INFERENCE IN FIRST-ORDER LOGIC IES 503 ARTIFICIAL INTELLIGENCE İPEK SÜĞÜT

CONTENTS PROPOSITIONAL VS. FIRST-ORDER LOGIC INFERENCE RULES FOR QUANTIFIERS UNIFICATION AND LIFTING FORWARD CHAINING BACKWARD CHAINING RESOLUTION CONCLUSION

PROPOSITIONAL VS. FIRST-ORDER LOGIC

INFERENCE RULES FOR QUANTIFIERS

REDUCTION TO PROPOSITIONAL INFERENCE

UNIFICATION AND LIFTING

FORWARD CHAINING

BACKWARD CHAINING This algorithm takes the first goal in the list and finds every clause in KB whose positive literal, or head, unifies with the goal. Each such clause creates a new recursive call in which the premise, or body, of the clause is added to the goal stack. Remember that the facts are clauses with a head but no body, so when a goal unifies with a known fact, no new subgoals are added to the stack and the goal is solved. The figure shows the proof tree:

RESOLUTION

The Resolution Inference Rule: First example is the crime example as shown: The structure: Beginning with the goal clause and, resolving against clauses from KB until the empty clause is generated. CNF SENTENCESCNF SENTENCES

RESOLUTION

The Paraphrased Proof: Suppose Curiosity did not kill Tuna. We know that either Jack or Curiosity did; thus Jack must have. Now, Tuna is a cat and cats are animals, so Tuna is an animal. Because anyone who kills an animal is loved by no one, we know that no one loves Jack. On the other hand, Jack loves all animals, so someone loves him. So we have a contradiction. Therefore, Curiosity killed the cat. Completeness of Resolution:

RESOLUTION

CONCLUSION A first approach uses inference rules for instantiating quantifiers in order to propositionalize the inference problem. Typically, this approach is very slow. The use of unification to identify appropriate substitutions for variables eliminates the instantiation step in first-order proofs, making the process much more efficient. A lifted version of Modus Ponens uses unification to provide a natural and powerful inference rule, generalized Modus Ponens. The forward chaining and backward chaining algorithms apply this rule to sets of definite clauses. Generalized Modus Ponens is complete for definite clauses. For Datalog programs consisting of function-free definite clauses, entailment is decidable. Forward chaining is used in deductive databases, where it can be combined with relational database operations. It is also used in production systems, which perform efficient updates with very large rule sets.

CONCLUSION Forward chaining is complete for Datalog programs and runs in polynomial time. Backward chaining is used in logic programming systems such as Prolog, which employ sophisticated compiler technology to provide very fast inference. Backward chaining suffers from redundant inferences and infinite loops; these can be alleviated by memorization. The generalized resolution inference rule provides a complete proof system for first-order logic, using KBs in conjunctive normal form. Several strategies exist for reducing the search space of a resolution system without compromising completeness.