### 12.3.1 Semantics of Ground Datalog

The first step in giving the semantics of Datalog is to give the semantics for the ground (variable-free) case.

An **interpretation** is a triple *I=⟨D,φ,π⟩*, where

*D*is a non-empty set called the**domain**. Elements of*D*are**individuals**.*φ*is a mapping that assigns to each constant an element of*D*.*π*is a mapping that assigns to each*n*-ary predicate symbol a function from*D*into^{n}*{*true*,*false*}*.

*φ* is a function from names into individuals in the world. The constant *c* is
said to **denote** the individual
*φ(c)*. Here *c* is
a symbol but *φ(c)* can be anything: a real physical object such as a person
or a virus, an abstract concept such as a course, love, or the number 2, or
even a symbol.
*π(p)* specifies whether the relation denoted by the *n*-ary predicate
symbol *p* is true or false for each *n*-tuple of individuals. If
predicate symbol *p* has no arguments, then *π(p)* is either true
or false. Thus, for predicate symbols with no arguments, this semantics reduces
to the semantics of propositional definite clauses.

**Example 12.3:**Consider the world consisting of three objects on a table:

These are drawn in this way because they are things in the world, not symbols. ✄ is a pair of scissors, ☎ is a telephone, and ✎ is a pencil.

Suppose the constants in our language are *phone*, *pencil*, and
*telephone*. We have the predicate symbols *noisy* and
*left_of*. Assume *noisy* is a unary predicate (it takes a single
argument) and that *left_of* is a binary predicate (it takes two
arguments).

An example interpretation that represents the objects on the table is

*D={✄, ☎, ✎}*.*φ(phone) = ☎*,*φ(pencil) = ✎*,*φ(telephone) = ☎*.*π(noisy)*:*⟨✄⟩**false**⟨☎⟩**true**⟨✎⟩**false**π(left_of)*:*⟨✄,✄⟩**false**⟨✄,☎⟩**true**⟨✄,✎⟩**true**⟨☎,✄⟩**false**⟨☎,☎⟩**false**⟨☎,✎⟩**true**⟨✎,✄⟩**false**⟨✎,☎⟩**false**⟨✎,✎⟩**false*Because

*noisy*is unary, it takes a singleton individual and has a truth value for each individual.Because

*left_of*is a binary predicate, it takes a pair of individuals and is true when the first element of the pair is left of the second element. Thus, for example,*π(left_of)(⟨✄,☎⟩)=true*, because the scissors are to the left of the telephone;*π(left_of)(⟨✎,✎⟩)=false*, because the pencil is not to the left of itself.

Note how the *D* is a set of things in the world. The relations are among the objects in the world, not
among the names. As *φ* specifies that *phone* and *telephone* refer to the same
object, exactly the same statements are true about them in this interpretation.

**Example 12.4:**Consider the interpretation of Figure 12.1.

*D* is the set with four elements: the person Kim, room 123, room
023, and the CS building. This is not a set of four
symbols, but it is the set containing the actual person, the actual rooms,
and the actual building. It is difficult to write down this set and,
fortunately, you never really have to. To remember the meaning and to
convey the meaning to another person, knowledge base designers
typically describe *D*, *φ*,
and *π* by pointing
to the physical individuals or a depiction of them (as is done in Figure 12.1) and
describe the meaning in natural language.

The constants are *kim*, *r123*, *r023*, and *cs_building*.
The mapping *φ* is defined by the gray arcs from each of these constants
to an object in the world in Figure 12.1.

The predicate symbols are *person*, *in*, and *part_of*. The meaning
of these are meant to be conveyed in the figure by the arcs from the predicate symbols.

Thus, the
person called Kim is in the room *r123* and is also in the CS
building, and these are the only instances of the *in* relation that are true.
Similarly, room *r123* and room *r023* are part of the CS building, and there are no other *part_of* relationships that are true in this interpretation.

It is important to emphasize that the elements of *D* are the real
physical individuals, and not their names. The name *kim* is not in the
name *r123* but, rather, the person denoted by *kim* is in the room
denoted by *r123*.

Each ground term denotes an individual in an interpretation. A
constant *c * denotes in *I* the individual *φ(c) *.

A ground atom is either *true* or *false* in an interpretation.
Atom *p(t _{1},...,t_{n}) * is

**in**

*true**I*if

*π(p)(⟨t*true, where

_{1}',...,t_{n}'⟩)=*t*is the individual denoted by term

_{i}'*t*, and is

_{i}**in**

*false**I*otherwise.

**Example 12.5:**The atom

*in(kim,r123)*is true in the interpretation of Example 12.4, because the person denoted by

*kim*is indeed in the room denoted by

*r123*. Similarly,

*person(kim)*is true, as is

*part_of(r123,cs_building)*. The atoms

*in(cs_building,r123)*and

*person(r123)*are false in this interpretation.