Transcript PowerPoint

CS 416
Artificial Intelligence
Lecture 12
First-Order Logic
Chapter 9
Midterm
October 25th
Up through chapter 9 (excluding chapter 5)
Review
Inference in First-Order Logic
• Could convert first-order logic (FOL) to propositional logic
(PL) and use PL inference
– Must convert (reduce) Universal and Existential
Quantifiers into PL
– Potential problem with infinite number of substitutions
• Could lift Modus Ponens to FOL
– Unification required to make expressions look identical
How to perform inference in FOL
Use a knowledge base
• Insert data with “STORE”
• Retrieve “unified” sentences using “FETCH”
– Search for substitutions that unite query with every
sentence in KB
How can se make FETCH fast?
• Only attempt unifications if they have a chance of succeeding
Predicate Indexing
Index the facts in the KB
• Example: unify Knows (John, x) with KB:
– Knows (John, Jane)
– Knows (y, Bill)
– Brother (Richard, John)
– Knows (y, Mother(y))
– Knows (x, Elizabeth)
No need to check
Predicate Indexing
• Predicate indexing puts all the Knows facts in one bucket and
all the Brother facts in another
– Might not be a win if there are lots of clauses for a
particular predicate symbol
 Consider how many people Know one another
– Instead index by predicate and first argument
 Clauses may be stored in multiple buckets
Subsumption lattice
How to construct indices for all possible queries
that unify with it
• Example: Employs (AIMA.org, Richard)
Subsumption Lattice
Subsumption lattice
• Each node reflects making one substitution
• The “highest” common descendent of any two nodes is the
result of applying the most general unifier
• Predicate with n arguments will create a lattice with O(2n)
nodes
• Benefits of indexing may be outweighed by cost of storing
and maintaining indices
Forward Chaining
Remember this from propositional logic?
• Start with atomic sentences in KB
• Apply Modus Ponens
– add new sentences to KB
– discontinue when no new sentences
• Hopefully find the sentence you are looking for in the
generated sentences
Lifting forward chaining
First-order definite clauses
– disjunction of literals with exactly one positive or
– implication whose antecedent is a conjunction of positive
literals and whose consequent is a single positive literal
Universal quantification
assumed
– all sentences are defined this way to simplify processing
Example
• The law says 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
• We will prove West is a criminal
Example
• It is a crime for an American to sell weapons to hostile nations
• Nono… has some missles
– Owns (Nono, M1)
– Missile (M1)
• All of its missiles were sold to it by Colonel West
Example
• We also need to know that missiles are weapons
• and we must know that an enemy of America counts as
“hostile”
• “West, who is American”
• The country Nono, an enemy of America
Forward-chaining
Starting from the facts
• find all rules with satisfied premises
• add their conclusions to known facts
• repeat until
– query is answered
– no new facts are added
First iteration of forward chaining
Look at the implication sentences first
• must satisfy unknown premises
• We can satisfy this rule
– by substituting {x/M1}
– and adding Sells(West, M1, Nono) to KB
First iteration of forward chaining
• We can satisfy
– with {x/M1}
– and Weapon (M1) is added
• We can satisfy
– with {x/Nono}
– and Hostile {Nono} is added
Second iteration of forward chaining
• We can satisfy
– with {x/West, y/M1, z/Nono}
– and Criminal (West) is added
Analyze this algorithm
Sound?
• Does it only derive sentences that are entailed?
– Yes, because only Modus Ponens is used and it is sound
Complete?
• Does it answer every query whose answers are entailed by the KB?
– Yes, if the clauses are definite clause NatNum(0)
Semi-decidable?
NatNum(n)  NatNum(S(n))
NatNum(S(0)), NatNum(S(S(0))),…
• Algorithm returns “yes” for every entailed sentence,
but does not return “no” for every nonentailed sentence
Proving completeness
Assume KB only has sentences with no function symbols
• What’s the most number of iterations through algorithm?
• Depends on the number of facts that can be added
– Let k be the arity, the max number of arguments of any predicate and
– Let p be the number of predicates
– Let n be the number of constant symbols
• At most pnk distinct ground facts
• Fixed point is reached after this many iterations
• A proof by contradiction shows that the final KB is complete
Complexity of this algorithm
Three sources of complexity
• inner loop requires finding all unifiers such that premise of
rule unifies with facts of database
– this “pattern matching” is expensive
• must check every rule on every iteration to check if its
premises are satisfied
• many facts are generated that are irrelevant to goal
Pattern matching
Conjunct ordering
• Missile (x) ^ Owns (Nono, x) => Sells (West, x, Nono)
– Look at all items owned by Nono, call them X
– for each element x in X, check if it is a missile
– Look for all missiles, call them X
– for each element x in X, check if it is owned by Nono
Optimal ordering is NP-hard, similar to matrix mult
• Good heuristic: group first according to whichever group is smaller
Incremental forward chaining
Pointless (redundant) repetition
• Some rules generate new information
– this information may permit unification of existing rules
• some rules generate preexisting information
– we need not revisit the unification of the existing rules
Every new fact inferred on iteration t must be derived
from at least one new fact inferred on iteration t-1
Irrelevant facts
Some facts are irrelevant and occupy computation
of forward-chaining algorithm
• What if Nono example included lots of facts about food
preferences?
– Not related to conclusions drawn about sale of weapons
– How can we eliminate them?
Magic Set
Rewriting the rule set
• We have learned earlier that backward chaining limits the
search space
• We’re still using forward chaining, but we will add elements to
premises of sentences that further restrict the forward
inferences we will draw
– added elements are based on goal (thus the backward
influence)
Magic Set
Rewriting the rule set
• Let goal = Criminal (West)
– American(x) ^ Weapon(y) ^ Sells(x,y,z) ^ Hostile(z)  Criminal(x)
• We know we are concerned with criminals and West
• Let’s distinguish West as important by placing it in a Magic set:
Magic(West)
• Let’s rewrite the rule to constrain our forward chaining to only include
sentences related to West
– Magic(x) ^ American(x) ^ Weapon(y) ^ Sells(x, y, z) ^ Hostile(z) =>
Criminal (x)
Backward Chaining
Start with the premises of the goal
• Each premise must be supported by KB
• Start with first premise and look for support from KB
– looking for clauses with a head that matches premise
– the head’s body (premise) must then be supported by KB
– Place premises on a stack with the goal
A recursive, depth-first, algorithm
• Suffers from repetition and incompleteness
Resolution
We saw earlier that resolution is a complete
algorithm for refuting statements
• Must put first-order sentences into conjunctive normal form
– conjunction of clauses, each is a disjunction of literals
 literals can contain variables (which are assumed to be
universally quantified)
Resolution - aside
Mathematical theorem provers
• Resolution has been used to prove mathematical theorems
• Kurt Gödel proved incompleteness of logic systems
containing induction (a building block of discrete math)
– There are sentences entailed by a knowledge base, but no
finite proof exists
 That is, generating all possible sentences, when
induction is one possible operator, will not derive all
sentences that are entailed
First-order CNF
• For all x, American(x) ^ Weapon(y) ^ Sells(x, y, z) ^ Hostile (z) =>
Criminal(x)
• ~American(x) V ~Weapon(y) V ~Sells(x, y, z) V ~Hostile(z) V Criminal(x)
Every sentence of first-order logic can be converted into
an inferentially equivalent CNF sentence (they are both
unsatisfiable in same conditions)
Example
Everyone who loves all animals is loved by someone
Example
Example
F and G are Skolem Functions
• arguments of function are universally quantified variables in
whose scope the existential quantifier appears
Example
• Two clauses
• F(x) refers to the animal potentially unloved by x
• G(x) refers to someone who might love x
Resolution inference rule
li and m are
complementary
literals
A lifted version of propositional resolution rule
• two clauses must be standardized apart
– no variables are shared
• can be resolved if their literals are complementary
– one is the negation of the other
– if one unifies with the negation of the other
Resolution
li and mj are
complementary
literals
Inference in first-order logic
Our goal is to prove that KB entails a fact, a
• We use logical inference
 Forward chaining
 Backward chaining
 Resolution
All three logical inference systems rely on search
to find a sequence of actions that derive the empty
clause