1 Propositional Logic Limits The expressive power of propositional logic is limited. The assumption...
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Propositional Logic Limits• The expressive power of propositional logic
is limited. The assumption is that everything can be expressed by simple facts.
• It is much easier to model real world objects using properties and relations.
• Predicate Logic provides these capabilities more formally and is used in many AI domains to represent knowledge.
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Inference by enumeration• Depth-first enumeration of all models is sound and complete
• For n symbols, time complexity is O(2n), space complexity is O(n)
••
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Logical equivalence• Two sentences are logically equivalent} iff true
in same models: α ≡ ß iff α╞ β and β╞ α••
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Validity and satisfiabilityA sentence is valid if it is true in all models,
e.g., True, A A, A A, (A (A B)) B
Validity is connected to inference via the Deduction Theorem:KB ╞ α if and only if (KB α) is valid
A sentence is satisfiable if it is true in some modele.g., A B, C
A sentence is unsatisfiable if it is true in no modelse.g., AA
Satisfiability is connected to inference via the following:KB ╞ α if and only if (KB α) is unsatisfiable
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Proof methods• Proof methods divide into (roughly) two kinds:
– Application of inference rules• Legitimate (sound) generation of new sentences from old• Proof = a sequence of inference rule applications
Can use inference rules as operators in a standard search algorithm
• Typically require transformation of sentences into a normal form
– Model checking• truth table enumeration (always exponential in n)• improved backtracking, e.g., Davis--Putnam-Logemann-
Loveland (DPLL)• heuristic search in model space (sound but incomplete)
e.g., min-conflicts-like hill-climbing algorithms•••
»••
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Forward and backward chaining• Horn Form (restricted)
KB = conjunction of Horn clauses– Horn clause =
• proposition symbol; or• (conjunction of symbols) symbol
– E.g., C (B A) (C D B)• Modus Ponens (for Horn Form): complete for Horn KBs
α1, … ,αn, α1 … αn ββ
• Can be used with forward chaining or backward chaining.
• These algorithms are very natural and run in linear time•••
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Forward chaining• Idea: fire any rule whose premises are
satisfied in the KB,– add its conclusion to the KB, until query is found
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Forward chaining algorithm
• Forward chaining is sound and complete for Horn KB
•
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Forward chaining example
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Forward chaining example
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Forward chaining example
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Forward chaining example
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Forward chaining example
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Forward chaining example
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Forward chaining example
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Forward chaining example
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Proof of completeness
• FC derives every atomic sentence that is entailed by KB
1. FC reaches a fixed point where no new atomic sentences are derived
2. Consider the final state as a model m, assigning true/false to symbols
3. Every clause in the original KB is true in m a1 … ak b
4. Hence m is a model of KB5. If KB╞ q, q is true in every model of KB,
including m6.
–•
1.2.–
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Backward chainingIdea: work backwards from the query q:
to prove q by BC,check if q is known already, orprove by BC all premises of some rule concluding q
Avoid loops: check if new subgoal is already on the goal stack
Avoid repeated work: check if new subgoal1. has already been proved true, or2. has already failed3.–
•
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Backward chaining example
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Forward vs. backward chaining
• FC is data-driven, automatic, unconscious processing,– e.g., object recognition, routine decisions
• May do lots of work that is irrelevant to the goal
• BC is goal-driven, appropriate for problem-solving,– e.g., Where are my keys? How do I get into a PhD
program?
• Complexity of BC can be much less than linear in size of KB
•»
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Efficient propositional inference
Two families of efficient algorithms for propositional inference:
Complete backtracking search algorithms• DPLL algorithm (Davis, Putnam, Logemann,
Loveland)• Incomplete local search algorithms
– WalkSAT algorithm–
•
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The DPLL algorithmDetermine if an input propositional logic sentence (in CNF) is
satisfiable.
Improvements over truth table enumeration:1. Early termination
A clause is true if any literal is true.A sentence is false if any clause is false.
2. Pure symbol heuristicPure symbol: always appears with the same "sign" in all
clauses. e.g., In the three clauses (A B), (B C), (C A), A and B
are pure, C is impure. Make a pure symbol literal true.
3. Unit clause heuristicUnit clause: only one literal in the clauseThe only literal in a unit clause must be true.
•
•
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The DPLL algorithm
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The WalkSAT algorithm
• Incomplete, local search algorithm• Evaluation function: The min-conflict heuristic
of minimizing the number of unsatisfied clauses
• Balance between greediness and randomness•••
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The WalkSAT algorithm
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Hard satisfiability problems• Consider random 3-CNF sentences.
e.g.,(D B C) (B A C) (C B E) (E D B) (B E C)
m = number of clauses n = number of symbols
– Hard problems seem to cluster near m/n = 4.3 (critical point)
–
••
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Hard satisfiability problems
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Hard satisfiability problems
• Median runtime for 100 satisfiable random 3-CNF sentences, n = 50
•
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Summary• Logical agents apply inference to a knowledge base to
derive new information and make decisions• Basic concepts of logic:
– syntax: formal structure of sentences– semantics: truth of sentences wrt models– entailment: necessary truth of one sentence given
another– inference: deriving sentences from other sentences– soundness: derivations produce only entailed sentences– completeness: derivations can produce all entailed
sentences• Resolution is complete for propositional logic
Forward, backward chaining are linear-time, complete for Horn clauses
• Propositional logic lacks expressive power•
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••