aima logo

© 2020. All rights reserved.

AIMA Artificial Intelligence a modern approch

AIMA-exercises is an open-source community of students, instructors and developers. Anyone can add an exercise, suggest answers to existing questions, or simply help us improve the platform. We accept contributions on this github repository.
Edit Question

Exercise 8.12

In each of the following we give an English sentence and a number of candidate logical expressions. For each of the logical expressions, state whether it (1) correctly expresses the English sentence; (2) is syntactically invalid and therefore meaningless; or (3) is syntactically valid but does not express the meaning of the English sentence.

  1. Every cat loves its mother or father.

    1. ${\forall\,x\;\;} {Cat}(x) {:\;{\Rightarrow}:\;}{Loves}(x,{Mother}(x)\lor {Father}(x))$.

    2. ${\forall\,x\;\;} \lnot {Cat}(x) \lor {Loves}(x,{Mother}(x)) \lor {Loves}(x,{Father}(x))$.

    3. ${\forall\,x\;\;} {Cat}(x) \land ({Loves}(x,{Mother}(x))\lor {Loves}(x,{Father}(x)))$.

  2. Every dog who loves one of its brothers is happy.

    1. ${\forall\,x\;\;} {Dog}(x) \land (\exists y\ {Brother}(y,x) \land {Loves}(x,y)) {:\;{\Rightarrow}:\;}{Happy}(x)$.

    2. ${\forall\,x,y\;\;} {Dog}(x) \land {Brother}(y,x) \land {Loves}(x,y) {:\;{\Rightarrow}:\;}{Happy}(x)$.

    3. ${\forall\,x\;\;} {Dog}(x) \land [{\forall\,y\;\;} {Brother}(y,x) {\;\;{\Leftrightarrow}\;\;}{Loves}(x,y)] {:\;{\Rightarrow}:\;}{Happy}(x)$.

  3. No dog bites a child of its owner.

    1. ${\forall\,x\;\;} {Dog}(x) {:\;{\Rightarrow}:\;}\lnot {Bites}(x,{Child}({Owner}(x)))$.

    2. $\lnot {\exists\,x,y\;\;} {Dog}(x) \land {Child}(y,{Owner}(x)) \land {Bites}(x,y)$.

    3. ${\forall\,x\;\;} {Dog}(x) {:\;{\Rightarrow}:\;}({\forall\,y\;\;} {Child}(y,{Owner}(x)) {:\;{\Rightarrow}:\;}\lnot {Bites}(x,y))$.

    4. $\lnot {\exists\,x\;\;} {Dog}(x) {:\;{\Rightarrow}:\;}({\exists\,y\;\;} {Child}(y,{Owner}(x)) \land {Bites}(x,y))$.

  4. Everyone’s zip code within a state has the same first digit.

    1. ${\forall\,x,s,z_1\;\;} [{State}(s) \land {LivesIn}(x,s) \land {Zip}(x)z_1] {:\;{\Rightarrow}:\;}{}$\ $[{\forall\,y,z_2\;\;} {LivesIn}(y,s) \land {Zip}(y)z_2 {:\;{\Rightarrow}:\;}{Digit}(1,z_1) {Digit}(1,z_2) ]$.

    2. ${\forall\,x,s\;\;} [{State}(s) \land {LivesIn}(x,s) \land {\exists\,z_1\;\;} {Zip}(x)z_1] {:\;{\Rightarrow}:\;}{}$\ $ [{\forall\,y,z_2\;\;} {LivesIn}(y,s) \land {Zip}(y)z_2 \land {Digit}(1,z_1) {Digit}(1,z_2) ]$.

    3. ${\forall\,x,y,s\;\;} {State}(s) \land {LivesIn}(x,s) \land {LivesIn}(y,s) {:\;{\Rightarrow}:\;}{Digit}(1,{Zip}(x){Zip}(y))$.

    4. ${\forall\,x,y,s\;\;} {State}(s) \land {LivesIn}(x,s) \land {LivesIn}(y,s) {:\;{\Rightarrow}:\;}{}$\ ${Digit}(1,{Zip}(x)) {Digit}(1,{Zip}(y))$.

Answer Improve This Solution
View Answer
Request Answer