For next Tuesday. Read chapter 8 No written homework Initial posts due Thursday 1pm and responses due by class time Tuesday

Transcription

1 For next Tuesday Read chapter 8 No written homework Initial posts due Thursday 1pm and responses due by class time Tuesday

2 Any questions? Program 1

3 Imperfect Knowledge What issues arise when we don t know everything (as in standard card games)?

4 State of the Art Chess Deep Blue, Hydra, Rybka Checkers Chinook (alpha-beta search) Othello Logistello Backgammon TD-Gammon (learning) Go Bridge Scrabble

5 Games/Mainstream AI

6 What about the games we play?

7 Knowledge Base Knowledge Inference mechanism (domain-independent) Information (domain-dependent) Knowledge Representation Language Sentences (which are not quite like English sentences) The KRL determine what the agent can know It also affects what kind of reasoning is possible Tell and Ask

8 Getting Knowledge We can TELL the agent everything it needs to know We can create an agent that can learn new information to store in its knowledge base

9 The Wumpus World Simple computer game Good testbed for an agent A world in which an agent with knowledge should be able to perform well World has a single wumpus which cannot move, pits, and gold

10 Wumpus Percepts The wumpus s square and squares adjacent to it smell bad. Squares adjacent to a pit are breezy. When standing in a square with gold, the agent will perceive a glitter. The agent can hear a scream when the wumpus dies from anywhere The agent will perceive a bump if it walks into a wall. The agent doesn t know where it is.

11 Wumpus Actions Go forward Turn left Turn right Grab (picks up gold in that square) Shoot (fires an arrow forward--only once) If the wumpus is in front of the agent, it dies. Climb (leave the cavern--only good at the start square)

12 Consequences Entering a square containing a live wumpus is deadly Entering a square containing a pit is deadly Getting out of the cave with the gold is worth 1,000 points. Getting killed costs 10,000 points Each action costs 1 point

13 Possible Wumpus Environment Stench Breeze Pit Stench Wumpus Stench Breeze Stench Gold Pit Breeze Breeze Agent Breeze Pit Breeze

14 Knowledge Representation Two sets of rules: Syntax: determines what atomic symbols exist in the language and how to combine them into sentences Semantics: Relationship between the sentences and the world --needed to determine truth or falsehood of the sentences

15 Reasoning Entailment Inference May produce new sentences entailed by KB May be used to determine which a particular sentence is entailed by the KB We want inference procedures that are sound, or truth-preserving.

16 What Is a Logic? A set of language rules Syntax Semantics A proof theory A set of rules for deducing the entailments of a set of sentences

17 Distinguishing Logics Language Ontological Commitment (what exists in the world) Epistemological Commitment (What an agent believes about facts) true/false/unknown Propositional facts Logic First-order logic facts, objects, true/false/unknown relations Temporal logic facts, objects, true/false/unknown relations, times Probability theory facts degree of belief 0 1 Fuzzy logic degree of truth degree of belief 0 1

18 Propositional Logic Simple logic Deals only in facts Provides a stepping stone into first order logic

19 Syntax Logical Constants: true and false Propositional symbols P, Q... are sentences If S is a sentence then (S) is a sentence. If S is a sentence then S is a sentence. If S 1 and S 2 are sentences, then so are: S 1 S 2 S 1 S 2 S 1 S 2 S 1 S 2

20 Semantics true and false mean truth or falsehood in the world P is true if its proposition is true of the world S is the negation of S The remainder follow standard truth tables S 1 S 2 : AND S 1 S 2 : inclusive OR S 1 S 2 : True unless S 1 is true and S 2 is false S 1 S 2 : bi-conditional, or if and only if

21 Vocabulary An interpretation is an assignment of true or false to each atomic proposition A sentence true under any interpretation is valid (a tautology or analytic sentence) Validity can be checked by exhaustive checking of truth tables A sentence that can be true is satisfiable

22 Rules of Inference Alternative to truth-table checking A sequence of inference rule applications leading to a desired conclusion is a logical proof We can check inference rules using truth tables, and then use to create sound proofs We can treat finding a proof as a search problem

23 Propositional Inference Rules Modus Ponens or Implication Elimination And Elimination And Introduction Unit Resolution Resolution

24 Building an Agent with Propositional Logic Propositional logic has some nice properties Easily understood Easily computed Can we build a viable wumpus world agent with propositional logic???

25 The Problem Propositional Logic only deals with facts. We cannot easily represent general rules that apply to any square. We cannot express information about squares and relate (we can t easily keep track of which squares we have visited)

26 More Precisely In propositional logic, each possible atomic fact requires a separate unique propositional symbol. If there are n people and m locations, representing the fact that some person moved from one location to another requires nm 2 separate symbols.

27 First Order Logic Predicate logic includes a richer ontology: objects (terms) properties (unary predicates on terms) relations (n ary predicates on terms) functions (mappings from terms to other terms) Allows more flexible and compact representation of knowledge Move(x, y, z) for person x moved from location y to z.

28 Syntax for First Order Logic Sentence AtomicSentence Sentence Connective Sentence Quantifier Variable Sentence Sentence (Sentence) AtomicSentence Predicate(Term, Term,...) Term=Term Term Function(Term,Term,...) Constant Variable Connective Quanitfier $ " Constant A John Car1 Variable x y z... Predicate Brother Owns... Function father of plus...

29 Terms Objects are represented by terms: Constants: Block1, John Function symbols: father of, successor, plus An n ary function maps a tuple of n terms to another term: father of(john), succesor(0), plus(plus(1,1),2) Terms are simply names for objects. Logical functions are not procedural as in programming languages. They do not need to be defined, and do not really return a value. Functions allow for the representation of an infinite number of terms.

30 Predicates Propositions are represented by a predicate applied to a tuple of terms. A predicate represents a property of or relation between terms that can be true or false: Brother(John, Fred), Left of(square1, Square2) GreaterThan(plus(1,1), plus(0,1)) In a given interpretation, an n ary predicate can defined as a function from tuples of n terms to {True, False} or equivalently, a set tuples that satisfy the predicate: {<John, Fred>, <John, Tom>, <Bill, Roger>,...}

31 Sentences in First Order Logic An atomic sentence is simply a predicate applied to a set of terms. Owns(John,Car1) Sold(John,Car1,Fred) Semantics is True or False depending on the interpretation, i.e. is the predicate true of these arguments. The standard propositional connectives ( ) can be used to construct complex sentences: Owns(John,Car1) Owns(Fred, Car1) Sold(John,Car1,Fred) Owns(John, Car1) Semantics same as in propositional logic.

32 Quantifiers Allow statements about entire collections of objects Universal quantifier: "x Asserts that a sentence is true for all values of variable x "x Loves(x, FOPC) "x Whale(x) Mammal(x) "x ("y Dog(y) Loves(x,y)) ("z Cat(z) Hates(x,z)) Existential quantifier: $ Asserts that a sentence is true for at least one value of a variable x $x Loves(x, FOPC) $x(cat(x) Color(x,Black) Owns(Mary,x)) $x("y Dog(y) Loves(x,y)) ("z Cat(z) Hates(x,z))

33 Use of Quantifiers Universal quantification naturally uses implication: "x Whale(x) Mammal(x) Says that everything in the universe is both a whale and a mammal. Existential quantification naturally uses conjunction: $x Owns(Mary,x) Cat(x) Says either there is something in the universe that Mary does not own or there exists a cat in the universe. "x Owns(Mary,x) Cat(x) Says all Mary owns is cats (i.e. everthing Mary owns is a cat). Also true if Mary owns nothing. "x Cat(x) Owns(Mary,x) Says that Mary owns all the cats in the universe. Also true if there are no cats in the universe.

34 Nesting Quantifiers The order of quantifiers of the same type doesn't matter: "x"y(parent(x,y) Male(y) Son(y,x)) $x$y(loves(x,y) Loves(y,x)) The order of mixed quantifiers does matter: "x$y(loves(x,y)) Says everybody loves somebody, i.e. everyone has someone whom they love. $y"x(loves(x,y)) Says there is someone who is loved by everyone in the universe. "y$x(loves(x,y)) Says everyone has someone who loves them. $x"y(loves(x,y)) Says there is someone who loves everyone in the universe.

35 Variable Scope The scope of a variable is the sentence to which the quantifier syntactically applies. As in a block structured programming language, a variable in a logical expression refers to the closest quantifier within whose scope it appears. $x (Cat(x) "x(black (x))) The x in Black(x) is universally quantified Says cats exist and everything is black In a well formed formula (wff) all variables should be properly introduced: $xp(y) not well formed A ground expression contains no variables.

36 Relations Between Quantifiers Universal and existential quantification are logically related to each other: "x Love(x,Saddam) $x Loves(x,Saddam) "x Love(x,Princess Di) $x Loves(x,Princess Di) General Identities "x P $x P "x P $x P "x P $x P $x P "x P "x P(x) Q(x) "x P(x) "x Q(x) $x P(x) Q(x) $x P(x) $x Q(x)