A simple persistent store

ChangeSafe contains a simple persistent store.  The code is here.

To get an idea of how this works, start with tests.lsp

First, when  open-persistent-store is called, the persistent store is (obviously) opened.  If the store doesn't exist, it is created.  It is easier to write code with the assumption that there is always a persistent state that can be updated rather than having special code deal with a once only initialization.

All interaction with the contents of the store takes place within an atomic transaction.  When the transaction ends, either all the updates to the store take place, or none do.  This allows the programmer to briefly ignore consistency requirements during computation so long as the end result is consistent. The interface is fairly simple:

call-with-transaction pstore transaction-type reason receiver

pstore - a persistent store
transaction-type - one of :read-only, :read-write, or :read-cons
reason - a human readable string describing the purpose of the transaction
receiver - a procedure of one argument (the transaction object) which is invoked during the transaction

In the usual, expected case, the receiver runs, possibly makes updates to the store, and returns a value. When the receiver returns, the transaction commits (finishes) and the changes to the store are applied atomically.

In the less usual case, the transaction may be aborted.  In this case, no (detectable) changes are made to the store.

The decision to commit or abort is usually made implicitly:  if the receiver procedure returns normally, the transaction commits, if it performs a non-local exit, via a throw or through error recovery, the transaction aborts.  However, the programmer can explicitly cause the transaction to commit or abort through a call to transaction/commit or transaction/abort.

The transaction-type indicates what kind of transaction is to be performed.  Naturally :read-write is used to update the store and :read-only is used if no update is necessary.  :read-cons is for the case of a transaction that only creates new persistent objects and does not change existing objects.  The implementation ensures that transactions are isolated (transaction in progress are not visible to each other) and serializable (committed transactions can be placed in a consistent order even if executed in parallel).  :read-only transactions only have to be consistent throughout execution, so many can (in theory) be run in parallel.  :read-write transactions can be run in parallel only if they do not interfere with each other.  :read-cons transactions are a special case of :read-write.  Since new objects are not visible outside the transaction until the transaction commits, there can be no interference between multiple :read-cons transactions. (Support for :read-cons is sketchy, though.  Although there should be no interference, each transaction must issue unique object ids and thus they do interfere implicitly.  There are ways to deal with this, but I didn't implement them.)

The reason is simply a string that is recorded along with the transaction.  This is to help the user understand the sequence of transactions when examining the history of the store.

The receiver is invoked on an object representing the transaction.  It is often ignored because the implicit commit or abort behavior is usually used, but the argument is there if desired.

During the transaction, the special variable *current-transaction* is bound to the transaction object.

So let's look at some code:

(let ((store (persistent-store/open pathname))
       id
       vid
       (object1 '(this is a list))
       (object2 #(this is a vector))
       (object3 (list 1.0d0 -1.0d0 0.125d0 -0.125d0 1024.0d0 -1024.0d0 .3d0 -.3d0))
       (object4 (list #x87654321 #x12345678 1 -1 2147483647 -2147483648
                      #*101010101010101010101010101
                      #*100000000000000000000000000
                      #*000000000000000000000000001))
       o1id
       o2id
       o3id
       o4id)
   (call-with-transaction
    store :read-write "Save some objects."
    (lambda (transaction)
      (declare (ignore transaction))
      (setf o1id (persistent-object/save object1 store)
            o2id (persistent-object/save object2 store)
            o3id (persistent-object/save object3 store)
            o4id (persistent-object/save object4 store)))))

This code saves some simple lisp objects to the store.  persistent-object/save writes the object to the store and returns a unique integer (the object-id). Obviously we'll want a :read-write (or :read-cons) transaction.

 Later on, we call

       (call-with-transaction
        store :read-only "Verify objects."
        (lambda (transaction)
          (declare (ignore transaction))
          (verify
           (verify-one-return-value (verify-value-equal object1))
           (persistent-object/find store o1id))
          (verify
           (verify-one-return-value (verify-value-equalp object2))
           (persistent-object/find store o2id))
          (verify
           (verify-one-return-value (verify-value-equal object3))
           (persistent-object/find store o3id))
          (verify
           (verify-one-return-value (verify-value-equalp object4))
           (persistent-object/find store o4id))))

persistent-object/find takes the integer returned by persistent-object/save and returns the object.  In this case we use a :read-only transaction, but we would use :read-write if we were going to modify any objects.

One has to be careful about equality.  The object returned by persistent-object/find may not be EQ to the object saved. It would be very difficult to implement this in general, and it isn't needed in most cases. Symbols are treated specially to preserve EQ-ness.

A little lisp

Several years ago I worked on ChangeSafe, a change management and version control system written in Common Lisp.  The CTO of ChangeSafe, LLC (me) has decided to open the source code of ChangeSafe.

Licensing

The code has been placed in my source code repository, which is published under the "new BSD" license.  You may take any of the code and use it under that license.  Do not be intimidated by the very scary words you might find.  Please feel free to contact me about different licensing if that would be useful.

Confusion about functional programming

Lawrence Botroff posted a set of questions and I thought I'd reply in my blog.

I've read some of the discussions here, as well as followed links to other explanations, but I'm still not able to understand the mathematical connection between "changing state" and "not changing state" as it pertains to our functional programming versus non-FP debate.

The concepts of "change" and "state" implicitly involve some notion of time. A mathematical function does not.

Mathematical functions are, however, an incredibly useful concept. So, too, are the concepts of change and state.

There are two trivial solutions: ignore the problem and avoid the problem. We can simply ignore the problem of change and state and not attempt to use mathematical functions to model our program. Or we can avoid the problem of change and state and avoid programming in a way that makes implicit use of time. There are more complex solutions as well.

The "debate", such as it is, is whether the benefits of functional programming are worth the effort of either avoiding change or explicitly modeling it.

Then it get foggy in my mind.

All we need now is to sow discord.

Here's an example. Let's say I want to display closed, block-like polygons on an x-y field. In typical GIS software I understand everything is stored as directed, closed graphs, i.e. a square is four "vectors," their heads and ends connected. The raw data representation is just the individual Cartesian start and end points of each vector. And, of course, there is a "function" in the GIS software that "processes" all these coordinate sets.

Can we avoid overloading the word "function"? It will be confusing. Let's consider function to be a mathematical concept. I'll call the programming structure a subroutine, just to be old fashioned. A program is a collection of subroutines. If we so choose, we can avoid the implicit use of time in our programs and subroutines. If we do that, then it is easy to model our subroutines as mathematical functions, and model our program through functional composition.

So let me rephrase your sentence: And, of course, there is "program" in the GIS software that "processes" all these coordinate sets.

Good. But what if we represent each polygon in a mathematical way, e.g., a rectangle in the positive x, negative y quadrant might be:

Z = {(x,y) | 3 <= x <= 5, -2 <= y <= -1}

So we might have many Z-functions, each one expressing an individual polygon -- and not being a whiz with my matrix math, maybe these "functions" could then be represented as matrices . . . but I digress.

You're making a subtle shift in your point of view, and that may be part of what is confusing you. The earlier paragraph addressed the idea of using functions to model computer subroutines. This paragraph addresses the idea of using computer subroutines to model functions.

These are not equivalent concepts. Imperative programs, for example, can also use subroutines to model functions. But naturally, imperative programmers don't generally want to do this, and people that prefer functional programming tend to do this all the time.

Now if we can use a mathematical function to model a subroutine, then the code itself may be able to use that subroutine to model the mathematical function (these would be first-class procedures.  It would be odd, but conceivable to have a functional language without first class procedures.). This is conditional on how we write our code. If we write in imperative style, then we may not be able to model subroutines as a mathematical functions. But if we write in functional style, we get both the ability to model subroutines as functions and to use those same routines to model the functions. You can ignore the distinction between code and the mathematical function it is equivalent to.  Functional programmers think this is really cool. Imperative programmers are not so impressed.

So with the usual raw vector-data method, I might have one function in my code that "changes state" as it processes each set of coordinates and then draws each polygon (and then deals with polygons changing), while the one-and-only-one-Z-function-per-polygon method would seem to hold to the "don't change state" rule exactly -- or at least not change memory state all that much. Right? Or am I way off here? It seems like the old-fashioned, one-function-processing-raw-coordinate-data is not mutating the domain-range purity law really. I'm confused....

Slow down a bit. We're talking about a program and we want to analyze it. (By "analyze", I don't mean "gain a deep understanding", I mean "any effort to try to understand anything at all".) So in your hypothetical code you have polygons that are modeled by functional subroutines. That's great. You can model these parts of the code as mathematical functions.

But apparently, there is a subroutine in your program that "changes state". This means that there is an implicit time component, and therefore you cannot trivially model this part of the program with a mathematical function. That's all there is to it.

But the many individual polygon Z-functions method would seem to hold a perfect picture of state in memory -- no?

No. You have shifted to a different abstraction. What you mean by "state" in this case is an abstract set of polygons. Since the polygons are immutable, you could write a program which deals with this set of polygons in a completely functional manner. But there is no requirement that you do so. You could write an imperative program that mutates the data structures that hold the polygons.

Then I was reminded of my days (1980s) in Cartography when an object-oriented GIS package (written in Smalltalk) was being discussed. Everyone dismissed it as ludicrous because it tried to have exactly that: all the cartographic objects (roads, areal, symbols, polygons, etc.) in live memory at once. But isn't this ultimately what FP wants?

This is yet another abstraction. One could implement this imperatively or functionally. It's also a much older idea than the 80s.

You may be confusing yourself by attempting to shoehorn too many ideas into the word "functional".

Another avenue of my inspiration came from reading about a new idea of image processing where instead of slamming racks of pixels, each "frame" would be represented by one big function capable of quasi "gnu-plotting" the whole image: edges, colors, gradients, etc. Now I ask, Is this germane? I guess I'm trying to fathom why I would want to represent, say, a street map of polygons (e.g. city blocks) one way or the other.

You might want to render in different sizes. It is likely that a high resolution bitmap for a large window would take far more memory than the specification of how to generate the bitmap. Polygons carry more information than bitmaps. You might want to do something other than rendering.

I keep hearing functional language advocates dance around the idea that a mathematical function is pure and safe and good and ultimately Utopian, while the non-FP software function is some sort of sloppy kludge holding us back from Borg-like bliss.

I don't hang around the places you do.

But again, more confusing is memory management vis-a-vis FP versus non-FP. What I keep hearing (e.g. parallel programming) is that FP isn't changing a "memory state" as much as, say, a C/C++ program does.

Again, if you program functionally, you avoid introducting implicit time dependencies. This makes it much, much easier to determine where you can safely program in parallel (basically anywhere). You can do it the hard way, too.

Is this like the Google File System (or my Smalltalk GIS example) where literally everything is just sitting out there in a virtual memory pool, rather than being data moved in and out of databases, bzw. memory locations? Somehow I sense all these things are related.

Well, there is a connection. Notice you said "data moved in and out of databases". This has an implicit time component: sometimes the data is in, other times it is out. If we want to model our database with mathematical functions, we'll have a problem. One solution is to simplify the model by abstracting away the actual location of the data, that is, make it appear as if "literally everything is just sitting out there in a virtual memory pool". (Unfortunately this only gets you so far because the data is mutable.)

Therefore, it seems like the perfect FP program is just one single function (possibly made up of many sub-functions like nesting Russian dolls) doing all tasks

Programs of this sort are quite easy to write. I wouldn't call them perfect.

-- although a quick glance at any elisp code seems to be a study of programming schizophrenia on this count.

Nor would I call typical elisp code functional.

Nasty details

John Cowan wanted more detail.  If device drivers and bus cycles are not up your alley, I suggest you skip this post.  Seriously.

---

Still here?  You were warned.

For various historic reasons, the LMI Lambda was built as a NuBus card set. The NuBus interface could be driven from the memory interface, and this microcode shows how:

((MD) a-map-scratch-block)      ;following inst gives maps time to settle.
 ((M-1) DPB C-PDL-BUFFER-POINTER-POP (BYTE-FIELD 8 24.) A-1)  ;FULL 32 BIT NUBUS ADR
 ((L2-MAP-CONTROL) (a-constant 1464))    ;no caching.
 ((L2-MAP-PHYSICAL-PAGE) LDB M-1 (BYTE-FIELD 22. 10.) A-ZERO)
 ((VMA-START-READ) LDB (BYTE-FIELD 8 2) M-1 a-map-scratch-block)

The first line loads the MD (memory data) register with the virtual address of the "scratch block". The second line pops the desired NuBus address off the stack. The third and fourth lines write the Level 2 maps (the maps are addressed by MD) to address NuBus space. The final line initiates the bus cycle, and the result can be read from MD when it arrives.

With some argument checking and boxing of the result, this is surfaced to the Lisp world as a primitive instruction called %NUBUS-READ:

(DEFMIC %NUBUS-READ 761 (NUBUS-SLOT SLOT-BYTE-ADR) T)
                                ;SLOT is really the high 8 bits.
                                ;the "top F" can be supplied via slot,
    ;avoiding bignums.

(SETF (DOCUMENTATION '%NUBUS-READ 'FUNCTION)
  "NUBUS-SLOT is top 8 bits of physical address (usually the top 4 bits are 1's).
SLOT-BYTE-ADR is the lower 24 bits of physical address.")
;;arglist = (NUBUS-SLOT SLOT-BYTE-ADR)

If there was a debug card installed on your machine, you could find out which NuBus slot it was in by reading the system configuration, and you could talk to it by calling %NUBUS-READ and %NUBUS-WRITE:

(defun assure-debug-board ()
  (when (null *my-debug-board*)
    (let ((slot (find-a-debug-board)))
      (setq *my-debug-board* (logior #xF0 slot)))))     ;put it in slot space

(defun read-debug-board (address)
  (%nubus-read *my-debug-board* address))

(defun write-debug-board (address data)
  (%nubus-write *my-debug-board* address data))

And if you know the offsets of the control registers on the card

(defregister debug-mode-register              *my-debug-board* #xFFF7FC :read-write)
(defregister debug-address-register           *my-debug-board* #xFFF7F8 :write-only)
(defregister debug-data-start-transfer        *my-debug-board* #xFFF7F4 :write-only)
(defregister debug-response-register          *my-debug-board* #xFFF7F4 :read-only)
(defregister debug-control-start-transfer     *my-debug-board* #xFFF7F0 :write-only)
(defregister debug-status-response            *my-debug-board* #xFFF7F0 :read-only)
(defregister debug-nubus-analyzer-ram-pointer *my-debug-board* #xFFF7EC :read-write)
(defregister debug-nubus-analyzer-ram-data    *my-debug-board* #xFFF7E8 :read-write)
(defregister debug-nubus-analyzer-ram-control *my-debug-board* #xFFF7E4 :read-write)
(defregister debug-nubus-analyzer-function    *my-debug-board* #xFFF7E0 :write-only)
(defregister debug-test-register              *my-debug-board* #xFFF7C4 :read-write)

and what the bit fields in the registers are

(defconstant %%mode-init                 (byte 1. 0.))
(defconstant %%mode-master               (byte 1. 1.))
(defconstant %%mode-led                  (byte 1. 2.))
(defconstant %%debug-mode-loopback       (byte 1. 3.))
(defconstant %%debug-mode-speed          (byte 1. 4.))
(defconstant %%debug-mode-txwait         (byte 1. 6.))
(defconstant %%debug-mode-response-ready (byte 1. 7.))

(defconstant %%debug-control-start-bit (byte 1. 0.))
(defconstant %%debug-control-ack-bit   (byte 1. 1.))
(defconstant %%debug-control-tm-bits   (byte 2. 2.))
(defconstant %%debug-control-axr-bit   (byte 1. 4.))

(defconstant %%debug-response-start-bit (byte 1. 0.))
(defconstant %%debug-response-ack-bit   (byte 1. 1.))
(defconstant %%debug-response-tm-bits   (byte 2. 2.))
(defconstant %%debug-response-axr-bit   (byte 1. 4.))

and some good values to stuff in

(defconstant $$init 1.)

(defconstant $$disable-mastership 0)
(defconstant $$enable-mastership  1)

(defconstant $$led-off 0)
(defconstant $$led-on  1)

(defconstant $$disable-loopback 0)
(defconstant $$enable-loopback  1)

(defconstant $$slow-transfers 0)
(defconstant $$fast-transfers 1)

(defconstant $$transmitter-idle 0)
(defconstant $$transmitter-busy 1)

(defconstant $$no-response    0)
(defconstant $$response-ready 1)

it is pretty straightforward to run the debug card. Here are some examples

(defun reset-debug-board ()
  (setf (debug-mode-register) (dpb $$init %%mode-init 0)))

(defun blink-debug-light ()
  (loop
    (sleep .5)
    (setf (debug-mode-register)
          (dpb $$led-on %%mode-led (debug-mode-register)))
    (sleep .5)
    (setf (debug-mode-register)
          (dpb $$led-off %%mode-led (debug-mode-register)))))

(defun reset-nubus-analyzer ()
  (setf (debug-nubus-analyzer-function)
        (dpb $$disable-analyzer %%nubus-analyzer-enable 0)))

(defun board-idle? ()
  (= $$transmitter-idle (ldb %%debug-mode-txwait (debug-mode-register))))

The debug card in the Lambda was connected via a serial cable to the debug slave card in the K machine test rack. Writing to the data, addr, and control registers on the card would cause the slave to perform bus cycles on the test rack.

(defun debug-read-word (addr)
  (write-debug-addr addr)
  (write-debug-control #x01)
  (wait-for-debug-response))

(defun debug-write-word (addr data)
  (write-debug-data data)
  (write-debug-addr addr)
  (write-debug-control #x09)
  (wait-for-debug-response))

Now that we can do bus cycles on the test rack, we can talk to other cards on the test rack. Some of the data and control registers of the K machine can be read directly from the bus.

(defvar k-mem-addr   nil "K Processor - Memory address base")
(defvar k-io-addr    nil "K Processor - I/O address base")
(defvar k-mode-addr  nil "K Processor - Mode register address")
(defvar k-hptr-addr  nil "K Processor - History RAM pointer address")
(defvar k-hram-addr  nil "K Processor - History RAM data address")
(defvar k-pc-addr    nil "K Processor - Program Counter address")
(defvar k-mmfio-addr nil "K Processor - MMFIO bus address")
(defvar k-spy0-addr  nil "K Processor - Low spy Instruction Register address")
(defvar k-spy1-addr  nil "K Processor - Hi spy Instruction Register address")
(defvar k-spyc-addr  nil "K Processor - Spy command register address")
(defvar k-int-addr   nil "K Processor - NUBUS interrupt register base address")

Of particular interest here is the Spy Command register. This can be written with one of these values

(defconstant $$spy-command-stop                0.)
(defconstant $$spy-command-run                 1.)
(defconstant $$spy-command-step                2.)
(defconstant $$spy-command-reload-instruction  3.)      ;called ICLOAD in spy-pal
(defconstant $$spy-command-clear-opc-clock     4.)
(defconstant $$spy-command-set-opc-clock       5.)
(defconstant $$spy-command-clear-spy-mode      6.)
(defconstant $$spy-command-set-spy-mode        7.)
(defconstant $$spy-command-stepmode-full-clock 8.)      ;called clr-stepmode
(defconstant $$spy-command-set-stepmode 9.)             ;       set-stepmode

The Spy Command register allows you to start, stop, reset, and debug the machine externally. For example,

(defun k-stop ()
  "Stop the processor clocks, and init spy modes"
  (k-spy-cmd $$spy-command-stop)
  (k-spy-cmd $$spy-command-stepmode-full-clock)
  (k-spy-cmd $$spy-command-set-spy-mode)     ; set spy mode
  (k-spy-cmd $$spy-command-clear-opc-clock)     ; clear opc clk
  (setq k-run-flag nil)
  (k-read-spy-pc))

(defun k-run ()
  "Start the processor running"
  (k-spy-cmd $$spy-command-stop)
  (k-spy-cmd $$spy-command-clear-spy-mode)
  (k-spy-cmd $$spy-command-set-opc-clock)
  (k-spy-cmd $$spy-command-reload-instruction)
  (k-spy-cmd $$spy-command-run)
  (setq k-run-flag t)
  (k-read-spy-pc))

Especially useful was this one:

(defun k-step ()
  "Step the processor one clock cycle"
  (k-stop-if-running)
  (k-spy-cmd $$spy-command-stop)
  (k-spy-cmd $$spy-command-clear-spy-mode)
  (k-spy-cmd $$spy-command-set-opc-clock)
  (k-spy-cmd $$spy-command-reload-instruction)
  (k-spy-cmd $$spy-command-step)
  (k-read-spy-pc))

You may have noticed that it takes multiple local NuBus cycles to instruct the debug card to issue a single remote cycle, and it takes multiple remote cycles to drive the K hardware through the spy interface. It was fast enough for debugging, but it is much faster to put the K processor in the same rack as the LMI Lambda and get rid of the debug board altogether. We used the debug board early on so we could turn off the test rack and remove the hardware.

Still here? I salute your intestinal fortitude. The above code was selected from the files at https://code.google.com/p/jrm-code-project/source/browse/#svn%2Ftrunk%2Fkmachine

Esoterica

I occasionally get email about things I never thought I would have to remember.  Recently, I was asked how the LMI K-machine booted.  The short answer is this:

In the K-machine schematics, in the NuBus Spy Data Paths, there is a switch near the upper right labeled AUTOBOOT.  If it is on, the machine boots from the boot prom.  Otherwise, the machine powers up in a halted state and some other device on the bus is expected to boot it externally.

---

Writing is hard.  My original post went into detail about we used a serial cable from an LMI Lambda to a NuBus debug card in the test rack.  You could write the K-machine memory from the NuBus to load a bootable program, then you could single step the machine through the debug path.

(Ref:  The code for the K-machine remote debugger.)

The nitty gritty details are there in the code.

About nothing

Here's a way to quickly compute the leftmost digit of a number:

(define (leftmost-digit radix n)
  (if (> radix n)
      n
      (let ((leftmost-digit-pair (leftmost-digit (* radix radix) n)))
 (if (> radix leftmost-digit-pair)
     leftmost-digit-pair
     (quotient leftmost-digit-pair radix)))))

It's harder to compute the remaining digits.

Memory management

My college professors constantly encouraged us to "Go back to first principles."

Consider a computing task that runs for some amount of time and then halts.  If a task dynamically allocates more memory than is available, it must re-use some (or crash!)  This is irrespective of the means of re-use, whether manual deallocation as in malloc/free or automatic deallocation with a garbage collector.

The amount of allocated memory at any point in the program is the difference between the total amount allocated up to that point and the total amount deallocated up to that same point.

allocation_final = allocation_total - free_total

or

free_total = allocation_total - allocation_final

If we free memory in discrete amounts, then the average amount freed each time is simply the total amount freed divided by the number of times we freed memory (by definition). The average amount retained each time is easily computed by subtracting the amount freed from the memory size. This is all simple arithmetic.

In particular,

free_total / deallocation count = free_mean (by definition)

and at deallocation,

memory size - free_mean = retained_mean (ditto)

The total amount of memory that a task uses might vary when the task runs at different times and different memory settings. A task could use reflection to adjust memory consumption according to resources, or the task might change resource consumption because of external circumstances such as time or memory alignment. But we expect that simple deterministic tasks will consume resources in a repeatable manner. If that is true, then the total allocation and the amount of reachable storage at any time should not depend upon the amount of memory. If you reduce the amount of memory, you'll just need to recycle it that much more to make up the difference.

For a fixed amount of freed memory, the deallocation count times free_mean is constant. So for some task that frees a certain amount of memory, the product of the deallocation count and free_mean will lie on a hyperbola. The product of the deallocation count and the memory size will not lie exactly on a hyperbola, but it will be pretty close, especially if the memory size is quite a bit larger than retained_mean. Again, this is simply the consequence of arithmetic. Whether we use a garbage collector to deallocate or some other memory management technique doesn't matter.

In the Economics of Garbage Collection, Singer and Jones investigate the issue of garbage collection from a microeconomics background. They introduce what they call the allocation curve of the program. A benchmark program is run with several different heap sizes and the number of garbage collections is counted for each run. The red curves in this figure show the results:

We see the expected pseudo-hyperbolas. (The blue lines are elasticity, which is not relevant here.)

Singer and Jones created these charts by empirically measuring execution of benchmark programs. But it is pointless to "measure" an arithmetic relationship. The positions of the points on the charts are deterministic located at the point where the deallocation count times the amount freed is constant. If a point does not fall on the expected line, this can only be because the total allocation or the retained_mean have changed. One of the desiderata of a good GC benchmark is having the total allocation and retained_mean be invariant across different heap sizes. The benchmarks used by Singer and Jones are not invariant across different heap sizes (or the pseudo-hyperbolas would be perfect), but the variations are generally small, so the charts are pretty close.

If we partition our allocation into "small" and "large" allocations, then we will have a pseudo-hyperbola for each case. Singer and Jones's figure 5 illustrates this:

Singer and Jones note that some benchmarks have a pronounced "knee" in the curve. In this blogpost I show how the appearance of a "knee" is an artifact of presentation, and not a property of the data.

Singer and Jones note that the upper extreme of the curve is the point at which the amount of memory a program has available is equal to what the program needs. No deallocation need be done. The extreme lower end is the point where the amount of memory given is just under the maximum amount of live storage needed. Singer and Jones note that the curve approaches an asymptote, but they do not identify the asymptote as the value of retained_mean. (Of course the curve does not reach the asymptote unless the peak memory usage is the same as the mean.)

In this post, I plot the GC count vs. the memory size for various runs of MIT Scheme. The plot is in log-log space, so the product of the GC count and memory size falls on a line rather than a hyperbola. The axes are swapped from the convention of Singer and Jones, so the asymptote is vertical rather than horizontal. A non-zero value of retained_mean displaces the hyperbola to the right and makes the upper end approach a vertical asymptote rather than intersect the axis. The blue "unadjusted" line is simply a plot of GC count * memory size. The green "adjusted" line is GC count * (memory size - retained_mean).