Exercise 4.35: Write a procedure an-integer-between that returns an integer between two given bounds. This can be used to implement a procedure that finds Pythagorean triples, i.e., triples of integers (i,j,k) between the given bounds such that i <= j and i^2 + j^2 = k^2, as follows:

Exercise 4.36: Exercise 3-69 discussed how to generate the stream of all Pythagorean triples, with no upper bound on the size of the integers to be searched. Explain why simply replacing an-integer-between by an-integer-starting-from in the procedure in Exercise 4-35 is not an adequate way to generate arbitrary Pythagorean triples. Write a procedure that actually will accomplish this. (That is, write a procedure for which repeatedly typing try-again would in principle eventually generate all Pythagorean triples.)

Exercise 4.37: Ben Bitdiddle claims that the following method for generating Pythagorean triples is more efficient than the one in Exercise 4-35. Is he correct? (Hint: Consider the number of possibilities that must be explored.)

Baker, Cooper, Fletcher, Miller, and Smith live on different floors of an apartment house that contains only five floors. Baker does not live on the top floor. Cooper does not live on the bottom floor. Fletcher does not live on either the top or the bottom floor. Miller lives on a higher floor than does Cooper. Smith does not live on a floor adjacent to Fletcher's. Fletcher does not live on a floor adjacent to Cooper's. Where does everyone live?

Exercise 4.38: Modify the multiple-dwelling procedure to omit the requirement that Smith and Fletcher do not live on adjacent floors. How many solutions are there to this modified puzzle?

Exercise 4.39: Does the order of the restrictions in the multiple-dwelling procedure affect the answer? Does it affect the time to find an answer? If you think it matters, demonstrate a faster program obtained from the given one by reordering the restrictions. If you think it does not matter, argue your case.

Exercise 4.40: In the multiple dwelling problem, how many sets of assignments are there of people to floors, both before and after the requirement that floor assignments be distinct? It is very inefficient to generate all possible assignments of people to floors and then leave it to backtracking to eliminate them. For example, most of the restrictions depend on only one or two of the person-floor variables, and can thus be imposed before floors have been selected for all the people. Write and demonstrate a much more efficient nondeterministic procedure that solves this problem based upon generating only those possibilities that are not already ruled out by previous restrictions. (Hint: This will require a nest of let expressions.)

Exercise 4.41: Write an ordinary Scheme program to solve the multiple dwelling puzzle.

Exercise 4.42: Solve the following ``Liars''puzzle (from Phillips 1934):

Five schoolgirls sat for an examination. Their parents---so they thought---showed an undue degree of interest in the result. They therefore agreed that, in writing home about the examination, each girl should make one true statement and one untrue one. The following are the relevant passages from their letters:

@item Betty: ``Kitty was second in the examination. I was only third.'' @item Ethel: ``You'll be glad to hear that I was on top. Joan was second.'' @item Joan: ``I was third, and poor old Ethel was bottom.'' @item Kitty: ``I came out second. Mary was only fourth.'' @item Mary: ``I was fourth. Top place was taken by Betty.''

What in fact was the order in which the five girls were placed?

Exercise 4.43: Use the amb evaluator to solve the following puzzle:@footnote{This is taken from a booklet called ``Problematical Recreations,'' published in the 1960s by Litton Industries, where it is attributed to the @cite{Kansas State Engineer}.}

Parsing natural language

Programs designed to accept natural language as input usually start by attempting to parse the input, that is, to match the input against some grammatical structure. For example, we might try to recognize simple sentences consisting of an article followed by a noun followed by a verb, such as ``The cat eats.'' To accomplish such an analysis, we must be able to identify the parts of speech of individual words. We could start with some lists that classify various words:@footnote{Here we use the convention that the first element of each list designates the part of speech for the rest of the words in the list.}

We also need a grammar , that is, a set of rules describing how grammatical elements are composed from simpler elements. A very simple grammar might stipulate that a sentence always consists of two pieces---a noun phrase followed by a verb---and that a noun phrase consists of an article followed by a noun. With this grammar, the sentence ``The cat eats'' is parsed as follows:

We can generate such a parse with a simple program that has separate procedures for each of the grammatical rules. To parse a sentence, we identify its two constituent pieces and return a list of these two elements, tagged with the symbol sentence:

A noun phrase, similarly, is parsed by finding an article followed by a noun:

At the lowest level, parsing boils down to repeatedly checking that the next unparsed word is a member of the list of words for the required part of speech. To implement this, we maintain a global variable *unparsed*, which is the input that has not yet been parsed. Each time we check a word, we require that *unparsed* must be non-empty and that it should begin with a word from the designated list. If so, we remove that word from *unparsed* and return the word together with its part of speech (which is found at the head of the list):@footnote{Notice that parse-word uses set! to modify the unparsed input list. For this to work, our amb evaluator must undo the effects of set! operations when it backtracks.}

To start the parsing, all we need to do is set *unparsed* to be the entire input, try to parse a sentence, and check that nothing is left over:

We can now try the parser and verify that it works for our simple test sentence:

The amb evaluator is useful here because it is convenient to express the parsing constraints with the aid of require. Automatic search and backtracking really pay off, however, when we consider more complex grammars where there are choices for how the units can be decomposed.

Let's add to our grammar a list of prepositions:

and define a prepositional phrase (e.g., ``for the cat'') to be a preposition followed by a noun phrase:

Now we can define a sentence to be a noun phrase followed by a verb phrase, where a verb phrase can be either a verb or a verb phrase extended by a prepositional phrase:@footnote{Observe that this definition is recursive---a verb may be followed by any number of prepositional phrases.}

While we're at it, we can also elaborate the definition of noun phrases to permit such things as ``a cat in the class.'' What we used to call a noun phrase, we'll now call a simple noun phrase, and a noun phrase will now be either a simple noun phrase or a noun phrase extended by a prepositional phrase:

Our new grammar lets us parse more complex sentences. For example

Observe that a given input may have more than one legal parse. In the sentence ``The professor lectures to the student with the cat,'' it may be that the professor is lecturing with the cat, or that the student has the cat. Our nondeterministic program finds both possibilities:

produces

Asking the evaluator to try again yields

We will construct the amb evaluator for nondeterministic Scheme by modifying the analyzing evaluator of section 4.1.7.@footnote{We chose to implement the lazy evaluator in section 4.2 as a modification of the ordinary metacircular evaluator of section 4.1.1. In contrast, we will base the amb evaluator on the analyzing evaluator of section 4.1.7, because the execution procedures in that evaluator provide a convenient framework for implementing backtracking.} As in the analyzing evaluator, evaluation of an expression is accomplished by calling an execution procedure produced by analysis of that expression. The difference between the interpretation of ordinary Scheme and the interpretation of nondeterministic Scheme will be entirely in the execution procedures.

Execution procedures and continuations

Recall that the execution procedures for the ordinary evaluator take one argument: the environment of execution. In contrast, the execution procedures in the amb evaluator take three arguments: the environment, and two procedures called continuation procedures . The evaluation of an expression will finish by calling one of these two continuations: If the evaluation results in a value, the success continuation is called with that value; if the evaluation results in the discovery of a dead end, the failure continuation is called. Constructing and calling appropriate continuations is the mechanism by which the nondeterministic evaluator implements backtracking.

It is the job of the success continuation to receive a value and proceed with the computation. Along with that value, the success continuation is passed another failure continuation, which is to be called subsequently if the use of that value leads to a dead end.

It is the job of the failure continuation to try another branch of the nondeterministic process. The essence of the nondeterministic language is in the fact that expressions may represent choices among alternatives. The evaluation of such an expression must proceed with one of the indicated alternative choices, even though it is not known in advance which choices will lead to acceptable results. To deal with this, the evaluator picks one of the alternatives and passes this value to the success continuation. Together with this value, the evaluator constructs and passes along a failure continuation that can be called later to choose a different alternative.

A failure is triggered during evaluation (that is, a failure continuation is called) when a user program explicitly rejects the current line of attack (for example, a call to require may result in execution of (amb), an expression that always fails---see section 4.3.1). The failure continuation in hand at that point will cause the most recent choice point to choose another alternative. If there are no more alternatives to be considered at that choice point, a failure at an earlier choice point is triggered, and so on. Failure continuations are also invoked by the driver loop in response to a try-again request, to find another value of the expression.

In addition, if a side-effect operation (such as assignment to a variable) occurs on a branch of the process resulting from a choice, it may be necessary, when the process finds a dead end, to undo the side effect before making a new choice. This is accomplished by having the side-effect operation produce a failure continuation that undoes the side effect and propagates the failure.

In summary, failure continuations are constructed by

@item amb expressions---to provide a mechanism to make alternative choices if the current choice made by the amb expression leads to a dead end; @item the top-level driver---to provide a mechanism to report failure when the choices are exhausted; @item assignments---to intercept failures and undo assignments during backtracking.

@item if the user program executes (amb); @item if the user types try-again at the top-level driver.

@item When the failure continuation created by an assignment finishes undoing a side effect, it calls the failure continuation it intercepted, in order to propagate the failure back to the choice point that led to this assignment or to the top level. @item When the failure continuation for an amb runs out of choices, it calls the failure continuation that was originally given to the amb, in order to propagate the failure back to the previous choice point or to the top level.

Structure of the evaluator

The syntax- and data-representation procedures for the amb evaluator, and also the basic analyze procedure, are identical to those in the evaluator of section 4.1.7, except for the fact that we need additional syntax procedures to recognize the amb special form:@footnote{We assume that the evaluator supports let (see Exercise 4-22), which we have used in our nondeterministic programs.}

We must also add to the dispatch in analyze a clause that will recognize this special form and generate an appropriate execution procedure:

The top-level procedure ambeval (similar to the version of eval given in section 4.1.7) analyzes the given expression and applies the resulting execution procedure to the given environment, together with two given continuations:

A success continuation is a procedure of two arguments: the value just obtained and another failure continuation to be used if that value leads to a subsequent failure. A failure continuation is a procedure of no arguments. So the general form of an execution procedure is

For example, executing

will attempt to evaluate the given expression and will return either the expression's value (if the evaluation succeeds) or the symbol failed (if the evaluation fails). The call to ambeval in the driver loop shown below uses much more complicated continuation procedures, which continue the loop and support the try-again request.

Most of the complexity of the amb evaluator results from the mechanics of passing the continuations around as the execution procedures call each other. In going through the following code, you should compare each of the execution procedures with the corresponding procedure for the ordinary evaluator given in section 4.1.7.

Simple expressions

The execution procedures for the simplest kinds of expressions are essentially the same as those for the ordinary evaluator, except for the need to manage the continuations. The execution procedures simply succeed with the value of the expression, passing along the failure continuation that was passed to them.

Notice that looking up a variable always ``succeeds.'' If lookup-variable-value fails to find the variable, it signals an error, as usual. Such a ``failure'' indicates a program bug---a reference to an unbound variable; it is not an indication that we should try another nondeterministic choice instead of the one that is currently being tried.

Conditionals and sequences

Conditionals are also handled in a similar way as in the ordinary evaluator. The execution procedure generated by analyze-if invokes the predicate execution procedure pproc with a success continuation that checks whether the predicate value is true and goes on to execute either the consequent or the alternative. If the execution of pproc fails, the original failure continuation for the if expression is called.

Sequences are also handled in the same way as in the previous evaluator, except for the machinations in the subprocedure sequentially that are required for passing the continuations. Namely, to sequentially execute a and then b, we call a with a success continuation that calls b.

Definitions and assignments

Definitions are another case where we must go to some trouble to manage the continuations, because it is necessary to evaluate the definition-value expression before actually defining the new variable. To accomplish this, the definition-value execution procedure vproc is called with the environment, a success continuation, and the failure continuation. If the execution of vproc succeeds, obtaining a value val for the defined variable, the variable is defined and the success is propagated:

Assignments are more interesting. This is the first place where we really use the continuations, rather than just passing them around. The execution procedure for assignments starts out like the one for definitions. It first attempts to obtain the new value to be assigned to the variable. If this evaluation of vproc fails, the assignment fails.

If vproc succeeds, however, and we go on to make the assignment, we must consider the possibility that this branch of the computation might later fail, which will require us to backtrack out of the assignment. Thus, we must arrange to undo the assignment as part of the backtracking process.@footnote{We didn't worry about undoing definitions, since we can assume that internal definitions are scanned out (section 4.1.6).}

This is accomplished by giving vproc a success continuation (marked with the comment ``*1*'' below) that saves the old value of the variable before assigning the new value to the variable and proceeding from the assignment. The failure continuation that is passed along with the value of the assignment (marked with the comment ``*2*'' below) restores the old value of the variable before continuing the failure. That is, a successful assignment provides a failure continuation that will intercept a subsequent failure; whatever failure would otherwise have called fail2 calls this procedure instead, to undo the assignment before actually calling fail2.

Procedure applications

The execution procedure for applications contains no new ideas except for the technical complexity of managing the continuations. This complexity arises in analyze-application, due to the need to keep track of the success and failure continuations as we evaluate the operands. We use a procedure get-args to evaluate the list of operands, rather than a simple map as in the ordinary evaluator.

In get-args, notice how cdring down the list of aproc execution procedures and consing up the resulting list of args is accomplished by calling each aproc in the list with a success continuation that recursively calls get-args. Each of these recursive calls to get-args has a success continuation whose value is the cons of the newly obtained argument onto the list of accumulated arguments:

The actual procedure application, which is performed by execute-application, is accomplished in the same way as for the ordinary evaluator, except for the need to manage the continuations.

Evaluating amb expressions

The amb special form is the key element in the nondeterministic language. Here we see the essence of the interpretation process and the reason for keeping track of the continuations. The execution procedure for amb defines a loop try-next that cycles through the execution procedures for all the possible values of the amb expression. Each execution procedure is called with a failure continuation that will try the next one. When there are no more alternatives to try, the entire amb expression fails.

Driver loop

The driver loop for the amb evaluator is complex, due to the mechanism that permits the user to try again in evaluating an expression. The driver uses a procedure called internal-loop, which takes as argument a procedure try-again. The intent is that calling try-again should go on to the next untried alternative in the nondeterministic evaluation. Internal-loop either calls try-again in response to the user typing try-again at the driver loop, or else starts a new evaluation by calling ambeval.

The failure continuation for this call to ambeval informs the user that there are no more values and re-invokes the driver loop.

The success continuation for the call to ambeval is more subtle. We print the obtained value and then invoke the internal loop again with a try-again procedure that will be able to try the next alternative. This next-alternative procedure is the second argument that was passed to the success continuation. Ordinarily, we think of this second argument as a failure continuation to be used if the current evaluation branch later fails. In this case, however, we have completed a successful evaluation, so we can invoke the ``failure'' alternative branch in order to search for additional successful evaluations.

The initial call to internal-loop uses a try-again procedure that complains that there is no current problem and restarts the driver loop. This is the behavior that will happen if the user types try-again when there is no evaluation in progress.

Based on Structure and Interpretation of Computer Programs, a work at https://mitpress.mit.edu/sicp/.