Minimizing Reference Count Updating with Deferred and Anchored Pointers for Functional Data Structures

Henry G. Baker
Nimble Computer Corporation, 16231 Meadow Ridge Way, Encino, CA 91436
(818) 986-1436 (818) 986-1360 (FAX)
This material is based upon work supported by the National Science Foundation under Grant No. III-9261682.
Copyright (c) 1993 by Nimble Computer Corporation

Reference counting can be an attractive form of dynamic storage management. It recovers storage promptly and (with a garbage stack instead of a free list) it can be made "real-time"--i.e., all accesses can be performed in constant time. Its major drawbacks are its inability to reclaim cycles, its count storage, and its count update overhead. Update overhead is especially irritating for functional (read-only) data where updates may dirty pristine cache lines and pages.

We show how reference count updating can be largely eliminated for functional data structures by using the "linear style" of programming that is inspired by Girard's linear logic, and by distinguishing normal pointers from anchored pointers, which indicate not only the object itself, but also the depth of the stack frame that anchors the object. An anchor for a pointer is essentially an enclosing data structure that is temporarily locked from being collected for the duration of the anchored pointer's existence by a deferred reference count. An anchored pointer thus implies a reference count increment that has been deferred until it is either cancelled or performed.

Anchored pointers are generalizations of "borrowed" pointers and "phantom" pointers. Anchored pointers can provide a solution to the "derived pointer problem" in garbage collection.


Reference counting [Collins60] can be an attractive form of dynamic storage management for functional (read-only) data structures because the cycles that cause trouble for reference counting do not exist in these data structures. Unfortunately, reference counting extracts a penalty not only when creating or destroying data structures, but also when simply traversing them. For example, when the length of a list is computed, the counts of the cells along the list are each incremented and then decremented. Although this traversal results in no net change to any count, the work to perform 2n count updates for a list of length n is substantial. In a modern RISC architecture with a data cache, these count references may cause lines to be brought into the cache that would not otherwise be touched, and worse still, these lines are also "dirtied", requiring that they be rewritten--even though the data eventually written out is identical to that brought in! In a shared-memory multiprocessor system, updates to counts require synchronization in order to avoid losing or gaining counts due to conflicting reads and writes. Thus, while reference count updating for the creation and deletion of structures may be tolerable, this continual updating for simple traversals is not.

In a classical reference count system, the reference count on a node indicates the exact number of references--whether from other nodes or local variables--that currently point to the node. The copying of a reference causes a reference count increment and the deletion of a reference causes a reference count decrement. The "linear style" of programming inspired by linear logic (see Appendix I) helps to minimize the reference count updating that is caused by the copying and deletion of references from local variables. In the linear style, the policy is that variable references are destructive reads--i.e., cons(a,b) does not have to change the reference counts on a,b because these references have simply moved from local variables into storage. The linear style makes the creation and deletion of references explicit--duplicating a local reference requires calling the function dup, while deleting a local reference requires calling the function kill. The function dup increments the reference count of the target object, while the function kill decrements the reference count, and may also reclaim the object, recursively killing its component references.

While the linear style can help organize and reduce reference count updating, it does not eliminate the updating that results from simple traversal. Thus, a linear length function will still increment and decrement reference counts on every cell it traverses. To understand this better, we show the code for a linear length:

(defun length (x)			; Length function in linear style.
  (if-null x (progn (kill x) 0)		; Kill x reference to nil.
    (dlet* (((carx . cdrx) x))		; Dissolve x into carx, cdrx.
      (kill carx)			; [footnote 1] Kill reference to carx.
      (1+ (length cdrx)))))		; Our result is one more than length of cdrx.

When length is entered, the reference count on x is almost certainly >1, or else length will consume its argument and return it to the freelist! Let us assume that the reference count of the head of the list is 2 and the rest of the list is unshared. A standard dlet* adds references to carx, cdrx and then deletes the reference to x. The reference to carx is immediately killed, and the reference to cdrx, which now has a reference count of 2, is passed to a recursive invocation of length. At list end, the extra reference from x to nil is killed, and 0 is returned. Thus, length performs 4n count updates (each element of the list has its reference count updated twice) for an n-element list.

Since length eventually returns the reference counts to their initial state, what would happen if we simply avoided all updates? A problem arises because the second and succeeding nodes on the list have reference counts of 1--the dlet*'s will reclaim all of these nodes and put them onto the freelist! We must therefore somehow suspend reclamation while length is in progress. Suppose that we use two types of pointers--a normal pointer and a deferred increment pointer. The normal pointer acts as we have described above, while the deferred increment pointer signals that reclamation has been turned off. In other words, a deferred increment pointer is a pointer with a deferred reference count increment. If we copy a deferred increment pointer using dup, we simply copy the bits, because n copies of a deferred increment implies a deferred increment of n. Similarly, if we delete a deferred increment pointer using kill, we simply delete the reference, since the decrement associated with kill cancels the deferred increment. When cons(x,y) is performed on a normal pointer x, the reference count on x does not have to be adjusted since the reference is simply transferred from the local variable x to the cons cell. However, when cons(x,y) is performed on a deferred increment pointer x, the reference count on x does have to be incremented, since a deferred reference means a deferred increment, and components of data structures have to be normal. The treatment of dlet* for a deferred increment pointer x is also elegant: the deferred increment on x cancels with dlet*'s decrement of x, and the local variables carx, cdrx are also deferred, so their increments are deferred. So deferred increment pointers appear to elegantly achieve our goals--executing length on a deferred increment pointer x performs no reference count updates!

If deferred increment pointers are so efficient, why not eliminate normal pointers altogether? The problem with this proposal is that no storage can ever be recovered with only deferred increment pointers--i.e., we are back in the realm of tracing garbage collection. Consider kill(cons(x,y)). If cons returns a deferred increment pointer, then kill does nothing, and the cons cell is lost. Thus, cons itself must always return a normal (non-deferred) pointer. [footnote 2]

We therefore propose a system of pointers that are dynamically typed as normal/deferred-increment. [footnote 3] Unfortunately, this system still does not quite work, as the third function demonstrates:

(defun third (x)			; Linearly return the third cons in list x.
  (dlet* (((carx . cdrx) x)		; Dissolve the first cons.
          ((cadrx . cddrx) cdrx))	; Dissolve the second cons.
    (kill carx) (kill cadrx) cddrx))	; Kill first 2 elements, then return 3rd cons.

If third is passed a deferred increment pointer x, it will traverse the list x without changing reference counts until it gets to the third cons cell. It then returns a deferred increment pointer to this third cons cell. Unfortunately, the caller to third can now kill the list, in which case the third cons will also be reclaimed, because the increment implied by the deferred pointer to it will never have been performed. We can fix this bug by making the policy that only normal pointers can be returned from a function. With this policy, the returned value from third will be coerced back to a normal pointer by performing the increment implied by the deferred pointer. [footnote 4] This policy will work correctly, because it is somewhat more prompt than the correct "anchored pointer" scheme described below, but is also more expensive, because it does not defer reference count updates as long as possible. However, if we have but one pointer bit to give to the cause, this normal/deferred scheme is still better than a classical reference count scheme--e.g., functions like length "win completely" with just a one-bit distinction.

A reference count updating scheme lazier than deferred increment pointers can be obtained at the cost of additional pointer bits by means of anchored pointers. An anchored pointer is a deferred increment pointer with an additional component which indicates its anchor. One implementation might use an integer component indicating the level in the stack at which the object is anchored. When a functional cell pointed at by a deferred increment pointer x is dissolved into its components, they each inherit the level number associated with x. An anchored pointer not only indicates deferredness, but also indicates the extent during which this deferral is valid. If a deferred increment pointer is returned from stack level n, it must have a level number that is strictly less than n. In other words, when returning a deferred increment pointer from stack level n+1, one first checks to see that the level is <=n, and if not, the pointer is coerced to normal--i.e., the reference count increment is performed.

The implementation of the anchored pointer scheme can be confined to a dlet1 special form, which dissolves a single list cell and binds its components. We identify the anchor level with the index of this dlet1 in the stack--i.e., each dlet1 form has its own level number. If dlet1 is given a normal pointer, then it checks the reference count. If the count is 1, then the cell is unshared, and dlet1 has the only pointer to this cell. The car and cdr are bound as normal pointers, and the cell is recycled. If the count is >1, then the cell is shared, so the car and cdr are bound as anchored pointers with a level number being the current dlet1 level. Later, when dlet1 must return values, these values are checked for deferral and normalized if their level numbers are >= the current level. Finally, the reference count on the cell input to the dlet1 is decremented. The last case involves an anchored pointer input to dlet1. In this case, the car and cdr bindings are also anchored with the same level number as their parent pointer. Since the parent pointer must have been anchored at a level strictly less than the current level, the return values need not be checked at all, since they will either already be normal, or will not require normalization until a lower level.

Working with anchored pointers can be extremely efficient. In the Boyer Benchmark [Gabriel85], for example, the (functional) database of rewrite rules is bound at "top-level"--i.e., stack level 0. Thus, traversals of these data structures can be performed entirely by anchored pointers without updating reference counts--an extremely important characteristic for a shared-memory multiprocessor implementation of this benchmark, where multiple processors have to concurrently access these shared read-only rules.

If our anchored pointer scheme is used in conjunction with a traditional tracing garbage collector, the deferred reference counts may give the collector fits. In order for the reference accounting to balance, a cell bound by a deferred dlet1 should be given a negative deferred reference by the dlet1, and this negative deferred reference is finally normalized before the dlet1 returns its values by causing its reference count to be decremented. Only dlet1 and the garbage collector ever see these negative deferred references, however.

The inclusion of a level number with every pointer in a local variable is probably prohibitive on today's 32-bit architectures, because the additional information would double the number of registers required to store local variables. However, with the newer 64-bit architectures--e.g., MIPS, DEC--a 16-bit level number would not constrain addressability, and would avoid the requirement for additional registers. Checking and masking this level data, however, could present efficiency problems if the architecture is not organized for it.


The dlet1 special form described above works, but it may be more complex than strictly needed to solve the problem. What we really need is a form that is given a normal pointer and provides an anchored pointer for use only within a dynamic context. In other words, it puts the normal pointer into "escrow", and provides an anchored pointer for temporary use, and when the construct is exited and its values normalized, the escrowed pointer can then be dropped. We suggest something like the with-anchored-pointer special form:

(with-anchored-pointer (ap-name) (<expression>)	; binds ap-name to copy of expression value.
  << use ap-name within this dynamic extent body >>
  < returned-value-expression >)

The with-anchored-pointer form evaluates <expression> to either a normal pointer or another anchored pointer. If it evaluates to an anchored pointer, then with-anchored-pointer acts just like a let-expression. However, if it evaluates to a normal pointer, then the pointer is saved internally, and an anchored pointer copy of the normal pointer is made and bound to the name ap-name. The body of the with-anchored-pointer form is evaluated with this new binding, and the values returned by the body are normalized if they depend upon the escrowed pointer. The normal pointer is killed just before the dynamic scope is exited, which involves decrementing its reference count.

Constructs like with-anchored-pointer allow the introduction of anchored pointers into a static type system. Within the body of the construct, the name is bound to a value which is guaranteed to be anchored, and thus many run-time checks may be omitted.


Programmers have been using temporary pointers to objects without updating their associated reference counts for decades--e.g., the Unix file system is reference counted, but it distinguishes between "hard" and "soft" links; soft links (created by ln -s) are uncounted. [Gelernter60] uses the distinction of owned/borrowed pointers; [Lemaitre86] uses the concept of phantom pointers. Our concept of anchored pointers, however, generalizes these techniques and makes more precise the situations under which they are safe. Maclisp "PDL numbers" [Steele77] are similar to anchored pointers in that they encode a stack level and can be used safely only within a dynamic context. Since PDL numbers have no explicit reference count, however, the only way to normalize them is by copying them, which is done whenever they are stored into a more global environment or returned as a value of a function.

Other schemes involving 1-bit reference counts include [Wise77] [Stoye84] [Chikayama87] [Inamura89] [Nishida90] and [Wise92]. Anchored pointers can be particularly useful when a reference count cannot be incremented for some particular reason--e.g., it has only 1 bit [Wise77] [Chikayama87] [Inamura89] [Nishida90]. Our linear scheme for avoiding reference count updates has goals similar to those of [Deutsch76] [Deutsch80] and especially [Barth77] and [Park91]. In particular, our scheme defers reference count updating like Barth's and Park's in the hope of a decrement cancelling an increment, leaving no net change. However, ours is simpler than theirs--requiring only static linearity checking instead of global flow analysis. Furthermore, linear style makes avoiding count updates more likely.

Anchored pointers are similar to generational reference counts [Goldberg89] in that they both attempt to minimize count updates, both explicitly identify their generation and both use escape analysis. However, anchored pointers do not require a count field, and can be slightly lazier than generational reference counts. Anchored pointers are closely related to the schemes described in [Baker92CONS] and [Baker93Safe]. The scheme of [Baker93Safe], which has different goals and works for mutable data, stores its stack level numbers in the objects themselves rather than in the pointers to the objects. Some reference count schemes [Deutsch76] [Deutsch80] avoid reference count updating overhead during traversal by never counting references from local variables on the stack. Such schemes require that the stack be scanned before cells can be reclaimed. It is becoming apparent, however, that scanning the stack of a compiled language like C [Yuasa90] or Ada [Baker93Safe] is becoming increasingly difficult, making stack-scanning schemes infeasible.

Anchored pointers can also be used to improve the efficiency of tracing non-copying garbage collection. In a real-time non-relocating garbage collector such as [Baker92Tread], anchored pointers can be accessed without a read barrier. Furthermore, since an anchored pointer contains within it a proof that the target object is not garbage, anchored pointers need not be traced by the garbage collector. While tracing the pointer will reveal that the target object has already been marked, it is more efficient to not trace the pointer, so that the target never has to be paged into main memory or the cache. Anchored pointers are similar to object declarations in Kyoto Common Lisp [Yuasa90], because object declarations are not traced by the garbage collector. KCL object declarations are not safe, however, because they can refer to mutable data, and carry no proof of accessibility.

Anchored pointers are similar to weak pointers, in that they are not traced by a garbage collector, but are used for a completely different purpose. A weak pointer is expected to escape from the context in which its target is protected from reclamation, but when its target is reclaimed, the escaped pointer is cleared to nil, so the application will know that the object is gone. Anchored pointers, on the other hand, are used primarily for improving efficiency, and should have no user-visible semantic differences except for improved performance.

Anchored pointers can also provide a solution to the garbage collection problems of derived pointers [Ellis88] [Nilsen88] [Boehm91] [Diwan92]--i.e., pointers into the interior of a large object such as an array. [Ellis88] suggests that a derived pointer have the form <base-ptr,derived-ptr>; a normal pointer to an array element would have the form <base-ptr,index>. A derived pointer can be normalized into a normal pointer by computing index=(derived-ptr-base-ptr)/element-size. Our anchored pointers and their use in a dynamic extent can be considered a generalization of the static scheme proposed in [Boehm91]. Our anchoring scheme avoids the tracing of derived pointers in a non-copying collector.

Our reference count scheme is safe because it defers increments only within an extent where a cell cannot be reclaimed anyway. The non-prompt incrementation of reference counts means that a garbage collection that restores reference counts may restore the reference counts improperly. Our fix for this problem is to create negative references when a cons is dissolved by dlet1. This negative reference is normalized when dlet1 exits. Negative references are also cleaned up when unwinding the stack--e.g., for C's setjmp/longjmp, or for Common Lisp's catch/throw.

Our reference count scheme is intended to support the efficient implementation of functional (read-only) objects--perhaps implemented by means of hash consing [Baker92BB] [Baker92LLL]--and so there is no attempt to handle mutable objects or cycles. The linear style of programming essentially requires functions to be total--i.e., they must return with a value. The explicit killing of references, and the normalization traps of anchored pointers requires that functions return in the normal way--the catch/throw of Common Lisp, the reified continuations of Scheme and the setjmp/longjmp of C will all cause significant problems with this storage management scheme.


Many thanks to David Wise and Kelvin Nilsen for their constructive criticisms on this paper.


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Linear Lisp is a style of Lisp in which every bound name is referenced exactly once. Thus, each parameter of a function is used just once, as is each name introduced via other binding constructs such as let, let*, etc. A linear language requires work from the programmer to make explicit any copying or deletion, but he is paid back by better error checking during compilation and better utilization of resources (time, space) at run-time. Unlike Pascal, Ada, C, and other languages providing explicit deletion, however, a linear language cannot have dangling references.

The identity function is already linear, but five must dispose of its argument before returning the value 5:

(defun identity (x) x)

(defun five (x) (kill x) 5)           ; a true Linear Lisp would use "x" instead of "(kill x)"

The kill function, which returns no values, provides an appropriate "boundary condition" for the parameter x. The appearance of kill in five signifies non-linearity. (See below for a definition of kill).

The square function requires two occurrences of its argument, and is therefore also non-linear. A second copy can be obtained by use of the dup function, which accepts one argument and returns two values--i.e., two copies of its argument. (See below for a definition of dup). The square function follows:

(defun square (x)
  (let* ((x x-prime (dup x)))         ; Use Dylan-style syntax for multiple values [Shalit92].
    (* x x-prime)))

Conditional expressions such as if-expressions require a bit of sophistication. Since only one "arm" of the conditional will be executed, we relax the "one-occurrence" linearity condition to allow a reference in both arms. [footnote 5] One should immediately see that linearity implies that an occurrence in one arm if and only if there is an occurrence in the other arm. (This condition is similar to that for typestates [Strom83] [Strom86]).

The boolean expression part of an if-expression requires more sophistication. Strict linearity requires that any name used in the boolean part of an if-expression be counted as an occurrence. However, many predicates are "shallow", in that they examine only a small (i.e., shallow) portion of their arguments (e.g., null, zerop), and therefore a modified policy is required. We have not yet found the best syntax to solve this problem, but provisionally use several new if-like expressions: if-atom, if-null, if-zerop, etc. These if-like expressions require that the boolean part be a simple name, which does not count towards the "occur-once" linearity condition. This modified rule allows for a shallow condition to be tested, and then the name can be reused within the arms of the conditional. [footnote 6]

We require a mechanism to linearly extract both components of a Lisp cons cell, since a use of (car x) precludes the use of (cdr x), and vice versa, due to the requirement for a single occurrence of x. We therefore introduce a "destructuring let" operation dlet*, which takes a series of binding pairs and a body, and binds the names in the binding pairs before executing the body. Each binding pair consists of a pattern and an expression; the expression is evaluated to a value, and the result is matched to the pattern, which consists of list structure with embedded names. The list structure must match to the value, and the names are then bound to the portions of the list structure as if the pattern had been unified with the value. Linearity requires that a name appear only once within a particular pattern. Linearity also requires that each name bound by a dlet* binding pair must occur either within an expression in a succeeding binding pair, or within the body of the dlet* itself. Using these constructs, we can now program the append and factorial (fact) functions:

(defun lappend (x y)				; append for "linear" lists.
  (if-null x (progn (kill x) y)			; trivial kill
    (dlet* (((carx . cdrx) x))			; disassociate top-level cons.
      (cons carx (lappend cdrx y)))))		; this cons will be optimized to reuse input cell x.

(defun fact (n)
  (if-zerop n (progn (kill n) 1)		; trivial kill.
    (let* ((n n-prime (dup n)))			; Dylan-style multiple-value syntax.
      (* n (fact (1- n-prime))))))

Below we show one way to program the kill and dup functions.

(defun kill (x)		; Return no values.  Expensive way to decrement a reference count.
  (if-atom x (kill-atom x)
    (dlet* (((carx . cdrx) x))
      (kill carx) (kill cdrx))))

(defun dup (x)		; Return 2 values.  Expensive way to increment a reference count.
  (if-atom x (dup-atom x)
    (dlet* (((carx . cdrx) x))
      (let* ((carx carx-prime (dup carx)) (cdrx cdrx-prime (dup cdrx)))
        (share-if-possible (cons carx cdrx) (cons carx-prime cdrx-prime))))))   ; reuse input.

[Footnote 1] Following logic languages, we could use syntax like dlet* (((_ . cdrx) x)) to immediately kill the car of x.

[Footnote 2] The Deutsch-Bobrow reference count scheme [Deutsch76] [Deutsch80] does not count references from local variables, and thereby avoids these count updates. Their scheme requires a stack scan before reclaiming storage, however, which scan is nearly impossible to do on stacks formatted by optimizing compilers for modern RISC architectures.

[Footnote 3] The normal/deferred-increment distinction is essentially identical to the real/phantom distinction of [Lemaitre86] and the owned/borrowed distinction of [Gelernter60].

[Footnote 4] [Lemaitre86] calls this normalization operation "materialization".

[Footnote 5] Any use of parallel or speculative execution of the arms of the conditional would require strict linearity, however.

[Footnote 6] Although this rule seems a bit messy, it is equivalent to having the shallow predicate return two values: the predicate itself and the unmodified argument. This policy is completely consistent with linear semantics.