lecture-inference-in-first-order-logic.pptx

Inference in First-Order
Logic
Artificial Intelligence: Inference in First-Order Logic

1
A Brief History of Reasoning
450B.C. Stoics propositional logic, inference (maybe)
322B.C. Aristotle “syllogisms” (inference rules), quantifiers
1565 Cardano probability theory (propositional logic + uncertainty)
1847 Boole propositional logic (again)
1879 Frege first-order logic
1922 Wittgenstein proof by truth tables
1930 Go¨ del ∃ complete algorithm for FOL
1930 Herbrand complete algorithm for FOL (reduce to propositional)
1931 Go¨ del ¬∃ complete algorithm for arithmetic systems
1960 Davis/Putnam “practical” algorithm for propositional logic
1965 Robinson “practical” algorithm for FOL—resolution
Artificial Intelligence: Inference in First-Order Logic 5 March 2024

2
The Story So Far
●Propositional logic
●Subset of propositional logic: horn clauses
●Inference algorithms
– forward chaining
– backward chaining
– resolution (for full propositional logic)
●First order logic (FOL)
– variables
– functions
– quantifiers
– etc.
●Today: inference for first order logic
Philipp Koehn Artificial Intelligence: Inference in First-Order Logic 5 March 2024

3
Outline
●Reducing first-order inference to propositional inference
●Unification
●Generalized Modus Ponens
●Forward and backward chaining
●Logic programming
●Resolution

4
reduction to
propositional inference

5
Universal Instantiation
●Every instantiation of a universally quantified sentence is entailed by it:
∀ v α
SUBST({v/g}, α)
for any variable v and ground term g
●E.g., ∀ x King(x) ∧ Greedy(x) =⇒ Evil(x) yields
King(John) ∧ Greedy(John) =⇒ Evil(John)
King(Richard) ∧ Greedy(Richard) =⇒ Evil(Richard)
King(Father(John)) ∧ Greedy(Father(John)) =⇒
Evil(Father(John))
⋮

6
Existential Instantiation
●For any sentence α, variable v, and constant symbol k
that does not appear elsewhere in the knowledge base:
∃ v
α
SUBST(Iv/k}, α)
●E.g., ∃ x Crown(x) ∧ OnHead(x, John) yields
Crown(C1) ∧ OnHead(C1, John)
provided C1 is a new constant symbol, called a Skolem constant

7
Instantiation
●Universal Instantiation
– can be applied several times to add new sentences
– the new KB is logically equivalent to the old
●Existential Instantiation
– can be applied once to replace the existential sentence
– the new KB is not equivalent to the old
– but is satisfiable iff the old KB was satisfiable

8
Reduction to Propositional
Inference
●Suppose the KB contains just the following:
∀ x King(x) ∧ Greedy(x) =⇒ Evil(x)
King(John)
Greedy(John)
Brother(Richard, John)
●Instantiating the universal sentence in all
possible ways, we have
King(John) ∧ Greedy(John) =⇒ Evil(John)
King(Richard) ∧ Greedy(Richard) =⇒ Evil(Richard)
King(John)
Greedy(John)
●The new KB is propositionalized:
proposition symbols are
King(John), Greedy(John),
Evil(John), Brother(Richard,
John), etc.

9
Reduction to Propositional
Inference
●Claim: a ground sentence is entailed by new KB iff entailed by original KB
●Claim: every FOL KB can be propositionalized so as to preserve entailment
●Idea: propositionalize KB and query, apply resolution, return result
●Problem: with function symbols, there are infinitely many ground terms,
e.g., Father(Father(Father(John)))
●Theorem: Herbrand (1930). If a sentence α is entailed by an FOL KB,
it is entailed by a finite subset of the propositional KB
●Idea: For n = 0 to ∞ do
create a propositional KB by instantiating with depth-n terms
see if α is entailed by this KB
●Problem: works if α is entailed, loops if α is not entailed
●Theorem: Turing (1936), Church (1936), entailment in FOL is
semidecidable

Practical Problems with
Propositionalization10
●Propositionalization seems to generate lots of irrelevant sentences.
●E.g., from ∀ x King(x) ∧ Greedy(x) =⇒ Evil(x)
King(John)
∀ y Greedy(y)
it seems obvious that Evil(John), but propositionalization produces lots of
facts such as Greedy(Richard) that are irrelevant
●With p k-ary predicates and n constants, there are p ⋅ nk instantiations
●With function symbols, it gets nuch much worse!

11
unification

12
Plan
●We have the inference rule
– ∀ x King(x) ∧ Greedy(x) =⇒ Evil(x)
●We have facts that (partially) match the precondition
– King(John)
– ∀ y Greedy(y)
●We need to match them up with substitutions: θ = Ix/John, y/John} works
– unification
– generalized modus ponens

13
Unification
●UNIFY(α, β) = θ if αθ =
βθ
p
q
θ
Knows(John, x)
Knows(John, x)
Knows(John, x)
Knows(John, x)
Knows(John, Jane)
Knows(y, Mary)
Knows(y, Mother(y))
Knows(x, Mary)

14
Unification
βθ
p
q
θ
Ix/Jane}
Knows(John, x)
Knows(John, x)
Knows(John, x)
Knows(John, x)
Knows(John, Jane)
Knows(y, Mary)
Knows(y, Mother(y))
Knows(x, Mary)

15
Unification
βθ
p
q
θ
Ix/Jane}
Ix/Mary,
y/John}
Knows(John, x)
Knows(John, x)
Knows(John, x)
Knows(John, x)
Knows(John, Jane)
Knows(y, Mary)
Knows(y, Mother(y))
Knows(x, Mary)

16
Unification
βθ
p
q
θ
Ix/Jane}
Ix/Mary, y/John}
Iy/John, x/Mother(John)}
Knows(John, x)
Knows(John, x)
Knows(John, x)
Knows(John, x)
Knows(John, Jane)
Knows(y, Mary)
Knows(y, Mother(y))
Knows(x, Mary)

17
Unification
βθ
p
q
θ
Knows(John, x)
Knows(John, x)
Knows(John, x)
Knows(John, x)
Knows(John, Jane)
Knows(y, Mary)
Knows(y, Mother(y))
Knows(x, Mary)
Ix/Jane}
Ix/Mary, y/John}
Iy/John, x/Mother(John)}
fail
●Standardizing apart eliminates overlap of variables, e.g., Knows(z17, Mary)
Knows(John, x) Knows(z17, Mary) Iz17/John, x/Mary}

18
generalized modus ponens

19
Generalized Modus Ponens
●Generalized modus ponens used with KB of definite clauses
(exactly one positive literal)
●All variables assumed universally quantified
p1
′, p2
′, . . . , pn
′, (p1 ∧ p2 ∧ . . . ∧ pn ⇒
q) q
θ
′
where pi θ = piθ for all
i
●Rule:
●Precondition of rule:
●Implication:
●Facts:
●Substitution:
⇒ Result of modus ponens:
King(x) ∧ Greedy(x) =⇒ Evil(x)
p2 is Greedy(x)
p1 is King(x)
q is Evil(x)
p1
′ is
King(John)
p2
′ is Greedy(y)
θ is Ix/John,
y/John}
qθ is Evil(John)

20
forward chaining

21
Example Knowledge
●The law says that it is a crime for an American to sell weapons to hostile
nations. The country Nono, an enemy of America, has some
missiles, and all of its missiles were sold to it by Colonel West, who is
American.
●Prove that Col. West is a criminal

22
Example Knowledge Base
●. . . it is a crime for an American to sell weapons to hostile nations:
American(x) ∧ Weapon(y) ∧ Sells(x, y, z) ∧ Hostile(z) =⇒ Criminal(x)
●Nono . . . has some missiles, i.e., ∃ x Owns(Nono, x) ∧
Missile(x): Owns(Nono, M1) and Missile(M1)
●. . . all of its missiles were sold to it by Colonel West
Missile(x) ∧ Owns(Nono, x) =⇒ Sells(West, x, Nono)
●Missiles are weapons:
Missile(x) ⇒ Weapon(x)
●An enemy of America counts as “hostile”:
Enemy(x, America) =⇒ Hostile(x)
●West, who is American . . .
American(West)
●The country Nono, an enemy of America . . .
Enemy(Nono, America)

23
Forward Chaining Proof

24
(Note: ∀ x Missile(x) ∧ Owns(Nono, x) =⇒ Sells(West, x, Nono))

25
(Note: American(x) ∧ Weapon(y) ∧ Sells(x, y, z) ∧ Hostile(z) =⇒ Criminal(x))

26
Properties of Forward Chaining
●Sound and complete for first-order definite clauses
(proof similar to propositional proof)
●Datalog (1977) = first-order definite clauses + no functions (e.g., crime example)
Forward chaining terminates for Datalog in poly iterations: at most p ⋅ nk
literals
●May not terminate in general if α is not entailed
●This is unavoidable: entailment with definite clauses is semidecidable

27
Efficiency of Forward Chaining
●Simple observation: no need to match a rule on iteration k
if a premise wasn’t added on iteration k − 1
=⇒ match each rule whose premise contains a newly
added literal
●Matching itself can be expensive
●Database indexing allows O(1) retrieval of known facts
e.g., query Missile(x) retrieves Missile(M1)
●Matching conjunctive premises against known facts is
NP-hard
●Forward chaining is widely used in deductive
databases

28
Hard Matching Example
Diff(wa, nt) ∧ Diff(wa, sa) ∧
Diff(nt, q)Diff(nt, sa) ∧
Diff(q, nsw) ∧ Diff(q, sa)
∧
Diff(nsw, v) ∧ Diff(nsw, sa) ∧
Diff(v, sa) =⇒ Colorable()
Diff(Red, Blue)
Diff(Green, Red)
Diff(Blue, Red)
Diff(Red, Green)
Diff(Green, Blue)
Diff(Blue, Green)
●Colorable() is inferred iff the constraint satisfaction problem has a solution
●CSPs include 3SAT as a special case, hence matching is NP-hard

29
Forward Chaining Algorithm
function FOL-FC-ASK(KB, α) returns a substitution or false
repeat until new is empty
new ← g
for each sentence r in KB do
( p1 ∧ . . . ∧ pn =⇒ q ) ←
STANDARDIZE-APART(r) 1 n
for each θ such that (p1 ∧ . . . ∧ pn)θ = (p′ ∧ . . . ∧ p
′ )θ 1 n
for some p′ , . . . , p′ in KB
q ′ ← SUBST(θ, q )
if q ′ is not a renaming of a sentence already in KB or new then do
add q ′ to new
φ ← UNIFY(q ′, α)
if φ is not fail then return φ
add new to KB
return false

30
backward chaining

31
Backward Chaining
●Start with query
●Check if it can be derived by given rules and facts
– apply rules that infer the query
– recurse over pre-conditions

32
Backward Chaining Example

33

34

35

36

37

38

39
Properties of Backward
Chaining
●Depth-first recursive proof search: space is linear in size of proof
●Incomplete due to infinite loops
=⇒ fix by checking current goal against every goal on
stack
●Inefficient due to repeated subgoals (both success and failure)
=⇒ fix using caching of previous results (extra space!)
●Widely used (without improvements!) for logic programming

40
Backward Chaining Algorithm
function FOL-BC-ASK(KB, goals, θ) returns a set of substitutions
inputs: KB, a knowledge base
goals, a list of conjuncts forming a query (θ already applied)
θ, the current substitution, initially the empty substitution g
local variables: answers, a set of substitutions, initially empty
if goals is empty then return Iθ}
q ′ ← SUBST(θ, FIRST(goals))
for each sentence r in KB
where STANDARDIZE-APART(r) = ( p1 ∧ . . . ∧ pn ⇒ q)
and θ′ ← UNIFY(q, q ′) succeeds
new goals ← [ p1, . . . , pn|REST(goals)]
answers ← FOL-BC-ASK(KB, new goals, COMPOSE(θ′, θ)) ∪
answers
return answers

41
logic programming

42
Logic Programming
●Computation as inference on logical KBs
Logic programming
1. Identify problem
2. Assemble information
3. Tea break
4. Encode information in KB
5. Encode problem instance as facts
6. Ask queries
7. Find false facts
Ordinary programming
Identify problem
Assemble information
Figure out solution
Program solution
Encode problem instance as data
Apply program to data
Debug procedural errors
●Should be easier to debug Capital(NewY ork, US) than x =
∶ x + 2 !

43
Prolog
●Basis: backward chaining with Horn clauses + bells & whistles
●Widely used in Europe, Japan (basis of 5th Generation project)
●Compilation techniques ⇒ approaching a billion logical inferences per second
●Program = set of clauses = head :- literal1, . . . literaln.
criminal(X) :- american(X), weapon(Y), sells(X,Y,Z), hostile(Z). missile(M1).
owns(Nono,M1).
sells(West,X,Nono) :- missile(X), owns(Nono,X).
weapon(X) :- missile(X).
hostile(X) :- enemy(X,America).
American(West).
Enemy(Nono,America).

44
Prolog Systems
●Depth-first, left-to-right backward chaining
●Built-in predicates for arithmetic etc., e.g., X is Y*Z+3
●Closed-world assumption (“negation as failure”)
e.g., given alive(X) :- not dead(X). alive(joe)
succeeds if dead(joe) fails

45
resolution

46
Resolution: Brief Summary
●Full first-order version:
l1 ∨ ⋯ ∨ lk , m1 ∨ ⋯ ∨ mn
(l1 ∨ ⋯ ∨ li−1 ∨ li+1 ∨ ⋯ ∨ lk ∨ m1 ∨ ⋯ ∨ mj−1 ∨ mj+1 ∨ ⋯ ∨
mn)θ
where UNIFY(li, ¬mj) = θ.
●For example, ¬Rich(x) ∨ Unhappy(x) Rich(Ken)
Unhappy(Ken)
with θ = Ix/Ken}
●Apply resolution steps to CNF (KB ∧ ¬α); complete for FOL

47
Conversion to CNF
Everyone who loves all animals is loved by someone:
∀ x [∀ y Animal(y) =⇒ Loves(x, y)] =⇒ [∃ y Loves(y, x)]
1. Eliminate biconditionals and implications
∀ x [¬∀ y ¬Animal(y) ∨ Loves(x, y)] ∨ [∃
y Loves(y, x)]
2. Move ¬ inwards: ¬∀ x, p ≡ ∃ x ¬p, ¬∃ x, p ≡ ∀ x ¬p:
∀ x [∃ y ¬(¬Animal(y) ∨ Loves(x, y))] ∨ [∃ y Loves(y, x)]
∀ x [∃ y ¬¬Animal(y) ∧ ¬Loves(x, y)] ∨ [∃ y Loves(y, x)]
∀ x [∃ y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃ y Loves(y, x)]

48
Conversion to CNF
3. Standardize variables: each quantifier should use a different one
∀ x [∃ y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃ z Loves(z, x)]
4. Skolemize: a more general form of existential instantiation.
Each existential variable is replaced by a Skolem
function of the enclosing universally quantified
variables:
∀ x [Animal(F (x)) ∧ ¬Loves(x, F (x))] ∨ Loves(G(x), x)
5. Drop universal quantifiers:
[Animal(F (x)) ∧ ¬Loves(x, F (x))] ∨ Loves(G(x), x)
6. Distribute ∧ over ∨:
[Animal(F (x)) ∨ Loves(G(x), x)] ∧ [¬Loves(x, F (x)) ∨
Loves(G(x), x)]

49
Our Previous Example
●Rules
– American(x) ∧ Weapon(y) ∧ Sells(x, y, z) ∧ Hostile(z) =⇒ Criminal(x)
– Missile(M1) and Owns(Nono, M1)
– Missile(x) ∧ Owns(Nono, x) =⇒ Sells(West, x, Nono)
– Missile(x) ⇒ Weapon(x)
– Enemy(x, America) =⇒ Hostile(x)
– American(West)
– Enemy(Nono, America)
●Converted to CNF
– ¬American(x) ∨ ¬Weapon(y) ∨ ¬Sells(x, y, z) ∨ ¬Hostile(z) ∨ Criminal(x)
– Missile(M1) and Owns(Nono, M1)
– ¬Missile(x) ∨ ¬Owns(Nono, x) ∨ Sells(West, x, Nono)
– ¬Missile(x) ∨ Weapon(x)
– ¬Enemy(x, America) ∨ Hostile(x)
– American(West)
– Enemy(Nono, America)
●Query: ¬Criminal(West)

50
Resolution Proof

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