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CSNB234 ARTIFICIAL INTELLIGENCE

This chapter discusses the early development of propositional logic, including the contributions of DeMorgan and Boole. It explores the syntax and semantics of propositional logic, covering sentences, negation, conjunction, disjunction, implication, and equivalence.

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CSNB234 ARTIFICIAL INTELLIGENCE

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  1. CSNB234ARTIFICIAL INTELLIGENCE Chapter 3 Propositional Logic & Predicate Logic UNIVERSITI TENAGA NASIONAL

  2. Early Development of Symbolic Logic • English mathematician DeMorgan criticised traditional logic because it was written in natural language. • He thought that the formal meaning of a syllogistic statement was confused by the semantics of natural language. • DeMorgan and Boole both contributed to the development of Propositional Logic (or Propositional Calculus). • Using familiar algebraic symbols, they showed how certain algebraic rules were equally applicable to numbers, set and truth values of propositions. UNIVERSITI TENAGA NASIONAL

  3. Propositional Logic (I) • Definition • Propositional Logic Sentences • Every propositional symbol and truth symbol is a sentence. • For example: true, P, Q, and R are four sentences • The negation of a sentence is a sentence • For example: P and  false are sentences • The conjunction (and) of two sentences is a sentence • For example: P  P is a sentence UNIVERSITI TENAGA NASIONAL

  4. Propositional Logic (II) • Propositional Logic Sentences • The disjunction (or) of two sentence s is a sentence • For example: P  P is a sentence • The implication of one sentence for another is a sentence • For example: P  Q is a sentence • The equivalence of two sentences is a sentence • for example: P  Q = R is a sentence UNIVERSITI TENAGA NASIONAL

  5. Propositional Logic Semantics • An interpretation of a set of propositions is the assignment of a truth value, either T of F, to each propositional symbol. • The interpretation or truth value for sentences is determined by: • The truth assignment of negation,  P, where P is any propositional symbol, is F if the assignment to P is T and T if the assignment to P is F. • The truth assignment of conjunction, , is T only when both conjuncts have truth value T; otherwise it is F. UNIVERSITI TENAGA NASIONAL

  6. Propositional Calculus Semantics • The truth assignment of disjunction, , is F only when both conjuncts have truth value F; otherwise it is T. • The truth assignment of implication, , is F only when the premise or symbol before the implication is T and the truth value of the consequent or symbol after the implication is F; otherwise it is always T. • The truth assignment of equivalence, =, is T only when both expressions have the same truth assignment for all possible interpretations; otherwise it is F. UNIVERSITI TENAGA NASIONAL

  7. Prove that ((PQR) = P  Q  R is a well-formed sentence in the propositional calculus. • Answer. Since: • P, Q and R are propositions and thus sentences • P  Q, the conjunction of two sentences, is a sentence • (P  Q) R, the implication of a sentence for another, is a sentence UNIVERSITI TENAGA NASIONAL

  8. P, Q and R are propositions and thus sentences P and Q , the negation of two sentences, are sentences P  Q, the disjunction of two sentences, is a sentence P  Q  R, the disjunction of two sentences, is a sentence ((P  Q) R) = P  Q  R, the equivalence of two sentences, is a sentence We get back the original sentence UNIVERSITI TENAGA NASIONAL

  9. Conclusion for the worked example The above is our original sentence, which has been constructed through a series of applications of legal rules and is therefore well-formed. UNIVERSITI TENAGA NASIONAL

  10. Constants refer to atomic propositions. raining snowing eating hungry wet • Compound sentences capture relationships among propositions. • raining  snowing  wet UNIVERSITI TENAGA NASIONAL

  11. Negations: ¬ raining • The argument of a negation is called the target . • Conjunctions: (raining  snowing ) • The arguments of a conjunction are called conjuncts . • Disjunctions: (raining  snowing ) • The arguments of a disjunction are called disjuncts . UNIVERSITI TENAGA NASIONAL

  12. Compound Sentences • Implications: (raining  cloudy ) • The left argument of an implication is the antecedent . • The right argument of an implication is called the consequent . • Reductions: cloudy  raining • The left argument of a reduction is the consequent . • The right argument of a reduction is called the antecedent . • Equivalences: raining  cloudy UNIVERSITI TENAGA NASIONAL

  13. Some Laws for Logic Use Rules of Algebraic Manipulation • x  y = y  x Commutativity • x  y = y  x • x  (y  z) = (x  y)  z Associativity • x  (y  z) = (x  y)  z • x  (y  z) = (x  y)  (x  z) Distributivity • x  (y  z) = (x  y) (x  z) UNIVERSITI TENAGA NASIONAL

  14. Semantics of Logical Operators • Negation: • Conjunction: P P T F F T P Q P  Q T T T T F F F T F F F F UNIVERSITI TENAGA NASIONAL

  15. Semantics of Logical Operators • Disjunction: P Q P  Q T T T T F T F T T F F F UNIVERSITI TENAGA NASIONAL

  16. More Semantics of Logical Operators Implication: Reverse Implication: Equivalence: P Q P  Q T T T T F F F T T F F T P Q Q  P T T T T F T F T F F F T P Q Q  P T T T T F F F T F F F T UNIVERSITI TENAGA NASIONAL

  17. Satisfaction • An interpretation i satisfies a sentence φ (written |=i φ ) if and only if φ i =T . • A sentence is satisfiable if and only if there is some interpretation that satisfies it. • A sentence is valid if and only if every interpretation satisfies it. • A sentence is unsatisfiable if and only if there is no interpretation that satisfies it. UNIVERSITI TENAGA NASIONAL

  18. Truth Tables A truth table is a table of all possible values for a set of propositional constants. p q r T T T T T F T F T T F F F T T F T F F F T F F F Each interpretation of a language is a row in the truth table for that language. For a propositional language with n logical constants,there are 2n interpretations. UNIVERSITI TENAGA NASIONAL

  19. Logical Equivalence Two sentences are logically equivalent if and only if they logically entail each other. Examples: ¬(¬p) p ¬(p  q ) ¬p  ¬q de Morgan’s law ¬(p  q ) ¬p  ¬q de Morgan’s law (p  q ) ¬p  q UNIVERSITI TENAGA NASIONAL

  20. Problems • There can be many, many interpretations for a propositional language. • Remember that, for a language with n constants, there are 2n possible interpretations. • Sometimes there are many constants among premises that are irrelevant to the conclusion. ---- Much work wasted. Solution: use other kind of proof theory, such as refutation proof (later part) UNIVERSITI TENAGA NASIONAL

  21. Example of validity: Problem to solve Problem: what sentence is this (p  q) (q  r)? Solution: p q r (p  q) (q  r) (p q )  (q  r ) T T T T T T T T F T F T T F T F T T T F F F T T F T T T T T F T F T F T F F T T T T F F F T T T It is a valid sentence! All values are “true” UNIVERSITI TENAGA NASIONAL

  22. Clausal Form • Propositional resolution works only on expressions in clausal form. • Fortunately, it is possible to convert any set of propositional calculus sentences into an equivalent set of sentences in clausal form. UNIVERSITI TENAGA NASIONAL

  23. Conversion to Clausal Form Implications Out: P  Q Ø P Ú Q P Q P Ú Ø Q P Q (Ø P Ú Q) Ù (P Ú ØQ ) Negations In: Ø ØP P Ø (P Ù Q) Ø PÚ Ø Q Ø (P ÚQ ) Ø P Ù Ø Q UNIVERSITI TENAGA NASIONAL

  24. Predicate Calculus(=Predicate Logic) UNIVERSITI TENAGA NASIONAL

  25. Predicate Calculus (I) In Proposition Logic, each atomic symbol (P, Q, etc) denotes a proposition of some complexity. There is no way to access the components of an individual assertion. Through inference rules we can manipulate predicate calculus expressions, accessing their individual components and inferring new sentences. UNIVERSITI TENAGA NASIONAL

  26. Predicate Calculus (II) • In Predicate Calculus, there are two ways variables may be used or quantified. In the first, the sentence is true for all constants that can be substituted for the variable under the intended interpretation. The variable is said to be universal quantified. • Variables may also be quantified existentially. In this case the expression containing the variable is said to be true for at least one substitution from the domain of definition. Several relationships between negation and the universal and existential quantifiers are given below: UNIVERSITI TENAGA NASIONAL

  27. Predicate Calculus (III) • Predicate calculus sentences • Every atomic sentence is a sentence • if s is a sentence, then so is its negation, s • if s1 and s2 are sentences, then so is their conjunction, s1  s2 • if s1 and s2 are sentences, then so is their disjunction, s1  s2 • if s1 and s2 are sentences, then so is their implication, s1  s2 • if s1 and s2 are sentences, then so is their equivalence, s1 = s2 UNIVERSITI TENAGA NASIONAL

  28. Predicate Calculus (IV) • If X is a variable and s is a sentence, then X sis a sentence • If X is a variable and s is a sentence, then X s is a sentence UNIVERSITI TENAGA NASIONAL

  29. English Sentences Represented In Predicate Calculus: • Some people like fried chicken. • X (people(X)  likes(X, fried_chicken)). • Nobody likes income taxes. •  X likes(X, income_taxes). • X  likes(X, income_taxes). UNIVERSITI TENAGA NASIONAL

  30. Rules: All students passed examination. X (student(X)  pass_exam(X)) All purple mushrooms are poisonous. X (purple(X)  mushroom(X)  poisonous(X)) Facts: Tom loves Jerry. loves(tom, Jerry). Ali is a student. student(ali). UNIVERSITI TENAGA NASIONAL

  31. Exercise #1: Translate The Following English Statements Into Predicate Expressions Batman is knowledgeable and he is wealthy. All people that are not poor are happy. Students who like to read books are intelligent. Tweety can fly if it is not fried and has wings. UNIVERSITI TENAGA NASIONAL

  32. Exercise #2 Everybody likes something. There is something whom everybody likes. UNIVERSITI TENAGA NASIONAL

  33. Answers to Exercise #1 • Everybody likes something. • "x.$y. likes(x,y) • There is something whom everybody likes. • $y."x. likes(x,y) UNIVERSITI TENAGA NASIONAL

  34. Exercise #3 For predicates p & q, and variables X and Y: Write the following in English  X p(X) =  X  p(X)  Y q(Y) =  Y  q(Y) UNIVERSITI TENAGA NASIONAL

  35. Exercise #4: Convert each of the following predicate logic to English sentences • food(laksa) • X loves(X, superman)  loves(superman, X) • X food(X)  like(arul, X) • X eat(haswan, X)  eat(hasman, X) • X Y eat(X, Y)  alive(X)  food(Y) UNIVERSITI TENAGA NASIONAL

  36. Exercise #5 • Convert each of the following into Predicate Calculus equivalence: • Marcus was a man • Marcus was a Pompeian • All Pompeians were Romans • Caesar was a ruler • All Romans were either loyal to Caesar or hated him • Everyone is loyal to someone • people only try to assassinate rulers they are not loyal to • Marcus tried to assassinate Caesar UNIVERSITI TENAGA NASIONAL

  37. Predicate logic for the 8 facts in Exercise #5 1. man(Marcus) 2. pompeian(Marcus) 3.  X. pompeian(X)  roman(X) 4. ruler(Caesar) 5.  X. roman(X)  loyalto(X, Caesar)  hate(X, Caesar) 6.  X. Y. loyalto(X,Y) 7.  X.  Y. person(X)  ruler(Y)  tryassassinate(X,Y)   loyalto(X,Y) 8. tryassasinate(Marcus, Caesar) 9.  X. man(X)  person(X) UNIVERSITI TENAGA NASIONAL

  38. Answers to Exercise #5 loyato(Marcus, Caesar) (using 7, substitution, & apply M.P) person(Marcus)  tryassassinate(Marcus, Caesar)  ruler(Caesar) using (4) person(Marcus)  tryassassinate(Marcus, Caesar) using (8) person(Marcus) (using 9, substitution & apply M.P) man(Marcus) using (1) nil UNIVERSITI TENAGA NASIONAL

  39. loyato(Marcus, Caesar) (using 7, substitution, & apply M.P) person(Marcus)  tryassassinate(Marcus, Caesar)  ruler(Caesar) using (4) person(Marcus)  tryassassinate(Marcus, Caesar) using (8) person(Marcus) (using 9, substitution & apply M.P) man(Marcus) using (1) nil

  40. Towards the Resolution and Refutation Proof UNIVERSITI TENAGA NASIONAL

  41. Stages involved in Proof Theory • Stage 1 • convert all axioms into prenex form • i.e. all quantifiers are at the front • Stage 2 • purge existential quantifiers • this process is known as skolemization • Stage 3 • drop universal quantifiers • as they convey no information UNIVERSITI TENAGA NASIONAL

  42. An Example Consider the arguments: All men are mortal (given premise) Superman is a man (given premise) Superman is mortal (goal to test) The argument gets formalised as: X man(X)  mortal(X) man(Superman) mortal(Superman) (goal) And has, as its conflict set in Clausal form:  man(X)  mortal(X) ---- (1) man(Superman) ---- (2)  mortal(Superman) ---- (3) Negation of goal UNIVERSITI TENAGA NASIONAL

  43. Apply resolution to derive at a contradiction: We get: • man(Superman) from (1) & (3) and, direct contradiction from (2) & (4) The conclusion is that “the goal is true” (i.e. superman is mortal) UNIVERSITI TENAGA NASIONAL

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