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                    GNU GENERAL PUBLIC LICENSE
                       Version 3, 29 June 2007

 Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>
 Everyone is permitted to copy and distribute verbatim copies
 of this license document, but changing it is not allowed.

                            Preamble

  The GNU General Public License is a free, copyleft license for
software and other kinds of works.

  The licenses for most software and other practical works are designed
to take away your freedom to share and change the works.  By contrast,
the GNU General Public License is intended to guarantee your freedom to
share and change all versions of a program--to make sure it remains free
software for all its users.  We, the Free Software Foundation, use the
GNU General Public License for most of our software; it applies also to
any other work released this way by its authors.  You can apply it to
your programs, too.

  When we speak of free software, we are referring to freedom, not
price.  Our General Public Licenses are designed to make sure that you
have the freedom to distribute copies of free software (and charge for
them if you wish), that you receive source code or can get it if you
want it, that you can change the software or use pieces of it in new
free programs, and that you know you can do these things.

  To protect your rights, we need to prevent others from denying you
these rights or asking you to surrender the rights.  Therefore, you have
certain responsibilities if you distribute copies of the software, or if
you modify it: responsibilities to respect the freedom of others.

  For example, if you distribute copies of such a program, whether
gratis or for a fee, you must pass on to the recipients the same
freedoms that you received.  You must make sure that they, too, receive
or can get the source code.  And you must show them these terms so they
know their rights.

  Developers that use the GNU GPL protect your rights with two steps:
(1) assert copyright on the software, and (2) offer you this License
giving you legal permission to copy, distribute and/or modify it.

  For the developers' and authors' protection, the GPL clearly explains
that there is no warranty for this free software.  For both users' and
authors' sake, the GPL requires that modified versions be marked as
changed, so that their problems will not be attributed erroneously to
authors of previous versions.

  Some devices are designed to deny users access to install or run
modified versions of the software inside them, although the manufacturer
can do so.  This is fundamentally incompatible with the aim of
protecting users' freedom to change the software.  The systematic
pattern of such abuse occurs in the area of products for individuals to
use, which is precisely where it is most unacceptable.  Therefore, we
have designed this version of the GPL to prohibit the practice for those
products.  If such problems arise substantially in other domains, we
stand ready to extend this provision to those domains in future versions
of the GPL, as needed to protect the freedom of users.

  Finally, every program is threatened constantly by software patents.
States should not allow patents to restrict development and use of
software on general-purpose computers, but in those that do, we wish to
avoid the special danger that patents applied to a free program could
make it effectively proprietary.  To prevent this, the GPL assures that
patents cannot be used to render the program non-free.

  The precise terms and conditions for copying, distribution and
modification follow.

                       TERMS AND CONDITIONS

  0. Definitions.

  "This License" refers to version 3 of the GNU General Public License.

  "Copyright" also means copyright-like laws that apply to other kinds of
works, such as semiconductor masks.

  "The Program" refers to any copyrightable work licensed under this
License.  Each licensee is addressed as "you".  "Licensees" and
"recipients" may be individuals or organizations.

  To "modify" a work means to copy from or adapt all or part of the work
in a fashion requiring copyright permission, other than the making of an
exact copy.  The resulting work is called a "modified version" of the
earlier work or a work "based on" the earlier work.

  A "covered work" means either the unmodified Program or a work based
on the Program.

  To "propagate" a work means to do anything with it that, without
permission, would make you directly or secondarily liable for
infringement under applicable copyright law, except executing it on a
computer or modifying a private copy.  Propagation includes copying,
distribution (with or without modification), making available to the
public, and in some countries other activities as well.

  To "convey" a work means any kind of propagation that enables other
parties to make or receive copies.  Mere interaction with a user through
a computer network, with no transfer of a copy, is not conveying.

  An interactive user interface displays "Appropriate Legal Notices"
to the extent that it includes a convenient and prominently visible
feature that (1) displays an appropriate copyright notice, and (2)
tells the user that there is no warranty for the work (except to the
extent that warranties are provided), that licensees may convey the
work under this License, and how to view a copy of this License.  If
the interface presents a list of user commands or options, such as a
menu, a prominent item in the list meets this criterion.

  1. Source Code.

  The "source code" for a work means the preferred form of the work
for making modifications to it.  "Object code" means any non-source
form of a work.

  A "Standard Interface" means an interface that either is an official
standard defined by a recognized standards body, or, in the case of
interfaces specified for a particular programming language, one that
is widely used among developers working in that language.

  The "System Libraries" of an executable work include anything, other
than the work as a whole, that (a) is included in the normal form of
packaging a Major Component, but which is not part of that Major
Component, and (b) serves only to enable use of the work with that
Major Component, or to implement a Standard Interface for which an
implementation is available to the public in source code form.  A
"Major Component", in this context, means a major essential component
(kernel, window system, and so on) of the specific operating system
(if any) on which the executable work runs, or a compiler used to
produce the work, or an object code interpreter used to run it.

  The "Corresponding Source" for a work in object code form means all
the source code needed to generate, install, and (for an executable
work) run the object code and to modify the work, including scripts to
control those activities.  However, it does not include the work's
System Libraries, or general-purpose tools or generally available free
programs which are used unmodified in performing those activities but
which are not part of the work.  For example, Corresponding Source
includes interface definition files associated with source files for
the work, and the source code for shared libraries and dynamically
linked subprograms that the work is specifically designed to require,
such as by intimate data communication or control flow between those
subprograms and other parts of the work.

  The Corresponding Source need not include anything that users
can regenerate automatically from other parts of the Corresponding
Source.

  The Corresponding Source for a work in source code form is that
same work.

  2. Basic Permissions.

  All rights granted under this License are granted for the term of
copyright on the Program, and are irrevocable provided the stated
conditions are met.  This License explicitly affirms your unlimited
permission to run the unmodified Program.  The output from running a
covered work is covered by this License only if the output, given its
content, constitutes a covered work.  This License acknowledges your
rights of fair use or other equivalent, as provided by copyright law.

  You may make, run and propagate covered works that you do not
convey, without conditions so long as your license otherwise remains
in force.  You may convey covered works to others for the sole purpose
of having them make modifications exclusively for you, or provide you
with facilities for running those works, provided that you comply with
the terms of this License in conveying all material for which you do
not control copyright.  Those thus making or running the covered works
for you must do so exclusively on your behalf, under your direction
and control, on terms that prohibit them from making any copies of
your copyrighted material outside their relationship with you.

  Conveying under any other circumstances is permitted solely under
the conditions stated below.  Sublicensing is not allowed; section 10
makes it unnecessary.

  3. Protecting Users' Legal Rights From Anti-Circumvention Law.

  No covered work shall be deemed part of an effective technological
measure under any applicable law fulfilling obligations under article
11 of the WIPO copyright treaty adopted on 20 December 1996, or
similar laws prohibiting or restricting circumvention of such
measures.

  When you convey a covered work, you waive any legal power to forbid
circumvention of technological measures to the extent such circumvention
is effected by exercising rights under this License with respect to
the covered work, and you disclaim any intention to limit operation or
modification of the work as a means of enforcing, against the work's
users, your or third parties' legal rights to forbid circumvention of
technological measures.

  4. Conveying Verbatim Copies.

  You may convey verbatim copies of the Program's source code as you
receive it, in any medium, provided that you conspicuously and
appropriately publish on each copy an appropriate copyright notice;
keep intact all notices stating that this License and any
non-permissive terms added in accord with section 7 apply to the code;
keep intact all notices of the absence of any warranty; and give all
recipients a copy of this License along with the Program.

  You may charge any price or no price for each copy that you convey,
and you may offer support or warranty protection for a fee.

  5. Conveying Modified Source Versions.

  You may convey a work based on the Program, or the modifications to
produce it from the Program, in the form of source code under the
terms of section 4, provided that you also meet all of these conditions:

    a) The work must carry prominent notices stating that you modified
    it, and giving a relevant date.

    b) The work must carry prominent notices stating that it is
    released under this License and any conditions added under section
    7.  This requirement modifies the requirement in section 4 to
    "keep intact all notices".

    c) You must license the entire work, as a whole, under this
    License to anyone who comes into possession of a copy.  This
    License will therefore apply, along with any applicable section 7
    additional terms, to the whole of the work, and all its parts,
    regardless of how they are packaged.  This License gives no
    permission to license the work in any other way, but it does not
    invalidate such permission if you have separately received it.

    d) If the work has interactive user interfaces, each must display
    Appropriate Legal Notices; however, if the Program has interactive
    interfaces that do not display Appropriate Legal Notices, your
    work need not make them do so.

  A compilation of a covered work with other separate and independent
works, which are not by their nature extensions of the covered work,
and which are not combined with it such as to form a larger program,
in or on a volume of a storage or distribution medium, is called an
"aggregate" if the compilation and its resulting copyright are not
used to limit the access or legal rights of the compilation's users
beyond what the individual works permit.  Inclusion of a covered work
in an aggregate does not cause this License to apply to the other
parts of the aggregate.

  6. Conveying Non-Source Forms.

  You may convey a covered work in object code form under the terms
of sections 4 and 5, provided that you also convey the
machine-readable Corresponding Source under the terms of this License,
in one of these ways:

    a) Convey the object code in, or embodied in, a physical product
    (including a physical distribution medium), accompanied by the
    Corresponding Source fixed on a durable physical medium
    customarily used for software interchange.

    b) Convey the object code in, or embodied in, a physical product
    (including a physical distribution medium), accompanied by a
    written offer, valid for at least three years and valid for as
    long as you offer spare parts or customer support for that product
    model, to give anyone who possesses the object code either (1) a
    copy of the Corresponding Source for all the software in the
    product that is covered by this License, on a durable physical
    medium customarily used for software interchange, for a price no
    more than your reasonable cost of physically performing this
    conveying of source, or (2) access to copy the
    Corresponding Source from a network server at no charge.

    c) Convey individual copies of the object code with a copy of the
    written offer to provide the Corresponding Source.  This
    alternative is allowed only occasionally and noncommercially, and
    only if you received the object code with such an offer, in accord
    with subsection 6b.

    d) Convey the object code by offering access from a designated
    place (gratis or for a charge), and offer equivalent access to the
    Corresponding Source in the same way through the same place at no
    further charge.  You need not require recipients to copy the
    Corresponding Source along with the object code.  If the place to
    copy the object code is a network server, the Corresponding Source
    may be on a different server (operated by you or a third party)
    that supports equivalent copying facilities, provided you maintain
    clear directions next to the object code saying where to find the
    Corresponding Source.  Regardless of what server hosts the
    Corresponding Source, you remain obligated to ensure that it is
    available for as long as needed to satisfy these requirements.

    e) Convey the object code using peer-to-peer transmission, provided
    you inform other peers where the object code and Corresponding
    Source of the work are being offered to the general public at no
    charge under subsection 6d.

  A separable portion of the object code, whose source code is excluded
from the Corresponding Source as a System Library, need not be
included in conveying the object code work.

  A "User Product" is either (1) a "consumer product", which means any
tangible personal property which is normally used for personal, family,
or household purposes, or (2) anything designed or sold for incorporation
into a dwelling.  In determining whether a product is a consumer product,
doubtful cases shall be resolved in favor of coverage.  For a particular
product received by a particular user, "normally used" refers to a
typical or common use of that class of product, regardless of the status
of the particular user or of the way in which the particular user
actually uses, or expects or is expected to use, the product.  A product
is a consumer product regardless of whether the product has substantial
commercial, industrial or non-consumer uses, unless such uses represent
the only significant mode of use of the product.

  "Installation Information" for a User Product means any methods,
procedures, authorization keys, or other information required to install
and execute modified versions of a covered work in that User Product from
a modified version of its Corresponding Source.  The information must
suffice to ensure that the continued functioning of the modified object
code is in no case prevented or interfered with solely because
modification has been made.

  If you convey an object code work under this section in, or with, or
specifically for use in, a User Product, and the conveying occurs as
part of a transaction in which the right of possession and use of the
User Product is transferred to the recipient in perpetuity or for a
fixed term (regardless of how the transaction is characterized), the
Corresponding Source conveyed under this section must be accompanied
by the Installation Information.  But this requirement does not apply
if neither you nor any third party retains the ability to install
modified object code on the User Product (for example, the work has
been installed in ROM).

  The requirement to provide Installation Information does not include a
requirement to continue to provide support service, warranty, or updates
for a work that has been modified or installed by the recipient, or for
the User Product in which it has been modified or installed.  Access to a
network may be denied when the modification itself materially and
adversely affects the operation of the network or violates the rules and
protocols for communication across the network.

  Corresponding Source conveyed, and Installation Information provided,
in accord with this section must be in a format that is publicly
documented (and with an implementation available to the public in
source code form), and must require no special password or key for
unpacking, reading or copying.

  7. Additional Terms.

  "Additional permissions" are terms that supplement the terms of this
License by making exceptions from one or more of its conditions.
Additional permissions that are applicable to the entire Program shall
be treated as though they were included in this License, to the extent
that they are valid under applicable law.  If additional permissions
apply only to part of the Program, that part may be used separately
under those permissions, but the entire Program remains governed by
this License without regard to the additional permissions.

  When you convey a copy of a covered work, you may at your option
remove any additional permissions from that copy, or from any part of
it.  (Additional permissions may be written to require their own
removal in certain cases when you modify the work.)  You may place
additional permissions on material, added by you to a covered work,
for which you have or can give appropriate copyright permission.

  Notwithstanding any other provision of this License, for material you
add to a covered work, you may (if authorized by the copyright holders of
that material) supplement the terms of this License with terms:

    a) Disclaiming warranty or limiting liability differently from the
    terms of sections 15 and 16 of this License; or

    b) Requiring preservation of specified reasonable legal notices or
    author attributions in that material or in the Appropriate Legal
    Notices displayed by works containing it; or

    c) Prohibiting misrepresentation of the origin of that material, or
    requiring that modified versions of such material be marked in
    reasonable ways as different from the original version; or

    d) Limiting the use for publicity purposes of names of licensors or
    authors of the material; or

    e) Declining to grant rights under trademark law for use of some
    trade names, trademarks, or service marks; or

    f) Requiring indemnification of licensors and authors of that
    material by anyone who conveys the material (or modified versions of
    it) with contractual assumptions of liability to the recipient, for
    any liability that these contractual assumptions directly impose on
    those licensors and authors.

  All other non-permissive additional terms are considered "further
restrictions" within the meaning of section 10.  If the Program as you
received it, or any part of it, contains a notice stating that it is
governed by this License along with a term that is a further
restriction, you may remove that term.  If a license document contains
a further restriction but permits relicensing or conveying under this
License, you may add to a covered work material governed by the terms
of that license document, provided that the further restriction does
not survive such relicensing or conveying.

  If you add terms to a covered work in accord with this section, you
must place, in the relevant source files, a statement of the
additional terms that apply to those files, or a notice indicating
where to find the applicable terms.

  Additional terms, permissive or non-permissive, may be stated in the
form of a separately written license, or stated as exceptions;
the above requirements apply either way.

  8. Termination.

  You may not propagate or modify a covered work except as expressly
provided under this License.  Any attempt otherwise to propagate or
modify it is void, and will automatically terminate your rights under
this License (including any patent licenses granted under the third
paragraph of section 11).

  However, if you cease all violation of this License, then your
license from a particular copyright holder is reinstated (a)
provisionally, unless and until the copyright holder explicitly and
finally terminates your license, and (b) permanently, if the copyright
holder fails to notify you of the violation by some reasonable means
prior to 60 days after the cessation.

  Moreover, your license from a particular copyright holder is
reinstated permanently if the copyright holder notifies you of the
violation by some reasonable means, this is the first time you have
received notice of violation of this License (for any work) from that
copyright holder, and you cure the violation prior to 30 days after
your receipt of the notice.

  Termination of your rights under this section does not terminate the
licenses of parties who have received copies or rights from you under
this License.  If your rights have been terminated and not permanently
reinstated, you do not qualify to receive new licenses for the same
material under section 10.

  9. Acceptance Not Required for Having Copies.

  You are not required to accept this License in order to receive or
run a copy of the Program.  Ancillary propagation of a covered work
occurring solely as a consequence of using peer-to-peer transmission
to receive a copy likewise does not require acceptance.  However,
nothing other than this License grants you permission to propagate or
modify any covered work.  These actions infringe copyright if you do
not accept this License.  Therefore, by modifying or propagating a
covered work, you indicate your acceptance of this License to do so.

  10. Automatic Licensing of Downstream Recipients.

  Each time you convey a covered work, the recipient automatically
receives a license from the original licensors, to run, modify and
propagate that work, subject to this License.  You are not responsible
for enforcing compliance by third parties with this License.

  An "entity transaction" is a transaction transferring control of an
organization, or substantially all assets of one, or subdividing an
organization, or merging organizations.  If propagation of a covered
work results from an entity transaction, each party to that
transaction who receives a copy of the work also receives whatever
licenses to the work the party's predecessor in interest had or could
give under the previous paragraph, plus a right to possession of the
Corresponding Source of the work from the predecessor in interest, if
the predecessor has it or can get it with reasonable efforts.

  You may not impose any further restrictions on the exercise of the
rights granted or affirmed under this License.  For example, you may
not impose a license fee, royalty, or other charge for exercise of
rights granted under this License, and you may not initiate litigation
(including a cross-claim or counterclaim in a lawsuit) alleging that
any patent claim is infringed by making, using, selling, offering for
sale, or importing the Program or any portion of it.

  11. Patents.

  A "contributor" is a copyright holder who authorizes use under this
License of the Program or a work on which the Program is based.  The
work thus licensed is called the contributor's "contributor version".

  A contributor's "essential patent claims" are all patent claims
owned or controlled by the contributor, whether already acquired or
hereafter acquired, that would be infringed by some manner, permitted
by this License, of making, using, or selling its contributor version,
but do not include claims that would be infringed only as a
consequence of further modification of the contributor version.  For
purposes of this definition, "control" includes the right to grant
patent sublicenses in a manner consistent with the requirements of
this License.

  Each contributor grants you a non-exclusive, worldwide, royalty-free
patent license under the contributor's essential patent claims, to
make, use, sell, offer for sale, import and otherwise run, modify and
propagate the contents of its contributor version.

  In the following three paragraphs, a "patent license" is any express
agreement or commitment, however denominated, not to enforce a patent
(such as an express permission to practice a patent or covenant not to
sue for patent infringement).  To "grant" such a patent license to a
party means to make such an agreement or commitment not to enforce a
patent against the party.

  If you convey a covered work, knowingly relying on a patent license,
and the Corresponding Source of the work is not available for anyone
to copy, free of charge and under the terms of this License, through a
publicly available network server or other readily accessible means,
then you must either (1) cause the Corresponding Source to be so
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n> will output the first letter of <CODE>:code</CODE> (Q in the example I gave earlier). But suppose the first input isn't an A. Then <CODE>codematch</CODE> has to solve a smaller subproblem: Find the input letter in the remaining 25 letters of the alphabet. Finding a smaller, similar subproblem means that we can use a recursive solution. <CODE>Codematch</CODE> invokes itself, but for its second and third inputs it uses the <CODE>butfirst</CODE>s of the original inputs because the first letter of the alphabet (A) and its corresponding coded letter (Q) have already been rejected. <P> Here is a trace of an example of <CODE>codematch</CODE> at work, to help you understand what's going on. <P><PRE>codematch &quot;e &quot;abcdefghijklmnopqrstuvwxyz &quot;qwertyuiopasdfghjklzxcvbnm codematch &quot;e &quot;bcdefghijklmnopqrstuvwxyz &quot;wertyuiopasdfghjklzxcvbnm codematch &quot;e &quot;cdefghijklmnopqrstuvwxyz &quot;ertyuiopasdfghjklzxcvbnm codematch &quot;e &quot;defghijklmnopqrstuvwxyz &quot;rtyuiopasdfghjklzxcvbnm codematch &quot;e &quot;efghijklmnopqrstuvwxyz &quot;tyuiopasdfghjklzxcvbnm codematch outputs &quot;t codematch outputs &quot;t codematch outputs &quot;t codematch outputs &quot;t codematch outputs &quot;t </PRE> <P>The fifth, innermost invocation of <CODE>codematch</CODE> succeeds at matching its first input (the letter E) with the first letter of its second input. That invocation therefore outputs the first letter of its third input, the letter T. Each of the higher-level invocations outputs the same thing in turn. <P>The pattern of doing something to the <CODE>first</CODE> of an input, then invoking the same procedure recursively with the <CODE>butfirst</CODE> as the new input, is a familiar one from recursive commands. If we only wanted to translate single letters, we could have written <CODE> codelet</CODE> and <CODE>codematch</CODE> as commands, like this: <P><PRE>to codelet :letter :code ;; command version codematch :letter &quot;abcdefghijklmnopqrstuvwxyz :code end to codematch :letter :clear :code ;; command version if emptyp :clear [print :letter stop] if equalp :letter first :clear [print first :code stop] codematch :letter butfirst :clear butfirst :code end </PRE> <P>You may find this version a little easier to understand, because it's more like the recursive commands we've examined in the past. But making <CODE>codelet</CODE> an operation is a much stronger technique. Instead of being required to print the computed code letter, we can make that letter part of a larger computation. In fact, we have to do that in order to encipher a complete word. Each word is made up of letters, and the task of <CODE>codeword</CODE> will be to go through a word, letter by letter, using each letter as input to <CODE> codelet</CODE>. The letters output by <CODE>codelet</CODE> must be combined into a new word, which will be the output from <CODE>codeword</CODE>. <P>We could write <CODE>codeword</CODE> using the higher order function <CODE>map</CODE>: <P><PRE>to codeword :word :code ;; using higher order function output map [codelet ? :code] :word end </PRE> <P>But to help you learn how to write recursive operations, in this chapter we'll avoid higher order functions. (As it turns out, <CODE>map</CODE> itself is a recursive operation, written using the techniques of this chapter.) <P>Recall the structure of a previous procedure that went through a word letter by letter: <P><PRE>to one.per.line :word if emptyp :word [stop] print first :word one.per.line butfirst :word end </PRE> <P>Compare this to the structure of the recursive <CODE>codeword</CODE>: <P><PRE>to codeword :word :code if emptyp :word [output &quot;] output word (codelet first :word :code) (codeword butfirst :word :code) end </PRE> <P>There are many similarities. Both procedures have a stop rule that tests for an empty input. Both do something to the <CODE> first</CODE> of the input (either <CODE>print</CODE> or <CODE>codelet</CODE>), and each invokes itself recursively on the <CODE>butfirst</CODE> of the input. (<CODE>Codeword</CODE> has an extra input for the code letters, but that doesn't really change the structure of the procedure. If that's confusing to you, you could temporarily pretend that <CODE>code</CODE> is a global variable and eliminate it as an input.) <P>The differences have to do with the fact that <CODE>codeword</CODE> is an operation instead of a command. The stop rule invokes <CODE>output</CODE> rather than <CODE>stop</CODE> and must therefore specify what is to be output when the stop condition is met. (In this case, when the input word is empty, the output is also the empty word.) But the main thing is that the action step (the <CODE>print</CODE> in <CODE>one.per.line</CODE>) and the recursive call (the <CODE>one.per.line</CODE> instruction) are not two separate instructions in <CODE>codeword</CODE>. Instead they are expressions (the two in parentheses) that are combined by <CODE>word</CODE> to form the complete output. Here's a picture: <P><CENTER><IMG SRC="codelet.gif" ALT="figure: codelet"></CENTER> <P>Remember what you learned in Chapter 2 about the way in which Logo instructions are evaluated. Consider the <CODE>output</CODE> instruction in <CODE>codeword</CODE>. Before <CODE>output</CODE> can be invoked, Logo must evaluate its input. That input comes from the output from <CODE>word</CODE>. Before <CODE>word</CODE> can be invoked, Logo must evaluate <EM> its</EM> inputs. There are two of them. The first input to <CODE>word</CODE> is the expression <P><PRE>codelet first :word :code </PRE> <P>This expression computes the coded version of the first letter of the word we want to translate. The second input to <CODE> word</CODE> is the expression <P><PRE>codeword butfirst :word :code </PRE> <P>This expression invokes <CODE>codeword</CODE> recursively, solving the smaller subproblem of translating a smaller word, one with the first letter removed. When both of these computations are complete, <CODE>word</CODE> can combine the results to form the translation of the complete input word. Then <CODE>output</CODE> can output that result. <P>Here's an example of how <CODE>codeword</CODE> is used. <P><PRE>? <U>print codeword &quot;hello &quot;qwertyuiopasdfghjklzxcvbnm</U> itssg </PRE> <P>Notice that we have to say <CODE>print</CODE>, not just start the instruction line with <CODE>codeword</CODE>; a complete instruction must have a command. Suppose you had the idea of saving all that typing by changing the <CODE>output</CODE> instruction in <CODE>codeword</CODE> to a <CODE>print</CODE>. What would happen? The answer is that <CODE>codeword</CODE> wouldn't be able to invoke itself recursively as an operation. (If you don't understand that, try it!) Also, it's generally a better idea to write an operation when you have to compute some result. That way, you aren't committed to printing the result; you can use it as part of a larger computation. <P>For example, right now I'd like to write a procedure <CODE>code</CODE> that translates an entire sentence into code. Like <CODE>codeword</CODE>, it will be an operation with two inputs, the second of which is a code (a word of 26 scrambled letters). The difference is that the first input will be a sentence instead of a word and the output will also be a sentence. <P>&raquo;Write <CODE>code</CODE> using a higher order function. Then see if you can write an equivalent recursive version. <P>Just as <CODE>codeword</CODE> works by splitting up the word into letters, <CODE>code</CODE> will work by splitting up a sentence into words. The structure will be very similar. Here it is: <P><PRE>to code :sent :code if emptyp :sent [output []] output sentence (codeword first :sent :code) (code butfirst :sent :code) end </PRE> <P>The main differences are that <CODE>code</CODE> outputs the empty list, instead of the empty word, for an empty input and that <CODE>sentence</CODE> is used as the combining operation instead of <CODE>word</CODE>. Here's an example of <CODE>code</CODE> at work. <P><PRE>? <U>print code [meet at midnight, under the dock.] ~</U> <U>&quot;qwertyuiopasdfghjklzxcvbnm</U> dttz qz dorfouiz, xfrtk zit rgea. </PRE> <P><H2>More Procedure Patterns</H2> <P><CODE>Code</CODE> and <CODE>codeword</CODE> are examples of a very common pattern in recursive operations: They are like using <CODE>map</CODE> with a particular function. Here is the pattern that they fit. <P><PRE><A NAME="map">to <U>procedure</U> :input</A> if emptyp :input [output :input] output <U>combiner</U> (<U>something</U> first :input) (<U>procedure</U> butfirst :input) end </PRE> <P>The <EM>combiner</EM> is often <CODE>word</CODE> or <CODE>sentence</CODE>, although others are possible. In fact, when working with lists, the most common combiner is not <CODE>sentence</CODE> but another operation that we haven't used before, <CODE>fput</CODE> (First PUT). <CODE>Fput</CODE> takes two inputs. The first can be any datum, but the second must be a list. The output from <CODE>fput</CODE> is a list that is equal to the second input, except that the first input is inserted as a new first member. In other words the output from <CODE>fput</CODE> is a list whose <CODE>first</CODE> is the first input and whose <CODE>butfirst</CODE> is the second input. <P><PRE>? <U>show sentence [hee hee hee] [ho ho ho]</U> [hee hee hee ho ho ho] ? <U>show fput [hee hee hee] [ho ho ho]</U> [[hee hee hee] ho ho ho] </PRE> <P><CODE>Fput</CODE> is a good combiner because the two things we want to combine are the <CODE>first</CODE> and the <CODE>butfirst</CODE> of a list, except that each has been modified in some way. But the <EM>shape</EM> of the final result (a list of so many members) should be the same as the shape of the input, and that's what <CODE>fput</CODE> ensures. <P>When you're working with sentences--lists of words rather than lists of lists--<CODE>sentence</CODE> and <CODE>fput</CODE> will work equally well as the combiner. For example, <CODE>code</CODE> could have been written using <CODE>fput</CODE> instead of <CODE>sentence</CODE>. Not until some of the later examples, when we use tree-structured lists, will the choice really be important. <P>&raquo;<CODE>Fput</CODE> is actually a &quot;more primitive&quot; operation than <CODE>sentence</CODE>, in the sense that the Logo interpreter actually constructs lists by doing the internal equivalent of <CODE>fput</CODE>. As an exercise, you might like to try writing your own versions of list combiners like <CODE>sentence</CODE> and <CODE>list</CODE> out of <CODE>fput</CODE>, <CODE>first</CODE>, and <CODE>butfirst</CODE>. You should also be able to write <CODE>last</CODE> and <CODE> butlast</CODE> using only those three building blocks. (Actually you'll also need <CODE>if</CODE>, <CODE>emptyp</CODE>, <CODE>wordp</CODE>, and <CODE>output</CODE>, but you won't need any other primitive combiners.) Give your versions distinct names, such as <CODE>my.sentence</CODE>, since Logo won't let you redefine primitives. <P>&raquo;Another &quot;less primitive&quot; primitive is <CODE>lput</CODE>, an operation that takes two inputs. As for <CODE>fput</CODE>, the first can be any datum but the second must be a list. The output from <CODE>lput</CODE> is a list whose <CODE>last</CODE> is the first input and whose <CODE>butlast</CODE> is the second. Write <CODE>my-lput</CODE> using <CODE>fput</CODE> and the selectors <CODE>first</CODE> and <CODE>butfirst</CODE>. <P>It may seem silly to learn a recursive pattern for problems that can be solved using <CODE>map</CODE>. But sometimes we run into a problem that's <EM> almost</EM> like a <CODE>map</CODE>, but not exactly. For example, how would you write the following operation: <P><PRE>? <U>show pairup [now here after glow worm hole]</U> [nowhere hereafter afterglow glowworm wormhole] </PRE> <P>Instead of the usual <CODE>map</CODE>-like situation in which each word in the result is a function of one word of the input, this time each word of the result is a function of <EM>two</EM> neighboring input words. <CODE>Map</CODE> won't solve this problem, but the <CODE>map</CODE>-like recursion pattern will. <P><PRE>to pairup :words if emptyp butfirst :words [output []] output (sentence (word first :words first butfirst :words) (pairup butfirst :words)) end </PRE> <P>Compare this procedure with the <A HREF="recops.html#map">general pattern</A>; look for similarities and differences. <P>&raquo;One difference is in the test for the base case. Why is the version in <CODE>pairup</CODE> different from the one in the pattern? <P>&raquo;Write an operation that interchanges pairs of words in a sentence, like this: <P><PRE>? <U>show swap [the rain in spain stays mainly on the plain]</U> [rain the spain in mainly stays the on plain] </PRE> <P>Don't forget to think about that leftover word in an odd-length sentence! <P><H2>The <CODE>Filter</CODE> Pattern</H2> <P>In Chapter 5 we saw this example: <P><PRE>? <U>show filter &quot;numberp [76 trombones, 4 calling birds, and 8 days]</U> [76 4 8] </PRE> <P>To write a recursive operation <CODE>numbers</CODE> with the same result, we must handle three cases: the base case, in which the input is empty; the case in which the first word of the input is a number; and the case in which the first word isn't a number. <P><PRE>to numbers :sent if emptyp :sent [output []] if numberp first :sent ~ [output sentence first :sent numbers butfirst :sent] output numbers butfirst :sent end ? <U>show numbers [76 trombones, 4 calling birds, and 8 days]</U> [76 4 8] </PRE> <P>Here's the general <CODE>filter</CODE> pattern: <P><PRE>to <U>procedure</U> :input if emptyp :input [output :input] if <U>predicate</U> first :input ~ [output <U>combiner</U> first :input <U>procedure</U> butfirst :input] output <U>procedure</U> butfirst :input end </PRE> <P>As in the case of the <CODE>map</CODE> pattern, this one is most useful in situations for which the higher order function won't quite do. <P>&raquo;Write an operation that looks for two equal words next to each other in a sentence, and outputs a sentence with one of them removed: <P><PRE>? <U>show unique [Paris in the the spring is a joy joy to behold.]</U> Paris in the spring is a joy to behold. </PRE> <P>What does your procedure do with <EM>three</EM> consecutive equal words? What should it do? <P><H2>The <CODE>Reduce</CODE> Pattern</H2> <P>Other examples from Chapter 5 introduced the <CODE>reduce</CODE> higher order function. <P><PRE>? <U>show reduce &quot;word [C S L S]</U> CSLS ? <U>show reduce &quot;sum [3 4 5 6]</U> 18 </PRE> <P>Recursive operations equivalent to these examples are very much like the <CODE>map</CODE> pattern except that the combiner function is applied to the members of the input directly, rather than to some function of the members of the input: <P><PRE>to wordify :sentence if emptyp :sentence [output &quot;] output word (first :sentence) (wordify butfirst :sentence) end to addup :numbers if emptyp :numbers [output 0] output sum (first :numbers) (addup butfirst :numbers) end </PRE> <P>What are the differences between these two examples? There are two: the combiner used and the value output in the base case. Here is the pattern: <P><PRE>to <U>procedure</U> :input if emptyp :input [output <U>identity</U>] output <U>combiner</U> (first :input) (<U>procedure</U> butfirst :input) end </PRE> <P>The identity in this pattern depends on the combiner; it's the value that, when combined with something else, gives that something else unchanged as the result. Thus, zero is the identity for <CODE>sum</CODE>, but the identity for <CODE>product</CODE> would be one. <P>&raquo;Write a <CODE>multiply</CODE> operation that takes a list of numbers as its input and returns the product of all the numbers. <P>&raquo;You can make your <CODE>multiply</CODE> procedure more efficient, in some situations, by having it notice when one of the numbers in the input list is zero. In that case, you can output zero as the overall result without looking at any more numbers. The resulting procedure will, in a sense, combine aspects of the <CODE>filter</CODE> and <CODE>reduce</CODE> patterns. <P><CODE>Addup</CODE> is one example of an important sub-category of <CODE>reduce</CODE>-like procedures in which the &quot;combining&quot; operation is arithmetic, usually <CODE>sum</CODE>. The simplest example is a procedure equivalent to the primitive <CODE>count</CODE>, which counts the members of a list or the letters of a word: <P><PRE>to length :thing if emptyp :thing [output 0] output 1+length butfirst :thing end </PRE> <P>In this procedure, as usual, we can see the reduction of a problem to a smaller subproblem. The length of any word or list is one more than the length of its <CODE>butfirst</CODE>. Eventually this process of shortening the input will reduce it to emptiness; the length of an empty word or list is zero. <P>Although <CODE>count</CODE> is a primitive, there are more complicated counting situations in which not every member should be counted. For example, here is a procedure to count the number of vowels in a word: <P><PRE>to vowelcount :word if emptyp :word [output 0] if vowelp first :word [output 1+vowelcount butfirst :word] output vowelcount butfirst :word end to vowelp :letter output memberp :letter [a e i o u] end </PRE> <P>(Actually, my predicate <CODE>vowelp</CODE> is somewhat oversimplified. The letter Y is a vowel in certain positions in the word, and even some other letters can sometimes play the role of a vowel. But this isn't a book on linguistics!) <P>You can see the similarities between <CODE>vowelcount</CODE> and <CODE> length</CODE>. The difference is that, in effect, <CODE>length</CODE> uses a predicate that is always <CODE>true</CODE>, so it always carries out the instruction inside the <CODE>if</CODE>. Here's the pattern: <P><PRE>to <U>procedure</U> :input if emptyp :input [output 0] if <U>predicate</U> first :input [output 1+<U>procedure</U> butfirst :input] output <U>procedure</U> butfirst :input end </PRE> <P>&raquo;Try writing a procedure that will accept as input a word like <CODE>21,997.00</CODE> and output the number of digits before the decimal point. (In this case the correct output is 5.) Don't assume that there <EM>is</EM> a decimal point; your program shouldn't blow up no matter what word it gets as input. <P>&raquo;Another counting problem is to output the position of a member in a list. This operation is the inverse to <CODE>item</CODE>, a Logo primitive, which outputs the member at a given position number. What I'm asking you to write is <CODE>index</CODE>, which works like this: <P><PRE>? <U>print index &quot;seven [four score and seven years ago]</U> 4 ? <U>print index &quot;v &quot;aardvark</U> 5 </PRE> <P><H2>The <CODE>Find</CODE> Pattern</H2> <P> A variation of the <CODE>filter</CODE> pattern is for <EM>selection</EM> operations: ones that pick a single element out of a list. The general idea looks like this: <P><PRE>to <U>procedure</U> :input if emptyp :input [output :input] if <U>predicate</U> first :input [output <U>something</U> first :input] output <U>procedure</U> butfirst :input end </PRE> <P>There will generally be extra inputs to these procedures, to indicate the basis for selection. For example, here is a program that translates English words into French. <P><PRE>to french :word output lookup :word [[book livre] [computer ordinateur] [window fenetre]] end to lookup :word :dictionary if emptyp :dictionary [output &quot;] if equalp :word first first :dictionary [output last first :dictionary] output lookup :word butfirst :dictionary end ? <U>print french &quot;computer</U> ordinateur </PRE> <P>The expression <P><PRE>first first :dictionary </PRE> <P>selects the English word from the first word-pair in the list. Similarly, <P><PRE>last first :dictionary </PRE> <P>selects the French version of the same word. (Of course, in reality, the word list in <CODE>french</CODE> would be much longer than the three word-pairs I've shown.) <P><CODE>Codematch</CODE>, in the example that started this chapter, follows the same pattern of selection. The only difference is that there are two inputs that are <CODE>butfirst</CODE>ed in parallel. <P> Somewhat similar to the selection pattern is one for a recursive <EM>predicate;</EM> the difference is that instead of <P><PRE>output <U>something</U> first :input </PRE> <P>for a successful match, the procedure simply says <P><PRE>output &quot;true </PRE> <P>in that case. This pattern is followed by predicates that ask a question like &quot;Does any member of the input do X?&quot; For example, suppose that instead of counting the vowels in a word, we just want to know whether or not there is a vowel. Then we're asking the question &quot;Is any letter in this word a vowel?&quot; Here's how to find out. <P><PRE>to hasvowelp :word if emptyp :word [output &quot;false] if vowelp first :word [output &quot;true] output hasvowelp butfirst :word end </PRE> <P>A more realistic example is also somewhat more cluttered with extra inputs and sometimes extra end tests. Here's a procedure that takes two words as input. It outputs <CODE>true</CODE> if the first word comes before the second in the dictionary. <P><PRE>to sort.beforep :word1 :word2 if emptyp :word1 [output &quot;true] if emptyp :word2 [output &quot;false] if (ascii first :word1) &lt; (ascii first :word2) [output &quot;true] if (ascii first :word1) &gt; (ascii first :word2) [output &quot;false] output sort.beforep butfirst :word1 butfirst :word2 end </PRE> <P>The procedure will end on one of the <CODE>emptyp</CODE> tests if one of the input words is the beginning of the other, like <CODE>now</CODE> and <CODE>nowhere</CODE>. Otherwise, the procedure ends when two letters are unequal. The recursion step is followed when the beginning letters are equal. (The operation <CODE>ascii</CODE> takes a one-character word as input, and outputs the numeric value for that character in the computer's coding system, which is called the American Standard Code for Information Interchange.) <P>A combination of the translation kind of operation and the selection kind is an operation that selects not one but several members of the input. For example, you sometimes want to examine the words in a sentence in various ways but have trouble because the sentence includes punctuation as part of some words. But the punctuation isn't <EM>really</EM> part of the word. In Chapter 4, for instance, I defined a predicate <CODE>about.computersp</CODE> and gave this example of its use: <P><PRE>? <U>print about.computersp [this book is about programming]</U> true </PRE> <P>But if the example were part of a larger program, carrying on a conversation with a person, the person would probably have ended the sentence with a period. The last word would then have been <CODE>programming.</CODE> (including the period). That word, which is different from <CODE>programming</CODE> without the period, isn't in the procedure's list of relevant words, so it would have output <CODE>false</CODE>. The solution is to write a procedure that strips the punctuation from each word of a sentence. Of course that's a straightforward case of the translation pattern, applying a subprocedure to each word of the sentence: <P><PRE>to strip :sent if emptyp :sent [output []] output sentence (strip.word first :sent) (strip butfirst :sent) end </PRE> <P><CODE>Strip.word</CODE>, though, is more interesting. It must select only the letters from a word. <P><PRE>to strip.word :word if emptyp :word [output &quot;] if letterp first :word ~ [output word (first :word) (strip.word butfirst :word)] output strip.word butfirst :word end to letterp :char output or (inrangep (ascii :char) (ascii &quot;A) (ascii &quot;Z)) ~ (inrangep (ascii :char) (ascii &quot;a) (ascii &quot;z)) end to inrangep :this :low :high output and (:this &gt; (:low-1)) (:this &lt; (:high+1)) end </PRE> <P><CODE>Strip.word</CODE> is like the translation pattern in the use of the combining operation <CODE>word</CODE> in the middle instruction line. But it's also like the selection pattern in that there are two different choices of output, depending on the result of the predicate <CODE>letterp</CODE>. <P>&raquo;You might want to rewrite <CODE>about.computersp</CODE> so that it uses <CODE> strip</CODE>. Consider an initialization procedure. <P> <H2>Numerical Operations: The <CODE>Cascade</CODE> Pattern</H2> <P>Certain mathematical functions are defined in terms of recursive calculations. It used to be that computers were used <EM>only</EM> for numerical computation. They're now much more versatile, as you've already seen, but sometimes the old numerical work is still important. <P>The classic example in this category is the <EM>factorial</EM> function. The factorial of a positive integer is the product of all the integers from 1 up to that number. The factorial of 5 is represented as 5! so <P><CENTER>5! = 1 &times; 2 &times; 3 &times; 4 &times; 5</CENTER> <P>We can use <CODE>cascade</CODE> to carry out this computation: <P><PRE>to fact :n ;; cascade version output cascade :n [? * #] 1 end ? <U>print fact 5</U> 120 </PRE> <P>In this example I'm using a feature of <CODE>cascade</CODE> that we haven't seen before. The template (the second input to <CODE>cascade</CODE>) may include a number sign (<CODE>#</CODE>) character, which represents the number of times the template has been repeated. That is, it represents 1 the first time, 2 the second time, and so on. <P>Here is a recursive version of <CODE>fact</CODE> that takes one input, a positive integer, and outputs the factorial function of that number. The input can also be zero; the rule is that 0!=1. <P><PRE>to fact :n if :n=0 [output 1] output :n * fact :n-1 end </PRE> <P>This procedure works because <P><CENTER>5! = 5 &times; 4!</CENTER> <P>That's another version of reducing a problem to a smaller subproblem. <P>&raquo;Chapter 5 gives the following example: <P><PRE>to power :base :exponent output cascade :exponent [? * :base] 1 end </PRE> <P>Write a version of <CODE>power</CODE> using recursion instead of using <CODE>cascade</CODE>. <P>Another classic example, slightly more complicated, is the Fibonacci sequence. Each number in the sequence is the sum of the two previous numbers; the first two numbers are 1. So the sequence starts <P><CENTER>1, 1, 2, 3, 4, 5, 13, &hellip;</CENTER> <P>A formal definition of the sequence looks like this: <P><CENTER><IMG SRC="fibonacci.gif" ALT="math display"></CENTER> <P>Here's an operation <CODE>fib</CODE> that takes a number <EM>n</EM> as input and outputs <EM>F<SUB>n</SUB></EM>. <P><PRE>to fib :n if :n&lt;2 [output 1] output (fib :n-1)+(fib :n-2) end </PRE> <P>That procedure will work, but it's quite seriously inefficient. The problem is that it ends up computing the same numbers over and over again. To see why, here's a trace of what happens when you ask for <CODE>fib 4</CODE>: <P><PRE>fib 4 fib 3 fib 2 fib 1 fib 0 fib 1 fib 2 fib 1 fib 0 </PRE> <P>Do you see the problem? <CODE>fib 2</CODE> is computed twice, once because <CODE>fib 4</CODE> needs it directly and once because <CODE>fib 4</CODE> needs <CODE>fib 3</CODE> and <CODE>fib 3</CODE> needs <CODE>fib 2</CODE>. Similarly, <CODE> fib 1</CODE> is computed three times. As the input to <CODE>fib</CODE> gets bigger, this problem gets worse and worse. <P>It turns out that a much faster way to handle this problem is to compute a <EM>list</EM> of all the Fibonacci numbers up to the one we want. Then each computation can take advantage of the work already done. Here's what I mean: <P><PRE>to fiblist :n if :n&lt;2 [output [1 1]] output newfib fiblist :n-1 end to newfib :list output fput (sum first :list first butfirst :list) :list end ? <U>print fiblist 5</U> 8 5 3 2 1 1 </PRE> <P>We can then define a faster <CODE>fib</CODE> in terms of <CODE> fiblist</CODE>: <P><PRE>to fib :n output first fiblist :n end </PRE> <P>Convince yourself that the two versions of <CODE>fib</CODE> give the same outputs but that the second version is much faster. I'm purposely not going through a detailed explanation of this example; you should use the analytical techniques you learned in Chapter 8. What problem is <CODE>fiblist</CODE> trying to solve? What is the smaller subproblem? <P>The hallmark of numerical recursion is something like <CODE>:n-1</CODE> in the recursion step. Sometimes this kind of recursion is combined with the <CODE> butfirst</CODE> style we've seen in most of the earlier examples. Logo has a primitive operation called <CODE>item</CODE>, which takes two inputs. The first is a positive integer, and the second is a list. The output from <CODE>item</CODE> is the <EM>n</EM>th member of the list if the first input is <EM>n.</EM> (Earlier I suggested that you write <CODE>index</CODE>, the opposite of <CODE>item</CODE>.) If Logo didn't include <CODE>item</CODE>, here's how you could write it: <P><PRE>to item :n :list if equalp :n 1 [output first :list] output item :n-1 butfirst :list end </PRE> <P><H2>Pig Latin</H2> <P>When I was growing up, every kid learned a not-very-secret &quot;secret&quot; language called Pig Latin. When I became a teacher, I was surprised to find out that kids apparently didn't learn it any more. But more recently it seems to have come back into vogue. Translating a sentence into Pig Latin is an interesting programming problem, so I'm going to teach it to you. <P>Here's how it works. For each word take any consonants that are at the beginning (up to the first vowel) and move them to the end. Then add &quot;ay&quot; at the end. So &quot;hello&quot; becomes &quot;ellohay&quot;; &quot;through&quot; becomes &quot;oughthray&quot;; &quot;aardvark&quot; just becomes &quot;aardvarkay.&quot; (Pig Latin is meant to be spoken, not written. You're supposed to practice so that you can do it and understand it really fast.) <P>By now you can write in your sleep the operation <CODE> piglatin</CODE>, which takes a sentence and outputs its translation into Pig Latin by going through the sentence applying a subprocedure <CODE>plword</CODE> to each word. (It's just like <CODE>code</CODE>, only different.) It's <CODE>plword</CODE> that is the tricky part. The stop rule is pretty straightforward: <P><PRE>if vowelp first :word [output word :word &quot;ay] </PRE> <P>If the first letter <EM>isn't</EM> a vowel, what we want to do is move that letter to the end and try again. Here's the complete procedure. <P><PRE>to plword :word if vowelp first :word [output word :word &quot;ay] output plword word butfirst :word first :word end </PRE> <P>What makes this tricky is that the recursion step doesn't seem to make the problem smaller. The word is still the same length after we move the first letter to the end. This would look more like all the other examples if the recursion step were <P><PRE>output plword butfirst :word </PRE> <P>That would make the procedure easier to understand. Unfortunately it would also give the wrong answer. What you have to see is that there <EM>is</EM> something that is getting smaller about the word, namely the &quot;distance&quot; from the beginning of the word to the first vowel. Trace through a couple of examples to clarify this for yourself. <P>By the way, this will work better if you modify <CODE>vowelp</CODE> (which we defined earlier) so that <CODE>y</CODE> is considered a vowel. You'll then get the wrong answer for a few strange words like <CODE>yarn</CODE>, but on the other hand, if you consider <CODE>y</CODE> a consonant, you'll get no answer at all for words like <CODE>try</CODE> in which <CODE>y</CODE> is the only vowel! (Try it. Do you understand what goes wrong?) <P>&raquo;Some people learned a different dialect of Pig Latin. According to them, if the word starts with a vowel in the first place, you should add &quot;way&quot; at the end instead of just &quot;ay.&quot; Modify <CODE>plword</CODE> so that it speaks that dialect. (I think the idea is that some words simply sound better with that rule.) Hint: You'll want an initialization procedure. <P>&raquo;The top-level procedure <CODE>piglatin</CODE>, which you wrote yourself, is a good candidate for careful thought about punctuation. You don't want to see <P><PRE>? <U>print piglatin [what is he doing?]</U> atwhay isway ehay oing?day </PRE> <P>A good first attempt would be to modify <CODE>piglatin</CODE> to use <CODE>strip</CODE>, to get rid of the punctuation altogether. But even better would be to remove the punctuation from each word, translate it to Pig Latin, then put the punctuation back! Then we could get <P><PRE>atwhay isway ehay oingday? </PRE> <P>That's the right thing to do for punctuation at the end of a word, like a period or a comma. On the other hand, the apostrophe inside a word like <CODE>isn't</CODE> should be treated just like a letter. <P>The project I'm proposing to you is a pretty tricky one. Here's a hint. Write an operation <CODE>endpunct</CODE> that takes a word as input and outputs a <EM>list</EM> of two words, first the &quot;real&quot; word full of letters, then any punctuation that might be at the end. (The second word will be empty if there is no such punctuation.) Then your new <CODE>plword</CODE> can be an initialization procedure that invokes a subprocedure with <CODE>endpunct</CODE>'s output as its input. <P><H2>A Mini-project: Spelling Numbers</H2> <P>Write a procedure <CODE>number.name</CODE> that takes a positive integer input, and outputs a sentence containing that number spelled out in words: <P><PRE>? <U>print number.name 5513345</U> five million five hundred thirteen thousand three hundred forty five ? <U>print number.name (fact 20)</U> two quintillion four hundred thirty two quadrillion nine hundred two trillion eight billion one hundred seventy six million six hundred forty thousand </PRE> <P>There are some special cases you will need to consider: <UL> <LI>Numbers in which some particular digit is zero. <LI>Numbers like 1,000,529 in which an entire group of three digits is zero. <LI>Numbers in the teens. </UL> <P>Here are two hints. First, split the number into groups of three digits, going from right to left. Also, use the sentence <P><PRE>[thousand million billion trillion quadrillion quintillion sextillion septillion octillion nonillion decillion] </PRE> <P>You can write this bottom-up or top-down. To work bottom-up, pick a subtask and get that working before you tackle the overall structure of the problem. For example, write a procedure that returns the word <CODE>fifteen</CODE> given the argument <CODE>15</CODE>. <P>To work top-down, start by writing <CODE>number.name</CODE>, freely assuming the existence of whatever helper procedures you like. You can begin debugging by writing <EM>stub</EM> procedures that fit into the overall program but don't really do their job correctly. For example, as an intermediate stage you might end up with a program that works like this: <P><PRE>? <U>print number.name 1428425</U> ;; intermediate version 1 million 428 thousand 425 </PRE> <P><H2>Advanced Recursion: <CODE>Subsets</CODE></H2> <P>We've seen that recursive operations can do the same jobs as higher order functions, and we've seen that recursive operations can do jobs that are similar to the higher order function patterns but not quite the same. Now we'll see that recursive operations can do jobs that are quite outside the bounds of any of the higher order functions in Chapter 5. <P>I'd like to write an operation <CODE>subsets</CODE> that takes a word as input. Its output will be a sentence containing all the words that can be made using letters from the input word, in the same order, but not necessarily using all of them. For example, the word <CODE>lit</CODE> counts as a subset of the word <CODE>lights</CODE>, but <CODE>hit</CODE> doesn't count because the letters are in the wrong order. (Of course the procedure won't know which words are real English words, so <CODE>iht</CODE>, which has the same letters as <CODE>hit</CODE> in the right order, <EM>does</EM> count.) <P>&raquo;How many subsets does <CODE>lights</CODE> have? Write them all down if you're not sure. (Or perhaps you'd prefer to count the subsets of a shorter word, such as <CODE>word</CODE>, instead.) <P>A problem that follows the <CODE>map</CODE> pattern is one in which the size of the output is the same as the size of the input, because each member of the input gives rise to one member of the output. A problem that follows the <CODE>filter</CODE> pattern is one in which the output is smaller than the input, because only some of the members are selected. And the <CODE>reduce</CODE> pattern collapses all of the members of the input into one single result. The <CODE>subsets</CODE> problem is quite different from any of these; its output will be much <EM>larger</EM> than its input. <P>If we can't rely on known patterns, we'll have to go back to first principles. In Chapter 8 you learned to write recursive procedures by looking for a smaller, similar subproblem within the problem we're trying to solve. What is a smaller subproblem that's similar to finding the subsets of <CODE>lights</CODE>? How about finding the subsets of its butfirst? This idea is the same one that's often worked for us before. So imagine that we've already found all the subsets of <CODE>ights</CODE>. <P>Some of the subsets of <CODE>lights</CODE> <EM>are</EM> subsets of <CODE>ights</CODE>. Which ones aren't? The missing subsets are the ones that start with the letter L. What's more, the other letters in such a subset form a subset of <CODE>ights</CODE>. For example, the word <CODE>lits</CODE> consists of the letter L followed by <CODE>its</CODE>, which is a subset of <CODE>ights</CODE>. <P><PRE>to subsets :word ;; incomplete local &quot;smaller make &quot;smaller subsets butfirst :word output sentence :smaller (map [word (first :word) ?] :smaller) end </PRE> <P>This procedure reflects the idea I've just tried to explain. The subsets of a given word can be divided into two groups: the subsets of its butfirst, and those same subsets with the first letter of the word stuck on the front. <P>The procedure lacks a base case. It's tempting to say that if the input is an empty word, then the output should be an empty sentence. But that isn't quite right, because every word is a subset of itself, so in particular the empty word is a subset (the only subset) of itself. We must output a sentence containing an empty word. That's a little tricky to type, but we can represent a quoted empty word as <CODE>&quot;</CODE> and so a sentence containing an empty word is <CODE>(sentence &quot;)</CODE>. <P><PRE>to subsets :word if emptyp :word [output (sentence &quot;)] local &quot;smaller make &quot;smaller subsets butfirst :word output sentence :smaller (map [word (first :word) ?] :smaller) end </PRE> <P>Why did I use the local variable <CODE>smaller</CODE> and a <CODE>make</CODE> instruction? It wasn't strictly necessary; I could have said <P><PRE>output sentence (subsets butfirst :word) ~ (map [word (first :word) ?] (subsets butfirst :word)) </PRE> <P>The trouble is that this would have told Logo to compute the smaller similar subproblem twice instead of just once. It may seem that that would make the program take twice as long, but in fact the problem is worse than that, because each smaller subproblem has a smaller subproblem of its own, and those would be computed four times--twice for each of the two computations of the first smaller subproblem! As in the case of the Fibonacci sequence we studied earlier, avoiding the duplicated computation makes an enormous difference. <P>Problems like this one, in which the size of the output grows extremely quickly for small changes in the size of the input, tend to be harder to program than most. Here are a couple of examples. Like <CODE>subsets</CODE>, each of these has a fairly short procedure definition that hides a very complex computation. <P>&raquo;On telephone dials, most digits have letters associated with them. In the United States, for example, the digit 5 is associated with the letters J, K, and L. (The correspondence is somewhat different in other countries.) You can use these letters to spell out words to make your phone number easier to remember. For example, many years ago I had the phone number 492-6824, which spells I-WANT-BH. Write a procedure that takes a number as its input, and outputs a sentence of all the words that that number can represent. You may want to test the program using numbers of fewer than seven digits! <P>&raquo;In the game of Boggle<SUP><SMALL>TM</SMALL></SUP>, the object is to find words by connecting neighboring letters in a four by four array of letters. For example, the array <P><PRE>BEZO URND AKAJ WEOE </PRE> <P>contains the words ZEBRA, DONE, and DARK, but not RADAR, because each letter can be used only once. Write a predicate procedure that takes a word and an array of letters (in the form of a sentence with one word for each row) as inputs, and outputs <CODE>true</CODE> if and only if the given word can be found in the given array. <P><PRE>? <U>print findword &quot;zebra [bezo urnd akaj weoe]</U> true ? <U>print findword &quot;radar [bezo urnd akaj weoe]</U> false </PRE> <P> <H2>A Word about Tail Recursion</H2> <P>What I want to talk about in the rest of this chapter isn't really very important, so you can skip it if you want. But <EM>some</EM> people think it's important, so this is for those people. <P>Every procedure invocation takes up a certain amount of computer memory, while the procedure remains active, to hold things like local variables. Since a recursive procedure can invoke itself many times, recursion is a fairly &quot;expensive&quot; technique to allow in a programming language. It turns out that if the only recursion step in a procedure is the very last thing the procedure does, the interpreter can handle that procedure in a special way that uses memory more efficiently. You can then use as many levels of recursive invocation as you want without running out of space. Such a procedure is called <EM>tail recursive.</EM> It doesn't make any difference to you as a programmer; it's just a matter of what's happening inside the Logo interpreter. <P>A tail recursive command is very easy to recognize; the last instruction is an invocation of the same procedure. Tail recursive commands are quite common; here are a couple of examples we've seen before. <P><PRE>to one.per.line :thing if emptyp :thing [stop] print first :thing one.per.line butfirst :thing end to poly :size :angle forward :size right :angle poly :size :angle end </PRE> <P>The thing is, many people are confused about what constitutes a tail recursive operation. It <EM>isn't</EM> one that is invoked recursively on the last instruction line! Instead, the rule is that the recursive invocation must be used <EM>directly</EM> as the input to <CODE>output</CODE>, not as part of a larger computation. For example, this is a tail recursive operation: <P><PRE>to lookup :word :dictionary if emptyp :dictionary [output &quot;] if equalp :word first first :dictionary [output last first :dictionary] output lookup :word butfirst :dictionary end </PRE> <P>But this is <EM>not</EM> tail recursive: <P><PRE>to length :thing if emptyp :thing [output 0] output 1+length butfirst :thing end </PRE> <P>It's that <CODE>1+</CODE> that makes the difference. <P>It's sometimes possible to change a nontail recursive operation into a tail recursive one by tricky programming. For example, look again at <CODE>fact</CODE>: <P><PRE>to fact :n if :n=0 [output 1] output :n * fact :n-1 end </PRE> <P>This is not tail recursive because the input to the final <CODE>output</CODE> comes from the multiplication, not directly from <CODE> fact</CODE>. But here is a tail recursive version: <P><PRE>to fact :n output fact1 :n 1 end to fact1 :n :product if :n=0 [output :product] output fact1 (:n-1) (:n*:product) end </PRE> <P>Indeed, this version can, in principle, compute the factorial of larger numbers than the simpler version without running out of memory. In practice, though, the largest number that most computers can understand is less than the factorial of 70, and any computer will allow 70 levels of recursion without difficulty. In fact, not every Logo interpreter bothers to recognize tail recursive operations. It's a small point; I only mention it because some people <EM>both</EM> make a big fuss about tail recursion <EM>and</EM> misunderstand what it means! <P><A HREF="../v1-toc2.html">(back to Table of Contents)</A> <P><A HREF="../v1ch10/v1ch10.html"><STRONG>BACK</STRONG></A> chapter thread <A HREF="../v1ch12/v1ch12.html"><STRONG>NEXT</STRONG></A> <P> <ADDRESS> <A HREF="../index.html">Brian Harvey</A>, <CODE>bh@cs.berkeley.edu</CODE> </ADDRESS> </BODY> </HTML>