this section outlines the main choices when designing loops. recursion, iteration, and higher-order forms like map or fold are interchangeable; the key is understanding when and how to use each based on termination, state, and structure.

each loop construct can be analyzed along the following axes:

• termination: how and when the loop stops

  • explicit condition (e.g. index bound, sentinel value)
  • implicit in recursion via base case
  • bounded iteration over structure (e.g. list, range)

• state management: what data is updated across iterations

  • single accumulator (e.g. in fold)
  • multiple accumulators or local bindings
  • in-place mutation (imperative) vs structural updates (functional)

• composition style: how the loop integrates with surrounding code

  • inline loop with side effects
  • returned data structure
  • higher-order combinators (map, filter, fold)
  • tail-recursive self-call

• control flow abstraction: how much of the control is abstracted

  • full control: explicit loop or recursion

  • partial abstraction: fold, unfold, etc.

  • total abstraction: lazy sequences, generators

• map: applies a function to each element and returns a new structure

  • one input, one output
  • no internal state across iterations

• fold/reduce: accumulates a result by applying a function across elements

  • single state variable updated per step
  • suitable for scalar summarization or structural transformation

• unfold: generates a structure from a seed by repeated transformation

  • inverse of fold
  • useful for stream generation or constructing lists recursively

• custom recursive loop: uses a named recursive binding (e.g. let loop)

  • full control over state and branching

  • allows multiple outputs and mixed control flows

in scheme

(let loop ((rest (list 1 2 3)))
  (if (null? rest) rest
    (cons (+ 1 (car rest)) (loop (cdr rest)))))

this implements a basic map operation by:

• checking termination via (null? rest)
• recursively calling loop with (cdr rest)
• producing a new list via cons and (+ 1 ...)

this style generalizes easily - for example, by accumulating additional values, branching conditionally, or logging state.

• identify the stop condition early

  • in recursion: base case
  • in loops: compare indices, flags, or structure traversal

• determine what state changes per iteration

  • single or multiple variables
  • structural updates or in-place mutation

• decide how much abstraction is acceptable

  • simple transformation -> use map, fold
  • multi-output or conditional branching -> use explicit recursion or let loop

• consider composition

  • is the result fed to another transformation?
  • does the loop perform side effects or produce a return value?