Knowledge Representation

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Knowledge Representation Knowledge Representation- 

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knowledge is acquired, it has to be organized and formaliz 1 then be put into the knowledge base. are two types of knowledge representations: Analysis Representation used for analysis and is usually gr Coding Representation for coding the knowledge Analysis Knowledge Representatio} acquisition 0 ار Knowledge Representation- 2 Logic 

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ll talk about coding representation methods and those that are: Logic Semantic Networks Productions Frames > in general is a subfield of philosophy and its development ted to ancient Greeks. bolic or mathematical logic is used in AI. In symbolic logic procedures are used to draw conclusions using various logi niques. Knowledge Representation- 3 Logic 

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he general form of a logical process Premises Inferences Or OR Facts Conclusions ence is a process that is used to derive new facts from the are known. Knowledge Representation- 4 Logic 

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Logic positional Logic A*B <->C st Order Logic (Predicate Logic or Predicate Calculus) V x (ODD(x)~EVEN(add(x,1))) Higher order Logics V P (P(x)) V £ (P(f))) Knowledge Representation 5 Logic 

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positional Logic and Automated Reasoning A proposition is a declaration: New Zealand is beautiful. The sun is made of chocolate. ymbols like A,B,C,... are used to denote propositions. These ۱1180 atoms. A: Today is Monday. B: It is raining now. Not A: Today is not Monday. Not B: It is not raining now. roposition can either be true or false but not both. This is | truth value and it is assigned not inherent to the propositi Knowledge Representation- 6 Logic 

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and the following five A Formula mula is built with atoms, parantheses al operators: Negation =~) NOt Conjunction ۸۱ 0 Disjunction Voor Implication -> if...then Double implication <-> ifand only if Order of precedence ۸ ۷ -> <-> GARB S VRP 7 Example formulae A (~A) (BVC)A~D) (((A -> B) AC) AD) Knowledge Representation- Logic 

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‘ecursive Definition of Formulae in Propositional Logic atom is a formula. ‘is a formula then ~F is a formula. ‘and G are formlae then 1 2 ‏با‎ ‎F ->G F<->G > formulae. sre are no formulae other than these just defined Knowledge Representation- 8 Logic 

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iterpretation of Formulae terpretation of a formula F is an assignment of truth values atom. ula having n atoms has 2“n interpretations (a table with 2*n ae have truth values. Truth value of a formula can be obta ts atoms and logical operators. (((BVC)A~C)\ true true false true false 56 fal fal tri ((BVC)A~C true. false false false false Knowledge Representation- Logic =e) false false true true false false true true (BVC) true [true D truel fals true fal: true| fals¢ true| falsd 0 true true false false true true false false 

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nula is a tautology (or a valid formula) if and only if it is tru all interpretations. ۱ ۱/ -8 ula is an inconsistency if and only if it is false under all etations. ula is a tautology if and only if its negation is an inconsiste! atisfiable formula). ~B/\B) ula is consistent (or satisfiable) if it is not inconsistent. r words, a formula is consistent if it is true under at least c etation. mula is a tautology it is consistent but the converse may nc ula F is equivalent to formula G if and only if the truth valu uivalent to the truth value of G under all interpretations: cate ue Reine en Ear Logic 

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A Proof nula G is said to be the logical consequence of formulae 121 Fn d only if every interpretation that satisfies (FIAF2/...AFn) satisfies G. We may also say, F1, F2,...,Fn imply G. called the Goal and the Fi are called the premisses. onstrate that G is a logical consequence of the Fs is to pro‘ ((F1/A\F2/...\Fn) -> G) is a theorem. smonstartion is the proof. rmulae we will work on will be the ones that are consisten Knowledge Representation 11 Logic 

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A proof may be a direct proof a refutation proof ct method: ((FIAF2/.../\Fn) -> G) is a tautology. tation method: ~((F1/AF2/\...\Fn) -> G) is an inconsistency. e the negation of a tautology is an inconsistency these are valent. IAF2/(.../\Fn) -> G) ~(FIAF2/\...\Fn) VG) ‘1/\F2/\...\Fn) /\ ~G) Knowledge Representation- 12 Logic 

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Example ve by both methods that G is a logical consequence of F and F->G need 4 rows because we have 2 atoms. c | ~d ->G |FA@->G)| (FA(F->G))->G (FA(F-> G))/ true| false true true true| falsel_trde false false true false true|_falbe true false true false false] true true false true false 1is is known as modus ponens. 10ws that there exists a finite procedure in propositional lo whether a given goal is a theorem or not. sitional logic is thus decidable. Knowledge Representation- 13 Logic 

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number of atoms (n) increases the number of rows in the t ses exponentially which makes the use of truth table for ‏ر‎ theorems impractical. There are other methods that may of inference ve proved using the table (modus ponens) is called a rule © 106 and is written as, f F and (F-> G) then G of inference applied to a set of premisses produces a formt case of modus ponens above we can say that G is deduced F-> G by modus ponens. are other rules of inference and these rules are used in a tl g method called the natural deduction. Knowledge Representation- 14 Logic 

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Natural Deduction of inference is applied to a set of given premisses to deduc il consequence formula which is added to the set of premis ormulae is deduced and added to a set by applying various rence until we deduce a formula that is identical to the gos n no longer deduce new formulae. One of these cases occu! time. Natural deduction is a direct method. Using this met! we may produce formulae that are not useful in reaching tl ; not a computationally desirable method is a refutation proof method called the resolution principle one rule of inference called the resolution which is effective yh to make other rules secondary. We will look at this rule ‏؛‎ 2m proving procedure. Knowledge Representation- 15 Logic 

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Example Proving D Given 1) A 2) B 3) AAC -> D 4) B->C modus ponens on 2 and 4 introducing conjunction on 1 and 5 modus ponens on 3 and 6 il has been deduce as a logical consequence of the premiss: il has been proved to be a theorem. Knowledge Representation- 16 Logic 

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Proving Theorems by Resolution rals and Clauses resolution method is applicable to a formula that is a conju auses. ause is a disjunction of n literals for any finite integer n>=( eral is an atom or the negation of an atom (Examples: A, B, tom and its negation are referred to as complementary lite mples: B and ~B, H and ~H, etc.) ises are constructed from literals. nple Clauses: AVBV~D ~AVB D NIL Knowledge Representation- 17 Logic 

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18 onverting a Formula to Clause Form Knowledge Representation- Logic