Agent that reason logically

1
Agent that reason logically

• ????

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Knowledge Base

• A set of representations of facts about the world
• Knowledge representation language
• tell what has been told to the knowledge base
  previously
• ask a question and the answer
• Inference what follows from what the KB has
  been Telled
• Background knowledge a knowledge base which may
  initially contained
• Sentence individual representation of a fact

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Knowledge base

• The knowledge level saying what it knows to KB
  ? Golden Gates Bridge links San Francisco and
  Marin Country
• The logical level the knowledge is encoding
  into sentences ? Links(GGBridge, SF, Marin)
• The implementation level the level that runs
  on the agent architecture (data structures to
  represent knowledge or facts)

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Knowledge

• declarative/procedural
• love(john, mary).
• can_fly(X) - bird(X), not(can_fly(X)), !.
• learning general knowledge about the
  environment given a series of percepts
• Commonsense knowledge

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Specifying the environment
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Domain specific knowledge

• Domain specific knowledge
• In the squares directly adjacent to a pit, the
  agent will perceive a breeze
• Commonsense knowledge
• logical reasoning
• stench(1,2) setnch(2,1) ? wumpus(2,2)
• wumpus(1,3) ?
• stench(2,1) stench(2,3) stench(1,4)

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Inference in Wumpus world(I)
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Inference in Wumpus world(II)
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Representation, Reasoning, and Logic

• Syntax the possible configurations that
  constitute sentences
• Semantics the facts in the world to which the
  sentences refer

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The logical reasoning
Figure 6.5 The connection between sentences and
facts is provided by the semantics of the
language. The property of one fact following from
some other facts is mirrored by the property of
one sentence being entailed by some other
sentences. Logical inference generates new
sentences that are entailed by existing sentences.
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Inference I

• Entailment generation of new sentences that
  are necessarily true, given that the old
  sentences are true
• Soundness, truth-preserving An inference
  procedure that generates only entailed sentences
  ? modus ponens lt-gt abduction
• KBi ?, ? is derived from KB by I
• Proof a sound inference procedure

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Inference II

• Completeness an inference procedure that can
  find a proof for any sentence that is entailed
• Proof specifying the reasoning steps that are
  sound
• Valid if and only if all possible
  interpretations in all possible worlds
• Tautologies, analytic sentences valid
  sentences
• Satisfiable if and only if there is some
  interpretation in some world for which it is true
• Unsatisfiable a sentence that is not
  satisfiable

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Logics

• Boolean logic
• Symbols represent whole propositions (facts)
• Boolean connectives
• First-order logic
• objects, predicates
• connectives, quantifiers

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Wrong logical reasoning

• FIRST VILLAGER We have found a witch. May we
  burn her?
• ALL A witch! Burn her!
• BEDEVERE Why do you think she is a witch?
• SECOND VILLAGER She turned me into a newt.
• BEDEVERE A newt?
• SECOND VILLAGER (after looking at himself for
  some time) I got better.
• ALL Burn her anyway.
• BEDEVERE Quiet! Quiet! There are ways of telling
  whether she is a witch.
• BEDEVERE Tell me What do you do with witches?
• ALL Burn them.
• BEDEVERE And what do you burn, apart from
  witches?
• FOURTH VILLAGER Wood?
• BEDEVERE So why do witches burn?
• SECOND VILLAGER (pianissimo) Because theyre
  made of wood?
• BEDEVERE Good.
• ALL I see. Yes, of course.
• BEDEVERE So how can we tell if she is made of
  wood?
• FIRST VILLAGER Make a bridge out of her.
• BEDEVERE Ah but can you not also make bridges
  out of stone?

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Ontological and epistemological commitments

• Ontological commitments to do with the nature
  of reality
• Propositional logic(true/false), Predicate logic,
  Temporal logic
• Epistemological commitments to do with the
  possible states of knowledge an agent can have
  using various types of logic
• degree of belief
• fuzzy logic

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Commitments
Formal languages and their and ontological and
epistemological commitments
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Propositional Logic

• logical constant true/false
• propositional symbols P, Q
• parentheses (P Q)
• logical connectives (conjuction),
  v(disjunction), -gt(implication),
  lt-gt(equivalence), (negation)

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Grammar

• Sentence ? AtomicSentence
  ComplexSentence
• AtomicSentence ? True False
• P Q R
• ComplexSentence ? ( Sentence )
• Sentence Connective Sentence
• ?Sentence
• Connective ? ? ? ? ?

Figure 6.8 A BNF (Backus-Naur Form) grammar of
sentences in propositional logic.
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Semantics
Truth table showing validity of a complex sentence
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Validity and Inference

• Truth tables for five logical connectives

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Models

• Any world in which a sentence is true under a
  particular interpretation
• Entailment a sentence ? is entailed by a
  knowledge base KB if the models of the KB are all
  models of ?
• The set of models of P Q is the intersection of
  the models of P and the models of Q

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Inference Rules for propositional logic

• Modus Ponens or Implication-Elimination (From an
  implication and the premise of the implication,
  you can infer the conclusion.)
• And-Elimination (From a conjunction, you can
  infer any of the conjuncts.)
• And-Introduction (From a list of sentences, you
  can infer their conjunction.)
• Or-Introduction (From a sentence, you can infer
  its disjunction with anything else at all.)
• Double-Negation Elimination (From a doubly
  negated sentence, you can infer a positive
  sentence.)
• Unit Resolution (From a disjunction, if one of
  the disjuncts is false, then you can infer the
  other one is true.)

? gt ?, ?
?
?1 ? ?2 ? ? ?n
?i
?1, ?2, , ?n
?1 ? ?2 ? ? ?n
?i
?1 ? ?2 ? ? ?n
???
?
? ? ?, ? ?
?

• Resolution (This is the most difficult. Because
  ? cannot be both true and false, one of the other
  disjucts must be true in one of the premises. Or
  equivalently, implication is transitive.)

? ? ?, ? ? ? ?
? ? gt ?, ? gt ?
or equivalently
? ? ?
? ? gt ?
Figure 6.13 Seven inference for propositional
logic. The unit resolution rule is a special case
of the resolution rule, which in turn is a
special case of the full resolution rule for
first-order logic discussed in Chapter 9.
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Complexity of propositional inference

• NP-complete
• Monotonicity
• If KB1 ? then (KB1 ? KB2) ?
• Horn clause logic
• polynomial time complexity
• P1?P2?.?Pn ? Q

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Wumpus world

• Initial state
• S1,1 B1,1
• S2,1 B2,1
• S1,2 B1,2
• Rule
• R1 S1,1 -gt W1,1 W1,2 W2,1
• R2 S2,1 -gt W1,1 W2,1 W2,2 W3,1
• R3 S1,2 -gt W1,1 W1,2 W2,2 W1,3
• R4 S1,2 -gt W1,3 V W1,2 V W2,2 V W1,2

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Finding the wumpus

• Inference process
• Modus ponens
• S1,1 and R1 ? W1,1 W1,2 W2,1
• And-Elimination
• W1,1 W1,2 W2,1
• Modus ponens and And-Elimination
• W2,2 W2,1 W3,1
• Modus ponens
• S1,2 and R4 ? W1,3 V W1,2 V W2,2 V W1,1

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Inference process(cont.)

• unit resolution
• W1,1 and W1,3 V W1,2 V W2,2 V W1,1
• ? W1,3 V W1,2 V W2,2
• unit resolution
• W2,2 and W1,3 V W1,2 V W2,2
• ? W1,3 V W1,2
• unit resolution
• W1,2 and W1,3 V W1,2 ? W1,3

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Translating knowledge into action

• A1,1 EastA W2,1 -gt Forward
• EastA facing east
• Propositional logic is not powerful enough to
  solve the wumpus problem easily

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??

• 6.3, 6.6, 6.7, 6.9, 6.10, 6.12, 6.15, 6.16

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First-order Logic
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Limitation of propositional logic

• A very limited ontology
• ? to need to the representation power
• ? first-order logic

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First-order logic

• A stronger set of ontological commitments
• A world in FOL consists of objects, properties,
  relations, functions
• Objects ? people, houses, number, colors, Bill
  Clinton
• Relations ? brother of, bigger than, owns, love
• Properties ? red, round, bogus, prime
• Functions ?father of, best friend, third inning of

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Examples

• One plus two equals three
• objects one, two, three, one plus two
• Relation equal
• Function plus
• Squares neighboring the wumpus are smelly
• Objects wumpus, square
• Property smelly
• Relation neighboring

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First order logics

• Objects? relations
• ??, ??, ???? ?? ???? ??
• ??? ?? ???? ??? ??? ? king? ??? property? ? ?
  ??, ??? ??? ???? relation? ? ?? ??
• ??????? ? ??? ??, ? ??? ??? ???

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Syntax and Semantics

• Sentence ? AtomicSentence
• Sentence Connective Sentence
• Auantifier Variable,Sentence
• ?Sentence
• (Sentence)
• AtomicSentence ? Predicate(Term,)
  TermTerm
• Term ?Function (Term,)
• Constant
• Variable
• Connective ? ? ? ? ?
• Quantifier ? ? ?
• Constant ? A X1 John
• Variable ? a x s
• Predicate ? Before HanColor
  Raining
• Function ? Mother LeftLegOf

Figure 7.1 The syntax of first-order logic (with
equality) in BNF (Backus-Naur Form).
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?

• Constant symbols A, B, John,
• Predicate symbols Round, Brother
• Function symbols Cosine, FatherOf
• Terms King John, Richards left leg
• Atomic sentences Brother(Richard,John),
  Married(FatherOf(Richard), MotherOf(John))
• Complex sentences Older(John,30)gtyounger(John
  ,30)

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Quantifiers

• World a, b, c
• Universal quantifier (?)
• ?x Cat(x) gt Mammal(x) ?
• Cat(a) gt Mammal(a)
• Cat(a) gt Mammal(a)
• Cat(a) gt Mammal(a)
• Existential quantifier (?)
• ?x Sister(x, Sopt) Cat(x)

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Nested quantifiers

• ?x,y Parent(x,y) gt Child(y,x)
• ?x,y Brother(x,y) gt Sibling(y,x)
• ?x?y Loves(x,y)
• ?y?x Loves(x,y)

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De Morgans Rule

• ?x P ? ?x P PQ ? (P v Q)
• ?x P ? ?x P (PQ) ? P v Q
• ?x P ? ?x P PQ ? (P v Q)
• ?x P ? ?x P P v Q ? (PQ)

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Equality

• Identity relation
• Father(John) Henry
• ?x,y Sister(Spot,x) Sister(Spot,y)
• (xy)
• ? ?x,y Sister(Spot,x) Sister(Spot,y)

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Higher-order logic

• ?x,y (xy) ? (?p p(x) ? p(y))
• ?f,g (fg) ? (?x f(x) ?g(x))

?
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?-expression

• ?x,y x2 y2
• ?-expression can be applied to arguments to yield
  a logical term in the same way that a function
  can be
• (?x,y x2 y2)(25,24) 252-242 49
• ?x,y Gender(x) ?Gender(y) Address(x)
  Address(y)

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?! (The uniqueness quantifier)

• ?!x King(x)
• ?x King(x) ?y King(y) gt xy
• world? ???? ???? gt object? 1, 2, 3?? ?
• a w0 ? king, w1 ? kinga ? w1? model
• a,b w0 ? king, w1 ? kinga,
• w2 ?b, w3 ? a,b ? w1, w2? model

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Representation of sentences by FOPL

• Ones mother is ones female parent
• ?m,c Mother(c)m ? Female(m) Parent(m)
• Ones husband is ones male spouse
• ?w,h Husband(h,w) ? Male(h) Spouse(h,w)
• Male and female are disjoint categories
• ?x Male(x) ? Female(x)
• A grandparent is a parent of ones parent
• ?g,c Grandparent(g,c) ? ?p parent(g,p)
  parent(p,g)

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Representation of sentences by FOPL

• A sibling is another child of ones parents ?x,y
  Sibling(x,y) ? x?y ?p Parent(p,x) Parent(p,y)
• Symmetric relations
• ?x,y Sibling(x,y) ? Sibling(y,x)

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The domain of sets (I)

• The only sets are the empty set and those made by
  adjoining something to a set
• ?s Set(s) ? (sEmptySet) v (?x,s2 Set(s2)
  sAdjoin(x,s2))
• The empty set has no elements adjoined into it.
• ?x,s Adjoin(x,s)EmptySet
• Adjoining an element already in the set has no
  effect
• ?x,s Member(x,s) ? sAdjoin(x,s)
• The only members of a set are the elements that
  were adjoined into it
• ?x,s Member(x,s) ?
• ?y,s2 (sAdjoin(y,s2) (xy v
  Member(x,s)))

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The domain of sets (II)

• A set is a subset of another if and only if all
  of the first sets are members of the second set
  
• ?s1,s2 Subset(s1,s2) ?
• (?x Member(x,s1) gt member(x,s2))
• Two sets are equal if and only if each is a
  subset of the other
• ?s1,s2 (s1s2) ? (Subset(s1,s2) Subset(s2,s1))

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The domain of sets (III)

• An object is a member of the intersection of two
  sets if and only if it is a member of each of
  sets
• ?x,s1,s2 Member(x,Intersection(s1,s2)) ?
• Member(x,s1) Member(x,s2)
• An object is a member of the union of two sets if
  and only if it is a member of either set
• ?x,s1,s2 Member(x,Union(s1,s2)) ?
• Member(x,s1) v Member(x,s2)

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Asking questions and getting answers

• Tell(KB, (?m,c Mother(c)m ? Female(m)
  Parent(m,c)))
• Tell(KB, (Female(Maxi) Parent(Maxi,Spot)
  Parent(Spot,Boots)))
• Ask(KB,Grandparent(Maxi,Boots)
• Ask(KB, ?x Child(x, Spot))
• Ask(KB, ?x Mother(x)Maxi)
• Substitution, unification, x/Boots