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Before moving on to algorithms for learning sets of Horn clauses, let us introduce some basic terminology from formal logic All expressions are composed of constants (eg, Bob, Louise), variables (eg, x, y), predicate symbols (eg, Married, Greater-Than), and function symbols (eg, age) The difference between predicates and functions is that predicates take on values of True or False, whereas functions may take on any constant as their value We will use lowercase symbols for variables and capitalized symbols for constants Also, we will use lowercase for functions and capitalized symbols for predicates From these symbols, we build up expressions as follows: A term is any constant, any variable, or any function applied to any term (eg, Bob, x, age(Bob)) A literal is any predicate or its negation applied to any term (eg, Married(Bob, symLouise), -Greater-Than(age(Sue), 20))If a literal contains a negation bol, we call it a negative literal, otherwise a positive literal A clause is any disjunction of literals, where all variables are assumed to be universally quantified A Horn clause is a clause containing at most one positive literal, such as

The first item in the Title section of the results list is our friend Document-Based Application Overview. Click on that, and the document appears.

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where H is the positive literal, and -Ll -Ln are negative literals Because of the equalities ( B v - A ) = ( B t A) and -(A A B) = (-A v - B ) , the above Horn clause can alternatively be written in the form

Once you have found a useful document, you might want to bookmark it. Use the Bookmarks toolbar button; the drop-down menu shows the current document is shown by default (Figure 8-5). You can add or remove bookmarks with the Manage Bookmarks window (in the same drop-down menu).

Every well-formed expression is composed of constants (eg, Mary, 23, or Joe), variables (eg, x), predicates (eg, Female, as in Female(Mary)), and functions (eg, age, as in age(Mary)) A term is any constant, any variable, or any function applied to any term Examples include Mary, x, age(Mary), age(x) A literal is any predicate (or its negation) applied to any set of terms Examples include Femal e(Mary), Female(x), Greaterf han (age(Mary), 20) A ground literal is a literal that does not contain any variables (eg, -Female(Joe)) A negative literal is a literal containing a negated predicate (eg, -Female(Joe)) A positive literal is a literal with no negation sign (eg, Female(Mary)) A clause is any disjunction of literals M1 v Mn whose variables are universally quantified A Horn clause is an expression of the form

where H, L1 Ln are positive literals H is called the head or consequent of the Horn clause The conjunction of literals L1 A L2 A A L, is called the body or antecedents of the Horn clause For any literals A and B, the expression (A t B) is equivalent to (A v -B), and the expression -(A A B) is equivalent to (-A v -B) Therefore, a Horn clause can equivalently be written as the disjunction Hv-L1 v v - L ,

NOTE: You can resize the results list by just mousing down over the junction between the list and the Table of Contents and dragging. There is no drag handle, so you have to position the mouse exactly on the junction. If you drag the junction to the left, the list suddenly disappears, which is a little startling the first time it happens. You can bring it back into view by grabbing the left edge of the window and dragging it in to the right. There doesn t seem to be an obvious way to remove the Spotlight-style selection buttons once they have appeared. Like all search-based information discovery, this presupposes that you know what you are looking for. What if you just want to see what is there Of course, there is a browser-based approach, though it is not immediately obvious. This comprises the Core and Developer Tools Reference Libraries. In the toolbar, clicking on the Home button provides a drop-down menu (see Figure 8-6).

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A substitution is any function that replaces variables by terms For example, the substitution {x/3, y/z) replaces the variable x by the term 3 and replaces the variable y by the term z Given a substitution 0 and a literal L we write L to denote the result of applying substitution 0 to L O A unrfying substitution for two literals L1 and L2 is any substitution 0 such that L10 = L1B

A L,,

Choose either the Mac OS X 10.6 Core Library or Xcode 3.2 Developer Tools Library item to go to the corresponding reference library home page.

Whatever the notation, the Horn clause preconditions L1 A A L, are called the clause body or, alternatively, the clause antecedents The literal H that forms the postcondition is called the clause head or, alternatively, the clause consequent For easy reference, these definitions are summarized in Table 103, along with other definitions introduced later in this chapter