Applications of Recursively Defined Data Structures.
Lloyd Allison, Australian Computer Journal, 25(1), pp.1420, arxiv:2206.12795, 1993
Abstract: A circular program contains a data structure whose definition is selfreferential or recursive. The use of such a definition allows efficient functional programs to be written and can avoid repeated evaluations and the creation of intermediate data structures that would have to be garbage collected. This paper uses circular programs in various ways, to implement memostructures and explicit searchtrees to hold solutions to constraintsatisfaction problems.
keywords: circular program, functional programming, list, recursion, tree.
Introduction.
A circular program contains a data structure whose definition is selfreferential or recursive. Such a program cannot be written in a conventional, strict, imperative programming language but it can be written in a functional language employing lazy evaluation[6,8] or call by need.
general schema: let rec ds = f(ds)  ds is some data structure eg. let rec posints = 1.(map succ posints)  list of all +ve integers
The list posints contains all the positive integers 1.(2.(3. ...)) or [1,2,3,...]. It begins with 1 and continues with the result of applying the successor function succ to each element of posints itself. Successor applied to the first element gives the second element, 2, and so on. It is the definition of the value of the data structure posints, not just its type (list), being recursive that makes this a circular program.
Under lazy evaluation, an expression is not evaluated unless it is needed. In particular, the right hand side of a definition, and the actual parameter of a function, are not evaluated until they are needed  if they are needed. When an expression is evaluated, the value is remembered to avoid recomputation later. (The conditional, `if', is the only nonstrict or lazy operator in many imperative languages.) Lazy evaluation permits recursive definitions of data structures and also allows some computations with infinite data structures. All of a potentially infinite data structure can be defined although only a finite part may be evaluated. If only a bounded part were defined and evaluated, a copy would have to be made if it had to be extended, wasting time and space, because functional languages do not permit sideeffects. Circular programs can, in certain cases, have it both ways  an expanding data structure with sideeffectfree programming. As a bonus, infinite data structures are sometimes easier to define because boundary cases are simpler or absent. Although posints represents an infinite list, a program using posints does not loop unless an attempt is made to print or otherwise evaluate all of the list. The program terminates provided that only finite parts of such structures are manipulated.
It is sometimes necessary to distinguish between the data structure as seen by the programmer and as implemented by the language system. If the definition simply incorporates the data structure directly, as in ones below, then a cyclic structure of cells and pointers is created in the computer memory.

let rec ones = 1 . ones  = [1,1,1, ...] data structure in memory: ones: > 1.> ^   v   <
Many functional programs compute their final result in stages, some data structure being operated on, often in small steps or passes, by functions such as map, filter, reduce, and so on. Each step produces an intermediate data structure which is eventually discarded and collected as garbage at some cost. Bird[3] used circular programs in program transformations to convert multipass algorithms into singlepass algorithms. He attributed knowledge of circular programs to Hughes and to Wadler and the technique is so useful that it has probably been discovered several times.
The objective of the paper is to promote this useful functional programming technique. The examples given here construct lists and trees in new ways. They are used to define memostructures and explicit searchtrees which remove the need to repeat tests in certain constraintsatisfaction problems. They have all been run on a small lazy interpreter which was instrumented to record program behaviour and they perform as predicted. The notation used is from a hypothetical, "generic" functional language and is explained when required.
Circular Lists.
The ones and posints examples are amongst the simplest circular programs. To introduce the technique more fully, some nontrivial but routine examples on lists are given here. A popular example computes the Hamming numbers. These are all numbers of the form 2^{i}×3^{j}×5^{k}, i, j, k>=0, i.e. 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, ... . In this section various circular programs are derived from the Hamming numbers program.
The simplest program for the Hamming numbers is the following:

let rec Hamming = 1 . (merge3 (map (× 2) Hamming) (map (× 3) Hamming) (map (× 5) Hamming) )
This is an example used in many books and papers[4,9,13]. Hamming is a selfreferential list. It begins with 1 followed by the result of merging three lists. These are the results of multiplying all members of Hamming by 2, 3 and 5 respectively giving [2,...], [3,...] and [5,...]. The second Hamming number is thus 2, so the three lists are [2,4,...], [3,...] and [5,...], which allows the computation to proceed to the next step. Many duplicate numbers are produced, for example 6 = 2×3 = 3×2, and all but one copy must be removed by the merge function. It is well known that this inefficiency can be removed by ensuring that the factors in a product are combined in ascending order:
let rec a = 1 . (map (× 2) a)  [1,2,4,8,...] and b = 1 . (merge (tl a) (map (× 3) b) )  [1,2,3,4,6,8,9,12,...] and Hamming = 1 . (merge (tl b) (map (× 5) Hamming) )
Note that hd (head) returns the first element of a list and tl (tail) returns a list minus its first element. A further operator null can be used to test if a list is empty.
The list a holds all powers of 2 and is defined in a very similar way to posints. List b holds all products of 2 and 3. No duplicates are produced because powers of 2 are multiplied by powers of 3 and then by powers of 5, in order. Consequently a simpler version of merge can be used that does not need to deal with duplicates.
It is a natural exercise to generalise the Hamming problem to find all products of an arbitrary list of factors. The factors are assumed to be coprime and in ascending order.

let rec products [] = [1]   no factors products (f.fs) =  at least one factor f let rec m = 1 . (merge (map (× f) m) (tl (products fs)) ) in m
If there are no factors the list [1] is returned. If there are factors, the products of the first factor f and of all other factors in fs must be combined. The result list m is selfreferential. First, multiplying any member of m by f is itself a product  map (× f) m. Second, products of members of fs are also members of m although the leading 1 is not needed  tl (products fs). Merging these two lists and putting 1 on the front gives m. It is easy to see that any valid product must be produced by this process.
The last program works correctly on finite lists of factors. It will not work when given an infinite list of factors because the merge operation requires the head of two lists before it can produce any output. For an infinite list of factors this would require the heads of an infinite number of lists to be assembled which is impossible. This drawback can be overcome by recognising that the second value in m must be the smallest factor f itself.

let rec products (f.fs) =  NB. not (null fs) let rec m = 1 . f . (merge (map (× f) (tl m)) (tl (products fs)) ) in m
Circular Trees.
The previous section gave examples of circular programs over lists. Here trees are defined in a similar way and used as memostructures to store the results of functions so that later calls can access them quickly without recomputation. In general, one or more functions and structures are defined using mutual recursion.

general schema: let rec  mutually recursive ds = g ds f  data structure(s) and and f x = h x ds f  function(s)

let rec slowfib n = if n<=2 then 1 else slowfib(n2)+slowfib(n1)

let fastfib n = let rec f n a b = if n=1 then b else f (n1) b (a+b) in f n 0 1
It is possible to define a circular program that builds a list of the Fibonacci numbers:

let rec fiblist = 1.1.(f fiblist)  [1,1,2,3,5,...] and f (a.t) = (a+(hd t)) . (f t)

let fib = let rec fiblist = 1.1.(f fiblist)  memo list and f (a.t) = (a+(hd t)) . (f t)  f builds fiblist and find n = index n fiblist  get nth element of fiblist in find
The list fiblist is a structure storing old results of fib n. On a first call of fib n the first n elements of fiblist are constructed. On a second call the result is just looked up in fiblist; the second call is faster but is still O(n) as index takes O(n) time. Actually, the result is looked up on the first call too, but it does not yet exist and so the list is built to the required length. Bird[2] discusses the use of arrays to store past values in the process of deriving the fast Fibonacci program in an imperative language. Hughes[10] describes a system that automatically stores past values of functions for fast recall; his system is implicit whereas the memostructure here is explicit.
If a tree of Fibonacci numbers were built, the results of later calls could be looked up in O(log n) time. The nodes of a complete binary tree can be numbered so that the children of node n are 2n and 2n+1:

1 . . . . 2 3 . . . . 4 5 6 7 . . . . ...
The value of fib n can be stored at node number n:

1 . . . . 1 2 . . . . 3 5 8 13 . . . . ...
Given this tree and an integer, for example n = 6_{10} = 110_{2}, the binary digits of n indicate whether to take left or right subtrees in locating the n^{th} node and this can be done in O(log n) time. The bits of n are read from the second most significant bit to the least significant bit; a 1 indicates go right and a 0 indicates go left. Therefore 110_{2} implies: start at the root, go right and then left. We first define an infinite binary tree type:

datatype tree = fork int tree tree  infinite binary tree type let element (fork e l r) = e  extract the element value and left (fork e l r) = l  extract left subtree and right (fork e l r) = r  extract right subtree
Fork is a constructor that builds a new tree given an element and left and right subtrees. Element, left and right return the components of a tree.
let fib = let rec fibtree = fork 1 (fork 1 (build 4) (build 5))  memo tree, (build 3)  fibtree :tree and build n = fork (f(n2)+f(n1))  build & f (build(2*n)) (build(2*n+1))  construct fibtree and f n = lookup n element  return fib n and lookup 1 g = g fibtree   lookup decodes n lookup n g =  n>1 lookup (n div 2) (g o (if even n then left else right)) in f
Graphically, fibtree is defined to be:

1 . . . . 1 build 3 . . . build 5 . build 4
The values in nodes one and two are both 1 and are provided to enable node three to be built. This allows node four to be built and so on. The function lookup extracts the n^{th} number from the tree. It does this by constructing a function g to follow left or right links according to the bits in n as described. Functional composition `o' is used to link the desired sequence of element, left and right operations together and these are finally applied to fibtree. ((p o q)(x) = p(q(x)))
Assuming fib has been previously called with a parameter greater than n, a second call fib n takes O(log n) time to scan down fibtree. It might appear that the first such call would take exponential time because of the two calls to f within build but this is not the case. The call f(n2) causes the tree to be evaluated and built up to node n2. The call f(n1) only causes one additional node to be evaluated using f(n3) and f(n2) which are just looked up, their corresponding nodes having been built already. On the first call, there are O(n) calls on build, f and lookup the latter being logarithmic, the total time taken is O(n log n). There is thus an increase in cost from O(n) to O(n log n) on first calls but a reduction from O(n) to O(log n) on subsequent calls over the "fast" Fibonacci program. There is also the cost of space to store the tree to be considered.
A subtle point should be noted: Our program let fib = let rec ... in f binds fib to f which has fibtree in its environment so fibtree persists for as long as fib does and is not recomputed. If we carelessly defined let fib2 n = let rec ... in f n then fibtree would persist for only as long as a call to fib2 remained unevaluated and would be recomputed on each call. Some optimising compilers would undo this unfortunate effect by effectively converting fib2 into fib. However this is a difficult issue because a programmer might deliberately write a function having the form of fib2 because he or she needs a temporary data structure but wants it to be destroyed to avoid tying up space.
It is natural to ask if the log(n) factor in the costs can be removed but it is caused by the use of a tree rather than an (unbounded) array with O(1) indexing. In an imperative language, and in some functional languages, one might use an array instead of the tree. However this would place a limit on the size of n.
If it is necessary both to have random access to the Fibonacci numbers in log(n) time and to have sequential access then it may be convenient to derive a list from the tree in breadthfirst order. Since the tree is complete and infinite the list is particularly simple to create:
let rec fibtree = as before and build = ... and f = ... and lookup = ... and fiblist = map element nodes  elements in breadthfirst order and nodes = bfirst fibtree  (sub)trees in breadthfirst order and bfirst t =  breadthfirst traversal of t let rec q = t . (traverse q) and traverse ((fork e l r) . q2) = l . r . (traverse q2) in q
Function bfirst returns a list or queue q of the subtrees of a tree t in breadthfirst order. The queue begins with t itself and the auxiliary function traverse produces the rest of the queue. Function traverse examines the first element in q and adds its subtrees to the end of q, i.e. in the second and third positions. This is repeated for successive elements of q. The definition of the queue q is selfreferential. Since q naturally grows as the tree is scanned, and since it is also infinite, it is hard to see how q could be created efficiently without a circular program.
The lists nodes and fiblist respectively contain the subtrees and the elements of fibtree in breadthfirst order. Accessing the n^{th} element of fiblist also causes fibtree to be evaluated to node number n.
Circular Search Trees.
Many search problems or constraintsatisfaction problems require finding a sequence of values <a, b, c, ...>, or just abc..., that satisfies certain constraints. A search program explores the search space, building an implicit or explicit tree of (partial) solutions. If the constraints are uniform in a certain sense then the solution tree may be defined recursively and explicitly. The advantage is that no test is performed twice as the results of previous tests are available from the structure of the tree. This is valuable if the cost of performing tests outweighs the cost of building and keeping the tree.
The uniformity required covers two conditions. Firstly, it must be possible to build all long solutions by extending short solutions. Secondly, the constraints on an element in the sequence must involve other members of the sequence only in ways that depend on their relative, not absolute, positions in the sequence. These conditions hold in many problems.
A suitable nary searchtree, in which a node contains an element of type 't, a subtree and the siblings of the node, can be defined as follows:

datatype tree 't = empty  node 't (tree 't) (tree 't)
Note that 't is a type parameter  an arbitrary type. There are two cases to tree  the empty tree and a node  separated by . Siblings are linked together via the third component of a node.
As an example, the complete infinite tree over {1, 2, 3} can be defined as:

let rec three = node 1 three (node 2 three (node 3 three empty))  :tree int
A simple generalisation of this example allows trees to be built over the range [1..n]:
let build n = let rec T = toplevel 1  :tree int and toplevel m = if m>n then empty else node m (f T) (toplevel (m+1)) and f T = T in T
The function toplevel builds the nodes at the top level of the tree and f fills in the subtrees. Here f is just the identity function and as such it is redundant but it is included to give a general schema. The program above creates a finite cyclic data structure. If we wanted to expand out the cyclic structure, into an infinite copy, f could be redefined as follows:

f empty = empty  f (node a subtree sibs) = let others = f sibs in node a (f subtree) others
Solutions to the various problems discussed below are formed only by redefining f in variations on the above. Apart from the changes that this entails the schema is unaltered in each case.
Permutations.
A common method of generating permutations is to extend partial permutations, beginning with the empty sequence. If abcde is a partial permutation it can be extended with X to abcdeX provided that X differs from a, b, c, d and e. Equivalently, it can be extended provided that bcdeX is a partial permutation and provided that X differs from a. Note that bcde is already a partial permutation because abcde is. If a tree is used to hold the permutations then bcdeX, being shorter than abcdeX, must occur in the tree at the previous level, if it is indeed a partial permutation. If we read permutations as paths from the root of the tree, and identify a node with the path to it, the subtree of abcde is a pruned version of the subtree of bcde with all occurrences of a filtered out or banned. We call bcde the shadow of abcde. The shadow of bcde is cde and so on. Coding these ideas into function f of the schema in the previous section gives the following program:
let build n = let rec T = toplevel 1 and toplevel m = if m>n then empty else node m (f m T) (toplevel (m+1)) and f banned empty = empty   f banned shadow f banned (node a subtree sibs) = let others = f banned sibs in if a = banned then others  prune a's subtree else node a (f banned subtree) others  no pruning in T
Function f has gained an extra parameter for the banned element and performs a filtering operation on the shadow tree. A single test, a=banned, tells if a permutation can be extended with a given value, all other exclusions being implicit in the tree structure.
As the permutation tree is traversed it is gradually evaluated. If for example the first permutation, 123, from build 3 were printed, the evaluated portion would be:

. . . . . . . . . . 1 2 3 . . . . . . . ? . . . . 2 1 3 1 2 . . . . . . ? . ? ? 3 1
The shadow of a path grows with the path, one step behind it. The structure of the tree stores the results of many past tests so that only a single extra test is performed to add a new node. (This is not a big issue in permutation generation but there are cases where it is.) Note that Topor[12] has examined the space complexity of functional programs for generating permutations represented as linear linked lists.
NQueens.
The well known nqueens problem is to place n queens on an n×n chess board so that no two queens threaten each other. Each queen must be on a separate row, column and diagonal and this property is an invariant that must be maintained as partial solutions are extended. The fastest imperative solutions[11] are based on permutation generators. A board is represented by the permutation of rows that the queens on the columns occupy. This representation automatically ensures the separate row and column parts of the invariant. Here we observe that a partial solution abcde can be extended to a partial solution abcdeX if and only if bcdeX is also a partial solution and a and X are on separate diagonals and rows. By using shadows, X need only be tested against a's diagonals as the results of the other diagonal tests against other queens are already encoded in the shadow tree and do not need to be repeated. Again, the required program is a modification of the general schema with f redefined. Function f gains a new parameter col, being the current column number.
let build n = let rec T = toplevel 1 and toplevel m = if m>n then empty else node m (f 1 m T) (toplevel (m+1)) and f col banned empty = empty  f col banned (node a subtree sibs) = let others = f col banned sibs in if member banned [a, a+col, acol] then others  prune else node a (f (col+1) banned subtree) others  no prune in T
The standard function member tests the membership of an element in a list. Note that the test member banned [a, a+col, acol] is an amalgam of the old permutation test, a=banned, and the new diagonal test.
Irreducible or Good Sequences.
Axel Thue[7] defined the notion of an irreducible sequence in a series of papers in the period 19061914. Dijkstra[5] later called these `good sequences' and used them in an exercise in structured programming. A sequence over the alphabet {1, 2, 3}, or in general over {1,...,n}, is irreducible if and only if it contains no adjacent subsequences that are identical. For example, 1213121 is irreducible but 12132131 is not because 213 is immediately repeated.
It is easy to see that (i) a sequence abcdeX of even length is irreducible iff the shorter sequences abcde and bcdeX are irreducible and the two halves abc and deX are unequal and (ii) a sequence abcdefX of odd length is irreducible iff abcdef and bcdefX are irreducible. This enables a circular program to be written for a tree representing all the irreducible sequences. For example, the shadow of abcde (of odd length) is bcde. Assuming that abcde is irreducible, abcdeX is irreducible if and only if bcdeX is irreducible, if and only if X is a descendant of bcde.
let build n = let rec T = toplevel 1 and toplevel m = if m>n then empty else node m (f 2 [m] T) (toplevel (m+1)) and f len seq empty = empty  f len seq (node a subtree sibs) = let others = f len seq sibs and seq2 = a.seq in if even len & repeated (len/2) seq2 then others prune else node a (f (len+1) seq2 subtree) others don't prune in T
A new parameter seq carries the particular sequence forward as f descends through the tree. When the length len is even, a test is made that the sequence a.seq is not the concatenation of two sequences of length len/2. Function repeated performs this test in O(len) time and has an obvious definition. It is not necessary to test for any shorter repeats. These are implicitly ruled out by the use of the shadow to generate subtrees. The test would be more complex without this information. That portion of the tree that is evaluated in order to print the first irreducible sequence of length five is shown below:

. . . . . . . . . . . . . 1 2 3 . . . . . . . . 2 3 1 1 . . . . . . 1 1 3 . . . . 3 1 . . 1
The shadow of 12131 is 2131 whose shadow is 131 whose shadow is 31 whose shadow is 1.
Conclusions.
Programmers are familiar with recursive functions but recursive or selfreferential data structures used in circular programs are rare. Circular programs are very powerful enabling many infinite structures to be efficiently defined. They often remove the need for intermediate structures and for repeated calculations. They can be used safely provided that later values depend only on earlier ones in the structure. Memostructures can be formed by a data structure and function defined using mutual recursion. Explicit circular searchtrees can reduce the number of tests performed in constraintsatisfaction problems. As usual, there is a tradeoff of time against space.
References.
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 [11] J. S. Rohl. A faster lexicographical N queens algorithm. Information Processing Letters, 17, 231233, 1983.
 [12] R. W. Topor. Functional programs for generating permutations. Computer Journal, 25(2), 257263, 1982.
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Mathematical Logic and Programming Languages,
C.A.R.Hoare and J.C.Shepherdson (eds), Prentice Hall 2954, 1984.
[14] L. Allison, Applications of Recursively Defined Data Structures, arxiv:2206.12795, June 2022,  appeared in the Australian Computer Journal (ISSN 00048917), vol.25, no.1, pp.1420, February 1993. The journal was published by the Australian Computer Society from 1967 to 1999.
Also see [Functional Programming] and [λcalculus examples].