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Artificial Intelligence

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Presentation on theme: "Artificial Intelligence"— Presentation transcript:

1 Artificial Intelligence
Chapter 13 Automated Reasoning Contents Week Methods in Theorem Proving General Problem Solver (GSP) Resolution Theorem Proving Resolution Refutations Answer Extraction CSC411 Artificial Intelligence

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) CSC411 Artificial Intelligence

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 CSC411 Artificial Intelligence

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 CSC411 Artificial Intelligence

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 CSC411 Artificial Intelligence

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 CSC411 Artificial Intelligence

7 Artificial Intelligence
Transformation rules for logic problems of LT CSC411 Artificial Intelligence

8 Artificial Intelligence
A proof of a theorem in propositional calculus CSC411 Artificial Intelligence

9 Artificial Intelligence
Flow chart and difference reduction table for the General Problem Solver CSC411 Artificial Intelligence

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: CSC411 Artificial Intelligence

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) CSC411 Artificial Intelligence

12 Artificial Intelligence
Resolution proof for the “dead dog” problem. CSC411 Artificial Intelligence

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 CSC411 Artificial Intelligence

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 CSC411 Artificial Intelligence

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 CSC411 Artificial Intelligence

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 CSC411 Artificial Intelligence

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? CSC411 Artificial Intelligence

18 Artificial Intelligence
CSC411 Artificial Intelligence

19 Artificial Intelligence
One refutation for the “happy student” problem. CSC411 Artificial Intelligence

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? CSC411 Artificial Intelligence

21 Artificial Intelligence
CSC411 Artificial Intelligence

22 Artificial Intelligence
Resolution proof for the “exciting life” problem. Answer: {Z/W}{X/Z}{Y/X}{john/Y}  W=john CSC411 Artificial Intelligence

23 Artificial Intelligence
Another resolution refutation for the “exciting lives” problem CSC411 Artificial Intelligence

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. CSC411 Artificial Intelligence

25 Artificial Intelligence
Complete state space for the “exciting life” problem generated by breadth-first search (to two levels). CSC411 Artificial Intelligence

26 Artificial Intelligence
The unit preference strategy and the linear input form strategy on the “exciting life” problem. CSC411 Artificial Intelligence

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 CSC411 Artificial Intelligence

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