Syntax of First-Order Predicate Calculus (FOPC): 1. Alphabet

1. Syntax of First-Order Predicate Calculus (FOPC): 1. Alphabet • Countable set of predicate symbols, each with specified arity  0. • Countable set of function symbols, each with specified arity  0. Function symbols with arity 0 are also called constants or individual symbols. • Countable set of variables.

2. 1. Alphabet (Continued) • (Consistent with Prolog, we will begin variables with an upper-case letter and predicate/function symbols with a lower-case letter.) • Logical symbols: ,,,,,,

3. 2. Terms • A variable is a term. • If f is a function symbol of arity n and t1,…,tn are terms then f(t1,…,tn) is a term.

4. Examples of Terms • 0 • s(s(s(0))) • nil • cons(1,nil) • cons(1,cons(2,nil)) • cons(1,cons(2,cons(3,nil)))

5. 3. Formulas • If p is a predicate symbol of arity n and t1,…,tn are terms, then p(t1,…,tn) is an atomic formula. • If a and b are formulas then so are a, ab, ab, ab, ab, ab. • If X is a variable and a is a formula then Xa and Xa are formulas. We say that X is quantified in the formulas Xa and Xa.

6. Some Notes • Predicates of arity 0 are also called propositions, the only atomic formulas allowed in propositional logic. • An expression is a term or formula. A formula with no free (unquantified) variables is a sentence.

17. Example: Models X(Y((mother(X)  child_of(Y,X))  loves(X,Y))) mother(mary) child_of(tom,mary)

28. Example: Models X(Y((mother(X)  child_of(Y,X))  loves(X,Y))) mother(mary) child_of(tom,mary)

30. Problem… • Difficult to compare two interpretations with different domains… e.g., one domain consists of apples and the other of oranges. • Could map one domain to another. Can be tricky to define… most domains are infinite. • Idea: for a given alphabet, pick a canonical domain and mapping. But how?

34. Some Notes • Typically, we are given a theory (set of sentences) T and wish to speak of Herbrand interpretations relative to T. • In this case we take the alphabet A to be the symbols in T. • If T has no constants, we introduce one.

35. Notes (continued) • Valuations with respect to a Herbrand interpretation may be thought of as grounding substitutions. • We’d like know it’s sufficient to consider only Herbrand interpretations… just ignore all others…

37. Herbrand Model Lemma • Let T be a theory (set of sentences) in Skolem Normal Form. • T has a model iff it has a Herbrand model.

41. Skolemization • Process is applied to one sentence at a time and applied only to the entire sentence (so outermost quantifier first). Each sentence initially has empty vector of free variables. • Replace X A(X) with A(X), and add X to vector of free variables. • Replace X A(X) with A(x(V)) where x is a new function symbol and V is the current vector of free variables.

42. Herbrand Model Lemma • Let T be a theory (set of sentences) in Skolem Normal Form. • T has a model iff it has a Herbrand model. Now recall our goal of identifying a unique simplest model.

43. Example: Models X(Y((mother(X)  child_of(Y,X))  loves(X,Y))) mother(mary) child_of(tom,mary)

48. Why no least Herbrand model? • Disjunctive “positive” information… creates uncertainty. We can satisfy the disjunction by satisfying either disjunct – a choice. • This is somewhat analogous to the uncertainty created by existential quantifiers. • This uncertainty also causes inefficiencies in deduction (recall prop. SAT is NP-complete but SAT for Horn CNFs is linear-time solvable).