1 / 18

CS 2710, ISSP 2610

CS 2710, ISSP 2610. Chapter 8, Part 2 First Order Predicate Calculus FOPC. Review. Sentence  AtomicSentence | (Sentence Connective Sentence) | Quantifier Variable, .. Sentence | ~Sentence AtomicSentence  Predicate(Term,…) | Term = Term Term  Function(Term,…) |

baina
Download Presentation

CS 2710, ISSP 2610

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CS 2710, ISSP 2610 Chapter 8, Part 2 First Order Predicate Calculus FOPC

  2. Review Sentence  AtomicSentence | (Sentence Connective Sentence) | Quantifier Variable, .. Sentence | ~Sentence AtomicSentence  Predicate(Term,…) | Term = Term Term  Function(Term,…) | Constant | Variable Connective   | ^ | v |  Quantifier  all, exists Constant  john, 1, … Variable  A, B, C, X Predicate  breezy, sunny, red Function  fatherOf, plus Knowledge engineering involves deciding what types of things Should be constants, predicates, and functions for your problem

  3. ReviewSemantics of FOPC • Interpretation: assignment of elements from the world and elements of the language • The world consists of a domain ofobjects D, a set of predicates, and a set of functions

  4. ReviewSemantics • Each constant is assigned an element of the domain • Each N-ary predicate is assigned a set of N-tuples • Each N-ary function symbol is assigned a set of N+1 tuples that does not include any pair of tuples with the first N elements and different (n+1)st elements

  5. Sample Domain • Richard I, the Lionhearted (1189-1199) • King John II, John Lackland, Evil King John • Their left legs • A crown • Picture in text (p. 291)

  6. Predicate of brotherhood: {<R,J>,<J,R>} Predicate of being on: {<C,J>} Predicate of being a person: {<J>,<R>} or just {J,R} Predicate of being the king: {J} Predicate of being a crown: {C} Function for left legs: {<J,JLL>,<R,RLL>} (book writes <J>  JLL, <R>  RLL) Predicate of being strong: {JLL} (Functions are total, so the crown and left legs have left legs too. Technical solution: it is the left leg of things that don’t have left legs. This is “invisible” and we will make no assertions about it, so we can ignore it)

  7. Interpretation • Specifies which objects, functions, and predicates are referred to by which constant symbols, function symbols, and predicate symbols. • Under the intended interpretation: • “richardI”, “richardLionhearted” refer to R; “johnII”,”johnLackland” refer to J; “crown” refers to the crown. • “onHead”,”brother”,”person”,”king”, “isCrown”, “leftLeg”, “strong” • [some changes 10/8; will be written on the board]

  8. Lots of other possible interpretations • 5 objects, so just for constants “richardI” and “johnII” there are 25 possibilities • Note that the legs don’t have their own names! • “johnII” and “johnLackland” may be assigned the same object, J

  9. Why isn’t the “intended interpretation” enough? • Vague notion. What is intended may be ambiguous (and often is, for non-toy domains) • Logically possible: square(x) ^ round(x). Your KB has to include knowledge that rules this out.

  10. Determining truth values of FOPC sentences • Assign meanings to terms: • “johnII”  J; “leftLeg(johnII)” JLL • Assign truth values to atomic sentences • “brother(johnII,richardI)” • “brother(johnlackland,richardI)” • Both True, because <J,R> is in the set assigned “brother” • “strong(leftleg(johnlackland))” • True, because JLL is in the set assigned “strong”

  11. Determining Truth Values (continued) • Connectives and negation are the same as in propositional logic • All X G is True if G is true with X assigned d, for all d in the domain. • Exists X G is True if G is true with X assigned d, for some d in the domain.

  12. Examples given the Sample Interpretation • All X,Y brother(X,Y) FALSE • All X,Y ((person(X) ^ person(Y))  brother(X,Y))FALSE • All X,Y ((person(X) ^ person(Y) ^ ~(X=Y))  brother(X,Y))TRUE • Exists X crown(X)TRUE • Exists X Exists Y sister(X,Y) FALSE

  13. Representational Schemes • What are the objects, predicates, and functions? you need to encode knowledge of specific problem instances and general knowledge. • In practice, consider interpretations just to understand what the choices are. The world and interpretation are defined, or at least constrained, through the logical sentences we write.

  14. Choices:Functions vs Predicates • Rep-Scheme 1: tall(fatherOf(bob)). • Rep-Scheme 2: Exists X (fatherOf(bob,X) ^ tall(X) ^ All Y (fatherOf(bob,Y)  X = Y)) • “fatherOf” in both cases is assigned a set of 2-tuples: {<b,bf>,<t,tf>,…} • But {<b,bf>,<t,tf>,<b,bff>,…} is possible if it is a predicate but not if it is a function

  15. Choices: Predicates versus Constants • Rep-Scheme 1: Let’s consider the world: D = {a,b,c,d,e}.red:{a,b,c}. pink: {d,e}. Some sentences that are satisfied by the intended interpretation: red(a). red(b). pink(d). ~(All X red(X)). All X (red(X) v pink(X)). But what if we want to say that red is a primary color and pink is not?

  16. Choices: Predicates versus Constants • Rep-Scheme 2: The world: D = {a,b,c,d,e,red,pink} colorof: {<a,red>,<b,red>,<c,red>,<d,pink>,<e,pink>} primary: {red} color: {red,pink} • Some sentences that are satisfied by the intended interpretation: colorOf(a,red). colorOf(b,red). colorOf(d,pink). ~(All X colorOf(X,red)). All X (colorOf(X,red) v colorOf(X,pink) v color(X)). ***~primary(pink). primary(red).*** We have reifiedpredicates pink and red: made them into objects

  17. Exercise • [Design a representation scheme for these sentences:] • Emily is either a surgeon or a lawyer • Joe is an actor, but he also holds another job • All surgeons are doctors • Joe does not have a lawyer (i.e., he isn’t the customer of a lawyer) • There exists a lawyer all of whose customers are doctors • Every surgeon has a lawyer • Note: imagine that implications are also added, so the system can reason about people and their jobs • job, customer, doctor, surgeon, lawyer, actor, emily, joe

  18. Wrap-up • Read everything in Chapter 8 carefully, including the examples in 8.4

More Related