secd
A silly implementation of the SECD machine, adapted from The Architecture of Symbolic Computers.
Ideas
Examples and exercises using the Clojure SECD
A Guide to SECD
The SECD machine is a virtual machine created by Peter Landin (of Next 700 Langs fame) a long, long time ago. The machine targets functional languages, and has a small and (relatively) simple instruction set. Implementing and understanding the SECD machine is an interesting exercise, and while it doesn't represent a "production-ready" compilation target, it will force you to think differently, and is damn fun to have around.
SECD stands for Stack, Environment, Code, and Dump, the four registers of the machine. Traditionally, each register points to the head of a linked list in memory.
Stack: The stack is used for ephemeral storage. Functions will take arguments from the stack and return values to the stack.
Environment: The environment contains arguments passed to functions, as well other values within the scope of that function.
Code: The code (or control) registerm contains instructions to the
machine. Executing one of these instructions can change any of the four
registers โ including the code register itself. The instruction on
the top of the code register is the next to be executed, and execution
continues until the STOP instruction is reached.
Dump: The dump is used to persist copies of the registers, which, as
we'll see, allows us to create JUMP-like instructions without "losing
our place" in execution.
Axiomatic Semantics
Axiomatic Semantics is an approach to describing programs in terms of changes in state โ not in terms of implementation. SECD instructions are defined axiomatically, which gives implementors total control over the way instructions work under the hood. (For instance, my implementation of the SECD machine is largely functional, and does not rely on linked-lists. Values and registers are, for the most part, immutable.)
An Axiomatic Notation for SECD
I will borrow Kogge's notation. Given a machine state, the next state of the machine can be described in terms of transformations to the registers.
s e c d => s' e' c' d'
where s, e, c, and d represent the initial state of each
register, and the primes represent the new states.
TODO: Formalize list dot notation
While the single letter s is taken to represent any possible
configuration of the stack register, we also need a way to match on more
specific states. Since each register is a list, we can destructure
registers to match on internal components.
(123.s) matches the configuration where the top item of the stack is
the value 123.
(x y.s) e (ADD.c) d matches the register configuration where the
ADD instruction is on the top of the code register, and at least two
items, x and y, are on the stack.
Note that the . (period) inside of a list can be read as, "and the
rest," such that (x y.s) can be read as, "any x, any y, and the rest
of the stack."
Given this, an instruction FOO can be described axiomatically using
something similar to the following:
(x.s) e (FOO.c) d => (bar.s) e c d
The Instruction Set
For reference, the instructions we will cover are as follows:
Basic instructions:
NIL - Nil
LDC - Load constant
LD - Load (from environment)
Built-ins:
CAR, CDR, ATOM, et al. - Unary built-in operations
ADD, SUB, CONS, et al. - Binary built-in operations
Branching instructions:
SEL - Select
JOIN - Join
Non-recursive function instructions:
LDF - Load function
AP - Apply
RTN - Return
Recursive function instructions:
DUM - Dummy
RAP - Recursive apply
Auxiliary instructions:
READC - Read character
WRITEC - Write character
STOP - Stop execution
Basic Instructions
NIL:
s e (NIL.c) d => (nil.s) e c d
NIL is the simplest SECD instruction. It pushes nil onto the stack.
In SECD-land, nil is also the empty list. We'll use NIL before we
have to build up an argument list, for example, and then CONS elements
onto it. Stay tuned.
LDC โ Load constant:
s e (LDC x.c) d => (x.s) e c d
LDC loads the next value from the code register onto the stack. For
instance, s e (LDC 1 . c) d would load the value 1 onto the stack. s e (LDC (1 2 3).c) d would load the list (1 2 3).
LD โ Load from environment:
s e (LD [y x].c) d => (locate([y x], e).s) e c d
LD loads a value from the current environment. I think now is a good
time for a quick aside about the environment register.
Like the other registers, the environment is a list. In fact, it's a list of lists, and each list contained represents a scope. A possible environment, for instance, may look like this:
((0 1 2)
(10 11 12)
(20 21 22)
(30 31 32))
The earlier in the environment a list of values appears, the "closer" it is in scope. During the execution of a function, arguments to that function will be at the top of the environment. If that function is inside another function, the arguments to that enclosing function will be at index 1, and so on and so forth.
The locate function used in LD, then, suddenly makes sense: it
accepts a pair of coordinates and indexes into the 2-dimensional
environment. If we called the previous example environment e,
locate([2 1], e) would return the number 21.
TODO: More clear explanation
Built-ins
Like all virtual machines, the SECD machine assumes a number of built-in operations which take their arguments from the stack.
TODO: Expand?
Unary:
(x.s) e (OP.c) d => (OP(x).s) e c d
Unary built-ins take one argument from the stack, evaluate the built-in with that argument, and push the return value back on. (Notice that the original argument as "disappeared" from the stack, and been replaced with the result of the operation.)
Examples of unary built-ins:
ATOM - Returns true if its argument isn't a composite value (like a list), else false
NULL - Returns true if its argument is nil or the empty list, else false
CAR - Returns the head of a list
CDR - Returns the tail of a list
Note: in the original SECD machine, the boolean values true and false
where represented as 1 and 0, respectively. I've chosen to instead use
Clojure's true and false for clarity and simplicity.
Binary:
(x y.s) e (OP.c) d => (OP(x,y).s) e c d
Binary built-ins are exactly like their unary counterparts, except that they take two arguments from the stack.
Examples of binary built-ins:
CONS - Prepend an item onto a list, returning the list
ADD - Addition
SUB - Subtraction
MTY - Multiplication
DIV - Division
EQ - Equality of atoms
GT - Greater than
LT - Less than
GTE - Greater than or equal to
LTE - Less than or equal to
Some Simple Examples
Some simple SECD examples using what we've learned, in Clojure:
;; do-secd accepts a vector representing the code register, and
;; recursively evaluates instructions until none are left. The value at
;; the top of the stack is returned.
(do-secd [NIL])
;;=> nil
(do-secd [LDC 1337])
;;=> 1337
;; (cons 1337 nil)
(do-secd [NIL LDC 1337 CONS])
;;=> (1337)
;; (cons 2448 (cons 1337 nil))
(do-secd [NIL
LDC 1337 CONS
LDC 2448 CONS])
;;=> (2448 1337)
;; (car (cons 2448 (cons 1337 nil)))
(do-secd [NIL
LDC 1337 CONS
LDC 2448 CONS
CAR])
;;=> 2448
;; (+ 5 5)
(do-secd [LDC 5 LDC 5 ADD])
;;=> 10
;; (- 20 (+ 5 5))
(do-secd [LDC 5 LDC 5 ADD
LDC 20
SUB])
;;=> 10
;; (atom 1)
(do-secd [LDC 1 ATOM])
;;=> trueBranching Instructions
Now that we know how to use some simple instructions, let's take a look
at some useful instructions: SEL and JOIN, which together give us
something akin to if-then-else.
SEL โ Select branch:
(x.s) e (SEL ct cf.c) d => s e c? (c.d)
where c? is (if (not= x false) ct cf)
SEL expects to be followed by two other code lists representing the
then and else branches of the statement (here seen as ct
and cf). SEL then checks the top item on the stack, replacing the
code register with ct if that item is not false and with cf if it
is. The remainder of the code register following the two branches, c,
is pushed onto the dump, where it can be recovered later.
JOIN โ Join with main control:
s e (JOIN.c) (c'.d) => s e c' d
JOIN is used to return control to the code register saved during the
SEL. It is expected, then, that the ct and cf code lists seen
above end with JOIN.
Trivial branching examples:
;; (if (= 0 (- 1 1)) true false)
(do-secd [LDC 1 LDC 1 SUB ;; (- 1 1)
LDC 0 EQ ;; (= 0 0)
SEL
[LDC true JOIN] ;; then-branch
[LDC false JOIN]] ;; else-branch
;;=> true
;; (+ 10 (if (nil? nil) 10 20))
(do-secd [NIL NULL
SEL
[LDC 10 JOIN]
[LDC 20 JOIN]
LDC 10 ADD]
;;=> 20Non-recursive Function Instructions
A function closure is a function that retains references to variables in-scope during that functions definition, even after the function has left that scope. Consider the following Clojure example:
(defn alwaysfn [x]
(fn [] x))
(def always-five (alwaysfn 5))
(always-five)
;;=> 5In this example, alwaysfn establishes a closure around x โ the
function returned from it "remembers" x's value, even though x is no
longer in scope. Because of this, we can say that the returned function
is closed over x.
This is a useful construct, and the SECD machine supports it. Since
local variables and function arguments are accessible through the
environment register, we can easily create closures by simply packing up
functions with the environment in which they were defined into a
[function context] pair.
LDF โ Load function:
s e (LDF f.c) => ([f e].s) e c d
LDF is used to create the [function context] closure, packing up
some function f (just a list of instructions) and the current
environment into a pair, and pushing it onto the stack.
AP โ Apply function:
([f e'] v.s) e (AP.c) d => nil (v.e') f (s e c.d)
AP does the dirty work, transferring control to the function
instructions and "installing" that functions arguments and context into
the environment register. The remaining stack, environment and code
registers are also pushed onto the dump so that execution can continue
after the function has executed.
TODO: More in depth here, specifically re: function args
RTN โ Return control:
(x.z) e' (RTN.q) (s e c.d) => (x.s) e c d
A RTN instruction is expected at the end of any function list, and is
used to return a value. When the RTN is encountered, the top value of
the stack (x) is saved, registers are restored from the dump, and that
saved value is pushed onto the restored stack.
Recursive Function Instructions
DUM โ Dummy environment:
s e (DUM.c) d => s (nil.e) c d
nil is replaced by an atom in the Clojure implementation
RAP โ Apply recursive function:
([f (nil.e)] v.s) (nil.e) (RAP.c) d
=>
nil (rplaca((nil.e), v).e) f (s e c.d)
Auxiliary Instructions
WRITEC READC STOP
Writing Simple SECD Programs
The SECD "Hello, world" โ Recursion
In many modern languages, Hello, world is basically the simplest a
program can get. In SECD-land, this is not so. We don't have a
println โ we only have WRITEC โ so we have to define our
own println first!
The function is quite simple, really. It will accept one argument, a list of integers (our model of characters), print the first character from the list, and recurse with the tail. Once the sequence is empty, we'll print a newline character.
TODO: Explain general recursive pattern
;; This is obviously not hello world, but it should be
(def hello-world [72 101 108 108 111 44 32 119 111 114 108 100 33])
(do-secd [NIL DUM
LDF [LD [0 0] NULL
TEST [LDC 10 WRITEC RTN]
LD [0 0] CAR WRITEC
NIL LD [0 0] CDR CONS
LD [1 0] AP
RTN]
CONS
LDF [NIL LDC hello-world CONS
LD [0 0] AP RTN]
RAP])
;; prints "Hello, world!"Optimizations โ Extending the Instruction Set
TEST DAP AA MA
Means of Abstraction
As mentioned earlier, the SECD machine is particularly well suited to support functional programming languages. It is no surprise, then, that common abstractions in such languages can be modeled. One such common abstraction is reduce, or fold.
reduce abstracts over recursion
recursion seems abstract in imperative languages, but can be seen as relatively "low level" in functional languages
this is because reduce requires a manual handling of a base-case, in the same way that a while loop would. Handling it poorly leads to infinite recursion.
reduce happens to be a particularly powerful function, and other common functions, like map and filter, can be trivially defined in terms of reduce.
Defining map and filter in terms of reduce in Clojure:
(defn reduce' [f acc sequence]
(if (seq sequence)
(recur f (f acc (first sequence)) (rest sequence))
acc))
(reduce' + 0 [0 1 2 3 4])
;;=> 10
(defn map' [f sequence]
(reduce' #(conj %1 (f %2)) [] sequence))
(map' inc [0 1 2 3 4])
;;=> [1 2 3 4 5]
(defn filter' [condition sequence]
(reduce' #(if (condition %2) (conj %1 %2) %1) [] sequence))
(filter' even? [0 1 2 3 4])
;;=> [0 2 4]Doing the same, but in SECD:
(def secd-reduce ;; fn args: [f acc sequence]
[LD [0 2] NULL ;; load sequence, check if null
TEST [LD [0 1] RTN] ;; if null, load acc and return
NIL ;; build args for recur
LD [0 2] CDR CONS ;; cons (rest sequence) onto recur argslist
NIL ;; build args for calling f
LD [0 2] CAR CONS ;; cons first(sequence) onto f argslist
LD [0 1] CONS ;; cons acc onto f argslist
LD [0 0] AP ;; load f and apply
CONS ;; cons result onto recur argslist
LD [0 0] CONS ;; cons f onto recur argslist
LD [1 0] DAP ;; load recursive fn and apply
RTN])
(def secd-add [LD [0 1] LD [0 0] ADD RTN])
(do-secd [DUM NIL
LDF secd-reduce CONS
LDF [NIL
LDC [0 1 2 3 4] CONS
LDC 0 CONS
LDF secd-add CONS
LD [0 0] AP RTN]
RAP])
;;=> 10
(def secd-reverse
[DUM NIL
LDF secd-reduce CONS
LDF [NIL
LD [1 0] CONS
NIL CONS
LDF [LD [0 0]
LD [0 1]
CONS
RTN]
CONS
LD [0 0] DAP RTN]
RAP])
(do-secd [NIL LDC [0 1 2 3 4] CONS
LDF secd-reverse AP])
;; TODO: Shouldn't have to reverse result?
(def secd-map ;; fn args: [f sequence]
[DUM NIL
LDF secd-reduce CONS ;; load recursive reduce
LDF [NIL ;; nil to be used for reverse call
NIL ;; build args for reduce call
LD [1 1] CONS ;; cons sequence onto args
NIL CONS ;; cons nil onto args
LDF [LD [0 0] ;; in fn to reduce, load accumulator
NIL LD [0 1] CONS ;; cons reduce item onto args for f
LD [2 0] AP ;; apply f
CONS ;; cons result onto accumulator
RTN]
CONS ;; cons reduce fn onto reduce args
LD [0 0] AP ;; load reduce and apply
CONS ;; cons reduce value onto list
LDF secd-reverse AP ;; reverse
RTN]
RAP])
(def secd-inc [LDC 1 LD [0 0] ADD RTN])
(do-secd [NIL
LDC [0 1 2 3 4] CONS
LDF secd-inc CONS
LDF secd-map AP])
;;=> (1 2 3 4 5)
(def secd-filter
[DUM NIL
LDF secd-reduce CONS
LDF [NIL
NIL
LD [1 1] CONS
NIL CONS
LDF [NIL LD [0 1] CONS ;; load reduce item and cons onto
;; args for condition
LD [2 0] AP ;; load condition and apply
TEST [LD [0 0]
LD [0 1]
CONS RTN]
LD [0 0] RTN]
CONS
LD [0 0] AP
CONS
LDF secd-reverse AP
RTN]
RAP])
(def secd-even? [LDC 0
LDC 2 LD [0 0] MOD
EQ RTN])
(do-secd [NIL
LDC [0 1 2 3 4] CONS
LDF secd-even? CONS
LDF secd-filter AP])
;;=> (0 2 4)SECD as a Compilation Target
References
Kogge, Peter M. 1991. The Architecture of Symbolic Computers.
License
Copyright ยฉ 2012 Zach Allaun
Distributed under the Eclipse Public License, the same as Clojure.