1. Artificial Intelligence
Chapter 13
Automated Reasoning
Contents
Week Methods in Theorem Proving
General Problem Solver (GSP)
Resolution Theorem Proving
Resolution Refutations
Answer Extraction
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2. Artificial Intelligence
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)
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3. 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
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4. Artificial Intelligence
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
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5. Artificial Intelligence
GPS Example
Goal (p¬p)¬p
Proof:
(AA)A 1 of 5 Known axioms
(¬A¬A)¬A Substitution
(A¬A)¬A Replacement
(p¬p)¬p Substitution
Issues:
Matching process
Search space
Control strategies
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6. Artificial Intelligence
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
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7. Artificial Intelligence
Transformation rules for logic problems of LT
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8. Artificial Intelligence
A proof of a theorem in propositional calculus
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9. Artificial Intelligence
Flow chart and difference reduction table for the General Problem Solver
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10. 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:
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11. Artificial Intelligence
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)
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12. Artificial Intelligence
Resolution proof for the “dead dog” problem.
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13. 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
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14. Artificial Intelligence
CNF Normalization
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
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15. 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
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16. Artificial Intelligence
One resolution proof for an example from the propositional calculus with the given clauses:
a¬b¬c b
c¬d¬e
ef ¬f
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17. “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?
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18. Artificial Intelligence
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19. Artificial Intelligence
One refutation for the “happy student” problem.
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20. Artificial Intelligence
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?
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21. Artificial Intelligence
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22. Artificial Intelligence
Resolution proof for the “exciting life” problem.
Answer: {Z/W}{X/Z}{Y/X}{john/Y}  W=john
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23. Artificial Intelligence
Another resolution refutation for the “exciting lives” problem
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24. 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.
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25. Artificial Intelligence
Complete state space for the “exciting life” problem generated by breadth-first search (to two levels).
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26. Artificial Intelligence
The unit preference strategy and the linear input form strategy on the “exciting life” problem.
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27. Artificial Intelligence
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
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