SOEN331 Winter Tutorial 4 on Logic

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1 SOEN331 Winter 2018 Tutorial 4 on Logic

2 Propositional Logic for the Quiz 3 Specify requirements in propositional logic Prove arguments Soundness and completeness 2

3 Propositional Logic: Satisfiability A propositional formula is a tautology if it is true in all interpretations A propositional formula is satisfiable if it is true in some interpretations A propositional formula is unsatisfiable (or contradictory) if it is false in all interpretations 3

4 Rewrite the following statements formally using conditional statements: 1. Being a Canadian citizen and of voting age is a necessary condition for being Prime Minister of Canada. 2. Successfully completing SOEN331 is necessary but not sufficient for completing the undergraduate program. 3. Divisibility by 2 is a criterion for being an even number.

5 1. Being a Canadian citizen and of voting age is a necessary condition for being Prime Minister of Canada. If one is Prime Minister of Canada, then one is a Canadian citizen and of legal voting age 5

6 2. Successfully completing SOEN331 is necessary but not sufficient for completing the undergraduate program. necessity: (completion of the undergraduate program) (successful completion of SOEN331) not sufficient: (successful completion of SOEN331) (completion of the undergraduate program) 6

7 3. Divisibility by 2 is a criterion for being an even number (Divisibility by 2) (even number). 7

8 Are these statements semantically true? For each one of the following arguments, prove that a set of premises P 1, P 2.. P n semantically entails a claim Q : Hint: Demonstrate the tautology by use of a truth table Premises If there is fire, then there is smoke. There is fire. Therefore, there is smoke. claims If there is a storm, then school will be closed. There is a storm. Therefore, there the school is closed.

9 Is this a tautology? f: fire, s: smoke ((f -> s) f)) -> s 9

10 Is this a tautology? s: storm, c: closed ((s -> c) s)) -> c 10

11 Proof A proof is a mechanism for showing that a given claim Q is a logical conclusion of some set S of premises A proof is presented in several steps, where each step logically follows from the preceding steps, and an axiom. The final step of the proof is the demonstration of the truth of the claim Q. 11

12 Proof by contradiction In a proof by contradiction it is shown that the denial of the claim results in such a contradiction. If P is the claim: P is assumed to be false, that is NOT P is true. It is shown that NOT P implies two mutually contradictory assertions, Q and NOT Q. Since these cannot both be true, the assumption that P is false must be wrong, and P must be true. 12

13 Are these statements provable? For each one of the following arguments, prove that a set of premises P 1, P 2.. P n syntactically entails a claim Q Hint: use proof by contradiction Premises If there is fire, then there is smoke. There is fire. Therefore, there is smoke. claims If there is a storm, then school will be closed. There is a storm. Therefore, there the school is closed.

14 f -> s, f - s 1. f v s 2. f 3. s 4. s (from 1 and 2) 5. NIL (from 3 and 4) 14

15 storm ->closed, storm - closed 1. storm v closed 2. storm 3. closed 4. storm (from 1 and 3) 5. NIL (from 4 and 2) 15

16 Propositional Logic: Soundness and Completeness Soundness All provable statements are semantically true. That is, if a set of premises S syntactically entails a claim P, then there is an interpretation in which P can be reasoned about from S. Completeness All semantically true statements are provable. That is, if a set of premises S semantically entails a claim P, then P can be derived formally (syntactically) within the formalism 16

17 Predicate Logic for the Exam 1 Specify requirements in predicate logic Quantifiers Existential Universal. Free and bound variables 17

18 Predicate logic Predicate is a verb phrase template that describes a property of objects, or a relationship among objects represented by the variables. Universe is the set of objects of interest 18

19 English to Predicate logic Consider the specification of a movie theater booking system, where B(p, s) denotes the predicate Person p has booked seat s. Rewrite the following sentences in symbolic form and for each one identify bound and free variables: (a)all the seats have been booked (b)no seats have been booked (c)some seats have been booked 19

20 English to Predicate logic (solution) Consider the specification of a movie theater booking system, where B(p, s) denotes the predicate Person p has booked seat s. Rewrite the following sentences in symbolic form and for each one identify bound and free variables: 1)Universes: Persons, Seats 2)Variables: p:person, s: Seat (a)all the seats have been booked (both variables are bound) (b)no seats have been booked (variable s is bound, variable p is free) (c)some seats have been booked (variable s is bound, variable p is free) 20

21 English to Predicate logic (2) Consider the predicate politician(x) which denotes x is a politician, and crooked(x) which denotes x is crooked. We assume that a politician is either crooked or honest. Rewrite the following statements in predicate logic: (cont) 21

22 English to Predicate logic (2) Consider the predicate politician(x) which denotes x is a politician, and crooked(x) which denotes x is crooked. We assume that a politician is either crooked or honest. Rewrite the following statements in predicate logic: (a) There is an honest politician: (b) All politicians are crooked: (c) No politician is crooked: (d) Some politician is crooked: (e) No politician is honest: (f) Not all politicians are crooked: 22

23 English to Predicate logic (2) solution Consider the predicate politician(x) which denotes x is a politician, and crooked(x) which denotes x is crooked. We assume that a politician is either crooked or honest. Rewrite the following statements in predicate logic: 23

24 English to Predicate logic (3) Let B(x) denote the predicate x is a bird and W(x) denote the s predicate x is white. Translate the following wff into English sentences : 24

25 English to Predicate logic (3) solution Let B(x) denote the predicate x is a bird and W(x) denote the s predicate x is white. Translate the following wff into English sentences : 25

26 Predicate Logic: Semantics Specification of the domain universe 1. Each constant, free variable and function is interpreted in a given universe 2. Truth values are assigned to the predicates In different universes (domains) the truth value might be different! A wff becomes a proposition when it is given an interpretation. 26

27 More exercises from Dr Constantinides tutorial? optional 27

28 Predicate Logic: Semantics Example 1 (see solutions in the textbook, Section , example 1 interpretations I1, I3) Well-formed formula: Interpretation 1: Interpretation 2: SOEN331W-L3: exercises on logic 28

29 Predicate Logic: Semantics Example 1 (solution) (see solutions in the textbook, Section , example 1 interpretations I1, I3) Well-formed formula: Interpretation 1: SOEN331W-L3: exercises on logic 29

30 Predicate Logic: Semantics Example 1 (see solutions in the textbook, Section , example 1 interpretations I1, I3) Well-formed formula: Interpretation 2: SOEN331W-L3: exercises on logic 30

31 Knowledge representation example 1 consider a snapshot of World Cup 2010 database predicate team(x) defines list of teams instantiating x over all participating countries, we get a collection of propositions to represent teams database Example: team(usa), team(canada) unary predicates city(x), date(x), player(x), and coach(x) represent knowledge on venues, dates, players, and coaches for games in World Cup 2010 SOEN331W-L3: exercises on logic 31

32 Knowledge representation example 1 memberof(x,y) indicates that the player x is a member of the team y. coach(x,y) indicates that the coach of the team y is x. game(x,y,z,w) indicates that the teams x and y play on the date z at the city w. schedule(x,y,z) indicates that the team x is scheduled to play on the date y at the venue z. plays(x,y) indicates that the player x is to play on the date y. SOEN331W-L3: exercises on logic 32

33 Knowledge representation example 1: Specification of the integrity constraints on data A player is a member of only one team in the league. A coach coaches only one team; a team has only one coach. A team plays at most one game a day. No player of a team can be the coach of the team. Every game played by a team should appear in the schedule. For every game, there are some players who do not play on the day of the game. There are players in every team who do not play consecutively scheduled games. SOEN331W-L3: exercises on logic 33

34 Knowledge representation example: solutions (in the textbook) SOEN331W-L3: exercises on logic 34

35 Questions? 35

Learning Goals of CS245 Logic and Computation

Learning Goals of CS245 Logic and Computation Alice Gao April 27, 2018 Contents 1 Propositional Logic 2 2 Predicate Logic 4 3 Program Verification 6 4 Undecidability 7 1 1 Propositional Logic Introduction