LECTURE 20 Unification: A Key Algorithm

1 Representing the Predicate Calculus in Scheme
2 Representing clauses in Scheme.
      2.1   Distinguishing Variables from Constants
3 Implementing the Unification Algorithm
      3.1   Binding variables to values; making frames
      3.2   The function unify
      3.3   The unify_var function unifies a variable with a term
      3.4   Implementing the occurs check with freefor?
      3.5   Examples of the use of unify

Note that the Scheme code used in this lecture is drawn from
the book "Structure and Interpretation of Computer Programs"
by Abelson and Sussman.

1 Representing the Predicate Calculus in Scheme

In the propositional calculus we have seen that we can perform inference using the resolution principle of "cancelling out" positive and negative literals occurring in the same clause. In this lecture we will develop the Predicate Calculus and extend the resolution principle to it by developing the unification algorithm to allow us to find out the most general conditions under which two literals can be made the same.

In the Predicate Calculus we replace the unstructured propositional variables of the propositional calculus by predicates applied to arguments. A predicate can be thought of as expressing a relation between its arguments. For example using the standard notation:


expresses the fact that an individual, John, is enamoured of another individual, Mary.

As well as constants such as the aforementioned lovers, we need variables which are subject to quantifiers. The universal quantifier, which I shall write ForAll allows us to state that a fact is true for all individuals. Thus it allows us to express the fact that all humans are mortal as:

    ForAll x. human(x) -> mortal(x)

The effect of the universal quantifier is that we may substitute any individual for x in the statement governed by the quantifier. For example, we may substitute Socrates, obtaining:

    human(Socrates) -> mortal(Socrates)

If we also know that Socrates is human, that is


we may infer, according to the rules of Propositional Logic, that Socrates is mortal:


Universal quantifiers are not enough of themselves to allow us to express all the ideas necessary for normal logical discourse. To express the fact that everybody has a mother, we may use an existential quantifier which I shall write Exists. Thus we would write:

    ForAll p.  Exists m. parent(m,p) & female(m)

The Predicate Calculus as described above, with predicates, constants, variables and quantifiers is adequate for logical discourse. However it is convenient to introduce function symbols as well. For example, one might have a function mother for which mother(p) is understood to denote the mother of an individual p.

In fact, for the formulation of logic that is normally used to support the Logic Paradigm, we put our logical statements in a conjunctive normal form in which all use of the existential quantifier has been replaced by the use of function symbols, and in which the scope of each universal quantifier is that of the clause in which it occurs. This being so, we are able to drop the explicit presence of universal quantifiers, the understanding being that all variables are implicitly quantified universally.

We can begin to see how we might do resolution in the Predicate Calculus. We represent the fact that all humans are mortal by

     ~ human(x) \/ mortal(x)

and the fact that Socrates is human as before by

Now we cannot immediately use the propositional principle of resolution, for human(x) is not the same as human(Socrates). Thus we have to make a substitution of Socrates for x as we did before. However, it is obvious that this substitution will work; indeed it is so obvious that we will write an algorithm to find the best possible substitution for the circumstances, called the unification algorithm.

2 Representing clauses in Scheme.

However, it is desirable to think about how we can represent clauses in the Scheme language so that we are not at a loss if we want to implement our algorithm.

The representation of clauses is quite straightforward. We can adopt the usual rule of Scheme that or is variadic - i.e. takes a variable number of arguments - and represent a clause as (or l1....ln) where l1...ln are literals. A literal is a predicate applied to arguments, possibly negated. So the allowable forms of a literal are:

       (p t1....tn)
       (not (p t1....tn))

And the arguments are terms each of which may be a variable, a constant or a function-symbol applied to arguments. In the first order predicate calculus, function and predicate variables are not allowed.

2.1 Distinguishing Variables from Constants

The range of possible constants can include Scheme symbols, so we have a problem in distinguishing between a symbol acting as a constant and as variable, just as we did in Scheme itself. In the logic paradigm, however, rather than use quotation to distinguish symbols acting as constants, it is more common to use some other convention. In the Lisp community it has been common to use a prefixed question mark. In this implementation we will treat a variable as a list (? v ) where v is (conventionally) a symbol.

(define (var? e)
    (and (pair? e) (eq? (car e) '?))

This means that any Scheme atom is a constant in our representation of logic:

(define constant? atom?)

and we may use equal? to determine if two constants are the same:

(define =_constant? equal?)

So, for example mortal(Socrates) gets encoded as

    (or (mortal socrates))

while ForAll x. mortal(x)->human(x) gets encoded as

    (or (not (mortal (? x))) (human (? x)))

3 Implementing the Unification Algorithm

3.1 Binding variables to values; making frames

To unify two terms t1 and t2 we must determining a mapping f from variables to constants under which f(t1) = f(t2). We will refer to our representation of such a mapping as a frame. However let us take an abstract approach, and suppose that we have the following operations:

make_binding      Makes a "binding" that is a pair associating
                  a variable with a value.

binding-in-frame  Finds a binding for a given variable in a frame

binding-value     Extracts the value component of a binding.

extend            Adds a new variable-value binding to a frame.

We may implement these operations using a-lists for frames as follows:

(define make-binding cons)

(define binding-in-frame assoc)

(define binding-value cdr)

(define (extend var val frame)
    (cons (make-binding var val) frame))

3.2 The function unify

To unify two terms p1 and p2 we write the function unify. It takes 3 arguments, p1, p2 frame. Here the frame serves the same role as a frame in the environment in our Scheme interpreter - it remembers what variables have been bound to and so avoids having to perform excessive substitution. However in the case of unification there are extra possibilities that have to be taken into account

So let us consider the unification function. Essentially there are the following cases to consider

(define (unify p1 p2 frame)
        ((eq? frame #f) #f)                             ;[1]
        ((var? p1) (unify_var p1 p2 frame))             ;[2]
        ((var? p2) (unify_var p2 p1 frame))             ;[3]
        ((constant? p1)                                 ;[4]
         (if (constant? p2)
             (if (=_constant? p1 p2) frame #f)
        ((constant? p2) #f)                             ;[5]
        (else                                           ;[6]
            (unify (cdr p1)                             ;[6.1]
                (cdr p2)
                (unify                                  ;[6.2]
                    (car p1)
                    (car p2)

3.3 The unify_var function unifies a variable with a term

Now we come to the definition of unify_var. There is a subtlety here that needs to be dealt with. What happens if we try to unify a variable with a term that already contains the variable. Suppose we want to unify a variable (? x) with the term (f (? x)); we can only do this if we systematically replace the variable (? x) with (f (? x)) everywhere in our terms, including inside the term (f (? x)). But this involves an infinite amount of work generating an infinite term. This kind of thing can happen if we try to unify:

    (g (? x) (f (? x)))
    (g (? y) (? y))

Standard logic does not allow such a circular substitution to happen, and so we have to put an occurs check into unify_var which makes sure that we do not unify a variable with a term in which it occurs. [It should be noted that the best known computer language based on the Logic Paradigm, Prolog, does not perform the occurs check because it is computationally expensive].

So let us now consider the definition of unify_var. Firstly [1] we have to allow for the possiblity that val is a variable, and indeed is the same variable as var. In this case, unification is trivial with the existing frame [1.1].

Otherwise we find the value-cell for the value of the variable in the frame [2].

If [3] it does not exist, then we may bind var to be val in a new frame made by extend [3.2]. However this is only legal if the "occurs-check" implemented by freefor? succeeds [3.1] - otherwise unification fails [3.3].

If [4] the variable var was bound in the frame, then we call unify to see if the value of var in frame and val can be unified.

(define (unify_var var val frame)
    (if (equal? var val)                                    ;[1]
        frame                                               ;[1.1]
        (let ((value-cell (binding-in-frame var frame)))    ;[2]
            (if (not value-cell)                            ;[3]
                (if (freefor? var val frame)                ;[3.1]
                    (extend var val frame)                  ;[3.2]
                    #f)                                     ;[3.3]
                (unify (binding-value value-cell)           ;[4]

3.4 Implementing the occurs check with freefor?

Finally let us look at freefor?. We write this with an internal recursive function freewalk to save us the trouble of passing more than one argument. So, we are asking, is exp free of occurrences of var? We recurse through freewalk, examining the various possible types of the expression e.

(define (freefor? var exp frame)
             (lambda (e)
                     ((constant? e) #t)                            ;[1]
                     ((var? e)                                     ;[2]
                      (if (equal? var e) #f                        ;[2.1]
                          (let ((b (binding-in-frame e frame)))    ;[2.2]
                              (if (not b) #t                       ;[2.3]
                                  (freewalk (binding-value b)))))) ;[2.4]
                     ( (freewalk (car e)) (freewalk (cdr e)))      ;[3]
                     (else #f))                                    ;[4]
                 )  ;end lambda
             ) ; end freewalk binding
         ) ;end letrec bindings
        (freewalk exp)
        ) ;end letrec

3.5 Examples of the use of unify

And here are some examples of the use of unify and associated functions.

    '(freefor? '(? x) '(f (? x)) '())
    '(unify 'Liz 'Phil '())
    '(unify '(+ a b) '(+ a b) '())
    '(unify '(+ a 2) '(+ a b) '())
    '(unify '(+ (? a) 4) '(+ b 4) '())
    '(((? a) . b)))
    '(unify '(+ (? a) (? a)) '(+ b b) '())
    '(((? a) . b)))
    '(unify '(+ (? a) (? a)) '(+ 4 3) '())
    '(unify '(+ (? a) 7) '(+ 4 (? b)) '())
    '(((? b) . 7) ((? a) . 4)))
    '(unify '(+ (? a) 4 ) '(+ 5 (? b)) '())
    '(((? b) . 4) ((? a) . 5))