Propagation of Constraints

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    Computer programs are traditionally organized as one-directional computations, which perform operations on prespecified arguments to produce desired outputs. On the other hand, we often model systems in terms of relations among quantities. For example, a mathematical model of a mechanical structure might include the information that the deflection d of a metal rod is related to the force F on the rod, the length L of the rod, the cross-sectional area A, and the elastic modulus E via the equation

    d A E = F L

    Such an equation is not one-directional. Given any four of the quantities, we can use it to compute the fifth. Yet translating the equation into a traditional computer language would force us to choose one of the quantities to be computed in terms of the other four. Thus, a procedure for computing the area A could not be used to compute the deflection d, even though the computations of A and d arise from the same equation.[*]

    In this section, we sketch the design of a language that enables us to work in terms of relations themselves. The primitive elements of the language are primitive constraints, which state that certain relations hold between quantities. For example, (adder a b c) specifies that the quantities a, b, and c must be related by the equation a+b=c, (multiplier x y z) expresses the constraint xy = z, and (constant 3.14 x) says that the value of x must be 3.14.

    Our language provides a means of combining primitive constraints in order to express more complex relations. We combine constraints by constructing constraint networks, in which constraints are joined by connectors. A connector is an object that ``holds'' a value that may participate in one or more constraints. For example, we know that the relationship between Fahrenheit and Celsius temperatures is


    9C = 5(F - 32)

    Such a constraint can be thought of as a network consisting of primitive adder, multiplier, and constant constraints (figure [*]). In the figure, we see on the left a multiplier box with three terminals, labeled m1, m2, and p. These connect the multiplier to the rest of the network as follows: The m1 terminal is linked to a connector C, which will hold the Celsius temperature. The m2 terminal is linked to a connector w, which is also linked to a constant box that holds 9. The p terminal, which the multiplier box constrains to be the product of m1 and m2, is linked to the p terminal of another multiplier box, whose m2 is connected to a constant 5 and whose m1 is connected to one of the terms in a sum.


      \begin{figure}\par\figcaption {The relation $9C = 5(F - 32)$ expressed as a constraint network.}\end{figure}

    Computation by such a network proceeds as follows: When a connector is given a value (by the user or by a constraint box to which it is linked), it awakens all of its associated constraints (except for the constraint that just awakened it) to inform them that it has a value. Each awakened constraint box then polls its connectors to see if there is enough information to determine a value for a connector. If so, the box sets that connector, which then awakens all of its associated constraints, and so on. For instance, in conversion between Celsius and Fahrenheit, w, x, and y are immediately set by the constant boxes to 9, 5, and 32, respectively. The connectors awaken the multipliers and the adder, which determine that there is not enough information to proceed. If the user (or some other part of the network) sets C to a value (say 25), the leftmost multiplier will be awakened, and it will set u to $25\cdot 9=225$. Then u awakens the second multiplier, which sets v to 45, and v awakens the adder, which sets F to 77.

    Using the constraint system

    To use the constraint system to carry out the temperature computation outlined above, we first create two connectors, C and F, by calling the constructor make-connector, and link C and F in an appropriate network: 

    (define C (make-connector))
(define F (make-connector))
(celsius-fahrenheit-converter C F)
ok
    The procedure that creates the network is defined as follows: 

    (define (celsius-fahrenheit-converter c f)
  (let ((u (make-connector))
        (v (make-connector))
        (w (make-connector))
        (x (make-connector))
        (y (make-connector)))
    (multiplier c w u)
    (multiplier v x u)
    (adder v y f)
    (constant 9 w)
    (constant 5 x)
    (constant 32 y)
    'ok))
    This procedure creates the internal connectors u, v, w, x, and y, and links them as shown in figure [*] using the primitive constraint constructors adder, multiplier, and constant. Just as with the digital-circuit simulator of section [*], expressing these combinations of primitive elements in terms of procedures automatically provides our language with a means of abstraction for compound objects.

    To watch the network in action, we can place probes on the connectors C and F, using a probe procedure similar to the one we used to monitor wires in section [*]. Placing a probe on a connector will cause a message to be printed whenever the connector is given a value: 

    (probe "Celsius temp" C)
(probe "Fahrenheit temp" F)
    Next we set the value of C to 25. (The third argument to set-value! tells C that this directive comes from the user.) 

    (set-value! C 25 'user)
Probe: Celsius temp = 25
Probe: Fahrenheit temp = 77
done
    The probe on C awakens and reports the value. C also propagates its value through the network as described above. This sets F to 77, which is reported by the probe on F.

    Now we can try to set F to a new value, say 212: 

    (set-value! F 212 'user)
Error! Contradiction (77 212)
    The connector complains that it has sensed a contradiction: Its value is 77, and someone is trying to set it to 212. If we really want to reuse the network with new values, we can tell C to forget its old value: 

    (forget-value! C 'user)
Probe: Celsius temp = ?
Probe: Fahrenheit temp = ?
done
    C finds that the user, who set its value originally, is now retracting that value, so C agrees to lose its value, as shown by the probe, and informs the rest of the network of this fact. This information eventually propagates to F, which now finds that it has no reason for continuing to believe that its own value is 77. Thus, F also gives up its value, as shown by the probe.

    Now that F has no value, we are free to set it to 212: 

    (set-value! F 212 'user)
Probe: Fahrenheit temp = 212
Probe: Celsius temp = 100
done
    This new value, when propagated through the network, forces C to have a value of 100, and this is registered by the probe on C. Notice that the very same network is being used to compute C given F and to compute F given C. This nondirectionality of computation is the distinguishing feature of constraint-based systems.

    Implementing the constraint system

    The constraint system is implemented via procedural objects with local state, in a manner very similar to the digital-circuit simulator of section [*]. Although the primitive objects of the constraint system are somewhat more complex, the overall system is simpler, since there is no concern about agendas and logic delays.

    The basic operations on connectors are the following:

    The connectors communicate with the constraints by means of the procedures inform-about-value, which tells the given constraint that the connector has a value, and inform-about-no-value, which tells the constraint that the connector has lost its value.

    Adder constructs an adder constraint among summand connectors a1 and a2 and a sum connector. An adder is implemented as a procedure with local state (the procedure me below): 

    (define (adder a1 a2 sum)
  (define (process-new-value)
    (cond ((and (has-value? a1) (has-value? a2))
           (set-value! sum
                       (+ (get-value a1) (get-value a2))
                       me))
          ((and (has-value? a1) (has-value? sum))
           (set-value! a2
                       (- (get-value sum) (get-value a1))
                       me))
          ((and (has-value? a2) (has-value? sum))
           (set-value! a1
                       (- (get-value sum) (get-value a2))
                       me))))
  (define (process-forget-value)
    (forget-value! sum me)
    (forget-value! a1 me)
    (forget-value! a2 me)
    (process-new-value))
  (define (me request)
    (cond ((eq? request 'I-have-a-value)  
           (process-new-value))
          ((eq? request 'I-lost-my-value) 
           (process-forget-value))
          (else 
           (error "Unknown request - ADDER" request))))
  (connect a1 me)
  (connect a2 me)
  (connect sum me)
  me)
    Adder connects the new adder to the designated connectors and returns it as its value. The procedure me, which represents the adder, acts as a dispatch to the local procedures. The following ``syntax interfaces'' (see footnote [*] in section [*]) are used in conjunction with the dispatch: 

    (define (inform-about-value constraint)
  (constraint 'I-have-a-value))

(define (inform-about-no-value constraint)
  (constraint 'I-lost-my-value))
    The adder's local procedure process-new-value is called when the adder is informed that one of its connectors has a value. The adder first checks to see if both a1 and a2 have values. If so, it tells sum to set its value to the sum of the two addends. The informant argument to set-value! is me, which is the adder object itself. If a1 and a2 do not both have values, then the adder checks to see if perhaps a1 and sum have values. If so, it sets a2 to the difference of these two. Finally, if a2 and sum have values, this gives the adder enough information to set a1. If the adder is told that one of its connectors has lost a value, it requests that all of its connectors now lose their values. (Only those values that were set by this adder are actually lost.) Then it runs process-new-value. The reason for this last step is that one or more connectors may still have a value (that is, a connector may have had a value that was not originally set by the adder), and these values may need to be propagated back through the adder.

    A multiplier is very similar to an adder. It will set its product to 0 if either of the factors is 0, even if the other factor is not known. 

    (define (multiplier m1 m2 product)
  (define (process-new-value)
    (cond ((or (and (has-value? m1) (= (get-value m1) 0))
               (and (has-value? m2) (= (get-value m2) 0)))
           (set-value! product 0 me))
          ((and (has-value? m1) (has-value? m2))
           (set-value! product
                       (* (get-value m1) (get-value m2))
                       me))
          ((and (has-value? product) (has-value? m1))
           (set-value! m2
                       (/ (get-value product) (get-value m1))
                       me))
          ((and (has-value? product) (has-value? m2))
           (set-value! m1
                       (/ (get-value product) (get-value m2))
                       me))))
  (define (process-forget-value)
    (forget-value! product me)
    (forget-value! m1 me)
    (forget-value! m2 me)
    (process-new-value))
  (define (me request)
    (cond ((eq? request 'I-have-a-value)
           (process-new-value))
          ((eq? request 'I-lost-my-value)
           (process-forget-value))
          (else
           (error "Unknown request - MULTIPLIER" request))))
  (connect m1 me)
  (connect m2 me)
  (connect product me)
  me)
    A constant constructor simply sets the value of the designated connector. Any I-have-a-value or I-lost-my-value message sent to the constant box will produce an error. 

    (define (constant value connector)
  (define (me request)
    (error "Unknown request - CONSTANT" request))
  (connect connector me)
  (set-value! connector value me)
  me)
    Finally, a probe prints a message about the setting or unsetting of the designated connector: 

    (define (probe name connector)
  (define (print-probe value)
    (newline)
    (display "Probe: ")
    (display name)
    (display " = ")
    (display value))
  (define (process-new-value)
    (print-probe (get-value connector)))
  (define (process-forget-value)
    (print-probe "?"))
  (define (me request)
    (cond ((eq? request 'I-have-a-value)
           (process-new-value))
          ((eq? request 'I-lost-my-value)
           (process-forget-value))
          (else
           (error "Unknown request - PROBE" request))))
  (connect connector me)
  me)

    Representing connectors

    A connector is represented as a procedural object with local state variables value, the current value of the connector; informant, the object that set the connector's value; and constraints, a list of the constraints in which the connector participates. 

    (define (make-connector)
  (let ((value false) (informant false) (constraints '()))
    (define (set-my-value newval setter)
      (cond ((not (has-value? me))
             (set! value newval)
             (set! informant setter)
             (for-each-except setter
                              inform-about-value
                              constraints))
            ((not (= value newval))
             (error "Contradiction" (list value newval)))
            (else 'ignored)))
    (define (forget-my-value retractor)
      (if (eq? retractor informant)
          (begin (set! informant false)
                 (for-each-except retractor
                                  inform-about-no-value
                                  constraints))
          'ignored))
    (define (connect new-constraint)
      (if (not (memq new-constraint constraints))
          (set! constraints 
                (cons new-constraint constraints)))
      (if (has-value? me)
          (inform-about-value new-constraint))
      'done)
    (define (me request)
      (cond ((eq? request 'has-value?)
             (if informant true false))
            ((eq? request 'value) value)
            ((eq? request 'set-value!) set-my-value)
            ((eq? request 'forget) forget-my-value)
            ((eq? request 'connect) connect)
            (else (error "Unknown operation - CONNECTOR"
                         request))))
    me))

    The connector's local procedure set-my-value is called when there is a request to set the connector's value. If the connector does not currently have a value, it will set its value and remember as informant the constraint that requested the value to be set.[*] Then the connector will notify all of its participating constraints except the constraint that requested the value to be set. This is accomplished using the following iterator, which applies a designated procedure to all items in a list except a given one: 

    (define (for-each-except exception procedure list)
  (define (loop items)
    (cond ((null? items) 'done)
          ((eq? (car items) exception) (loop (cdr items)))
          (else (procedure (car items))
                (loop (cdr items)))))
  (loop list))

    If a connector is asked to forget its value, it runs the local procedure forget-my-value, which first checks to make sure that the request is coming from the same object that set the value originally. If so, the connector informs its associated constraints about the loss of the value.

    The local procedure connect adds the designated new constraint to the list of constraints if it is not already in that list. Then, if the connector has a value, it informs the new constraint of this fact.

    The connector's procedure me serves as a dispatch to the other internal procedures and also represents the connector as an object. The following procedures provide a syntax interface for the dispatch: 

    (define (has-value? connector)
  (connector 'has-value?))

(define (get-value connector)
  (connector 'value))

(define (set-value! connector new-value informant)
  ((connector 'set-value!) new-value informant))

(define (forget-value! connector retractor)
  ((connector 'forget) retractor))

(define (connect connector new-constraint)
  ((connector 'connect) new-constraint))

    Exercise. Using primitive multiplier, adder, and constant constraints, define a procedure averager that takes three connectors a, b, and c as inputs and establishes the constraint that the value of c is the average of the values of a and b.

    Exercise. Louis Reasoner wants to build a squarer, a constraint device with two terminals such that the value of connector b on the second terminal will always be the square of the value a on the first terminal. He proposes the following simple device made from a multiplier: 

    (define (squarer a b)
  (multiplier a a b))
    There is a serious flaw in this idea. Explain.  

    Exercise. Ben Bitdiddle tells Louis that one way to avoid the trouble in exercise [*] is to define a squarer as a new primitive constraint. Fill in the missing portions in Ben's outline for a procedure to implement such a constraint: 

    (define (squarer a b)
  (define (process-new-value)
    (if (has-value? b)
        (if (< (get-value b) 0)
            (error "square less than 0 - SQUARER" (get-value b))
            alternative1)
        alternative2))
  (define (process-forget-value) body1)
  (define (me request) body2)
  rest of definition
  me)

    Exercise. Suppose we evaluate the following sequence of expressions in the global environment: 

    (define a (make-connector))
(define b (make-connector))
(set-value! a 10 'user)
    At some time during evaluation of the set-value!, the following expression from the connector's local procedure is evaluated: 

    (for-each-except setter inform-about-value constraints)
    Draw an environment diagram showing the environment in which the above expression is evaluated.

    Exercise. The celsius-fahrenheit-converter procedure is cumbersome when compared with a more expression-oriented style of definition, such as 

    (define (celsius-fahrenheit-converter x)
  (c+ (c* (c/ (cv 9) (cv 5))
          x)
      (cv 32)))

(define C (make-connector))
(define F (celsius-fahrenheit-converter C))
    Here c+, c*, etc. are the ``constraint'' versions of the arithmetic operations. For example, c+ takes two connectors as arguments and returns a connector that is related to these by an adder constraint: 

    (define (c+ x y)
  (let ((z (make-connector)))
    (adder x y z)
    z))
    Define analogous procedures c-, c*, c/, and cv (constant value) that enable us to define compound constraints as in the converter example above.[*]

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