Planning and search
Logical agents; FOL
Logical agents; FOL
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Outline
♦ Knowledge-based agents
♦ Wumpus world knowledge in propositional logic
Logical agents; FOL
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Knowledge bases
Inference engine
domain−independent algorithms
Knowledge base
domain−specific content
Knowledge base = set of sentences in a formal language
Declarative approach to building an agent (or other system):
Tell it what it needs to know
Then it can Ask itself what to do—answers should follow from the KB
Agents can be viewed at the knowledge level
i.e., what they know, regardless of how implemented
Or at the implementation level
i.e., data structures in KB and algorithms that manipulate them
Logical agents; FOL
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A simple knowledge-based agent
function KB-Agent( percept) returns an action
static: KB, a knowledge base
t, a counter, initially 0, indicating time
Tell(KB, Make-Percept-Sentence( percept, t))
action ← Ask(KB, Make-Action-Query(t))
Tell(KB, Make-Action-Sentence(action, t))
t←t + 1
return action
The agent must be able to:
Represent states, actions, etc.
Incorporate new percepts
Update internal representations of the world
Deduce hidden properties of the world
Deduce appropriate actions
Logical agents; FOL
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Wumpus World
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
4
Breeze
Stench
Breeze
3
Stench
PIT
Breeze
PIT
Gold
2
Breeze
Stench
Breeze
1
Breeze
PIT
START
1
2
3
4
Actions Left turn, Right turn,
Forward, Grab, Release, Shoot
Percepts Breeze, Glitter, Smell (in this square)
Logical agents; FOL
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Wumpus world characterization
Observable??
Deterministic??
Logical agents; FOL
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Wumpus world characterization
Observable?? No—only local perception
Deterministic?? Yes—outcomes exactly specified
Logical agents; FOL
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Exploring a wumpus world
OK
OK
OK
A
Logical agents; FOL
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Exploring a wumpus world
B
OK
A
OK
OK
A
Logical agents; FOL
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Exploring a wumpus world
P?
B
OK
P?
OK
OK
A
A
Logical agents; FOL
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Exploring a wumpus world
P?
B
OK
P?
OK S
OK
A
A
A
Logical agents; FOL
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Exploring a wumpus world
P?
P
B
OK
P?
OK
OK S
OK
A
A
A
W
Logical agents; FOL
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Exploring a wumpus world
P?
P
B
OK
A
P?
OK
A
OK S
A
OK
A
W
Logical agents; FOL
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Exploring a wumpus world
P?
OK
OK
P?
OK
P
B
A
A
OK S
A
OK
OK
A
W
Logical agents; FOL
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Exploring a wumpus world
P?
OK
P
B
OK
A
P? BGS OK
OK
A
A
OK S
A
OK
A
W
Logical agents; FOL
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Logic in general
Logics are formal languages for representing information
such that conclusions can be drawn
Syntax defines the sentences in the language
Semantics define the “meaning” of sentences;
i.e., define truth of a sentence in a world
E.g., the language of arithmetic
x + 2 ≥ y is a sentence; x2 + y > is not a sentence
x + 2 ≥ y is true iff the number x + 2 is no less than the number y
x + 2 ≥ y is true in a world where x = 7, y = 1
x + 2 ≥ y is false in a world where x = 0, y = 6
Logical agents; FOL
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Entailment
Entailment means that one thing follows from another:
KB |= α
Knowledge base KB entails sentence α
if and only if
α is true in all worlds where KB is true
E.g., x + y = 4 entails 4 = x + y
Entailment is a relationship between sentences (i.e., syntax)
that is based on semantics
Logical agents; FOL
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Models
Logicians typically think in terms of models, which are formally
structured worlds with respect to which truth can be evaluated (in propositional logic - assignments)
We say m is a model of a sentence α if α is true in m
M (α) is the set of all models of α
Then KB |= α if and only if M (KB) ⊆ M (α)
x
E.g. KB = it is raining and it is cold
α = it is cold
x
x
x
x
M(
x
x
)
x
x
x
x
x
xx
x x
x
x
x
x
x
x
x
x
x
xx
x
x
x
x
x
x
x
x
M(KB)
x
x
x
x
x
x
x
x
x
x
x
x
Logical agents; FOL
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Entailment in the wumpus world
Situation after detecting nothing in [1,1],
moving right, breeze in [2,1]
? ?
B
Consider possible models for ?s
assuming only pits
A
A
?
3 Boolean choices ⇒ 8 possible models
Logical agents; FOL
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Wumpus models
PIT
2
2
Breeze
Breeze
1
1
PIT
1
1
2
2
3
3
2
PIT
PIT
2
2
Breeze
PIT
1
Breeze
1
Breeze
1
1
2
1
2
3
3
1
2
3
2
PIT
PIT
PIT
2
Breeze
Breeze
1
1
PIT
2
PIT
PIT
1
1
2
2
3
3
Breeze
1
1
2
PIT
3
Logical agents; FOL
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Wumpus models
PIT
2
2
Breeze
Breeze
1
1
PIT
1
1
KB
2
2
3
3
2
PIT
PIT
2
2
Breeze
PIT
1
Breeze
1
Breeze
1
1
2
1
2
3
3
1
2
3
2
PIT
PIT
PIT
2
Breeze
Breeze
1
1
PIT
2
PIT
PIT
1
1
2
2
3
3
Breeze
1
1
2
PIT
3
KB = wumpus-world rules + observations
Logical agents; FOL
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Wumpus models
PIT
2
2
Breeze
Breeze
1
1
PIT
1
1
KB
2
2
3
3
1
2
PIT
PIT
2
2
Breeze
PIT
1
Breeze
1
Breeze
1
1
2
1
2
3
3
1
2
3
2
PIT
PIT
PIT
2
Breeze
Breeze
1
1
PIT
2
PIT
PIT
1
1
2
2
3
3
Breeze
1
1
2
PIT
3
KB = wumpus-world rules + observations
α1 = “[1,2] is safe”, KB |= α1, proved by examining all models of KB
Logical agents; FOL
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Wumpus models
PIT
2
2
Breeze
Breeze
1
1
PIT
1
1
KB
2
2
3
3
2
PIT
PIT
2
2
Breeze
PIT
1
Breeze
1
Breeze
1
1
2
1
2
3
3
1
2
3
2
PIT
PIT
PIT
2
Breeze
Breeze
1
1
PIT
2
PIT
PIT
1
1
2
2
3
3
Breeze
1
1
2
PIT
3
KB = wumpus-world rules + observations
Logical agents; FOL
23
Wumpus models
PIT
2
2
Breeze
Breeze
1
1
2
PIT
1
1
KB
2
2
3
3
2
PIT
PIT
2
2
Breeze
PIT
1
Breeze
1
Breeze
1
1
2
1
2
3
3
1
2
3
2
PIT
PIT
PIT
2
Breeze
Breeze
1
1
PIT
2
PIT
PIT
1
1
2
2
3
3
Breeze
1
1
2
PIT
3
KB = wumpus-world rules + observations
α2 = “[2,2] is safe”, KB 6|= α2
Logical agents; FOL
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Wumpus world sentences
Let Pi,j be true if there is a pit in [i, j].
Let Bi,j be true if there is a breeze in [i, j].
¬P1,1
¬B1,1
B2,1
“Pits cause breezes in adjacent squares”
Logical agents; FOL
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Wumpus world sentences
Let Pi,j be true if there is a pit in [i, j].
Let Bi,j be true if there is a breeze in [i, j].
¬P1,1
¬B1,1
B2,1
“Pits cause breezes in adjacent squares”
B1,1
B2,1
⇔
⇔
(P1,2 ∨ P2,1)
(P1,1 ∨ P2,2 ∨ P3,1)
“A square is breezy if and only if there is an adjacent pit”
Can reason about Wumpus world using truth tables (as for 2-queens)
Logical agents; FOL
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Pros and cons of propositional logic
Propositional logic is declarative: pieces of syntax correspond to facts
Propositional logic allows partial/disjunctive/negated 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
Logical agents; FOL
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