CS 416
Artificial Intelligence
Lecture 10
Logic
Chapters 7 and 8
Homework Assignment
First homework assignment is out
Due October 20th
Midterm
October 25th
Up through chapter 9 (excluding chapter 5)
Review
Store information in a knowledge base
• Backus-Naur Form (BNF)
• Use equivalences to isolate what you want
• Use inference to relate sentences
– Modus Ponens
– And-Elimination
Constructing a proof
Proving is like searching
• Find sequence of logical inference rules that lead to desired
result
• Note the explosion of propositions
– Good proof methods ignore the countless irrelevant
propositions
The two inference rules are sound but not complete!
Completeness
Recall that a complete inference algorithm is one
that can derive any sentence that is entailed
Resolution is a single inference rule
• Guarantees the ability to derive any sentence that is entailed
– i.e. it is complete
• It must be partnered with a complete search algorithm
Resolution
Resolution is a single inference rule
• Guarantees the ability to derive any sentence that is entailed
– i.e. it is complete
• It must be partnered with a complete search algorithm
Resolution
Unit Resolution Inference Rule
• If m and li are
complementary
literals
Resolution Inference Rule
Also works with clauses
But make sure each literal appears only once
We really just want the resolution to return A
Resolution and completeness
Any complete search algorithm, applying only the
resolution rule, can derive any conclusion
entailed by any knowledge base in propositional
logic
• More specifically, refutation completeness
– Able to confirm or refute any sentence
– Unable to enumerate all true sentences
What about “and” clauses?
Resolution only applies to “or” clauses (disjunctions)
• Every sentence of propositional logic can be transformed to a
logically equivalent conjunction of disjunctions of literals
Conjunctive Normal Form (CNF)
• A sentence expressed as conjunction of disjunction of literals
CNF
An algorithm for resolution
We wish to prove KB entails a
• That is, will some sequence of inferences create new
sentences in knowledge base equal to a?
• If some sequence of inferences created ~ a, we would know
the knowledge base could not also entail a
– a ^ ~ a is impossible
An algorithm for resolution
To show KB cannot resolve a
• Must show (KB ^ ~a) is unsatisfiable
– No possible way for KB to entail (not a)
– Proof by contradiction
An algorithm for resolution
Algorithm
• (KB ^ ~a) is put in CNF
• Each pair with complementary literals is resolved to produce
new clause which is added to KB (if novel)
– Cease if no new clauses to add (~a is not entailed)
– Cease if resolution rule derives empty clause
 If resolution generates a… a ^ ~a = empty clause
 (~a is entailed)
Example of resolution
Proof that there is not a pit in P1,2: ~P1,2
• KB ^ P1,2 leads to empty clause
• Therefore ~P1,2 is true
Formal Algorithm
Horn Clauses
Horn Clause
• Disjunction of literals where at most one is positive
– (~a V ~b V ~c V d)
– (~a V b V c V ~d) Not a Horn Clause
Horn Clauses
Why?
•
Every Horn clause can be written as an implication:
– (a conjunction of positive literals) => single positive literal
– Inference algorithms can use forward chaining and
backward chaining
– Entailment computation is linear in size of KB
Horn Clauses
Can be written as a special implication
• (~a V ~b V c) becomes (a ^ b) => c
– (~a V ~b V c) == (~(a ^ b) V c) … de Morgan
– (~(a ^ b) V c) == ((a ^ b) => c
… implication elimination
Forward Chaining
Does KB (Horn clauses) entail q (single symbol)?
• Keep track of known facts (positive literals)
• If the premises of an implication are known
– Add the conclusion to the known set of facts
• Repeat process until no further inferences or q is added
Forward Chaining
Forward Chaining
Properties
• Sound
• Complete
– All entailed atomic sentences will be derived
Data Driven
• Start with what we know
• Derive new info until we discover what we want
Backward Chaining
Start with what you want to know, a query (q)
• Look for implications that conclude q
– Look at the premises of those implications
 Look for implications that conclude those premises…
Goal-Directed Reasoning
• Can be less complex search than forward chaining
Making it fast
Example
• Problem to solve is in CNF
– Is Marvin a Martian given --- M == 1 (true)?
 Marvin is green --- G=1
 Marvin is little
--- L=1
 (little and green) implies Martian --- (L ^ G) => M
~(L^G) V M
~L V ~G V M
– Proof by contradiction… are there true/false values for G, L, and M that are
consistent with knowledge base and Marvin not being a Martian?
 G ^ L ^ (~L V ~G V M) ^ ~M == empty?
Searching for variable values
Want to find values such that:
G ^ L ^ (~L V ~G V M) ^ ~M == 0
• Randomly consider all true/false assignments to variables
until we exhaust them all or find match (model checking)
– (G, L, M) = (1, 0, 0)… no
= (0, 1, 0)… no
= (0, 0, 0)… no
= (1, 1, 0)… no
• Alternatively…
Backtracking Algorithm
Davis-Putnam Algorithm (DPLL)
• Search through possible assignments to (G, L, M) via depth-first search
(0, 0, 0) to (0, 0, 1) to (0, 1, 0)…
G
– Each clause of CNF must be true
L
M
 Terminate consideration when clause evaluates to false
– Use heuristics to reduce repeated computation of propositions
 Early termination
 Pure symbol heuristic
 Unit clause heuristic
Cull branches from tree
Searching for variable values
Other ways to find (G, L, M) assignments for:
G ^ L ^ (~L V ~G V M) ^ ~M == 0
• Simulated Annealing (WalkSAT)
– Start with initial guess (0, 1, 1)
– With each iteration, pick an unsatisfied clause and flip one
symbol in the clause
– Evaluation metric is the number of clauses that evaluate to true
– Move “in direction” of guesses that cause more clauses to be
true
– Many local mins, use lots of randomness
WalkSAT termination
How do you know when simulated annealing is
done?
• No way to know with certainty that an answer is not possible
– Could have been bad luck
– Could be there really is no answer
– Establish a max number of iterations and go with best
answer to that point
So how well do these work?
Think about it this way
• 16 of 32 possible assignments are models (are satisfiable)
for this sentence
• Therefore, 2 random guesses should find a solution
• WalkSAT and DPLL should work quickly
What about more clauses?
If # symbols (variables) stays the same, but
number of clauses increases
• More ways for an assignment to fail (on any one clause)
• More searching through possible assignments is needed
• Let’s create a ratio, m/n, to measure #clauses / # symbols
• We expect large m/n causes slower solution
What about more clauses
Higher m/n means fewer
assignments will work
If fewer assignments work
it is harder for DPLL and
WalkSAT
Combining it all
4x4 Wumpus World
• The “physics” of the game
–
–
• At least one wumpus on board
–
• A most one wumpus on board (for any two squares, one is free)
–
n(n-1)/2 rules like: ~W1,1 V ~W1,2
• Total of 155 sentences containing 64 distinct symbols
Wumpus World
Inefficiencies as world becomes large
• Knowledge base must expand if size of world expands
Preferred to have sentences that apply to all
squares
• We brought this subject up last week
• Next chapter addresses this issue
Chapter 8: First-Order Logic
What do we like about propositional
logic?
It is:
• Declarative
– Relationships between variables are described
– A method for propagating relationships
• Expressive
– Can represent partial information using disjunction
• Compositional
– If A means foo and B means bar, A ^ B means foo and bar
What don’t we like about
propositional logic?
Lacks expressive power to describe the
environment concisely
• Separate rules for every square/square relationship in
Wumpus world
Natural Language
English appears to be expressive
• Squares adjacent to pits are breezy
But natural language is a medium of communication, not
a knowledge representation
• Much of the information and logic conveyed by language is dependent
on context
• Information exchange is not well defined
• Not compositional (combining sentences may mean something different)
• It is ambiguous
But we borrow representational
ideas from natural language
Natural language syntax
• Nouns and noun phrases refer to objects
– People, houses, cars
• Properties and verbs refer to object relations
– Red, round, nearby, eaten
• Some relationships are clearly defined functions where there is only one
output for a given input
– Best friend, first thing, plus
We build first-order logic around objects and relations
Ontology
• a “specification of a conceptualization”
• A description of the objects and relationships that can exist
– Propositional logic had only true/false relationships
– First-order logic has many more relationships
• The ontological commitment of languages is different
– How much can you infer from what you know?
 Temporal logic defines additional ontological
commitments because of timing constraints