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 SEMANTICS OF PREDICATE CALCULUS

Semantics of Predicate Calculus Predicate calculus semantics provide a formal basis, for determining the truth value of well-formed expressions. The truth of expressions depends on the mapping of constants, variables, predicates, and functions into objects and relations in the domain of discourse.

Semantics of Predicate Calculus The truth of relationships in the domain determined the truth of the corresponding expressions. For example, owns(john, ford) friends(john, bill) If it is indeed that John owns Ford (car) and he and Bill are friends both expressions would each have the truth value (assignment) T.

Semantics of Predicate Calculus To use the predicate calculus as a representation for problem solving, we describe objects and relations in the domain of interpretation with a set of well-formed expressions. The terms and predicates of these expressions denote objects and relations in the domain.

Semantics of Predicate Calculus This knowledge base of predicate calculus expressions, each having truth value T, describes the “state of the world”.

Semantics of Predicate Calculus Interpretation Let the domain D be a nonempty set. An interpretation over D is an assignment of the entities of D to each of the constant, variable, predicate and function symbols of a predicate calculus expression.

Semantics of Predicate Calculus Interpretation (continued) 1.Each constant is assigned an element of D. 2.Each variable is assigned to a nonempty subset of D. These are the allowable substitutions for that variable.

Semantics of Predicate Calculus Interpretation (continued) 3.Each function f of arity m is defined on m arguments of D and defines a mapping from D m into D. 4.Each predicate p of arity n is defined on n arguments from D and defines a mapping from D n into {T, F}.

Semantics of Predicate Calculus Truth value of expressions Given an interpretation, the meaning of an expression is a truth value assignment over the interpretation. Let E be an expression and I an interpretation for E over a nonempty domain D. The truth value for E is determined by:

Semantics of Predicate Calculus Truth value of expressions (continued) 1. The value of constant is the element of D it is assigned to by I. 2. The value of a variable is the set of elements of D it is assigned to by I. 3. The value of a function expression is that element of D obtained by evaluating the function for the parameter values assigned by the interpretation.

Semantics of Predicate Calculus Truth value of expressions (continued) 4. The value of truth symbol “true” is T and “false” is F. 5. The value of an atomic sentence is either T or F, as determined by the interpretation I. 6. The value of the negation of a sentence is T if the value of the sentence is F and is F if the value of the sentence is T.

Semantics of Predicate Calculus Truth value of expressions (continued) 7. – 10. The truth value of expressions using , , , and  is determined from the values of their operands as defined by truth table (the same as in propositional calculus).

Semantics of Predicate Calculus Truth value of expressions (continued) For a variable X and a sentence S containing X: 11. The values of  XS is T for all assignments to X under I, and it is F otherwise. 12. The values of  XS is T if there is an assignment to X in the interpretation under which S is T, otherwise it is F.

Semantics of Predicate Calculus Problems of computing the truth value Universal quantification introduces problems in computing the truth value of a sentence, because all the possible values of a variable symbol must be tested to see whether the expression remains true.

Semantics of Predicate Calculus Problems of computing the truth value (continued) If the domain of interpretation is infinite, exhaustive testing of all substitutions to a universally quantified variable is computationally impossible: the algorithm may never halt. That is why the predicate calculus is said to be undecidable.

Semantics of Predicate Calculus Problems of computing the truth value (continued) Evaluating the truth value of a sentence containing an existentially quantified variable may not be easier. If the domain of the variable is infinite and the sentence is false under all substitutions, the algorithm will never halt.

Semantics of Predicate Calculus Relations between quantifiers For predicates p and q and variables X and Y:  Xp(X)   X  p(X)  Xp(X)   X  p(X)

Semantics of Predicate Calculus Relations between quantifiers (continued)  Xp(X)   Yp(Y)  Xp(X)   Yp(Y)  X(p(X)  q(X))   Xp(X)  Xq(X)  X(p(X)  q(X))   Xp(X)  Xq(X)

Why First-Order Logic? In the language defined, universally and existentially quantified variables may refer only to objects (constants) in the domain of discourse. This language is called the first-order logic (first-order predicate calculus). Predicate and function names may not be replaced by quantified variables (it is allowed in high-order predicate calculi).

Knowledge Representation in First-Order Logic Almost any grammatically correct sentence in any natural language may be expressed in first-order logic using the symbols, connectives, and variable symbols. Important ! – there is no unique mapping of sentences into predicate calculus expressions.

Knowledge Representation in First-Order Logic Example: “All basketball players are tall”. Mappings: 1.  X(basketball player(X)  tall(X)) 2.  X(basketball player(X)  tall(X))

Knowledge Representation in First-Order Logic Blocks world B C T A Knowledge base: 1. Block(A) 2. Block(B) 3. Block(C) 4. Table(T) 5. On(A, T) 6. On(C, T) 7. On(B, C) 8. Clear Top(A) 9. Clear Top(B)