First-Order Logic
CISC 6525
Artificial Intelligence
First-Order Logic
Russell & Norvig, Chapter 8
Outline
Why FOL?
Syntax and semantics of FOL
Using FOL
Wumpus world in FOL
Knowledge engineering in FOL
Last Week: Propositional Logic
Propositions; Syntax, Semantics
Entailment: α ╞ iff M(α) M()
Model checking for Wumpus world
KB ╞ g iff M(KB) M(g)
Inference: α ├i = sentence can be derived from α (sound?, complete?)
Resolution: show KBα unsatisfiable
Horn clauses; Back/Forward Chaining
Propositional Logic: Pros & Cons
Propositional logic is declarative
Propositional logic allows partial/disjunctive/negated information
(unlike many 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
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, …
Syntax of FOL: Basic elements
Constants KingJohn, 2, NUS,…
Predicates Brother, >,…
Functions Sqrt, LeftLegOf,…
Variables x, y, a, b,…
Connectives , , , ,
Equality =
Quantifiers ,
Atomic sentences
Atomic sentence predicate (term1,…,termn) or term1 = term2
Term function (term1,…,termn) or constant
or variable
E.g.,
Brother( KingJohn, RichardTheLionheart )
E.g.,
>( Length( LeftLegOf( Richard) ),
Length( LeftLegOf( KingJohn) ) )
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 )
E.g.,
>( 1, 2 ) ≤ ( 1, 2)
>( 1,2 ) >( 1, 2)
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
Models for FOL: Example
Universal quantification
Everyone at Fordham is smart:
x At(x, Fordham ) 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,Fordham) Smart(KingJohn)
At(Richard, Fordham) Smart(Richard)
At(Fordham,Fordham) Smart(Fordham)
…
A common mistake to avoid
Typically, is the main connective with
Common mistake: using as the main connective with –
x At(x,Fordham) Smart(x)
means “Everyone is at Fordham and everyone is smart”
Existential quantification
Someone at Fordham is smart:
x At(x,Fordham) 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,Fordham) Smart(KingJohn)
At(Richard, Fordham) Smart(Richard)
At(Fordham,Fordham) Smart(Fordham)
…
Another common mistake to avoid
Typically, is the main connective with
Common mistake: using as the main connective with –
x At(x,Fordham) Smart(x)
is true if there is anyone who is not at Fordham!
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)
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 ) ]
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)
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)
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
Interacting with FOL KBs
Knowledge base for the wumpus world
Perception
t,s,b Percept([s,b,Glitter],t) Glitter(t)
Reflex
t Glitter(t) BestAction(Grab,t)
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) r Adjacent(r,s) Pit(r)
Causal rule—infer effect from cause
r Pit(r) [s Adjacent(r,s) Breezy(s) ]
Knowledge engineering in FOL
Identify the task
Assemble the relevant knowledge
Decide on a vocabulary of predicates,
functions, and constants
Encode general knowledge about the domain
Encode a description of the specific problem instance
Pose queries to the inference procedure and get answers
Debug the knowledge base
Summary
First-order logic:
objects and relations are semantic primitives
syntax: constants, functions, predicates, equality, quantifiers
Increased expressive power: sufficient to define wumpus world
The electronic circuits domain
One-bit full adder
The electronic circuits domain
Identify the task
Does the circuit actually add properly? (circuit verification)
Assemble the relevant knowledge
Composed of wires and gates; Types of gates (AND, OR, XOR, NOT)
Irrelevant: size, shape, color, cost of gates
Decide on a vocabulary
Alternatives:
Type(X1) = XOR
Type(X1, XOR)
XOR(X1)
The electronic circuits domain
Encode general knowledge of the domain
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))
The electronic circuits domain
Encode the specific problem instance
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))
The electronic circuits domain
Pose queries to the inference procedure
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
Debug the knowledge base
May have omitted assertions like 1 ≠ 0
Summary
First-order logic:
objects and relations are semantic primitives
syntax: constants, functions, predicates, equality, quantifiers
Increased expressive power: sufficient to define wumpus world