cs236501 introduction to ai tutorial 8 resolution tutorial 8 resolution
TRANSCRIPT
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CS 236501Introduction to AI
CS 236501Introduction to AI
Tutorial 8Resolution
Tutorial 8Resolution
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Problem DefinitionProblem Definition
Input1. Database containing formally represented facts: First-order
logic sentences converted into clause form.2. Inference rule: Resolution principle (MP & MT)
Goal: An inference procedure
Requirements:1. Soundness – every sentence produced by the procedure will
be “true”.2. Completeness – every “true” sentence can be produced by the
procedure
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DefinitionsDefinitions
• Terms:
– Constants (e.g. “c1”, “c2”)
– Variables (e.g. “x1”, “x2”)
– Functions (e.g. “f(x1, x2)”)
• Predicate – Indicator function on terminals.– e.g. EVEN(t) : Numbers {TRUE, FALSE}
• Atom – the application of a predicate on a literal.– e.g. EVEN(t)
• Literal – A predicate or its negation– e.g. EVEN(t), ¬EVEN(t)
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DefinitionsDefinitions
• Formulae - Recursively defined:– Every Atom is a formula
– If w1, w2 are formulae, then so are:
• Clause – Disjunction (or) of literals.
– e.g. L1 V L2 V ¬L3 (can be written as: {L1, L2 ,¬L3})
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The Resolution PrincipleThe Resolution Principle
• Given:– A clause Φ containing the literal: φ – A clause Ψ containing the literal: ¬φ
• We can conclude:– (Φ – {φ}) U (Ψ – {¬φ})
• Or in the generalized version…
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The Resolution PrincipleThe Resolution Principle
• Given:– A clause Φ containing the literal: φ – A clause Ψ containing the literal: ¬ψ
– A most general unifier g of φ and ¬ψ
• We can conclude:– ((Φ – {φ}) U (Ψ – {¬ψ})) | g
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The Resolution ProcedureThe Resolution Procedure
• Let DB be a set of true sentences without contradictions, and C be a sentence we want to prove.
The Idea - proof by negation:• Assume ¬C and try to find a contradiction.
Intuition• If all DB sentences are true, and assuming ¬C
creates a contradiction then C must be inferred from DB.
•
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The Resolution ProcedureThe Resolution Procedure
1. Convert: DB U {¬C} to clause form.
2. If there is a contradiction in DB, C was proved. Terminate.
3. Select two clauses and add their resolvents to the current DB. If there are no resolvable clauses – the procedure fails, terminate. Else, go to step 2.
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Conversion to Clause FormConversion to Clause Form
1. Eliminate all :– Replace AB with ¬A V B
2. Distribute negations:– Replace ¬¬A with A– with– …
3. Eliminate existential quantifiers by replacing with Skölem constants or functions:– e.g.
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Conversion to Clause FormConversion to Clause Form
4. Rename variables to avoid duplicates between different quantifiers.
5. Drop all universal quantifiers
6. Put expression into conjunctive normal form (CNF).
7. Convert to clauses (sets of literals).
8. Rename variables to avoid duplicates between different clauses.
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Conversion to Clauses - ExampleConversion to Clauses - Example
• Initial expression:
• Remove implications:
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Intro. to AI – Tutorial 8 11
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Conversion to Clauses - ExampleConversion to Clauses - Example
• Previous step:
• Move negations inwards:
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Intro. to AI – Tutorial 8 12
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Conversion to Clauses - ExampleConversion to Clauses - Example
• Previous step:
• Remove existential quantifiers:
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Intro. to AI – Tutorial 8 13
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Conversion to Clauses - ExampleConversion to Clauses - Example
• Previous step:
• Rename variables:
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Intro. to AI – Tutorial 8 14
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Conversion to Clauses - ExampleConversion to Clauses - Example
• Previous step:
• Remove universals quantifiers:
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Intro. to AI – Tutorial 8 15
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Conversion to Clauses - ExampleConversion to Clauses - Example
• Previous step:
• Convert to CNF:
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Intro. to AI – Tutorial 8 16
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Conversion to Clauses - ExampleConversion to Clauses - Example
• Previous step:
• Convert to clauses:
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Intro. to AI – Tutorial 8 17
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Conversion to Clauses - ExampleConversion to Clauses - Example
• Previous step:
• Rename variables:
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Simple ExampleSimple Example
• The problem:– “Heads I win, tails you lose.” – Use resolution to show I always win.
• Facts representation:
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Simple ExampleSimple Example
• Proof:
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Intro. to AI – Tutorial 8 20
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