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T h e J o y o f
C o n c u r r e n t L o g i c P r o g r a m m i n g
Introduction
This article is an attempt to describe my experiences with concurrent
logic programming languages, a programming paradigm that has been
largely forgotten but has a number of interesting properties that
make it worthwhile to explore. Especially today, where CPU designers
have run out of ideas and have settled on throwing more and more
processor cores at us, parallel and concurrent programming remain
the areas where performance gains seem still be achievable, yet,
our programming models are not at all suited to this - coarse-grained
processes with shared state is still the most widely used strategy
and parallelizing application remains hard and error prone.
One concept that promises an easier model for thinking about
concurrency are actors, lightweight processes communicating via
messages. Depending on the implementation this can solve many of
the synchronization problems in a clean manner and allows composing
concurrent functionality conveniently. Erlang [10] would be the
prime example, being probably the first well engineered attempt to
embrace concurrent software while at the same time admitting that
software can break. This humility to accept that processes fail and
putting thought into how to handle such situations and how to recover
is probably one of the great achievements of the folks who invented
and implemented Erlang.
Yet, there is a programming paradigm that precedes Erlang and which
is, in a sense, even more parallel while providing similar mechanisms
for synchronization and composition, but in a more elegant and
consistent way. The family of concurrent logic programming languages
was once a fertile field of research and was the basis of Japan's
Fifth Generation Computing [11] project, which unfortunately died,
together with a lot of other "AI" research during the AI winter.
This is a shame, since, logic programming languages (just as LISP)
can be used for a lot more than what is or was usually called
"Artificial Intelligence".
All of these languages originate in PROLOG, the most well known
logic programming notation, but differ in the way they deviate from
the basic execution model. Starting with Concurrent Prolog (CP),
further language developments removed more and more of the distinctive
features of Prolog while becoming more streamlined and easier and
more efficient to implement. Dialects that where spawned by CP are
Parlog, FCP, GHC, FGHC, KL1, Flat Parlog, Strand, FLENG, Janus, PCN
and many others (as I said, the research was fertile). Most of these
are gone now, where only theoretical models or where never fully
implemented. KL1 [8] was a dialect of (Flat) GHC used on custom
hardware designed in Japan. Strand [1] is probably the most
streamlined and elegant of the family. One could say that Strand
is to Erlang what Scheme is to Lisp, the cleaned up and generalized
version of a more pragmatic engineering effort to implement reliable
concurrent systems. Note that Strand precedes Erlang, in fact,
Erlang was initially implemented on top of Strand, but deemed
impractical due to the very fine grained parallelism that Strand
provides. What must be noted is that, as the dialects develop
further, they remove more and more classic Prolog features. This
is explained in detail in [5].
As to my knowledge there exist only very few implementations of
concurrent logic languages. There is a compiler for KL1 to C available
[2], which seems to work, but suffers from bit rot. See also [7]
for an FCP implementation, the status of which I can say nothing
about. There exists a simple Strand implementation (by the author)
[3]. Strand was once a commercial product but it doesn't seem to
be available anymore. The Strand book [1] demonstrates very
sophisticated industrial applications in its section on case studies,
which show that it was used in a commercial context. The book is
highly recommended, by the way, as it gives a very gentle and clear
introduction to the concepts and ideas that are fundamental to
programming in these languages.
In what follows, I will use the FGHC dialect, for which I also
provide an implementation [3], in case you are interested.
Syntactically, the various dialects have a lot in common and the
techniques that are used apply to most of them. It is in particular
the techniques that I find fascinating, as the programming model
is fundamentally different from the mainstream paradigms and which
lets you see software from a new viewpoint, something that always
enlivens and refreshes ones conception of the practice of software
construction.
A Crash Course in FGHC
So, let's begin. Some basic knowledge of Prolog may be helpful to
avoid difficulties understanding the jargon.
Programs in FGHC consist of sets of clauses of the following form:
predicate(Argument, ...) :- Guard, ... | Term, ... .
Predicates of the same name and arity (number of Arguments) together
form a process definition and should be grouped acordingly. Invocation
of a process with the given name and arity will select one of the
predicates with matching arguments and execute the body terms when
all guards succeed. Guards are conditional expressions and must
be selected from a predefined set of primitives (that is what makes
FGHC "flat"). Once the clause is selected, the terms in the body
are executed in parallel - there is no predefined order. That's not
a problem, in case this worries you, as there is a simple method
of synchronizing execution. As a matter of fact, you merely restrict
the parallelism by forcing synchronization implicitly or explicitly,
the execution model is geared towards as much parallelism as possible.
This can be surprising, as we will see later.
I wrote that a predicate clause is selected by matching. Matching
is not the classic unification of Prolog: if a logic variable is
unbound, then matching a pattern containing such a variable suspends
and does not continue until the variable has been bound. This is
different from Prolog, where access to an unbound variable does not
influence the control flow. Unification does take place in the body
part of a clause and works similar to Prolog. FGHC has logic variables
like Prolog, they can be bound to other variables and dereferencing
chains of variables is automatic, so the unifications in
A = B, B = C, D = 123, C = foo(D)
result in A holding the value "foo(123)" - the order in which these
unifications take place is immaterial, any order is fine, as unbound
variables can be thought of as "holes" that can be filled at any
time. Variables are distinguished by starting with an uppercase
letter or with "_". The variable "_" is the anonymous variable,
multiple occurrences in a clause stand for multiple distinct
variables. Unification is restricted to body terms, all accesses
to unbound variables in head matching, guards and calls to builtin
predicates will suspend the current process until some other process
binds the variable. Then the suspended process will resume and
perform a new attempt at selecting a matching clause or exeuting
the builtin predicate.
Another subtle point is that the clauses of a process definition
are not matched in order - any clause can be matched first, the
order is arbitrary and doesn't even have to be the same from
invocation to invocation, it is non-deterministic. To control the
order one can use suitable guards.
A clause without guards omits the " ... |" part. A clause without
guards and body looks thus:
done.
If no clause matches, then the process fails, which terminates the
program. There is no backtracking as in Prolog.
And, curiously enough, this is nearly all. There are no loops, you
use recursion instead: a clause that (directly or indirectly) spawns
a single process invoking the same predicate again is tail-recursive
and so has constant memory requirements. There are a number of
built-in guards and primitive predicates, "is/2" for expression
evaluation, "=/2" for unification and that is about all you need
to know about the semantics of FGHC: There exists a pool of processes
executing concurrently, processes are taken from that pool, executed
and spawn new processes to be added to the pool until all processes
are finished or a process fails. How the concurrency is implemented
is currently secondary, but usually you have multiple processors
each holding a pool of quasi-parallel processes.
Here is a example FGHC program, a predicate to reverse a list:
/* FGHC */
-initialization(main).
rev([], Y) :- Y = [].
rev([X|Xs], Y) :-
rev(Xs, Ys),
append(Ys, [X], Y).
append([], Ys, Z) :- Z = Ys.
append([X|Xs], Ys, Z) :-
append(Xs, Ys, Z2),
Z = [X|Z2].
main :- rev([1, 2, 3, 4, 5], R), writeln(R).
This is basically equivalent to the Prolog version, with the exception
that one has to unify result (or "output") variables manually. The
first clause of "rev/2" shows this clearly: the body ("Y = []") can
only be executed when the head is fully matched. Different dialects
handle this in different ways, FCP and Parlog have "mode" declarations,
FGHC and Strand do not allow output-unification in clause heads,
Strand furthermore does not allow unification at all, only direct
assignment to unbound variables.
One distinctive property of this program is that the reversal takes
place concurrently, for different parts of the input list. As long
as processes do not need to suspend, execution continues by spawning
sub-processes without restraint, until an unbound variable argument
is matched with a non-variable element in a clause head or guard.
So, in the example above, since the argument list is fully ground
(contains no unbound variables), processes of "rev/2" for all
sub-lists are spawned, as are "append/3" processes, but the latter
will suspend instantly until the various instances of "Ys" are bound
to some result. "writeln/1" is a built-in and will suspend until
the result is fully ground and then prints the reversed list.
Note that synchronization between all the processes takes place
automatically by waiting for variables to be bound. There is no
need to artificially orchestrate how the processes execute in
parallel, this is done implicitly like the cells of a spreadsheet
- once a required result is available, it may trigger further
variables to be bound and processes to be resumed, and so on, until
the network of processes settles.
Declarative Programming
A word about syntax: I find Prolog syntax quite pleasing, it reduces
special characters mostly to punctuation, similar to Lisp, Pascal
or Ada, which makes programs more readable. Operators are restricted
to expressions where one is used to them but do not clutter up the
code, giving subtle differences in meaning that confuse more than
that they help to abbreviate. Also, the meta-syntax of Prolog lends
itself to automated manipulation by code, not necessarily using
macros, but with the option to generate or process code in a useful
manner, an advantage of languages that are "homoiconic", like Lisp
or Prolog or derived languages. If you prefer a more traditional
syntax, see below for PCN, which provides a more familiar look but
offers the same facilities for concurrent programming.
The declarative nature of logic programs may make them suitable to
automated and manual proof, but this is something that someone more
familiar with this subject than me must decide. I personally find
Prolog code to be shorter than equivalent programs in other languages.
Also, the declarative notation of specifying clauses to match
arguments makes it straightforward to cleanly encode special cases,
provided one properly factors subsequent code. It is a bit of an
art to describe certain complex algorithms elegantly and dumb
imperative code may look awkward, but that is perhaps because it
is dumb and imperative and doesn't look much nicer in any language.
In classic Prolog, modelling a problem to take maximum advantage
of backtracking, failure driven loops or unification needs a lot
of experience and the same will apply to other languages which have
a powerful internal resolution scheme. In concurrent logic
programming, you will use different techniques, but I claim that
they provide actually more flexibility and power in a way that
classic Prolog can not. Moreover we have to start looking for
suitable programming paradigms if we want to truly master multicore
processors and distributed computing that is more than just C++ and
locks, message passing libraries like MPI or HTTP based lowest
common denominators like REST. Textbooks on multithreaded programming
mostly present a messy amalgam of locks, shared state and an absurd
obsession with low-level details, which becomes so complex and
obscure that one wants just to give up because the attempt to produce
anything highly parallel seems to be inhumanly hard (because, using
these techniques, it is). The languages I am talking about here
will not make parallel or concurrent programs trivial, but they
will let you concentrate on the important parts and will reduce
boilerplate code and bookkeeping tasks.
Concurrent logic languages usually have no side effects, or when,
very little, in a controlled manner. I have seen descriptions of
stream based I/O for some implementations, but prefer the pragmatic
view of the previous generation of languages that started functional
programming: by accepting that side effects are unavoidable sometimes
and to let the programmer decide when to use them and then let her
use it without cumbersome abstractions or "monad" pseudoscience.
Streams as Communication Channels
So, how do we actually design code that takes advantage of parallel
processes and concurrent execution? By, at the same time, generalizing
and simplifying the concept of what shared state is. The logic
variable is the key concept - a first class object that can be bound
to other unbound variables and where referencing it dereferences
any existing chain of bindings automatically. Accessing (matching
a variable) will suspend until it is bound. Unification will bind
variables. Taken together we can view a list where the tail is an
unbound variable as a stream of values, being produced by some
process and consumed by another process. The consumer suspends
(blocks) when a tail is not bound yet, the producer extends the
stream by binding the tail. This makes producer and consumer run
in parallel, without any further explicit synchronization:
/* FGHC */
-initialization(main).
producer(1000, Out) :- Out = []. % terminate stream
producer(N, Out) :-
otherwise |
Out = [N|Out2], % Create a list cell
N2 is N + 1,
producer(N2, Out2).
consumer([N|In]) :- % Match and deconstruct the list cell
writeln(N),
consumer(In).
main :- producer(1, S), consumer(S).
The stream "S" (which is "In" for the consumer and "Out" for the
producer) is the communication channel between the two processes
and the program will endlessly print (usually) increasing numbers.
It may also run out of memory, because the producer may run faster
than the consumer and so may spawn many child processes before the
consumer is able to process them. This should always be kept in
mind. If the producer runs slower than the consumer, the latter
will patiently wait. One way to control the flow of messages (numbers,
in this case) is to make the consumer provide the place where it
expects the input, using the "bounded buffer" technique:
/* FGHC */
-initialization(main).
producer(N, [P|Places], Out) :-
P = N,
Out = [P|Out2],
N2 is N + 1,
producer(N2, Places, Out2).
consumer([N|In], P) :-
P = [_|Places],
writeln(N),
consumer(In, Places).
main :- P = [_|P2], producer(1, P, S), consumer(S, P2).
By initializing P to having one element we set the capacity of the
buffer. We could have increased the initial size and the number of
empty variables in "consumer/2" to give the producer more space to
fill up at a time, thereby increasing throughput.
We have effectively turned it around: the consumer produces a stream
of unbound variables and passes it to the producer to fill them in.
Running this program will print numbers, not necessarily strictly
increasing, as the invocations of "writeln/1" may not take place
in strict order - everything is parallel. Depending on the actual
language there may be facilities to enforce completion of a side
effecting operation, either by having a sequencing operator or by
having "writeln/2" return a variable that is bound to some value
once the operation is complete. The classic problems of concurrent
programming have not magically vanished, race conditions and deadlocks
still exist, but at least you can concentrate on the logic instead
of dealing with low-level details.
You may have noticed that both producer and consumer are tail
recursive, they invoke themselves before terminating and thus run
in a loop with bounded storage consumption. This is similar but not
equivalent to tail-recursion in procedural languages, where the
position of the recursive call designates whether a process has
constant storage requirements. In a concurrent logic language, a
recursive call adds the current process to the process pool, possibly
with different arguments, and the old one will vanish once all
sub-goals have been processed.
This example already demonstrates how having the building blocks
for communication channels can be seen to be more powerful than
having built-in channels themselves, as we can construct more
elaborate devices, like the "short circuit" solving the problem of
how to tell if N processes have all finished?
/* FGHC */
-initialization(main).
main :- one(ok, R), done(R).
done(ok).
one(L, R) :- two(L, R2), wait(R2, R).
two(L, R) :- three(L, R2), wait(R2, R).
three(L, R) :- wait(L, R).
wait(R1, R) :- R = R1.
Here we start 3 processes, each having two variables ("L"eft and
"R"ight). We pass a value into the left for the first process, and
once a process is done, it assigns the left side to the right. At
the end of the chain we know that all processes have finished once
the right side of the last process in the chain has been bound. In
this example the processes do no real work, but in real code, the
different processes could start arbitrary complex jobs. The Strand
book [2] explains these and many other techniques using often
fancyful analogies.
The ease with which communication channels can thus be created
allows for a straightforward decoupling of components. As list
(stream) elements can be arbitrarily complex objects that are easily
destructured by pattern matching, and can contain themselves unbound
variables as a back-channel for communicating results, different
parts of an application can exchange information, commands and
results through a single stream:
/* FGHC */
-initialization(main).
logger([X|S]) :- log(X, Ok), logger(Ok, S).
logger([], S) :- logger(S).
filter([bogus(_)|S1], S) :- filter(S1, S).
filter([_|S1], S) :- otherwise | S = [S1|S2], filter(S1, S2).
main :- start_task(S), logger(S), filter(S, S2), consume_work(S2).
The logger has been wedged between "start_task/1" and "consume_work/1"
to log all communication, without the other processes being bothered
having to know any of this. We use a second predicate to continue
logging when the actual log operation has completed (signified by
"Ok" being bound to "[]", a common idiom). "filter/2" on the other
hand removes certain messages and passes all others on to an
intermediate stream. Here we use the "otherwise/0" guard, which
marks the second clause of "filter/2" to be applicable for matching
only if all preceding clauses fail to match. "start_task/1" and
"consume_work/1" are unaware of the operations that have been put
in between or any communication that may be taking place between
auxiliary processes. That way components are kept strictly separate,
have a high degree of compositionality but few restrictions on how
to communicate, if they require to do so.
Streams are also a convenient API to functionality that requires
sequential access: as the messages are ordered, their processing
is, too. Streams can be merged as well, preserving the order of
elements in each sub-stream:
/* FGHC */
merge([], B, C) :- C = B.
merge(A, [], C) :- C = A.
merge([X|A], B, C) :- C = [X|C2], merge(A, B, C2).
merge(A, [X|B], C) :- C = [X|C2], merge(A, B, C2).
Here we take advantage of the non-deterministic nature of clause
selection: some clauses match both, we intentionally don't care as
long as one will finally be selected and continues the merge
operation.
As in Prolog, strings are lists of character codes which makes them
automatically suitable for stream processing. Apart from avoiding
the tedious duplication of list and string manipulation operations,
this allows for efficient, asynchronous handling of streams of
characters, so the extra space needed for lists is partially balanced
by manipulating them in a lazy manner, one piece at a time, by
communicating processes.
Streams even allow a certain style of object-orientation. If we
consider a stream as a message-handling entity, class hierarchies
can be mirrored as hierarchies of message handlers:
/* FGHC */
transport([move(Distance)|S], Position) :-
update_position(Position, Distance, NewPosition),
transport(S, NewPosition).
transport([get_info(T)|S], Position) :-
T = transport(Position),
transport(S, Position).
transport([_|S], Position) :-
otherwise | transport(S, Position).
car([refuel(Amount)|S], Transport, Fuel) :-
fill_tank(Fuel, Amount, NewFuel),
car(S, Transport, NewFuel).
car([get_info(T)|S], Transport, Fuel) :-
Transport = [get_info(T2)|Tr2],
T = car(Fuel, T2), % delegate and intercept
car(S, Tr2, Fuel).
car([M|S], Transport, Fuel) :-
Transport = [M|T2], % delegate to parent
car(S, T2, Fuel).
make_car(Car, Position) :-
transport(T, Position),
car(Car, T, 0).
Note how we keep local state by passing it along in the tail-recursive
message loop, updating it as needed, depending on messages. There
is nothing special about this, we just use streams of messages
between entities that perform message processing in a loop, we
create processes and then plumb them together with streams. Note
in the example above how we can pass a variable in a message
("get_info/1") to receive a reply, exposing internal state just as
much as is desired and necessary. "Car" and "transport" could run
concurrently on different processors, all synchronization is implicit
and automatic.
You may have noticed in the examples above that streams represent
general, first-class communication channels. This is quite similar,
or even a generalized variant of the "Actor" model[5]. Actors are
independent processes that communicate solely by messages and by
creating new actors. One difficulty may be the sharing of an actors
identity among multiple "client" actors that want to share messages,
but there are solutions like "ports"[12] or my using an intermediary
"merger" stream.
Resource Allocation
How all these processes are mapped to hardware is something that
depends on the implementation of the language and nothing has been
said about this so far. Strand has the concept of "virtual machines",
that is, a network of processors connected in some topology. In
Strand, you explicitly spawn a process on a different processing
node using the "@" notation, where
goal(...)@<peer>
designates that the invocation of "goal" shall take place on the
processor given by "<peer>". What "<peer>" stands for depends on
the topology, so for a ring, "<peer>" may be "fwd" or "bwd", meaning
the next or previous node. For a torus topology you would have
"north", "west", etc. All nodes share the available code and, in a
circular topology, spawning a process on the "next" node may wrap
around.
Here is some code that models a "mapreduce"-like computation over
a number of processors in a ring topology:
/* FGHC */
mapreduce([], _, L, R) :- R = L.
mapreduce([X|List], Call, L, R) :-
% Call = "foo(1, 2)" will result in a call to "foo(1, 2, X, L, L2)"
apply(Call, [X, L, L2]),
mapreduce(List, Call, L2, R)@fwd.
"apply/2" creates a goal to call dynamically, where the list of
arguments in the second position is appended to already existing
arguments of the goal designated by "Call". Each invocation of the
call takes the list element and some state and passes it on via the
final argument to the next iteration of "mapreduce" which runs on
the processor in forward direction from the current processor. The
number of processors is fully transparent, it may be just one, or
it may be many, the workload will be spread among the available
processors. Communication by sending data from one processor to
another is seamless and requires no further extra effort.
Provided the implementation allows interfacing to native code or
provides means to easily run native sub-processes, one might want
to consider using these languages just as a means to coordinate
parallel computations spread over multiple processors (or machines,
if supported).
PCN
A perhaps more traditional way of writing concurrent logic programs
is "PCN" (Program Composition Notation)[13, 14]. It shares a lot
of syntax with C and makes interfacing to native functions written
in C quite easy. Here is the producer/consumer program in PCN:
/* PCN */
main()
{||
producer(1, s),
consumer(s)
}
producer(n, out)
int n;
{;
{? n < 1000 ->
{; out = [n|out2], /* Create a list cell */
n := n + 1,
producer(n, out2)
},
default -> out = [] /* terminate stream */
}
}
consumer(inp)
{;
{? inp ?= [n|inp2] -> /* Match and deconstruct the list cell */
{; writeln(n),
consumer(inp2)
}
}
}
PCN programs consist of procedures ("program definitions") that
represent one of three types of compositions: sequential ("{; ...}"),
parallel ("{|| ...}") and choice ("{? ... }"). Compositions can be
arbitrarily nested and execute the constituents of their body one
after the other, all in parallel, in no particular order, or select
one statement, depending on a "guard", which is a comparison or a
pattern match ("?=").
PCN supports the usual first-class, single-assignment variables of
logic languages with the addition of mutable numeric variables that
can be assigned multiple times using the assigment (":=") statement.
Arrays of numerical type are provided as well, as is a very
straightforward integration of C code.
PCN makes a good choice for integrating C libraries into applications
requiring complex concurrent behaviour. It also serves as an
introductory step into the world of concurrent logic programming
due to its familiar syntax.
Conclusion
So, where to start? How can I try out this magical stuff? As you
might have guessed, this whole text is a bit of pitch for my own
FGHC implementation [3], a compiler of FGHC, Strand, PCN (and a low
level dialect called FLENG, on which it is based) to native code
for several platforms. The project is relatively new, but not so
experimental that one wouldn't want to give it a serious look. FLENG
provides a few additional features for convenience, for example the
"&/2" operator to ensure sequential execution of goals, tasks
(controlled process trees), ports (identity-preserving streams) and
the "idle/0" guard to execute code when the current processor is
idle. FLENG takes advantage of native threads, has a basic facility
for interprocess communication, a number of useful libraries and
compiles to native code. The garbage collection scheme is based on
reference counting and thus does not cause GC pauses. Convenient
use of single-shot and periodic timers, handling of UNIX signals
and execution of child processes is provided, which makes concurrent
logic programming an elegant tool to implement games, servers,
coordinate parallel workloads, or for infrastructure applications.
An interface to allow writing primitives in C/C++ is available as
well.
Give it a try, or at least learn more about this paradigm. It is
definitely worth the effort to have a look at a completely different
way of thinking about software architecture for concurrent tasks.
Information about these languages is not easy to find, the research
seems to have stopped and what is available is buried in scientific
papers and a small number of books. The Strand book [1] is an
excellent introduction, Shapiro [9] and Tick [5] provide an overview
over the different dialects and their development.
Does concurrent and parallel programming have to suck? I think not,
but we may have to try to look at it from a different angle. This
is an attempt to revive thinking about a paradigm, due to some
unfortunate events regarding the failure of the early "AI" hype,
has been lost in time.
--
[1] "Strand: new concepts in parallel programming"
Ian Foster, Stephen Taylor
https://dl.acm.org/doi/book/10.5555/100638
Also available online here:
http://www.call-with-current-continuation.org/files/strand-book.pdf
[2] KLIC
https://web.archive.org/web/20100724120623/http://www.klic.org/software/klic/index.en.html
[3] FLENG
A FLENG and FGHC implementation compiling to native code
http://www.call-with-current-continuation.org/fleng/fleng.html
[4] https://en.wikipedia.org/wiki/Actor_model
[5] "The Deevolution of Concurrent Logic Programming Languages"
Evan Tick
https://www.sciencedirect.com/science/article/pii/0743106694000388/pdf?md5=c34f030442a47297654dd824682780c0&pid=1-s2.0-0743106694000388-main.pdf&_valck=1
[6] "Guarded Horn Clauses"
K. Ueda
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.44.4015&rep=rep1&type=pdf
[7] Emulated Flat Concurrent Prolog
https://www.nongnu.org/efcp/
[8] "Design of the Kernel Language for the Parallel Inference Machine"
U. Kazunori et al., 1990
[9] "The Family of Concurrent Logic Programming Languages"
E. Shapiro
https://apps.dtic.mil/sti/pdfs/ADA213958.pdf
[10] http://www.erlang.org
[11] https://en.wikipedia.org/wiki/Fifth_generation_computing
[12] "Ports of Objects in Concurrent Logic Languages"
https://people.kth.se/~johanmon/papers/port.pdf
[13] "Parallel programming with PCN"
Ian Foster and Steven Tuecke
https://digital.library.unt.edu/ark:/67531/metadc1310344/
[14] "A Primer for Program Composition Notation."
Chandy, K. Mani and Taylor, Stephen (1990),
http://resolver.caltech.edu/CaltechCSTR:1990.cs-tr-90-10