First-Order Logic
LESSON 8
Reading
Chaper 8
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Outline
Why FOL?
Syntax and semantics of FOL
Using FOL
Wumpus world in FOL
Knowledge engineering in FOL
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Pros and cons of propositional
logic
Propositional logic is declarative
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
◦
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First-order logic
Whereas propositional logic assumes the world contains facts,
first-order logic (like natural language) assumes the world contains
◦ 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, …
◦
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Syntax of FOL: Basic elements
Constants KingJohn, 2, NUS,...
Predicates Brother, >,...
Functions Sqrt, LeftLegOf,...
Variables x, y, a, b,...
Connectives , , , ,
Equality =
Quantifiers ,
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Atomic sentences
Atomic sentence = predicate (term1,...,termn)
or term1 = term2
Term = function (term1,...,termn)
or constant or variable
E.g., Brother(KingJohn,RichardTheLionheart) >
(Length(LeftLegOf(Richard)), Length(LeftLegOf(KingJohn)))
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Complex sentences
Complex sentences are made from atomic sentences using connectives
S, S1 S2, S1 S2, S1 S2, S1 S2,
E.g. Sibling(KingJohn,Richard) Sibling(Richard,KingJohn)
>(1,2) ≤ (1,2)
>(1,2) >(1,2)
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Truth in first-order logic
Sentences are true with respect to a model and an interpretation
Model contains objects (domain elements) and relations among them
Interpretation specifies referents for
constant symbols → objects
predicate symbols → relations
function symbols → functional relations
An atomic sentence predicate(term1,...,termn) is true
iff the objects referred to by term1,...,termn
are in the relation referred to by predicate
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Models for FOL: Example
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Universal quantification
<variables> <sentence>
Everyone at NUS is smart:
x At(x,NUS) 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,NUS) Smart(KingJohn)
At(Richard,NUS) Smart(Richard)
At(NUS,NUS) Smart(NUS)
...
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A common mistake to avoid
Typically, is the main connective with
Common mistake: using as the main connective
with :
x At(x,NUS) Smart(x)
means “Everyone is at NUS and everyone is smart”
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Existential quantification
<variables> <sentence>
Someone at NUS is smart:
x At(x,NUS) 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,NUS) Smart(KingJohn)
At(Richard,NUS) Smart(Richard)
At(NUS,NUS) Smart(NUS)
...
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Another common mistake to
avoid
Typically, is the main connective with
Common mistake: using as the main connective with
:
x At(x,NUS) Smart(x)
is true if there is anyone who is not at NUS!
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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)
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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)]
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Using FOL
The kinship 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)
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Using FOL
The set domain:
s Set(s) (s = {} ) (x,s2 Set(s2) s = {x|s2})
x,s {x|s} = {}
x,s x s s = {x|s}
x,s x s [ y,s2} (s = {y|s2} (x = y x s2))]
s1,s2 s1 s2 (x x s1 x s2)
s1,s2 (s1 = s2) (s1 s2 s2 s1)
x,s1,s2 x (s1 s2) (x s1 x s2)
x,s1,s2 x (s1 s2) (x s1 x s2)
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Interacting with FOL KBs
Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t=5:
Tell(KB,Percept([Smell,Breeze,None],5))
Ask(KB,a BestAction(a,5))
I.e., does the KB entail some best action at t=5?
Answer: Yes, {a/Shoot} ← substitution (binding list)
Given a sentence S and a substitution σ,
Sσ denotes the result of plugging σ into S; e.g.,
S = Smarter(x,y)
σ = {x/Hillary,y/Bill}
Sσ = Smarter(Hillary,Bill)
Ask(KB,S) returns some/all σ such that KB╞ σ
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Knowledge base for the
wumpus world
Perception
◦ t,s,b Percept([s,b,Glitter],t) Glitter(t)
◦
Reflex
◦ t Glitter(t) BestAction(Grab,t)
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Deducing hidden properties
x,y,a,b Adjacent([x,y],[a,b])
[a,b] {[x+1,y], [x-1,y],[x,y+1],[x,y-1]}
Properties of squares:
s,t At(Agent,s,t) Breeze(t) Breezy(s)
Squares are breezy near a pit:
◦ Diagnostic rule---infer cause from effect
s Breezy(s) \Exi{r} Adjacent(r,s) Pit(r)$
◦ Causal rule---infer effect from cause
r Pit(r) [s Adjacent(r,s) Breezy(s)$ ]
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Knowledge engineering in FOL
1. Identify the task
2.
2. Assemble the relevant knowledge
3.
3. Decide on a vocabulary of predicates, functions, and constants
4.
4. Encode general knowledge about the domain
5.
5. Encode a description of the specific problem instance
6.
6. Pose queries to the inference procedure and get answers
7.
7. Debug the knowledge base
8.
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The electronic circuits domain
One-bit full adder
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The electronic circuits domain
1. Identify the task
2.
◦ Does the circuit actually add properly? (circuit verification)
◦
2. Assemble the relevant knowledge
3.
◦ Composed of wires and gates; Types of gates (AND, OR, XOR, NOT)
◦
◦ Irrelevant: size, shape, color, cost of gates
◦
3. Decide on a vocabulary
4.
◦ Alternatives:
◦
Type(X1) = XOR
Type(X1, XOR)
XOR(X1)
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The electronic circuits domain
4. Encode general knowledge of the domain
5.
◦ t1,t2 Connected(t1, t2) Signal(t1) = Signal(t2)
◦ t Signal(t) = 1 Signal(t) = 0
◦
◦ 1≠0
◦
◦ t1,t2 Connected(t1, t2) Connected(t2, t1)
◦
◦ g Type(g) = OR Signal(Out(1,g)) = 1 n Signal(In(n,g)) = 1
◦
◦ g Type(g) = AND Signal(Out(1,g)) = 0 n Signal(In(n,g)) = 0
◦
◦ g Type(g) = XOR Signal(Out(1,g)) = 1 Signal(In(1,g)) ≠ Signal(In(2,g))
◦
◦ g Type(g) = NOT Signal(Out(1,g)) ≠ Signal(In(1,g))
◦
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The electronic circuits domain
5. Encode the specific problem instance
6.
Type(X1) = XOR Type(X2) = XOR
Type(A1) = AND Type(A2) = AND
Type(O1) = OR
Connected(Out(1,X1),In(1,X2)) Connected(In(1,C1),In(1,X1))
Connected(Out(1,X1),In(2,A2)) Connected(In(1,C1),In(1,A1))
Connected(Out(1,A2),In(1,O1)) Connected(In(2,C1),In(2,X1))
Connected(Out(1,A1),In(2,O1)) Connected(In(2,C1),In(2,A1))
Connected(Out(1,X2),Out(1,C1)) Connected(In(3,C1),In(2,X2))
Connected(Out(1,O1),Out(2,C1)) Connected(In(3,C1),In(1,A2))
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The electronic circuits domain
6. Pose queries to the inference procedure
7.
What are the possible sets of values of all the terminals for the adder circuit?
i1,i2,i3,o1,o2 Signal(In(1,C_1)) = i1 Signal(In(2,C1)) = i2
Signal(In(3,C1)) = i3 Signal(Out(1,C1)) = o1
Signal(Out(2,C1)) = o2
7. Debug the knowledge base
8.
May have omitted assertions like 1 ≠ 0
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Summary
First-order logic:
◦ objects and relations are semantic primitives
◦ syntax: constants, functions, predicates, equality, quantifiers
◦
Increased expressive power: sufficient to define wumpus world
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