ARTIFICIAL INTELLIGENCE [INTELLIGENT AGENTS PARADIGM] Professor Janis Grundspenkis Riga Technical University Faculty of Computer Science and Information Technology Institute of Applied Computer Systems Department of Systems Theory and Design KNOWLEDGE PROCESSING IN FIRST ORDER LOGIC

Knowledge Processing in First-Order Logic The semantics of first-order logic provide a basis for a formal theory of logical inference. The ability to infer new correct expressions from a set of true assertions is very important feature of first-order logic. These new expressions are correct in that they are consistent with all previous interpretations of the original set of expressions.

Knowledge Processing in First-Order Logic For a first-order predicate calculus sentence S and an interpretation I: An interpretation I that makes a sentence (expression) S true is said to satisfy S. An interpretation I that satisfies every member of a set of expressions is said to satisfy the set.

Knowledge Processing in First-Order Logic An expression X logically follows from a set of predicate calculus expressions S if every interpretation that satisfies S also satisfies X. If I satisfies S for all variable assignments, then I is a model of S.

Knowledge Processing in First-Order Logic In the blocks world example, the blocks world is a model for its logical description because all sentences are true under this interpretation. When a knowledge base is implemented as a set of true assertions about a problem domain, that domain is a model for the knowledge base.

Knowledge Processing in First-Order Logic S is satisfiable if and only if there exist an interpretation and variable assignment that satisfy it; otherwise it is unsatisfiable. A set of expressions is satisfiable if and only if there exist an interpretation and variable assignment that satisfy every element.

Knowledge Processing in First-Order Logic If a set of expressions is not satisfiable, it is said to be inconsistent. For example:  X(p(X)   p(X)) If S has a value T for all possible interpretations, S is said to be valid. For example:  X(p(X)   p(X))

Knowledge Processing in First-Order Logic How to test validity? Truth table method “+” any expression not containing variables can be tested. “-” for expressions containing variables it is not always possible to decide the validity (process may not terminate).

Knowledge Processing in First-Order Logic Complete proof procedures “+” can produce any expression that logically follows from a set of expressions A predicate calculus expression X logically follows from a set S of predicate calculus expressions if every interpretation and variable assignment that satisfies S also satisfies X.

Knowledge Processing in First-Order Logic Determining what follows from what is captured in the knowledge base is the job of the inference mechanism. The terms “inference” and “reasoning” are generally used to cover any process by which conclusions are reached.

Knowledge Processing in First-Order Logic Logical inference or deduction is one way of sound reasoning. Logical inference is a process that implements the entailment between sentences. The process by which the soundness of an inference is established using truth tables can be extended to entire classes of inference.

Knowledge Processing in First-Order Logic There are certain patterns of inferences that occur many times, but their soundness can be proved only once and for all. The pattern can be captured in an inference rule. Once a rule is established, it can be used infinite times to make inferences without going validity test.

Knowledge Processing in First-Order Logic An inference rule is sound if every predicate calculus expression produced by the rule from a set S of predicate calculus expressions also logically follows from S.

Knowledge Processing in First-Order Logic An inference rule is complete if, given a set S of predicate calculus expressions, the rule can infer every expression that logically follows from S.

Knowledge Processing in First-Order Logic Notations used for inference rules  |=  Meaning:  can be derived from  by inference (Meta-form) , , etc. are intended to match any sentence, not just individual proposition symbols.

Knowledge Processing in First-Order Logic An inference rule is sound, if the conclusion is true in all cases where the premises are true. To prove the soundness, the truth table must be constructed with one line for each possible model of the proposition symbols in the premises. In all models where the premise is true, the conclusion must be also true.

Knowledge Processing in First-Order Logic Example:  TTFT TFTT FTFT FFTF