(More) Propositional Logic and an Intro to Predicate Logic. CSCI 3202, Fall 2010

Transcription

1 (More) Propositional Logic and an Intro to Predicate Logic CSCI 3202, Fall 2010

2 Assignments Next week: Guest lectures (Jim Martin and Nikolaus Correll); Read Chapter 9 (but you can skip sections on logic programming, ) Problem Set 1 due Thursday! Problem Set 2 posted Thursday

3 Recurring Themes in Logic (not just propositional logic, by the way) Satisfiability Contradiction Tautology Entailment (implication) Equivalence

4 A First (Basic) Algorithmic Idea for Working with Propositional Logic Represent all of our knowledge in one particular standard syntax: conjunctive normal form To show that our knowledge entails a given statement A, show that adding NOT(A) to our knowledge would result in a contradiction: that is, KNOWLEDGE AND NOT(A) is unsatisfiable. Use the resolution rule repeatedly, producing new clauses, until either we arrive at the empty clause (FALSE) or until we can produce no more new clauses. In the former case, A is entailed by our knowledge.

5 Getting Statements into CNF Step 1: Replace <--> with two --> statements, then replace all --> with the equivalent OR form: A --> B becomes (NOT A OR B) Step 2: Move NOT inward until it only applies to literals (atomic propositions), using DeMorgan s laws and double negation NOT (A AND B) becomes (NOT A) OR (NOT B) Step 3: Use distributivity law to get AND forms outside OR forms: A OR (B AND C) becomes (A OR B) AND (A OR C)

6 Resolution Just one rule, really: (A OR B) AND (NOT B OR C) --> (A OR C)

7 Modus Ponens is just resolution (A --> B) AND A B is just an instance of resolution: (NOT A OR B) AND A B

8 Forward and Backward Chaining These are strategies that are less general than resolution, but still extremely useful When sentences in our knowledge base are in the form of Horn clauses, we can represent our knowledge as an AND-OR graph and use forward or backward chaining.

9 Horn Clauses are of the form: (P AND Q AND R) --> S P A --> B

10 Forward Chaining: the Basic Idea Suppose we have a given knowledge base and then assert one additional fact (like P ). Find all Horn clauses that have P among their premises, and if all premises have been asserted, go ahead and assert the conclusion. (Example: P --> Q means that we can now assert Q as well.) Repeat this process until no new assertions can be added.

11 Backward Chaining: the Basic Idea Suppose we have a given knowledge base and wish to see whether we can prove a particular assertion (say, Q ). Look to see whether Q is the consequent ( head ) of any Horn clause (e.g., P-->Q) and see if the body of that Horn clause has been asserted. If not, continue with this process by seeing if the body of the Horn clause can itself be proven by backward chaining. (For instance, we may find a clause of the form (A AND B)--> P where both A and B have already been asserted.)

12 Forward and Backward Chaining, Revisited Forward chaining looks at the data and sees what we can discover: it s a bottom-up or data-driven process Backward chaining tries to see if we can prove a particular statement: it s a goaldriven process

13 Propositional Logic: Where We ve Come So Far Basic objects are sentences with T/F values Connectors (AND, OR, NOT, etc.) are used to make compound sentences Basic rules of inference (Modus Ponens, etc.) are used to derive new sentences from a knowledge base of existing sentences Resolution as a general, all-purpose rule of inference Forward and backward chaining as efficient techniques for more special-purpose (AND/OR graph) situations

14 First-Order Predicate Logic We introduce a world of objects. Our logical sentences will refer to these objects. Bob Fred We also introduce the idea of relations. An atomic sentence now states a relation: Father-of(Bob, Fred)

15 First-Order Predicate Logic (continued) A relation can be viewed as a set of n-tuples of objects for which the relation happens to hold: The father relation: (Bob, Fred), (Joe, Jill), (Fred, Jane) Relations that are 1-tuples can be thought of as properties. The property Male: (Bob), (Fred), (Joe) Some relations are functions: there is exactly one value for each possible input object (or, sometimes, input objects).

16 First-Order Predicate Logic: A Bit More Terminology We still have all the usual connectors (AND, OR, and so forth) We have an equality symbol: Day-after(Monday) = Tuesday We also have two quantifiers: THERE-EXISTS and FOR- ALL FOR-ALL (x) [Human(x) --> (Male(x) OR Female(x))] FOR-ALL(s)[Breezy(s) --> THERE-EXISTS(p) (Adjacent(p,s) AND PIT(p))]

17 Quantifiers: An Example You can fool all of the people some of the time, and some of the people all of the time, but you can t fool all of the people all of the time. FOR-ALL (x) [Person(x) --> THERE-EXISTS(t) [Time(t) AND Fool-at-time(x, t)]] THERE-EXISTS(x) [Person(x) AND (FOR-ALL(t) [Time(t) --> Fool-at-time(x, t)]] NOT(FOR-ALL(x, t)[(person(x) AND Time(t)) --> Fool-at-time(x,t)]

18 Quantifiers and Dean Martin Everybody loves somebody sometime. FOR-ALL(x)[Person(x) --> THERE-EXISTS(y, t) [Person(y) and Time(t) and Loves-at-time(x,y,t)]] THERE-EXISTS(t,y)[Time(t) AND Person(y) AND FOR-ALL(x)[Person(x) --> Loves-at-time(x,y, t)]]

19 THERE-EXISTS(t)[Time(t) AND FOR-ALL(x) [Person(x) --> THERE-EXISTS(y) THERE-EXISTS(y)[Person(y) AND [Person(y) AND Loves-at-time(x,y,t)]} FOR-ALL(x) [Person (x) --> THERE-EXISTS(t)[Time(t) AND Loves-at-time(x, y, t)]]

20 What Can t We Do in FOPL? We can t take anything back once we ve asserted it. We can t make statements about relations themselves (e.g., Brother is a commutative relation ) We can t distinguish between types of truth We can t express degrees of belief

21 Some Sample Wumpus World Statements FOR-ALL(s) Breezy(s) <--> THERE-EXISTS(p)[Adjacent(p,s) and Pit(p)] FOR-ALL(x,y) (Wumpus(x) AND NOT(x = y)) --> NOT(Wumpus(y)) FOR-ALL(x,y)(Wumpus(x) AND Wumpus(y)) --> (x=y) THERE-EXISTS(x) Wumpus(x)

22 FOPL Version of Sudoku Square(S111) Square(S121) Digit(1) Digit (2) Row(S111) = 1 Column(S111)= 1 Block(S111) = 1

23 Sample Sudoku Constraints FOR-ALL(s) (Square(s) THERE-EXISTS(n) (Contents(s,n) AND Digit (n))) FOR-ALL (s1, s2) ((Row(s1) = Row (s2) AND NOT(s1=s2)) NOT (Contents(s1) = Contents(s2)))

24 FOR-ALL (n, b) (Digit(n) AND THERE-EXISTS(s)(Block(s) = b)) (THERE-EXISTS(s) (Block(s) = b AND Contents(s) = n)))