Intelligent Systems Lecture 8 Knowledge Representation in FirstOrder Logic

1
Intelligent Systems Lecture 8Knowledge
Representation in First-Order Logic
WS 2008
2
Outline

• Motivation
• Syntax and semantics of FOL
• Using FOL
• Wumpus world in FOL

3
Pros and cons of propositional logic

• ? Propositional logic is declarative
• ? Propositional logic allows partial/disjunctive/n
  egated information
• (unlike most data structures and databases)?
• Propositional logic is compositional
• meaning of B1,1 ? P1,2 is derived from meaning of
  B1,1 and of P1,2
• ? Meaning in propositional logic is
  context-independent
• (unlike natural language, where meaning depends
  on context)?
• ? Propositional logic has very limited expressive
  power
• (unlike natural language)?
• E.g., cannot say "pits cause breezes in adjacent
  squares
• except by writing one sentence for each square

4
First-order logic (FOL)?

• FOL (like propositional logics) is declarative,
  compositional, and the meaning of its sentences
  is context-independent
• Propositional logic assumes the world contains
  facts, FOL (like natural language) assumes the
  world contains (ontological commitment)?
• Objects people, houses, numbers, colors,
  baseball games, wars,
• Relations red, round, prime, brother of, bigger
  than, part of, comes between,
• Functions father of, best friend, one more than,
  plus,
• FOL can express facts about some or all the
  objects in the universe
• Every sentence in FOL is either true, false or
  unknown to an agent (epistemological commitment)

5
Syntax of FOL Basic elements

• Constants KingJohn, 2, NUS,...
• Predicate symbols Brother, gt,...
• Function symbols Sqrt, LeftLegOf,...
• Variables x, y, a, b,...
• Connectives ?, ?, ?, ?, ?
• Quantifiers ?, ?

6
FOL sentences
Atomic sentence predicate (term1,...,termn)
or term1 term2 Term
function (term1,...,termn) or
constant or variable Complex sentences are made
from atomic sentences using connectives ?S, S1 ?
S2, S1 ? S2, S1 ? S2, S1 ? S2,
7
Truth in FOL

• Sentences are true with respect to a model and an
  interpretation
• A model contains objects (domain elements) and
  relations among them
• Interpretation specifies referents for
• constant symbols ? objects
• predicate symbols ? relations
• function symbols ? functions
• An atomic sentence predicate(term1,...,termn) is
  true
• iff the objects referred to by term1,...,termn
• are in the relation referred to by predicate

8
Models for FOL Example
9
Truth in the example
10
Universal quantification

• ?ltvariablesgt ltsentencegt
• "Everyone at UIBK is smart"
• ?x At(x,UIBK) ? Smart(x)?
• ?x P is true in a model m iff P is true with x
  being each possible object in the model
• Roughly speaking, equivalent to the conjunction
  of instantiations of P
• At(KingJohn, UIBK) ? Smart(KingJohn)
• ? At(Richard, UIBK) ? Smart(Richard)
• ? At(UIBK, UIBK) ? Smart(UIBK)
• ? ...

11
A common mistake to avoid

• Typically, ? is the main connective with ?
• Common mistake using ? as the main connective
  with ?
• ?x At(x,UIBK) ? Smart(x)?
• means Everyone is at UIBK and everyone is
  smart
• Correct ?x At(x,UIBK) ? Smart(x)?

12
Existential quantification

• ?ltvariablesgt ltsentencegt
• "Someone at UIBK is smart"
• ?x At(x,UIBK) ? Smart(x)?
• ?x P is true in a model m iff P is true with x
  being some possible object in the model
• Roughly speaking, equivalent to the disjunction
  of instantiations of P
• At(KingJohn,UIBK) ? Smart(KingJohn)
• ? At(Richard,UIBK) ? Smart(Richard)
• ? At(UIBK,UIBK) ? Smart(UIBK)
• ? ...

13
Another common mistake to avoid

• Typically, ? is the main connective with ?
• Common mistake using ? as the main connective
  with ?
• ?x At(x,UIBK) ? Smart(x)?
• is true if there is anyone who is not at UIBK!
• Usually used in Queries
• "Is there someone in UIBK who is smart?"
• Correct ?x At(x,UIBK) ? Smart(x)?

14
Properties of quantifiers

• ?x ?y is the same as ?y ?x
• ?x ?y is the same as ?y ?x
• ?x ?y is not the same as ?y ?x
• ?x ?y Loves(x,y)?
• There is a person who loves everyone in the
  world
• ?y ?x Loves(x,y)?
• Everyone in the world is loved by at least one
  person
• Quantifier duality each can be expressed using
  the other
• ?x Likes(x,IceCream) ??x ?Likes(x,IceCream)?
• ?x Likes(x,Broccoli) ??x ?Likes(x,Broccoli)?

15
Equality

• term1 term2 is true under a given
  interpretation if and only if term1 and term2
  refer to the same object
• E.g., definition of Sibling in terms of Parent
• ?x,y Sibling(x,y) ? ?(x y) ? ?m,f ? (m f) ?
  Parent(m,x) ? Parent(f,x) ? Parent(m,y) ?
  Parent(f,y)

16
Using FOL

• The family domain
• Brothers are siblings
• ?x,y Brother(x,y) ? Sibling(x,y)?
• One's mother is one's female parent
• ?m,c Mother(c) m ? (Female(m) ? Parent(m,c))?
• Sibling is symmetric
• ?x,y Sibling(x,y) ? Sibling(y,x)?

17
Interacting with FOL KBs
18
Wumpus world revisited

• PEAS
• Performance measure
• gold 1000, death -1000
• -1 per step, -10 for using the arrow
• Environment
• Squares adjacent to wumpus are smelly
• Squares adjacent to pit are breezy
• Glitter iff gold is in the same square
• Shooting kills wumpus if you are facing it
• Shooting uses up the only arrow
• Grabbing picks up gold if in same square
• Releasing drops the gold in same square
• Actuators
• Left turn, Right turn, Forward, Grab, Release,
  Shoot
• Sensors
• Stench, Breeze, Glitter, Bump, Scream

19
Wumpus world in FOL

• Smell and breeze (but no glitter) at t5
• Tell(KB,Percept(Smell,Breeze,None,5))?
• Ask(KB,?a BestAction(a,5))?
• Does the KB entail some best action at t5?
• Answer Yes, a/Shoot ? substitution (binding
  list)?
• Given a sentence S and a substitution s, Ss
  denotes the result of plugging s into S e.g.,
• S Smarter(x,y)?
• s x/Hillary,y/Bill
• Ss Smarter(Hillary,Bill)?
• Ask(KB,S) returns some/all s such that KB s

20
Knowledge base for the wumpus world - Percepts
and actions

• Perception
• ?t,s,b Percept(s,b,Glitter,t) ? Glitter(t)?
• Reflex
• ?t Glitter(t) ? BestAction(Grab,t)?

21
Representing the Environment

• ?x,y,a,b Adjacent(x,y,a,b) ?
• a,b ? x1,y, x-1,y,x,y1,x,y-1
• Pit(1,3), Pit (2,5), etc.
• Wumpus Home(Wumpus)?
• Properties of squares
• ?s,t At(Agent,s,t) ? Breeze(t) ? Breezy(s)?

22
Deducing hidden properties

• Squares are breezy near a pit
• Diagnostic rule---infer cause from effect
• ?s Breezy(s) ? ?r Adjacent(r,s) ? Pit(r)?
• ?s ? Breezy(s) ? ? ?r Adjacent(r,s) ? Pit(r)?
• ?s Breezy(s) ? ?r Adjacent(r,s) ? Pit(r)?
• Causal rule---infer effect from cause
• ?r Pit(r) ? ?s Adjacent(r,s) ? Breezy(s)
• ?s ?r Adjacent(r,s) ? ? Pit(r) ? ?Breezy(s)?
• Diagnostic reasoning systems vs. model-based
  reasoning systems (based on causal rules)?

23
Summary

• First-order logic
• objects, relations and functions are semantic
  primitives
• syntax constants, function symbols, predicate
  symbols, equality, quantifiers
• Increased expressive power sufficient to define
  wumpus world, with time stamps, etc.
• Next week inference in FOL

24

• Questions?

25
See you in two weeks!