ARTIFICIAL INTELLIGENCE [INTELLIGENT AGENTS PARADIGM] INFERENCE RULES Professor Janis Grundspenkis Riga Technical University Faculty of Computer Science and Information Technology Institute of Applied Computer Systems Department of Systems Theory and Design E-mail: Janis.Grundspenkis@rtu.lv

Inference Rules Modus Ponens or Implication Elimination From two sentences    and  that are true (so called axioms) the new true sentence  can be concluded (a theorem is proved with respect to the axioms, i.e. the theorem logically follows from the axioms).

A – “sun shines”; B – “it is warm”. Inference Rules Example: Sentence: If sun shines it is warm A – “sun shines”; B – “it is warm”. Axioms: A  B A Theorem: B, i.e., “it is warm”.

Inference Rules AND-Elimination From a conjunction of sentences any of conjuncts can be inferred.

Inference Rules AND-Introduction From a list of sentences their conjunction can be inferred.

Inference Rules OR-Introduction From a sentence its disjunction with anything else at all can be inferred.

Inference Rules Double-Negation Elimination From a double negated sentence a positive sentence can be inferred.

Inference Rules Unit Resolution From a disjunction, if one of the disjuncts is false it can be inferred that the other one is true.

Inference Rules or equivalently Since  cannot be both true and false, one of the other disjuncts must be true in one of the premises. Or equivalently, implication is transitive.

Inference Rules Modus Tolens      T F or equivalently

Inference Rules It is not a sound inference rule! Abduction Rule    T F It is not a sound inference rule!

Inference Rules PROOF THEORY AND PROCEDURE The proof theory is a set of rules for logical inferencing the entailments of a set of sentences. The way to prove a theorem is to use a proof procedure. A proof procedure is a combination of an inference rule and an algorithm for applying that rule to a set of logical expressions to generate new sentences.

Inference Rules PROOF PROCEDURE (continued) Proof procedures use manipulations called sound rules of inference that produce new expressions from old expressions. More precisely, models of the old expressions are guaranteed to be models of the new ones too.

Inference Rules PROOF PROCEDURE (continued) The most straightforward proof procedure is to apply sound rules of inference to the axioms, and to the results of applying sound rules of inference, until the desired theorem appears.

Inference Rules PROOF PROCEDURE (continued) A logical proof consists of a sequence of applications of inference rules, starting with sentences initially in the knowledge base, and ending with the generation of the sentence whose proof is desired. The job of an inference procedure is to construct proof by finding appropriate sequences of applications of inference rules.