Transcript Document

Chapter 13
Automated Reasoning
Contents
Week Methods in Theorem Proving
General Problem Solver (GSP)
Resolution Theorem Proving
Resolution Refutations
Answer Extraction
CSC411
Artificial Intelligence
1
Automated Reasoning
Week methods
– Focus on techniques/strategies, instead of
knowledge base
Automated reasoning
– Employs an unambiguous and exacting
notation for representing information, precise
inference rules for drawing conclusions, and
carefully delineated strategies to control those
inference rules
– Monotonic reasoning
– LT – Logic Theorist, the first program for
automated reasoning to prove many of
theorems in Russell and Whitehead’s Principia
Mathematica (1950)
CSC411
Artificial Intelligence
2
General Problem Solver
Rooted at LT
Address mechanic process of proof
Three inference rules:
– Substitution: substitute an expression for all
occurrences of a symbol in a already-true
proposition
E.g. BBB to ¬A¬A¬A
– Replacement: equivalent replacement between
propositions
E.g. AB  ¬AB
– Detachment: modus ponens
E.g. A, AB to B
CSC411
Artificial Intelligence
3
GPS Executive Routine
Four steps:
– Substitute the current goal to match against all
know axioms and theorems
– If fails, apply detachments and replacements
to the goal to obtain a list of subgoals
– Use chaining method to find a new
subproblem: if ac is a problem and bc is
found, then ab is set up as a new
subproblem
– If above fails on the original problem, go to the
subproblem list and select the next untried
subproblem
CSC411
Artificial Intelligence
4
GPS Example
Goal (p¬p)¬p
Proof:
–
–
–
–
(AA)A
(¬A¬A)¬A
(A¬A)¬A
(p¬p)¬p
1 of 5 Known axioms
Substitution
Replacement
Substitution
Issues:
– Matching process
– Search space
– Control strategies
CSC411
Artificial Intelligence
5
Means-Ends Analysis
Method: the operators for difference
reduction are indexed by the differences
they can reduce
Heuristic search – difference table
Difference table: list the symbol difference
between the goal and the expression that
the operator creates
E.g. pq and ¬pq, the difference table
should contain  and , as well as p and
¬p
CSC411
Artificial Intelligence
6
Transforma
tion rules
for logic
problems of
LT
CSC411
Artificial Intelligence
7
A proof of
a theorem
in
proposition
al calculus
CSC411
Artificial Intelligence
8
Flow chart
and
difference
reduction
table for the
General
Problem
Solver
CSC411
Artificial Intelligence
9
Resolution Theorem Proving
Root of Prolog
Resolution refutation principle: to prove A
– assume A is false,
– add A to known axioms and theorems,
– show a contradiction
Resolution refutation proof steps:
CSC411
Artificial Intelligence
10
An Example
Problem
– Statements:
Fido is a dog
All dogs are animals
All animals will die
– goal:
Fido will die
Predicates representation
– Statements
dog(fido)
(X) (dog(X)animal(X))
(X) (animal(X)die(X))
– Goal:
die(fido)
CSC411
Artificial Intelligence
11
Resolution proof for the “dead dog” problem.
CSC411
Artificial Intelligence
12
Conjunctive Normal Form
Clause form: conjunction of disjuncts
Any expression can be transformed into
conjunctive normal form
Horn class is a special case of conjunctive
normal form
Conjunctions are “,”
Knowledge base is a set of expressions in
CNF
E.g. AB, CDE, ¬A¬C
Literals: letters or their negates
CSC411
Artificial Intelligence
13
CNF Normalization
1.
2.
3.
4.
5.
6.
7.
8.
9.
CSC411
Eliminate 
Reduce the scope of negation
Variable renaming
Move all qualifiers left without changing
their order
Eliminate all existential quantifiers using
Skolemization
Drop all universal quantifiers
Convert to conjunct of disjuncts
Separate into a set of disjuncts
Standardize the variables apart so that
different clauses contain different
variable names
Artificial Intelligence
14
Binary Resolution Proof Procedure
Given two disjunct expressions,
merge them by eliminating all literals
with their negates
P1: a1a2…an
P2: b1b2…bm
If ai=¬bj, then P1 and P2 can be merged
by eliminating ai and ¬bj:
P: a1a2…ai-1ai+1…an
b1b2…bj-1bj+1…bm
E.g. a¬b and bc  ac
CSC411
Artificial Intelligence
15
One resolution proof
for an example from
the propositional
calculus with the
given clauses:
a¬b¬c
b
c¬d¬e
ef
¬f
CSC411
Artificial Intelligence
16
“Happy Student” Problem
Anyone passing his history exams
and winning the lottery is happy. But
anyone who studies or is lucky can
pass all his exams. John did not
study but he is lucky. Anyone who is
lucky wins the lottery.
Is John happy?
CSC411
Artificial Intelligence
17
CSC411
Artificial Intelligence
18
One refutation for the “happy student” problem.
CSC411
Artificial Intelligence
19
Answer Extraction
Extract correct answer to a problem from
a resolution refutation by retaining
information on the unification
substitutions made in the resolution
refutation
“Exciting lives problem”
– All people who are not poor and are smart are
happy. Those people who read are not stupid.
John can read and is wealthy. Happy people
have exciting lives.
– Who can be found with an exciting life?
CSC411
Artificial Intelligence
20
CSC411
Artificial Intelligence
21
Resolution
proof for
the
“exciting
life”
problem.
Answer: {Z/W}{X/Z}{Y/X}{john/Y}  W=john
CSC411
Artificial Intelligence
22
Another resolution refutation for the “exciting lives”
problem
CSC411
Artificial Intelligence
23
Strategies and Simplification
Problem:
– More than one resolution refutations exist, which is the
best?
– When there are N clauses in the clause space, there are
N2 ways of combining them or checking to see whether
they can be combined at just the first level
Strategies
– Breadth-first strategy: exhaustive search to find the best
binary resolution each step
– Set of support strategy: for a set of input clauses S,
specify a subset T, the set of support. In each resolution
one of the resolvents have an ancestor in the use of
support
– Unit preference strategy: Unit clause is a clause of one
literal. Each resolution contains at least one unit clause
– Linear input form strategy: Start from the negated goal,
resolve the result of previous step with one of input
clauses, until empty clauses is produced.
CSC411
Artificial Intelligence
24
Complete state space for the “exciting life” problem
generated by breadth-first search (to two levels).
CSC411
Artificial Intelligence
25
The unit preference strategy and the linear input form
strategy on the “exciting life” problem.
CSC411
Artificial Intelligence
26
Complete Strategies
Complete strategy
– A set of clauses is unsatisfiable if no interpretation exists
that establishes the set as satisfiable
– An inference rule is refutation complete if, given an
unsatisfiable set of clauses, the unsatisfiability can be
established by use of this inference rule alone
– A strategy is refutation complete if by its use with a
refutation-complete inference rule we can guarantee
finding a refutation whenever a set of clauses is
unsatisfiable.
Strategies
– The Breadth-first strategy is refutation complete
– The set of support strategy is refutation complete if
input set S is unsatisfiable but S-T (support set) is
satisfiable
– The unit preference strategy is not complete
– The linear input form strategy is not complete
CSC411
Artificial Intelligence
27