNF):* +左边 (L):$((F or D) => P) or (M => P)$ +$ equiv (not(F or D) or P) or (not M or P) $ +$ equiv (not F and not D) or P or not M or P $ +$ equiv (not F or not M or P) and (not D or not M or P) $ + +右边 (R):$(F and M) => P equiv not F or not M or P$ + +全式 $L => R equiv not L or R$: +由于 $R$ 已经是 $L$ 的一个合取项,根据 $(A and B) => A$ 永远为真,转换结果支持了有效性结论。 + + += 一阶逻辑翻译 (基础谓词) +*题目要求:使用给定谓词表示语句。* + +1. *有一门课所有老师都教授:* + $ exists c ("Course"(c) and forall t ("Teacher"(t) => "Teach"(t, c))) $ + +2. *恰有一门课由 "Alice" 教授且被 "Bob" 选修:* + $ exists c ("Course"(c) and "Teach"("Alice", c) and "Study"("Bob", c) and forall c' (("Teach"("Alice", c') and "Study"("Bob", c')) => c = c')) $ + +3. *所有老师教授的课程都至少有一个学生选修:* + $ forall t forall c (("Teacher"(t) and "Teach"(t, c)) => exists s ("Student"(s) and "Study"(s, c))) $ + +4. *有一个学生选修了 "Alice" 教授的所有课程:* + $ exists s ("Student"(s) and forall c ("Teach"("Alice", c) => "Study"(s, c))) $ + + += 一阶逻辑翻译 (自定义谓词) +*自定义谓词:* $"Fail"(s, c)$ 表示 $s$ 考试缺考;$"Pass"(s, c)$ 表示 $s$ 通过课程 $c$;$"Borrow"(s, b)$ 表示学生 $s$ 借阅书 $b$。 + +1. *有老师只教授了一门课程:* + $ exists t ("Teacher"(t) and exists c ("Teach"(t, c) and forall c' ("Teach"(t, c') => c' = c))) $ + +2. *如果一名学生缺考,他就无法通过该课程:* + $ forall s forall c (("Student"(s) and "Course"(c) and "Fail"(s, c)) => not "Pass"(s, c)) $ + +3. *没有一本书被两名学生同时借出:* + $ not exists b exists s_1 exists s_2 ("Book"(b) and "Student"(s_1) and "Student"(s_2) and s_1 != s_2 and "Borrow"(s_1, b) and "Borrow"(s_2, b)) $ + + += 推理有效性论证 (归结证明法) +*前提:* 所有的智能手机都是电子设备。 $forall x ("Smartphone"(x) => "ElectronicDevice"(x))$ +*结论:* 任何智能手机的制造商都是电子设备制造商。 $forall x forall y (("Smartphone"(x) and "ManufacturedBy"(x, y)) => exists z ("ElectronicDevice"(z) and "ManufacturedBy"(z, y)))$ + +*归结证明步骤:* +1. *前提子句化:* + $C_1: not "Smartphone"(x) or "ElectronicDevice"(x)$ +2. *结论否定并子句化:* + $not [forall x forall y (("Smartphone"(x) and "ManufacturedBy"(x, y)) => exists z ("ElectronicDevice"(z) and "ManufacturedBy"(z, y)))]$ + 引入常量 $S, M$ (Skolemization): + $C_2: "Smartphone"(S)$ + $C_3: "ManufacturedBy"(S, M)$ + $C_4: not "ElectronicDevice"(z) or not "ManufacturedBy"(z, M)$ +3. *推导矛盾:* + - 由 $C_1$ 和 $C_2$ (置换 $\{x/S\}$): $"ElectronicDevice"(S) (C_5)$ + - 由 $C_4$ 和 $C_3$ (无法直接归结,但由于 $S$ 满足 $"ManufacturedBy"(S, M)$): + 将 $C_4$ 中的 $z$ 置换为 $S$: $not "ElectronicDevice"(S) (C_6)$ + - $C_5$ 与 $C_6$ 归结产生空子句 $square$。 + +*结论:* 推理有效。