Unit III - KNOWLEDGE REPRESENTATION SCHEMES and LOGICgood.pdf
- 1. Unit III - KNOWLEDGE REPRESENTATION SCHEMES and LOGIC Approaches to knowledge representation, Propositional logic, predicate logic, Resolution in proportional logic Nature cares nothing for logic, our human logic: she has her own, which we do not recognize and acknowledge until we are crushed under its wheel. -Ivan Turgenev (1818-1883), Russian novelist and playwright
- 2. Knowledge Representation Earlier it was believed that the best approach to solutions was through the development of General Purpose Problem Solver, that is, systems powerful to prove a theorem in geometry, perform a complex robotic task, or to develop a plan to complete a sequence of operations. But it was deduced that the systems became effective only when the solution methods incorporated domain specific rules and facts, i.e. after gaining specific knowledge. It eventually led to knowledge based systems.
- 3. Knowledge Representation Importance of Knowledge Knowledge can be defined as the body of facts and principles accumulated by human-kind or the act, fact or state of knowing. In actuality it is more than this, it also includes having a familiarity with language, concepts, procedures, rules, ideas, abstractions , places, customs, facts associations along with ability to use these notions effectively in modeling different aspects of the world. How is knowledge stored in biological organisms and computers? Human brain weighs 3.3 pounds - estimated number of neurons 1012 - potential storage – 1014 In computers, knowledge is also stored as symbolic structures, in the form of collections of magnetic spots and voltage states.
- 4. Knowledge Representation Let’s take the following examples A is tall. A Likes B. C has learned to use recursion to manipulate linked lists in several programming languages. 1st one represents a fact, an attribute possessed by a person. 2nd expresses a complex binary relation between two persons. 3rd is most complex, expressing relations between a person and more abstract programming concepts.
- 5. Knowledge Representation Knowledge can be of following types Declarative (statements) Procedural (facts) Heuristics (rule of thumb / experience) We should not confuse Knowledge with data. – Physician example Belief v/s Hypothesis Belief is any meaningful and coherent expression that can be represented. Hypothesis is a justified belief that is not known to be true. Epistemology – study of nature of knowledge. Metaknowldge – knowledge about knowledge.
- 6. Knowledge Representation Knowledge alone cannot serve the purpose, we need to include understanding, learning, thinking, remembering & reasoning. Knowledge based systems get their power from expert knowledge that has been coded into facts, rules, heuristics and procedures. Knowledge is stored in a knowledge base separate from the control and inferencing components , this makes it possible to add new knowledge or refine existing knowledge without recompiling the control and inferencing programs. This simplifies the construction and maintenance of knowledge-based systems.
- 7. Mappings between Facts and Representation Facts: Truths in some relevant world. These are the things we want to represent Representation of fact in some chosen formalism. These are the things we will actually be able to manipulate
- 8. Knowledge Representation Some mathematical ways of representing the sentences Spot is a dog. dog(spot) All dogs have tails. x : dog(x) hastail(x) Then using the deduction method of logic, we may generate they new representation object : hastail(spot) So we can , using backward mapping function : Spot has a tail.
- 10. Knowledge Representation Approaches to Knowledge representation A good system for the representation of knowledge in a particular domain should possess the following properties : Representational Adequacy – the ability to represent all of the kinds of knowledge that are needed in that domain Inferential Adequacy – The ability to manipulate the representational structures in such a way as to derive new structures corresponding to new knowledge inferred from old. Inferential Efficiency – The ability to incorporate into the knowledge structure additional information that can be used to focus the attention of the inference mechanisms in the most promising direction. Acquisitional Efficiency - Acquiring new information easily.
- 11. Knowledge Representation Simple Relational Knowledge Player Height Weight Runs Dhoni 6-0 150 3000 Sachin 5-4 140 15000 Zaheer 6-2 160 1000 This even does not tell us that who is the heaviest player? A procedure should be defined to figure out the result.
- 12. Knowledge Representation Inheritable Knowledge It is possible to enhance the basic representation with inference mechanism that operate on the structure of the representation. The most useful form of inference is Property Inheritance, in which elements of specific classes inherit attributes and values from ore general classes in which they are included. In order to do this, objects must be organized into classes, classes must be arranged into generalization hierarchy.
- 14. 85 Algorithm : Property Inheritance To retrieve a value V for attribute A of an instance object O: 1. Find O in the knowledge base. 2. If there is a value there for the attribute A, report that value. 3. Otherwise, see if there is a value for the attribute instance. If not, then fail. 4. Otherwise, move to the node corresponding to that value and look for a value for the attribute A. If one is found, report it. 5. Otherwise, do until there is no value for the is a attribute or until an answer is found: (a)Get the value of the is a attribute and move to that node. (b) See if there is a value for the attribute A. If there is, report it.
- 15. 84 Viewing a Node as a Frame
- 16. Knowledge Representation Inferential Knowledge Sometimes all the power of traditional logic is necessary to describe he inferences that are needed.We can represent inferential knowledge about a domain using first order predicate-logic. But all this knowledge is useless unless there is also an inference procedure that can exploit it. The required inference procedure now is one that implements the standard logical rules of inference. There are many procedures, some of which reason forward from given facts to conclusions, some reason backward from desired conclusions to given facts. One most common procedure is RESOLUTION, which uses contradiction strategy.
- 18. Knowledge Representation Procedural Knowledge The previous forms deal with Static, Declarative facts. This knowledge specifies what to do and when. This knowledge can be represented in programs in many ways. The most common way is simply as code for doing something. The machine uses the knowledge when it executes the code to perform a task. Eg : If – then else etc.
- 19. Representing Simple facts in logic • The logical formalism is appealing because it immediately suggests a powerful way of deriving new knowledge from old - mathematical deduction. • Propositional logic is appealing because it is simple to deal with and a decision procedure for it exists. We can easily represent real-world facts as logical propositions written as well-formed formulas (wff's) in propositional logic
- 20. Socrates is a man. SOCRATESMAN Plato is a man. PLATOMAN Better representations : MAN(SOCRATES) MAN(PLATO) All men are mortal. MORTALMAN Better representation : Limitations of Propositional Logic
- 21. A Predicate Logic Example 1. Marcus was a man. 2. Marcus was a Pompeian. 3. All Pompeian were Romans. 4. Caesar was a ruler. 5. All Romans were either loyal to Caesar or hated him. 6. Everyone is loyal to someone. 7. People only try to assassinate rulers they are not loyal to. 8. Marcus tried to assassinate Caesar.
- 22. A Predicate Logic Example
- 23. Points to consider in predicate logic representation • Statement 1: This representation captures the critical fact of Marcus being a man. It fails to capture some of the information in the English sentence, namely the notion of past tense. Whether this omission is acceptable or not depends on the use to which we intend to put the knowledge. For this simple example, it will be all right. • Statement 4: Here we ignore the fact that proper names are often not references to unique individuals, since many people share the same name. Sometimes deciding which of several people of the same name is being referred to in a particular statement may require a fair amount of knowledge and reasoning. • Statement 5: In English, the word "or" sometimes means the logical inclusive-or and sometimes means the logical exclusive-or (XOR). Here we have used the inclusive interpretation. Some people will argue, however, that this English sentence is really stating an exclusive-or. • Statement 6:A major problem that arises when trying to convert English sentences into logical statements is the scope of quantifiers. Does this sentence say that for each person there exists someone to whom he or she is loyal, possibly a different someone for everyone? Or does it say that there exists someone to whom everyone is loyal (which would be written as y : x : loyalto(x,y))? Often only one of the two interpretations seems likely, so people tend to favor it.
- 24. Points to consider in predicate logic representation • Statement 7: This sentence, too, is ambiguous. Does it mean that the only rulers that people try to assassinate are those to whom they are not loyal (the interpretation used here), or does it mean that the only thing people try to do is to assassinate rulers to whom they are not loyal? In representing this sentence the way we did, we have chosen to write ''try to assassinate" as a single predicate. This gives a fairly simple representation with which we can reason about trying to assassinate. But using this representation, the connections between trying to assassinate and trying to do other things and between trying to assassinate and actually assassinating could not be made easily. If such connections were necessary, we would need to choose a different representation.
- 25. An Attempt to Prove Backward reasoning
- 26. Another Predicate Logic Example 1. Marcus was a man. 2. Marcus was a Pompeian. 3. Marcus was born in 40 A.D. 4. All men are mortal. 5. All Pompeians died when the volcano erupted in 79 A.D. 6. No mortal lives longer than 150 years. 7. It is now 1991. 8. Alive means not dead. 9. If someone dies, then he is dead at all later times.
- 27. A Set of Facts about Marcus This example shows how the ideas of computable functions and predicates can be useful. Here gt is a computational function.
- 28. One Way of Proving That Marcus Is Dead
- 29. Another Way of Proving That Marcus Is Dead • Even very simple conclusions can require many steps to prove. • A variety of processes, such as matching, substitution, and application of modus ponens are involved in the production of a proof. This is true even for the simple statements we are using. It would be worse if we had implications with more than a single term on the right or with complicated expressions involving ands and ors on the left.
- 30. Resolution: Conversion to Clause Form Problem : Solution : Flatten Separate out Quantifiers Conjunctive Normal Form : Clause Form : Conjunctive Normal Form No instances of
- 31. Algorithm : Convert to Clause Form
- 32. Examples of Conversion to Clause Form Example : 1 Eliminate 2 Reduce scope of 3 Standardize Variables.
- 33. Examples of Conversion to Clause Form 4 Move quantifiers. 5 Eliminate existential quantifiers. will be converted to while will be converted to 6 Drop the prefix. 7 Convert to a conjunction of disjuncts.
- 34. Examples of Conversion to Clause Form The Formula becomes and then becomes
- 35. The Basis of Resolution Given : The resolution procedure is a simple iterative process: at each step, two clauses, called the parent clauses, are compared (resolved), yielding a new clause that has been inferred from them. The new clause represents ways that the two parent clauses interact with each other precisely one of winter and ¬ winter will be true at any point. If winter is true, then cold must be true to guarantee the truth of the second clause. If ¬ winter is true, then summer must be true to guarantee the truth of the first clause. Thus we see that from these two clauses we can deduce
- 36. Algorithm : Propositional Resolution 1. Convert all the propositions of F to clause form. 2. Negate P and convert the result to clause form. Add it to the set of clauses obtained in step 1. 3. Repeat until either a contradiction is found or no progress can be made: (a) Select two clauses. Call these the parent clauses. (b) Resolve them together. The resulting clause, called the resolvent, will be the disjunction of all of the literals of both of the parent clauses with the following exception: If there are any pairs of literals L and ¬L such that one of the parent clauses contains L and the other contains ¬L, then select one such pair and eliminate both L and ¬L from the resolvent. (c) If the resolvent is the empty clause, then a contradiction has been found. If it is not, then add it to the set of clauses available to the procedure.
- 37. A Few Facts in Propositional Logic Let's look at a simple example. Suppose we are given the axioms shown in the following figure and we want to prove R. First convert the axioms to clause form, as shown in the second column of the figure. Then we negate R, producing ¬ R, which is already in clause form.
- 38. Resolution in Propositional Logic
- 39. Resolution in predicate logic: Unification predicate names are different substitute x for y
- 40. Finding General Substitutions Given : We could produce :
- 41. Algorithm : Unify (L1, L2) 1. If L1 or L2 are both variables or constants, then: (a) If L1 and L2 are identical, then return NIL. (b) Else if L1 is a variable, then if L1 occurs in L2 then return {FAIL}, else return (L2/L1). (c) Else if L2 is a variable then if L2 occurs in L1 then return {FAIL}, else return (L1/L2). (d) Else return {FAIL}. 2. If the initial predicate symbols in L1 and L2 are not identical, then return {FAIL). 3. If LI and L2 have a different number of arguments, then return {FAIL}. 4. Set SUBST to NIL. 5. For i ← 1 to number of arguments in L1: (a) Call Unify with the /th argument of L1 and the ith argument of L2, putting result in S. (b) If S contains FAIL then return {FAIL}. (c) If S is not equal to NIL then: (i) Apply S to the remainder of both L1 and L2. (ii) SUBST : = APPEND(S, SUBST). 6. Return SUBST.
- 42. Why Do the Occur Check?: Reason for step 1.b and 1.c in Unify Algo Example : If we accepted g(x) as a substitution for x, then we would have to substitute it for x in the remainder of the expressions. But this leads to infinite recursion since it will never be possible to eliminate x.
- 43. Resolution in Predicate Logic Example : Yield the substitution : Marcus/x1 So it does not yield the resolvent : mortal/x1 It does yield : mortal(Marcus)
- 44. Algorithm : Resolution 1. Convert all the statements of F to clause form. 2. Negate P and convert the result to clause form. Add it to the set of clauses obtained in 1. 3. Repeat until either a contradiction is found, no progress can be made, or a predetermined amount of effort has been expended. (a) Select two clauses. Call these the parent clauses. (b) Resolve them together. The resolvent will be the disjunction of all the literals of both parent clauses with appropriate substitutions performed and with the following exception: If there is one pair of literals T1 and ¬T2 such that one of the parent clauses contains T2 and the other contains T1 and if T1 and T2 are unifiable, then neither T1 nor T2 should appear in the resolvent. If there is more than one pair of complimentary literals, only one pair shold be omitted from the resolvent. (c) If the resolvent is the empty clause, then a contradiction has been found. If it is not, then add it to the set of clauses available to the procedure.
- 46. An Unsuccessful Attempt at Resolution
- 47. The Need to Standardize Variables
- 48. Using Resolution with Equality and Reduce
- 50. Answers Extraction Using Resolution
- 51. The Need to Change Representations “What happened in 79 A.D.?” But we have erupted(volcano, 79)
- 53. Resolution Example John likes all kind of food. Apples are food. Chicken is food. Anything anyone eats and isn’t killed by is food. Bill eats peanuts and is still alive. Sue eats everything Bill eats.
- 54. Resolution Example The members of the Elm St. Bridge Club are Joe, Sally, Bill and Ellen. Joe is married to Sally. Bill is Ellen’s brother. The spouse of every married person in the club is also in the club. The last meeting of the club was at Joe’s house.
- 55. Resolution Example Steve only likes easy courses. Science courses are hard. All the courses in the basket weaving department are easy. BK301 is a basket weaving course.
- 57. A Problem Given : Prove : What’s wrong with :
- 58. Some Rules of Inference
- 60. Example –Resolution Refutation in predicates Consider the following axioms: • All hounds howl at night. • Anyone who has any cats will not have any mice. • Light sleepers do not have anything which howls at night. • John has either a cat or a hound. • (Conclusion) If John is a light sleeper, then John does not have any mice. • Basic predicates are: HOUND(x), HOWL(x), HAVE(x,y), CAT(x), MOUSE(x), LS(x)