Real-World Knowledge Representation and Reasoning

Abstract

Apart from its theoretical significance, the AI must represent real-world knowledge, and produce reasoning using that. The real-world things are collections of entities in different classes. This chapter presents the representations structures for such knowledge, e.g., taxonomies and reasoning based on that. Other phenomena in real-world, that are presented are, action and change, commonsense reasoning, ontology structures for different domains, like, language, and world. The Sowa’s ontology for objects, and processes, both concrete and abstract, is explained. The situation calculus is presented in its formal details, along with worked exercises. The more prevalent real-world reasoning, like nonmonotonic and default reasoning are also treated in sufficient details, along with supporting worked exercises. This is followed, with summary of the chapter, and exhaustive list of practice exercises.

Exercises

• 1. Give the top level ontology of following structures, represent the concepts using relations, and attributes.

  • a. University system—consisting of faculties, departments, teachers, classes, students, courses, etc.

  • b. Organizational ontology of a manufacturing firm.

  • c. Organizational ontology of a project based software company.

  • d. Government system ontology with various bodies and their responsibilities.

• 2. Suggest some applications of Sowa’s ontology.

• 3. Compare and contrast the Wordnet and Sowa’s ontology, and explain the reasoning performed in both with small examples.

• 4. Represent the ontologies of the following worlds, and explain, how you will perform the question-answering using each of these ontology?

  • a. Ontology of Shirt.

  • b. Ontology of Dining table.

  • c. Ontology of University system.

• 5. Suggest the approach, how you will generate e-learning exercises using the ontology.

• 6. What is unnatural deduction in reference to situation calculus? Give examples to justify your claims.

• 7. How the inheritance works in a world of contexts? For example, in space-craft, on earth, and when context changes from one to other?

• 8. Show some similarities between “contexts” and “properties” in expressing situation calculus axioms.

• 9. Consider a robotic-hand which can move between several bins, pickup an object from the bin if the hand is above the bin and the hand is empty. The hand can drop an object into a bin if the hand is holding an object and the hand is above the bin. Moving of hand from any bin to any other bin is always possible, it does not require any preconditions. The actions are:

  • drop(x, y) (drop object x into bin y)

  • move(y) (move hand to be above the bin y)

  • grab(x, y) (pickup object x from bin y).

  The fluents are:

  • holding(x, s) (the hand is holding x in situation s)

  • over(y, s) (the hand is over bin y in situation s)

  • in(x, y, s) (object x is in the bin y in a situation s).

  • a. Write the axioms for move, drop and grab actions.

  • b. Write the successor state axioms for all the fluents.

• 10. A robot is to pickup n ($n=10$) cuboid lying on the table and drop one-by-one in a bucket, available nearby the table. Write the statements for sequential calculus. What are the situations, actions, and fluents here?

• 11. Consider the eight puzzle shown in Fig. 6.8 with initial and goal states (situations).  

  The objective is to go from a initial situation to the goal situation. We are allowed to move a tile into the empty space if that tile is adjacent to the empty space (e.g. in the initial situation tile 6 and 8 are adjacent to empty space). The locations are numbered as 1-9, as shown in initial situation, with number 9 as empty tile. The tiles are numbered 1-8. There is a single action move(t, l) which indicates moving tile t to location l. Assume a predicate $adjacent(l_1 , l_2)$, which is true when location $l_2$ is one move from $l_1$. It is only possible to do a move action if a tile is in a location adjacent to an empty location. The only fluent is location(t, l, s) meaning tile t is in location l in situation s. Given this, write down:

  • initial conditions,

  • effect axioms,

  • precondition axioms, and

  • from the effect axioms derive the successor state axioms.

• 12. Given the following set of facts and default rules:

  • a. People typically live in the same city where they work (default: $d_1$)

  • b. People typically live in the same city where their spouses are (default: $d_2$)

  • c. John works in New Delhi (fact: $f_1$)

  • d. John’s spouse works in Mumbai (fact: $f_2$)

  Answer these questions:

  • a. Where does John live according to default logic?

  • b. Where does John live according to your intuition?

• 13. Formalize these set of facts and default rules:

  • a. Bob usually speaks the truth ($d_1$).

  • b. John usually speaks the truth ($d_2$).

  • c. Bob says that the suspect stabbed the victim to death ($f_1$).

  • d. John says that the suspect shot the victim to death ($f_2$).

  • e. Nobody can be both stabbed and shot to death ($f_3$).

  • f. Stabbing or shooting to death is killing ($f_4$).

  Answer these questions:

  • a. Did the suspect kill the victim according to default logic?

  • b. Did the suspect kill the victim according to your intuitions?

• 14. Is the formula (6.10) sufficient and correct form of default reasoning, with X as variable? Justify your answer.

• 15. Translate the following into first-order predicate logic, and check whether the given conclusion follow from it?

  Typically, the computer science students like computers. Female students who like computers are typically interested in cognitive science. The computer science students are typically female: for example Anita, Babita, Cathy; but Dorthy is an exception to this rule. Conclusion: Anita, Babita, Cathy are interested in cognitive science; Dorthy is not interested in cognitive science.

• 16. Compute the default extensions of following theories $T = (M, D)$:

  • a. $M=\{a\}, D = \{\frac{a: \lnot b}{c}, \frac{:\lnot c}{d}, \frac{:\lnot d}{e}\}$

  • b. $M=\{a\rightarrow c, b \rightarrow c \}, D = \{\frac{: \lnot b}{a}, \frac{:\lnot a}{b}, \frac{:\lnot d}{e}\}$

  • c. $M=\{ \}, D = \{ \frac{: \lnot b}{a}, \frac{:\lnot a}{b}, \frac{:\lnot d}{d}\}$

  • d. $M=\{p\wedge q\}, D = \{ \frac{b : a}{a}, \frac{:\lnot a}{a}, \frac{:\lnot a}{\lnot c}, \frac{:\lnot q}{b},\frac{:\lnot p}{q}\}$

• 17. Compute the default extensions of $T = \langle M, D\rangle$, where

  $$\begin{aligned}&M = \{\forall x [mynah(x) \rightarrow \lnot nests(x)],\\&\qquad \forall x [penguin(x) \rightarrow \lnot flies(x)],\\&\qquad \forall x [birds(x) \equiv mynah(x) \vee penguin(x) \vee canary(x)], \\&\qquad bird(Tweety)\},\\&D = \{\frac{bird(x): nests(x)}{nest(x)}, \frac{bird(x): flies(x)}{flies(x)}\} \end{aligned}$$

• 18. Find the extensions of the following default theories:

  • a. $T=\langle \{ \}, \{\frac{ :\lnot p}{p}, \frac{p\vee q: \lnot p}{\lnot p}\}\rangle$

  • b. $T=\langle \{\lnot Sun\text {-}shining \wedge Summer\}, \{\frac{Summer:\lnot Rain}{Sun\text {-}shining}\}\rangle$

  • c. $T=\langle \{ \}, \{\frac{r:\exists x~P(x)}{\exists x~p(x)}, \frac{:r \wedge \lnot p(x)}{r \wedge \lnot p(x)}\}\rangle$

  • d. $T=\langle \{p\vee q\}, \{\frac{ :\lnot p}{p}, \frac{p\vee q: \lnot p}{\lnot p}\}\rangle$

• 19. Assume that $\langle D, W\rangle$ be a propositional default theory, and $D^\prime$ be a set of normal defaults such that $D \subseteq D^\prime$. If E is an extension of $\langle D, W\rangle$, then show that there exists an extension $E^\prime$ of $\langle D^\prime , W\rangle$ such that $E \subseteq E^\prime$.