The Anti-Cons February 27th, 2012
Patrick Stein

Motivation

I no longer remember what led me to this page of synthetic division implemented in various languages. The author provides a common lisp implementation for taking a list representing the coefficients of a polynomial in one variable x and a number \alpha and returning the result of dividing the polynomial by (x-\alpha).

The author states: I’m very sure this isn’t considered Lispy and would surely seem like an awkward port from an extremely Algol-like mindset in the eyes of a seasoned Lisper. In the mood for the exercise, I reworked his code snippet into slightly more canonical lisp while leaving the basic structure the same:

(defun synthetic-division (polynomial divisor)                                  
  (let* ((result (first polynomial))                                            
         (quotient (list result)))                                              
    (dolist (coefficient (rest polynomial))                                    
      (setf result (* result divisor))                                          
      (incf result coefficient)                                                
      (push result quotient))                                                  
    (let ((remainder (pop quotient)))                                          
      (list :quotient (nreverse quotient) :remainder remainder))))

From there, I went on to implement it using tail recursion to get rid of the #'setf and #'incf and #'push:

(defun synthetic-division-2 (polynomial divisor)                                
  (labels ((divide (coefficients remainder quotient)                            
             (if coefficients                                                  
                 (divide (rest coefficients)                                    
                         (+ (* divisor remainder) (first coefficients))        
                         (list* remainder quotient))                            
               (list :quotient (reverse quotient) :remainder remainder))))      
    (divide (rest polynomial) (first polynomial) nil)))

What I didn’t like about this was the complexity of calling the tail-recursive portion. If I just called it like I wished to (divide polynomial 0 nil) then I ended up with one extra coefficient in the answer. This wouldn’t do.

The Joke

There’s an old joke about a physicist, a biologist, and a mathematician who were having lunch at an outdoor café. Just as their food was served, they noticed a couple walk into the house across the street. As they were finishing up their meal, they saw three people walk out of that very same house.

The physicist said, We must have miscounted. Three must have entered before.

The biologist said, They must have pro-created.

The mathematician said, If one more person enters that house, it will again be empty.

The Anti-Cons

What I needed was a more-than-empty list. I needed a negative cons-cell. I needed something to put in place of the nil in (divide polynomial 0 nil) that would annihilate the first thing it was cons-ed to.

I haven’t come up with the right notation to make this clear. It is somewhat like a quasigroup except that there is only one inverse element for all other elements. Let’s denote this annihilator: \omega. Let’s denote list concatenation with \oplus.

Only having one inverse-ish element means we have to give up associativity. For lists a and b, evaluating (a \oplus \omega) \oplus b equals b, but a \oplus (\omega \oplus b) equals a.

Associativity is a small price to pay though for a prettier call to my tail-recursive function, right?

The Basic Operations

For basic operations, I’m going to need the anti-cons itself and a lazy list of an arbitrary number of anti-conses.

(defconstant anti-cons 'anti-cons)

(defclass anti-cons-list ()
  ((length :initarg :length :reader anti-cons-list-length))
  (:default-initargs :length 1))

(defmethod print-object ((obj anti-cons-list) stream)
  (print-unreadable-object (obj stream)
    (prin1 (loop :for ii :from 1 :to (anti-cons-list-length obj)
              :collecting 'anti-cons)
           stream)))

Then, I’m going to make some macros to define generic functions named by adding a minus-sign to the end of a Common Lisp function. The default implementation will simply be the common-lisp function.

(defmacro defun- (name (&rest args) &body methods)
  (let ((name- (intern (concatenate 'string (symbol-name name) "-"))))
    `(defgeneric ,name- (,@args)
       (:method (,@args) (,name ,@args))
       ,@methods)))

I’m even going to go one step further for single-argument functions where I want to override the body for my lazy list of anti-conses using a single form for the body:

(defmacro defun1- (name (arg) a-list-form &body body)
  `(defun- ,name (,arg)
     (:method ((,arg anti-cons-list)) ,a-list-form)
     ,@body))

I need #'cons- to set the stage. I need to be able to cons an anti-cons with a normal list. I need to be able to cons an anti-cons with a list of anti-conses. And, I need to be able to cons something other than an anti-cons with a list of anti-conses.

(defun- cons (a b)
  (:method ((a (eql 'anti-cons)) (b list))
    (if (null b)
        (make-instance 'anti-cons-list)
        (cdr b)))
 
  (:method ((a (eql 'anti-cons)) (b anti-cons-list))
    (make-instance 'anti-cons-list :length (1+ (anti-cons-list-length b))))
 
  (:method (a (b anti-cons-list))
    (let ((b-len (anti-cons-list-length b)))
      (when (> b-len 1)
        (make-instance 'anti-cons-list :length (1- b-len))))))

Now, I can go on to define some simple functions that can take either anti-cons lists or regular Common Lisp lists.

(defun1- length (a) (- (anti-cons-list-length a))

(defun1- car (a) :anti-cons)

(defun1- cdr (a)
  (let ((a-len (anti-cons-list-length a)))
    (when (> a-len 1)
      (make-instance 'anti-cons-list :length (1- a-len)))))

(defun1- cadr (a) (when (> (anti-cons-list-length a) 1) :anti-cons))
(defun1- caddr (a) (when (> (anti-cons-list-length a) 2) :anti-cons))

To give a feel for how this all fits together, here’s a little interactive session:

ANTI-CONS> (cons- anti-cons nil)
#<(ANTI-CONS)>

ANTI-CONS> (cons- anti-cons *)
#<(ANTI-CONS ANTI-CONS)>

ANTI-CONS> (cons- :a *)
#<(ANTI-CONS)>

ANTI-CONS> (length- *)
-1

Denouement

Only forty or fifty lines of code to go from:

(divide (rest polynomial) (first polynomial) nil)

To this:

(divide polynomial 0 (cons anti-cons nil))

Definitely worth it.

My Favorite Macro Patterns February 17th, 2012
Patrick Stein

I read Jorge Tavares’s article on Macro Patterns a few days ago.  I was thinking about replying to mention a few of my favorites:

  • The with- pattern which makes sure a special variable is bound for the body and makes sure the tied resources are released at the end of the block.
  • Macros which collect content (usually into a special variable) so they can do something with the content at the end of the close of the macro.

Then, I was working on something tonight when I re-discovered a favorite pattern that I’d forgotten about: Putting multiple wrappers on the same body.

I am working on an HTML+JavaScript+CSS project. In the end, I need static files. But, I thought I would use the opportunity to really experience CL-Who, Parenscript, and CSS-Lite.

I have now made a macro called define-web-file which takes a CL-Who or Parenscript body and wraps it up as both a Hunchentoot handler and a write-to-file wrapper. Now, I can test interactively with Hunchentoot and generate the whole web application when I’m ready.

Dusting off my Growl Library December 21st, 2011
Patrick Stein

I’ve spent the last few hours dusting off my Common Lisp Growl client library. The last time I worked on it was before the Mac Growl Application supported GNTP (Growl Notification Transport Protocol).

Today, working on it, I’m not quite sure what’s up, but I am not succeeding in communicating with the server using encryption. I’ll have to look more closely. Last time that I worked on it, I extended Ironclad, but I never got those changes pushed fully into Ironclad’s main line. But, I think I’m using the same version of Ironclad that I was using when I tested against the Windows Growl Application. *shrug*

I’ve also run into a snag with the Callbacks. Essentially, your Lisp program could get a callback when the user has clicked on your Growl notification. This actually works except for the fact that I am calling READ-SEQUENCE into a buffer that is longer than the message. The server, I believe, is supposed to close the socket after the callback. But, it does not. So, I am stuck waiting for more bytes that will never come.

Now, I either have to do one of the following:

  • refactor it to use READ-LINE instead
  • switch from using USocket to using IOLib (and hope that :dont-wait works as expected)
  • extend USocket to support SOCKET-RECEIVE even on TCP sockets

Anyone have a preference?

If it quacks like a parabola… September 21st, 2011
Patrick Stein

I am working on a game idea that involves (special) relativistic mechanics instead of classical mechanics. Working out the details that I needed was easy enough if I assumed that:

  • ships had a maximum speed at which they could move relative to my base frame
  • ships could instantly go from stopped to maximum speed or vice-versa

I didn’t like those assumptions at all. So, I started playing with the equations for relativity. In classical mechanics, the rate-of-change of velocity equals the force you’re applying divided by your mass: \frac{dv}{dt} = \frac{F}{m}.

In special relativity, your mass increases with velocity. So, that equation becomes: \frac{d\left(\frac{v}{\sqrt{1-v^2}}\right)}{dt} = \frac{F}{m_0} (assuming units where the speed of light is 1 unit of distance per 1 unit of time and m_0 is your rest-mass).

For the purposes of this post, I’m going to assume the simplest initial conditions: you start motionless and at the origin. For ease of notation, let a = \frac{F}{m_0}. Solving the above differential equation to get a formula for velocity and solving the resulting differential equation to get the distance x you’ve travelled in my base frame by time t, the answer comes out to: x(t) = \frac{1}{a}\left(-1 + \sqrt{1+a^2t^2}\right).

I have solved this problem at least thirty times in the past two months. Sometimes I used the simple initial conditions as above. Sometimes I did it in all of its gory details (including the messy case where the applied force is not aligned with the current velocity).

I got the same answer (well, plus the extra mess when I did the full-on problem) every way that I tried it.

So, why did I do it over and over again?

If this were classical mechanics, the end equation would have been x(t) = \frac{1}{2}at^2. And, I know that for low velocities, the classical mechanics answer should be almost identical to the special relativity answer. And, there was no way that I thought \frac{1}{a}\left(-1 + \sqrt{1+a^2t^2}\right) \approx \frac{1}{2}at^2.

I knew what the graph x = \sqrt{a^2t^2} looked like when t \ge 0. It is a straight line. It doesn’t look much like the parabola x = \frac{1}{2}a t^2 at all.

My assumption was that since x = \sqrt{a^2t^2} was a straight line for t \ge 0, then x = \sqrt{1 + a^2t^2} would be a straight line shifted up one unit and bent (concave-down) a little bit like the graph of x = \sqrt{at} is bent.

Boy was I wrong. Here is a plot of the two together (created with fooplot). The red line is the classical mechanics version. The black line is the relativistic version. Here, the force is such that the body is accelerating at a rate of the speed of light per second so they’ve already gotten up to around 28,000 miles per second before you can see any separation in the graphs here.

distance (in light-seconds) vs. time (in seconds)

Definitely, I can see the resemblance. Now, to fix my intuition about square-roots.

Keeping Server and Client Separate September 8th, 2011
Patrick Stein

The problem

Whenever I write client-server applications, I run into the same problem trying to separate the code. To send a message from the server to the client, the server has to serialize that message and the client has to unserialize that message. The server doesn’t need to unserialize that message. The client doesn’t need to serialize that message.

It seems wrong to include both the serialization code and the unserialization code in both client and server when each side will only be using 1/2 of that code. On the other hand, it seems bad to keep the serialization and unserialization code in separate places. You don’t want one side serializing A+B+C and the other side trying to unserialize A+C+D+B.

One approach: data classes

Some projects deal with this situation by making every message have its own data class. You take all of the information that you want to be in the message and plop it into a data class. You then serialize the data class and send the resulting bytes. The other side unserializes the bytes into a data class and plucks the data out of the data class.

The advantage here is that you can have some metaprogram read the data class definition and generate a serializer or unserializer as needed. You’re only out-of-sync if one side hasn’t regenerated since the data class definition changed.

The disadvantage here is that I loathe data classes. If my top-level interface is going to be (send-login username password), then why can’t I just serialize straight from there without having to create a dippy data structure to hold my opcode and two strings?

Another approach: suck it up

Who cares if the client contains both the serialization and unserialization code? Heck, if you’re really all that concerned, then fmakunbound half the universe before you save-lisp-and-die.

Of course, unless you’re using data classes, you’re either going to have code in your client that references a bunch of functions and variables that only exist in your server or your client and server will be identical except for:

(defun main ()
  #+server (server-main)
  #-server (client-main))

Now, of course, your server is going to accidentally depend on OpenGL and OpenAL and SDL and a whole bunch of other -L’s it never actually calls. Meanwhile, your client is going to accidentally depend on Postmodern and Portable-Threads and a whole bunch of other Po-‘s it never actually calls.

Another approach: tangle and weave, baby

Another way that I’ve got around this is to use literate programming tools to let me write the serialiization and unserialization right next to each other in my document. Then, anyone going to change the serialize code would be immediately confronted with the unserialize code that goes with it.

The advantage here is that you can tangle the client code through an entirely separate path than the server code keeping only what you need in each.

The disadvantage here is that now both your client code and your server code have to be in the same document or both include the same sizable chunk of document. And, while there aren’t precisely name-capturing problems, trying to include the “serialize-and-send” chunk in your function in the client code still requires that you use the same variable names that were in that chunk.

How can Lisp make this better?

In Lisp, we can get the benefits of a data-definition language and data classes without needing the data classes. Here’s a snippet of the data definition for a simple client-server protocol.

;;;; protocol.lisp
(userial:make-enum-serializer :opcode (:ping :ping-ack))
(defmessage :ping     :uint32 ping-payload)
(defmessage :ping-ack :uint32 ping-payload)

I’ve declared there are two different types of messages, each with their own opcode. Now, I have macros for define-sender and define-handler that allow me to create functions which have no control over the actual serialization and unserialization. My functions can only manipulate the named message parameters (the value of ping-payload in this case) before serialization or after unserialization but cannot change the serialization or unserialization itself.

With this protocol, the client side has to handle ping messages by sending ping-ack messages. The define-sender macro takes the opcode of the message (used to identify the message fields), the name of the function to create, the argument list for the function (which may include declarations for some or all of the fields in the message), the form to use for the address to send the resulting message to, and any body needed to set fields in the packet based on the function arguments before the serialization. The define-handler macro takes the opcode of the message (again, used to identify the message fields), the name of the function to create, the argument list for the function, the form to use for the buffer to unserialize, and any body needed to act on the unserialized message fields.

;;;; client.lisp
(define-sender  :ping-ack send-ping-ack (ping-payload) *server-address*)
(define-handler :ping     handle-ping   (buffer) buffer
  (send-ping-ack ping-payload))

The server side has a bit more work to do because it’s going to generate the sequence numbers and track the round-trip ping times.

;;;; server.lisp

(defvar *last-ping-payload* 0)
(defvar *last-ping-time*    0)

(define-sender :ping send-ping (who) (get-address-of who)
  (incf *last-ping-payload*)
  (setf *last-ping-time*    (get-internal-real-time)
        ping-payload        *last-ping-payload*))

(define-handler :ping-ack handle-ping-ack (who buffer) buffer
  (when (= ping-payload *last-ping-payload*)
    (update-ping-time who (- (get-internal-real-time) *last-ping-time*))))

Problems with the above

It feels strange to leave compile-time artifacts like the names and types of the message fields in the code after I’ve generated the functions that I’m actually going to use. But, I guess that’s just part of Lisp development. You can’t (easily) unload a package. I can makunbound a bunch of stuff after I’m loaded if I don’t want it to be convenient to modify senders or handlers at run-time.

There is intentional name-capture going on. The names of the message fields become names in the handlers. The biggest problem with this is that the defmessage calls really have to be in the same namespace as the define-sender and define-handler calls.

I still have some work to do on my macros to support &key and &optional and &aux and &rest arguments properly. I will post those macros once I’ve worked out those kinks.

Anyone care to share how they’ve tackled client-server separation before?

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