### 3` `Using Conventional Racket Expressions in Racklog

The arguments of Racklog predicates can be any Racket objects. In particular, composite structures such as lists, vectors, strings, hash tables, etc can be used, as also Racket expressions using the full array of Racket’s construction and decomposition operators. For instance, consider the following goal:

(%member x '(1 2 3))

Here, %member is a predicate, x is a logic variable, and '(1 2 3) is a structure. Given a suitably intuitive definition for %member, the above goal succeeds for x = 1, 2, and 3.

Now to defining predicates like %member:

(define %member |

(%rel (x y xs) |

[(x (cons x xs))] |

[(x (cons y xs)) |

(%member x xs)])) |

Ie, %member is defined with three local variables: x, y, xs. It has two clauses, identifying the two ways of determining membership.

The first clause of %member states a fact: For any x, x is a member of a list whose head is also x.

The second clause of %member is a rule: x is a member of a list if we can show that it is a member of the tail of that list. In other words, the original %member goal is translated into a subgoal, which is also a %member goal.

Note that the variable y in the definition of %member occurs only once in the second clause. As such, it doesn’t need you to make the effort of naming it. (Names help only in matching a second occurrence to a first.) Racklog lets you use the expression (_) to denote an anonymous variable. (Ie, _ is a thunk that generates a fresh anonymous variable at each call.) The predicate %member can be rewritten as

(define %member |

(%rel (x xs) |

[(x (cons x (_)))] |

[(x (cons (_) xs)) |

(%member x xs)])) |

#### 3.1` `Constructors

We can use constructors – Racket procedures for creating structures – to simulate data types in Racklog. For instance, let’s define a natural-number data-type where 0 denotes zero, and (succ x) denotes the natural number whose immediate predecessor is x. The constructor succ can be defined in Racket as:

(define succ |

(lambda (x) |

(vector 'succ x))) |

Addition and multiplication can be defined as:

(define %add |

(%rel (x y z) |

[(0 y y)] |

[((succ x) y (succ z)) |

(%add x y z)])) |

(define %times |

(%rel (x y z z1) |

[(0 y 0)] |

[((succ x) y z) |

(%times x y z1) |

(%add y z1 z)])) |

We can do a lot of arithmetic with this in place. For instance, the factorial predicate looks like:

(define %factorial |

(%rel (x y y1) |

[(0 (succ 0))] |

[((succ x) y) |

(%factorial x y1) |

(%times (succ x) y1 y)])) |

#### 3.2` `%is

The above is a very inefficient way to do arithmetic, especially when the underlying language Racket offers excellent arithmetic facilities (including a comprehensive number “tower” and exact rational arithmetic). One problem with using Racket calculations directly in Racklog clauses is that the expressions used may contain logic variables that need to be dereferenced. Racklog provides the predicate %is that takes care of this. The goal

(%is X E)

unifies X with the value of E considered as a Racket expression. E can have logic variables, but usually they should at least be bound, as unbound variables may not be palatable values to the Racket operators used in E.

We can now directly use the numbers of Racket to write a more efficient %factorial predicate:

(define %factorial |

(%rel (x y x1 y1) |

[(0 1)] |

[(x y) (%is x1 (- x 1)) |

(%factorial x1 y1) |

(%is y (* y1 x))])) |

A price that this efficiency comes with is that we can use %factorial only with its first argument already instantiated. In many cases, this is not an unreasonable constraint. In fact, given this limitation, there is nothing to prevent us from using Racket’s factorial directly:

(define %factorial |

(%rel (x y) |

[(x y) |

(%is y (racket-factorial |

x))])) |

or better yet, “in-line” any calls to %factorial with %is-expressions calling racket-factorial, where the latter is defined in the usual manner:

(define racket-factorial |

(lambda (n) |

(if (= n 0) 1 |

(* n (factorial |

(- n 1)))))) |

#### 3.3` `Lexical Scoping

One can use Racket’s lexical scoping to enhance predicate definition. Here is a list-reversal predicate defined using a hidden auxiliary predicate:

(define %reverse |

(letrec |

([revaux |

(%rel (x y z w) |

[('() y y)] |

[((cons x y) z w) |

(revaux y |

(cons x z) w)])]) |

(%rel (x y) |

[(x y) (revaux x '() y)]))) |

(revaux X Y Z) uses Y as an accumulator for reversing X into Z. (Y starts out as (). Each head of X is consed on to Y. Finally, when X has wound down to (), Y contains the reversed list and can be returned as Z.)

revaux is used purely as a helper predicate for %reverse, and so it can be concealed within a lexical contour. We use letrec instead of let because revaux is a recursive procedure.

#### 3.4` `Type Predicates

Racklog provides a couple of predicates that let the user probe the type of objects.

succeeds if X is an atomic object.

The predicate %compound, the negation of %constant, checks if its argument is not an atomic object.

The above are merely the logic-programming equivalents of corresponding Racket predicates. Users can use the predicate %is and Racket predicates to write more type checks in Racklog. Thus, to test if X is a string, the following goal could be used:

User-defined Racket predicates, in addition to primitive Racket predicates, can be thus imported.