1. Chapter 7. Propositional and Predicate Logic Fall 2013 Comp3710 Artificial Intelligence Computing Science Thompson Rivers University

2. TRU-COMP3710 Propositional/Predicate Logic2 Course Outline Part I – Introduction to Artificial Intelligence Part II – Classical Artificial Intelligence Knowledge Representation Searching Knowledge Represenation and Automated Reasoning Propositinoal and Predicate Logic Inference and Resolution for Problem Solving Rules and Expert Systems Part III – Machine Learning Part IV – Advanced Topics Genetic Algorithms

3. TRU-COMP3710 Propositional/Predicate Logic3 Chapter Objectives

4. TRU-COMP3710 Propositional/Predicate Logic4 Chapter Outline Propositional logic 1. Introduction Introduction 2. What is logic? Why is logic used in Artificial Intelligence? What is logic? Why is logic used in Artificial Intelligence? 3. How to use logical operators How to use logical operators 4. How to translate an English statement with logic notations How to translate an English statement with logic notations 5. Let’s recall complex truth tables Let’s recall complex truth tables 6. Let’s recall tautology and contradictory Let’s recall tautology and contradictory 7. How to use equivalent propositions How to use equivalent propositions 8. How to logically use propositions – propositional logic How to logically use propositions – propositional logic 9. Introduction to predicate calculus Introduction to predicate calculus 10. Summary Summary

5. 5 1. Introduction [Q] What is reasoning? [Wikipedia] Reason is the capacity for consciously making sense of things, applying logic, for establishing and verifying facts, and changing or justifying practices, institutions, and beliefs based on new or existing information. To form conclusions, inferences, or judgments [Q] How to automate reasoning? Need to know how to represent information, knowledge, facts and beliefs, and how to apply logic, …

6. 6 [Q] How to formalize/validate our arguments? Argument = premises (proposition or statement) + conclusion To have confidence in the conclusion in your argument, the premises should be acceptable on their own merits or follow from other statements that are known to be true. [Q] Any logical forms for valid arguments? Examples Argument 1: If the program syntax is faulty or if program execution results in division by zero, then the computer will generate an error message. Therefore, if the computer does not generate, then the program syntax is correct and program execution does not result in division by zero. Argument 2: If x is a real number such that x 2, then x 2 > 4. Therefore, if x 2 /> 4, then x / 2. The common logical form of both of the above arguments: If p or q, then r. Therefore, if not r, then not p and not q. Is this logical form valid?

7. TRU-COMP3710 Propositional/Predicate Logic7 You may recall “Logic and Truth Tables” in COMP 1380. In this unit, Boolean logic Propositional logic Introduction to predicate calculus – first-order predicate logic Propositional logic will be used in the following unit to solve some problems. Based on rules, knowledge, and facts, Decide if a given query is valid. Topics

8. TRU-COMP3710 Propositional/Predicate Logic8 2. What is Logic? Logic: reasoning about the validity of arguments. An argument is valid if its conclusions follow logically from its premises (proposition or statement) – even if the argument doesn’t actually reflect the real world: Mary is a lemon. All lemons are blue. Therefore, Mary is blue. Logic is widely used as a representation method of AI, and allows us to easily reason about negatives (i.e., “NOT”) and disjunctions (,i.e., “OR”) One of the main weaknesses of traditional logic (i.e., Boolean logic) is its inability to deal with uncertainty. Later probabilistic method and fuzzy logic will be discussed to deal with uncertainty. Topics

9. TRU-COMP3710 Propositional/Predicate Logic9 3. How to Use Logical Operators Definition of statement A statement (or proposition) is a sentence that is true or false but not both. Examples Two plus two equals four. 2 + 2 = 4 I am a TRU student. x + y > 0???

10. TRU-COMP3710 Propositional/Predicate Logic10 Compound Statements Symbols used in complicated logical statements: ~not~pnegation of p  andp  qconjunction of p and q  orp  qdisjunction of p and q  exclusive orp  q Order of operations: ( ) and ~ have the higher precedence. ~p  q = (~p)  q ~(p  q)

11. TRU-COMP3710 Propositional/Predicate Logic11 And (conjunction)  Or (disjunction)  Not (negation)  or~ Implies (conditional)  (if… then…) Iff (biconditional)  (if and only if) Exclusive OR? Topics

12. TRU-COMP3710 Propositional/Predicate Logic12 4. Translating between English and Logic Facts and rules need to be translated into logical notation. For example: It is Raining and it is Thursday: R  T, where R represents “It is Raining”, T represents “it is Thursday”.

13. TRU-COMP3710 Propositional/Predicate Logic13 More complex sentences need predicates. That part of a proposition that is affirmed or denied about the subject. For example, in the proposition We are mortal, mortal is the predicate. E.g., It is raining in New York: R(N) Could also be written N(R), or even just R. [Q] How to express “It is not raining in New York”??? It is important to select the correct level of detail for the concepts you want to reason about.

14. TRU-COMP3710 Propositional/Predicate Logic14 Example It is not hot but it is sunny.It is neither hot nor sunny. ->It is not hot, and it is sunny.It is not hot, and it is not sunny. Let h = “it is hot” and s = “it is sunny.” Then the above statements can be translated as ~h  s~h  ~s Example Suppose x is a particular real number. Let p, q, and r symbolize “0 < x,” “x < 3,” and “x = 3.” respectively. Then the following inequalities x  30 < x < 30 < x  3 can be translated as q  rp  qp  (q  r) Topics

15. TRU-COMP3710 Propositional/Predicate Logic15 5. Truth Tables Tables that show truth values for all possible inputs to a logical operator. For example: [Q] Truth table for implies (  ) ??? A  B: A is the antecedent, and B is the consequent. A  B   A  B[Q] Can you prove it? How? A  B   B   A[Q] Can you prove it? How? [Q] Truth table for iff (  ) ???

16. TRU-COMP3710 Propositional/Predicate Logic16 We can produce truth tables for complex logical expressions, which show the overall value of the expression for all possible combinations of variables: Topics

17. TRU-COMP3710 Propositional/Predicate Logic17 6. Tautology and Contradictory A tautology is true under any interpretation. The expression A ˅ ¬A is a tautology. This means it is always true, regardless of the value of A. P is a tautology: this is written ╞ P An expression which is false under any interpretation is contradictory (or unsatisfiable). A  ¬A Some expressions are satisfiable, but not valid. This means that they are true under some interpretation, but not under all interpretations. A  B Topics

18. TRU-COMP3710 Propositional/Predicate Logic18 7. How to Use Equivalent Propositions Two expressions are equivalent if they always have the same logical value under any interpretation: A ˄ B  B ˄ A [Q] How to prove the above equivalence? Equivalences can be proven by examining truth tables. [Q] Are there more equivalences?

19. TRU-COMP3710 Propositional/Predicate Logic19 A ˅ A  ??? A ˄ A  ??? A ˄ (B ˄ C)  (A ˄ B) ˄ C A ˅ (B ˅ C)  (A ˅ B) ˅ C A ˄ (B ˅ C)  (A ˄ B) ˅ (A ˄ C) A ˅ (B ˄ C)  (A ˅ B) ˄ (A ˅ C) A ˄ (A ˅ B)  ??? A ˅ (A ˄ B)  ??? A ˄ true  ???A ˄ false  ??? A ˅ true  ???A ˅ false  ??? [Q] DeMorgans’ Laws ??? [Q] Why do we need these equivalences? By using the above and other equivalences, logical expressions can be simplified.

20. TRU-COMP3710 Propositional/Predicate Logic20 Some more Topics

21. TRU-COMP3710 Propositional/Predicate Logic21 8. How to Use Propositions Propositional logic in this chapter is a logical system. It deals with propositions. Propositional calculus is the language we use to reason about propositional logic. A sentence in propositional logic is called a well-formed formula (wff) (or sentence).

22. TRU-COMP3710 Propositional/Predicate Logic22 Propositional calculus: BNF (Backus-Naur Form) – The following are wff’s: P, Q, R…propositional symbols true, false (A) ¬A A ˄ B A ˅ B A → B A ↔ B Any combination of wff’s is a wff. [Q] Is P  Q  (B   C)  A  B  D  (  E) a wff ???

23. TRU-COMP3710 Propositional/Predicate Logic23 Deduction: the process of deriving a conclusion from a set of assumptions. Will be discussed in the following unit again to solve some problems. If we deduce a conclusion C from a set of assumptions, we write: {A 1, A 2, …, A n } ├ C If C can be concluded without any assumption ├ C The inference rule A ├ B is expressed as A B Given A, B is deduced (or concluded). It is like if A is true, then B is true.

24. TRU-COMP3710 Propositional/Predicate Logic24  introduction Given A and B, we can deduce A  B. A, B{A, B} ├ A ˄ B A ˄ B  introduction A _ A  B  elimination A  B A B  elimination (called Modus Ponens) A, A  B[Q] Can you prove? B   elimination   A A Some valid inference rules

25. TRU-COMP3710 Propositional/Predicate Logic25 Reduction to absurdity: Reductio Ad Absurdum (proof by contradiction)  A. _ Contradiction, i.e., false A  Induction (called deduction theorem) A. C __ A  C

26. TRU-COMP3710 Propositional/Predicate Logic26  Introduction A __ B  A Modus Tollens ~B, A  B ~A Topics

27. TRU-COMP3710 Propositional/Predicate Logic27 9. Introduction to Predicate Calculus Predicate Calculus extends the syntax of propositional calculus with predicates and quantifiers: P(X) – P is a predicate. First Order Predicate Calculus (FOPC) allows predicates to apply to objects or terms, but not functions or predicates. Just introduction in this unit. Predicate calculus is used in solving more complex problems.

28. TRU-COMP3710 Propositional/Predicate Logic28 Quantifiers  and   - For all:  x P(x) is read “For all x’es, P (x) is true”. E.g., for all pine TRU COMP students, they are smart.  - There Exists:  x P(x) is read “there exists an x such that P(x) is true”. E.g., there is a TRU COMP student who is not smart. Relationship between the quantifiers:  x P(x)  ¬ (  x)¬P(x) “If There exists an x for which P holds, then it is not true that for all x P does not hold”.

29. TRU-COMP3710 Propositional/Predicate Logic29 Properties of Logical Systems Four factors to consider of: Soundness: Is every theorem valid? Completeness: Is every tautology a theorem? Decidability: Does an algorithm exist that will determine if a wff is valid? Monotonicity: Can a valid logical proof be made invalid by adding additional premises or assumptions? Topics

30. TRU-COMP3710 Propositional/Predicate Logic30 10. Summary Propositional logic Propositions Boolean logic Deduction; inference rules Topics