Recursion with no base case?

Here's my solution:

(define (all-descendants node-children root)
  (cons root
        (append-map (lambda (child)
                      (all-descendants node-children child))
                    (node-children root))))

I wasn't measuring, but it took me more than a minute. I spent most of that time thinking about transitive closures, queues, iterating over elements, etc., before I realized that since I'm working with a tree rather than a directed graph, I need not worry about shared structure or duplicate elements.

This is an example of a well-founded recursive program that does not have an explicit base case. append-map (mapappend in Common Lisp) implicitly does not call the mapped function if the list argument is empty, so the recursion ends at the nodes of the tree that have no children.

1 minute puzzle

Given a procedure that returns a list of the children of a given node of a tree, write a procedure that returns a list of all the descendants of that node.

jrm bloviates

I recently received an email asking me, “ Do you know of any good books about developing software or thinking in abstraction? I have books, but how did you train your mind?”

I'm not sure I even have a clue. I think that some of the things I've read and worked on were influential, but it isn't clear to me whether anyone else would agree. I've met people that have read the same books and their thought processes are completely alien to me. I know people that are very good at what they do, and reach solid, well-engineered solutions, but I cannot talk with them because they seem to have an insane basic approach to understanding. I've also been on the other end of this where I cannot seem to get the other person to understand a very simple idea on which everything hinges. But since this is a blog, I hardly need to be informed to express an opinion.

College was a huge influence on my thinking. I was fortunate enough to learn from some of the luminaries not only in Computer Science, but also in Math (Gian-Carlo Rota, Alar Toomre), Material Science (August Witt), Physics (I forget), Film (Richard Leacock), etc. What I noticed was that these really smart people spent an awful lot of their time and energy thinking about and talking about the basics and fundamentals. Over and over again they would stress going back to first principles. They invariably would lecture about “creating a model” to help in understanding. In the introductory courses, we'd focus on the commonly used models, but the professors would usually show us alternative models that had interesting properties. In the advanced courses, we would often use several different models. Sometimes the models were apparently contradictory, but were nonetheless useful.

6.001, Structure and Interpretation of Computer Programs, was heavily oriented towards controlling complexity with abstraction. To a large extent, it wasn't a programming course, but a survey course that explored a huge number of abstractions that have been found useful. The fact that it used Scheme as an expository language is irrelevant. The real point of the course was to look at the many different techniques of abstraction and to illustrate the commonalities and differences between them. However, the difficulty of the course tended to obscure this for many students. When you're trying to learn the syntax and semantics of a language, you can easily get bogged down in the minutiae and miss the big picture. For example, the “Triangle Language” problem set made heavy use of higher-order procedures.

The pedagogic goal was twofold: to become comfortable manipulating procedures as first-class objects, but also to be able to create an abstract model of picture composition through triangle-oriented primitives. At the abstract level, we had triangular `pictures' that could be decomposed by cutting larger triangles into sections, or composed by pasting smaller triangles together with scaling and rotation.

The Triangle Language is not a very complex language. It is not very general or useful. It is simply an example of a trivial domain-specific language. It is possible for a student to do the entire problem set, learn a huge amount about higher-order procedures, ace the quiz, and completely miss the point about solving a problem through creating a domain-specific toolkit. Many did. It was ok because every problem set contained some more examples of this. Eventually it will sink in.

6.004 also spent a lot of time talking about different abstractions, but it proceeded differently. We started with the linear model of a transistor, then developed the digital abstraction, from there we had switches. With a couple of more electronic elements, we could implement combinational logic. The combinational logic could be used to implement flip-flops and finite state machines. The state machines, combined with some memory, could implement simple push-down automata and stack machines. A stack machine is a good target for a higher level language, so we could write a compiler. The really interesting part of this course was the lab work where we'd wire these things up. Each week we'd augment the circuits built the previous week and we had built a small computer by the time we got to the end of the course. What was good about this was that you could see all the levels of abstraction at once. The little black transistors over on that side of the board were implementing a gate in a state machine that drove the clock for the microcode unit. You could look at the transistor as switching bits, computing part of a trivial function, or throttling electrons, whichever abstraction was the most relevant at the time.

I guess that the old maxim “practice makes perfect” is true. I trained my mind — They trained my mind by constantly teaching about various abstractions.

As for books, I obviously recommend Structure and Interpretation of Computer Programs, but there aren't many books that just talk about abstractions. It is well worth learning alternative abstractions to things you already know. Hamiltonian and Lagrangian physics has a different view than Newtonian physics, for example. (Unfortunately, it is hard to just get a simple introduction to Lagrangian and Hamiltonian physics.) It unfortunate that many useful alternative abstractions are very poorly taught. Try to find alternative abstractions that make sense to you. Try recasting familiar ideas in different abstractions (this can be fun and bizarre if the abstraction is from an unrelated subject). Rewrite a program based on a completely different abstract model. Try to simplify a complex program to the minimal abstraction possible. Just keep on practicing!

ILC 2012 Call for Papers

+----------------------------------------------------------------------+
|                                                                      |
|                  INTERNATIONAL LISP CONFERENCE 2012                  |
|                                                                      |
|             http://www.international-lisp-conference.org             |
|                                                                      |
|       Campus Plaza Kyoto, Kyoto, Japan -  October 21-24, 2012        |
|                                                                      |
|            Sponsored by:  The Association of Lisp Users              |
|                                                                      |
+----------------------------------------------------------------------+

   General Information:

     The Association of Lisp Users is pleased to announce the 2012
     International Lisp Conference will be held in Kyoto, Japan at
     Campus Plaza Kyoto from October 21st to 24th, 2012.

     This year's program consists of tutorials at beginners' and
     advanced levels, prominent invited speakers from the Lisp
     communities, an excellent technical session, tours of
     Jidai-Matsuri: festival enjoyed by people of all ages,
     participating in its historical reenactment parade dressed in
     authentic costumes representing various periods, and characters
     in Japanese feudal history.

     General conference announcements are made on a very occasional
     basis to the low-volume mailing list
     ilc12-announce. http://www.alu.org/mailman/listinfo/ilc12-announce

   Technical Program:

     Original submissions in all areas related to the conference themes
     are invited for the following categories:

     Papers: Technical papers of up to 15 pages that describe original
     results.

     Demonstrations: Abstracts of up to 2 pages for demonstrations of
     tools, libraries and applications.

     Workshops: Abstracts of up to 2 pages for groups of people who
     intend to work on a focussed topic for half a day.

     Tutorials: Abstracts of up to 2 pages for indepth presentations
     about topics of special interest for 90 - 180 minutes.

     Panel discussions: Abstracts of up to 2 pages for discussions about
     current themes. Panel discussion proposals must mention panel
     member who are willing to partake in a discussion.

     Lightning talks: Abstracts of up to one page for talks to last
     for no more than 5 minutes.


   Important Dates:

     Please send contributions before the submission deadline, including
     abstracts of 4 pages for technical papers and abstracts of 2 pages
     for all other categories.

     Deadline for abstract submissions: July 15, 2012
     Notification of acceptance or rejection: July 31, 2012
     Deadline for final paper submissions: August 31, 2012

     Papers to be presented should be submitted electronically at
     easychair
     (https://www.easychair.org/account/signin.cgi?conf=ilc2012)
     and need to use the ACM format
     (http://www.acm.org/sigs/publications/proceedings-templates)

   Scope:

    Lisp is one of the greatest ideas from computer science and a
    major influence for almost all programming languages and for all
    sufficiently complex software applications.

    The International Lisp Conference is a forum for the discussion of
    Lisp and, in particular, the design, implementation and
    application of any of the Lisp dialects.  We encourage everyone
    interested in Lisp to participate.

    We invite high quality submissions in all areas involving Lisp
    dialects and any other languages in the Lisp family, including,
    but not limited to, ACL2, AutoLisp, Clojure, Common Lisp,
    ECMAScript, Dylan, Emacs Lisp, ISLISP, Racket, Scheme, SKILL, etc.

    Topics may include any and all combinations of Lisp and:

      * Language design and implementation
      * Language integration, inter-operation and deployment
      * Applications (especially commercial)
      * Reflection, meta-object protocols, meta-programming
      * Domain-specific languages
      * Programming paradigms and environments
      * Parallel and distributed computing
      * Theorem proving
      * Scientific computing
      * Data mining
      * Semantic web


   Organizing Committee:

     General Chair: KURODA Hisao (Mathematical Systems Inc. / ALU)
     Members: Daniel Herring (ALU)
              Jon L White (ALU)
              Rusty Johnson (ALU)

     Program Chair: Hiroshi Okuno (Kyoto Univ.)
     Members: Keith Corbett (Clozure Associates)
              Alex Fukunaga (University of Tokyo)
              Antonio Leitao (INESC-ID)
              Joe Marshall (MIT)
              Scott Mckay (ITA software)
              Nancy Reed (University of Hawaii)
              Kent Pitman (nhplace.com)
              Duane Rettig (Franz Inc.)
              Didier Verna (EPITA)
              Takuo Watanabe (Tokyo Institute of Technology)
              Edi Weitz (weitz.de)
              Taiichi Yuasa (Kyoto University)

     Local chair: Tetsuya Ogata (Kyoto Univ.)
     Members: CHIBA Masaomi
              SANO Masatoshi

  Contacts:

    * General Questions: ilc12-organizing-committee at alu.org
    * Program Committee: ilc2012 at easychair.org

For more information, see http://www.international-lisp-conference.org

A bit more challenging

In my previous post, I gave the puzzle of taking a pattern and generating code that matches it. The tricky part was making sure that the pattern is completely traversed at compile time. If the object to be tested matches the pattern, the pattern variables and their values were to be returned in an alist.

It would be better, however, to generate code where the pattern variables become bindings of Scheme variables. Instead of generating code that stuffs the values into an alist, like this code does at the highlighted points:

(make-matcher '((? one) and (? two))) =>

(lambda (object)
  (and (pair? object)
       (let ((left-submatch
              ((lambda (object) (list (cons 'one object))) (car object)))
             (right-submatch
              ((lambda (object)
                 (and (pair? object)
                      (let ((left-submatch
                             ((lambda (object)
                                (and (eqv? object 'and)
                                     '()))
                              (car object)))
                            (right-submatch
                             ((lambda (object)
                                (and (pair? object)
                                     (let ((left-submatch
                                            ((lambda (object)
                                               (list (cons 'two object)))
                                             (car object)))
                                           (right-submatch
                                            ((lambda (object)
                                               (and (eqv? object '())
                                                    '()))
                                             (cdr object))))
                                       (and left-submatch
                                            right-submatch
                                            (append left-submatch
                                                    right-submatch)))))
                              (cdr object))))
                        (and left-submatch
                             right-submatch
                             (append left-submatch right-submatch)))))
               (cdr object))))
         (and left-submatch
              right-submatch
              (append left-submatch right-submatch)))))

We'd generate something more like this:

(make-matcher '((? one) and (? two)) <user code goes here>) =>

(lambda (object)
  (and (pair? object)
       (let ((one (car object))
             (tail1 (cdr object)))
         (and (pair? tail1)
              (eq? (car tail1) 'and)
              (let ((tail2 (cdr tail1)))
                (and (pair? tail2)
                     (let ((two (car tail2)))
                       (and (null? (cdr tail2))
                            <user code goes here>))))))))

This is more challenging for two reasons. First, we need to ensure that the pattern variable names become bound in a scope that encloses the user's code so that free references to pattern variables are correctly captured. In addition, we need to ensure that other “helper” bindings, like tail1 and tail2 do not capture free references by accident. (That is to say, watch your macro hygiene.) Second, you have to be sure that subpattern bindings are visible to the entire body of the user code. This will throw a monkey wrench into the simple recursive solution.

Slightly trickier

In my previous post, I gave a small pattern matching problem. It is easily solved by recursively descending the pattern and the object to match. This is analogous to interpreting the pattern because you walk the pattern each time you want to try to match an object.

If the pattern is constant, though, you can walk the pattern once and generate code that can match against an object much more quickly:

(define my-matcher
  (eval (make-matcher '(a (? var1) (nested (c (? var2)))))
        user-initial-environment))

(my-matcher '(a b (nested (c d)))) => ((var1 . b) (var2 . d))

I'd rate this as an intermediate puzzle. It isn't very different from the previous one, but you have to pay more attention to the phase of evaluation. As a hint, here are a pair of possible matchers:

(make-matcher '((? car) . (? cdr))) =>

(lambda (object)
  (and (pair? object)
       (let ((p1 ((lambda (object) (list (cons 'car object))) (car object)))
             (p2 ((lambda (object) (list (cons 'cdr object))) (cdr object))))
         (and p1
              p2
              (append p1 p2)))))

(make-matcher '((? one) and (? two))) =>

(lambda (object)
  (and (pair? object)
       (let ((left-submatch
              ((lambda (object) (list (cons 'one object))) (car object)))
             (right-submatch
              ((lambda (object)
                 (and (pair? object)
                      (let ((left-submatch
                             ((lambda (object)
                                (and (eqv? object 'and)
                                     '()))
                              (car object)))
                            (right-submatch
                             ((lambda (object)
                                (and (pair? object)
                                     (let ((left-submatch
                                            ((lambda (object)
                                               (list (cons 'two object)))
                                             (car object)))
                                           (right-submatch
                                            ((lambda (object)
                                               (and (eqv? object '())
                                                    '()))
                                             (cdr object))))
                                       (and left-submatch
                                            right-submatch
                                            (append left-submatch
                                                    right-submatch)))))
                              (cdr object))))
                        (and left-submatch
                             right-submatch
                             (append left-submatch right-submatch)))))
               (cdr object))))
         (and left-submatch
              right-submatch
              (append left-submatch right-submatch)))))

Astute readers will notice that this latter matcher is doing more work than necessary. If the match against part of the pattern fails, it still attempts to match the rest of the pattern. It only notices just before assembling the final result. Also, using append to assemble the sub-matches is a terrible waste.

A small puzzle

Here's a quick little puzzle that isn't too hard.

A pattern is:

1. A symbol, number, boolean, or null (an atom).
2. A pattern variable, which is a two element list where the first element is the symbol ? and the second is a symbolic (a symbol) name.

   ;;   Examples:
   ;;     (pattern-variable? '(? foo))                      => #t
   ;;     (pattern-variable? '(? another-pattern-variable)) => #t
   ;;
   ;;     (pattern-variable? '(not a (pattern variable)))   => #f
   ;;     (pattern-variable? '(?))                          => #f
   ;;     (pattern-variable? '(foo ?))                      => #f
   ;;     (pattern-variable? '(? foo . bar))                => #f
   ;;     (pattern-variable? '(? foo quux))                 => #f
   
   (define (pattern-variable? thing)
     (and (pair? thing)
          (eq? (car thing) '?)
          (pair? (cdr thing))
          (symbol? (cadr thing))
          (null? (cddr thing))))

3. A pair (a cons) of two patterns.

Write a program that given a pattern and some list structure (an object composed of pairs, numbers, symbols, nulls, etc.) returns an association list (an alist) of the pattern variable names and the associated matching elements, or #F if the pattern does not match.

;;  Examples:
;;   (pmatch '(foo (? pvar1) (? pvar2) bar) '(foo 33 #f bar))
;;      => ((pvar2 . #f) (pvar1 . 33))
;;
;;   (pmatch '(foo (? pvar) bar) '(quux 33 bar))
;;      => #f
;;
;;   (pmatch '(a (? var1) (nested (c (? var2)))) '(a b (nested (c d))))
;;      => ((var2 . d) (var1 . b))
;;
;;  Edge cases:
;;
;;   (pmatch '(a b c) '(a b c))
;;      => '()
;;
;;   (pmatch '(foo (? pvar1) (? pvar2) bar) '(foo 33 (xyzzy #f) bar))
;;      => ((pvar2 xyzzy #f) (pvar1 . 33))
;;
;;   (pmatch '(foo . (? pvar)) '(foo bar baz))
;;      => ((pvar bar baz))
;;
;;   (pmatch '((? ?) quux) '(foo quux))
;;      => ((? . foo))
;;
;;   (pmatch '(? ?) '(foo quux))
;;      => ((? foo quux))
;;
;;   (pmatch '(? ? ?) '(foo quux))
;;      => #f

Please be careful and obfuscate your solution if you want to post it.