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Continuation-passing style

Continuation-passing style (CPS) is a programming technique used primarily in , where s are transformed to accept an additional argument known as a continuation—a that represents the remaining to be performed upon receiving the function's result. Rather than directly returning values, s in CPS invoke the continuation with their computed result, thereby making explicit and avoiding implicit returns or stack-based mechanisms. This style ensures that the order of evaluation can be precisely controlled without relying on the underlying environment's . The origins of CPS trace back to the early in the development of formal semantics for programming languages. John C. Reynolds introduced the concept in his 1972 paper on definitional interpreters for higher-order programming languages, where was employed to eliminate dependencies on the order of in interpreters, allowing for order-independent while preserving higher-order features. Building on this, and Christopher Wadsworth formalized in 1974 as state transformations in , enabling the modeling of jumps and control structures like exceptions in pure functional settings by representing the "rest of the program" as a discardable continuation. Key features of CPS include its ability to express arbitrary control flows, such as non-local exits or coroutines, within a purely functional without side effects. For instance, in , a simple arithmetic operation like might be rewritten as add_cps(x, y, k) = k(x + y), where k is the that processes the sum. This transformation, often called CPS conversion, is systematic and can be applied mechanically to direct-style code, resulting in a form that is easier to analyze or optimize, particularly for tail calls which become direct jumps. CPS has found wide application in compiler design as an , facilitating optimizations like conversion and defunctionalization. In , it underpins the implementation of the call/cc (call-with-current-continuation) , which captures and manipulates at runtime; CPS also serves as a key intermediate form in compilers for languages like . More broadly, CPS influences modern paradigms, including the for handling computational effects in functional languages and asynchronous programming patterns in , where promises can be viewed as resembling delimited . Its mathematical foundations connect to , notably the , underscoring its role in embedding classical control into via translations.

Fundamentals

Definition

Continuation-passing style (CPS) is a technique in which are structured to accept an extra argument known as the , representing the remaining computation to be performed after the yields its result. Instead of returning values directly via implicit mechanisms like the , a in CPS explicitly invokes its argument with the computed result, thereby making the program's transparent and compositional. In this style, a that normally takes input x and produces output v is transformed into a form f(x, k), where k denotes the —a that accepts v and proceeds with the rest of the program, typically by calling k(v). This approach ensures that all potential return points and control transfers are represented as values passed explicitly. A key advantage of is its ability to eliminate hidden stack-based returns, allowing for explicit management of computation sequences and enabling optimizations like tail-call elimination in supporting environments.

Direct style comparison

In direct-style programming, functions implicitly return values through the call stack, allowing control to flow naturally from the caller to the callee without explicitly managing post-computation actions. This approach relies on the runtime environment to handle evaluation order and returns, making it intuitive for sequential code but dependent on the underlying , such as call-by-value or call-by-name. Continuation-passing style (CPS) transforms this implicit into an explicit mechanism by converting direct-style code such that every receives an additional argument representing its —a that processes the result once completes. For instance, a direct-style f(x) that returns a value becomes f(x, k) in CPS, where k is the , often starting as the for the top-level result. This transformation, formalized in the between direct and CPS semantics, ensures that evaluation order is independent of the host language's strategy, as is passed explicitly via continuations. The explicit handling of continuations in CPS offers advantages over direct style, such as facilitating the implementation of advanced control features like coroutines by treating pending computations as reified objects that can be suspended or resumed. However, CPS introduces drawbacks, including increased code verbosity due to the pervasive passing of continuation arguments and potential loss of tail recursion optimization benefits unless the implementation explicitly supports tail calls to continuations. Tail calls, which enable loop-like efficiency in both styles, are preserved in CPS through its inherent structure of final continuation invocations. To illustrate the contrast, consider a simple . In direct style: 
add(a, b) = a + b
This returns the sum directly via the call stack. In CPS: 
add(a, b, k) = k(a + b)
Here, the continuation k receives the sum and determines the next step, making explicit at the cost of additional parameters.

Examples

In

In , the primitive call-with-current-continuation, commonly abbreviated as call/cc, captures the current of a and passes it as an argument to a given . This takes one argument, a of one argument, and invokes it with the captured , which is itself a first-class that, when applied to a value, restores the control context at the point of capture and delivers that value as the result of the call/cc expression. 's treatment of as first-class objects enables their and manipulation, facilitating the implementation of (CPS) by allowing explicit passing and invocation of without relying on implicit stack management. A straightforward example of CPS in is the function, which avoids traditional returns by passing an explicit to handle the result. The CPS version is defined as follows: 
(define fact-cps
  (lambda (n k)
    (if (= n 0)
        (k 1)
        (fact-cps (- n 1) ([lambda](/page/Lambda) (v) (k (* n v)))))))
Invoking (fact-cps 5 ([lambda](/page/Lambda) (v) v)) computes 120 by chaining nested continuations, ensuring tail-recursive calls that prevent in implementations supporting proper tail calls. This transforms recursive computations into iterative ones mediated by continuations, leveraging Scheme's functional nature. The call/cc also supports non-local control transfers, such as implementing handling akin to try-catch mechanisms. For instance, consider a that may abort on : 
(define (divide x y)
  (call/cc
    (lambda (escape)
      (if (= y 0)
          ([escape](/page/Escape) 'division-by-zero)
          (/ x y)))))
Here, if y is zero, ([escape](/page/Escape) 'division-by-zero) invokes the captured , immediately returning 'division-by-zero from the call/cc expression and bypassing the of the computation, thus providing a dynamic route without unwinding the manually. This capability underscores how Scheme's first-class continuations empower to model exceptional explicitly.

In Haskell

In Haskell, continuation-passing style (CPS) provides a mechanism for explicitly managing in pure functional programs, where functions accept an additional continuation representing the computation to perform with the result, rather than returning values directly. This approach is particularly useful in Haskell's statically typed environment, enabling optimizations without relying on runtime continuation capture mechanisms. A in typically has the type signature a -> (b -> r) -> r, where a is the input type, b is the intermediate result type passed to the , and r is the overall result type of the enclosing computation. An illustrative example of CPS in Haskell is a continuation-based reversal, which can be implemented recursively to ensure tail and linear-time performance: 
haskell
reverseCPS :: [a] -> ([a] -> r) -> r
reverseCPS []     k = k []
reverseCPS (x:xs) k = reverseCPS xs (\ys -> k (ys ++ [x]))
This formulation passes the accumulated reversed prefix ys to the outer k, avoiding stack growth through 's tail-call optimization in CPS form. CPS also underpins 's handling of effects, particularly through monad transformers like ContT, which sequence computations in continuation-passing style within broader monadic , allowing delimited control over effectful operations.

As first-class objects

In languages supporting continuation-passing style (CPS) with first-class s, such as , a can be captured and treated as a value that is assignable to variables, storable in data structures, or passed as arguments to other s. This is typically achieved through a like call-with-current-continuation (abbreviated call/cc), which packages the current as an escape procedure and passes it to a provided . The resulting is a first-class procedure with unlimited extent, meaning it can be invoked at any time to resume the captured computation from the point of capture, abandoning the current in favor of the saved one. For instance, in , a can be stored for later invocation as follows: 
(define cont (call/cc ([lambda](/page/Lambda) (k) k)))
Here, cont holds the , and invoking (cont value) later resumes the computation by passing value to the original context, effectively returning value as if from the call/cc point. This demonstrates how continuations become manipulable objects rather than hidden runtime mechanisms. Treating continuations as first-class objects enables explicit manipulation for advanced control structures, such as implementing threads via multiple interleaved continuations, exceptions through non-local jumps to handler continuations, and by reinstating prior states in search algorithms. In contrast, direct-style programming keeps continuations implicit and non-manipulable, limiting programmers to the language's built-in without such flexibility. Delimited continuations offer a bounded variant of this capability, restricting captures to a specified context rather than the full program .

Control Flow

Tail calls

In continuation-passing style (CPS), a tail call occurs when a function's final action is to invoke a continuation with its result, leaving no additional computation to perform in the current activation frame. This structure explicitly represents the "rest of the computation" as the continuation parameter, ensuring that passes directly to the successor without intermediate steps. Tail call optimization (TCO) leverages this property by allowing compilers to reuse the current frame for the invocation, preventing stack growth and converting recursive calls into iterative loops with space usage. For instance, in a recursive function written in , each step accumulates the result and tails to the next , enabling the optimizer to avoid allocating new frames and thus supporting unbounded without overflow. Consider the following pseudocode for a CPS version of a list summation function: 
define sum-cps xs k =
  if null? xs then
    k 0
  else
    let x = [car](/page/Car) xs
    in sum-cps ([cdr](/page/CDR) xs) ([lambda](/page/Lambda) y . k (+ x y))
Here, the recursive call to sum-cps is in tail position, passing a new that adds the current element; TCO transforms this into a , preserving space across iterations. The CPS transformation process preserves tail positions from direct-style code by mapping them directly to applications, ensuring no extra stack frames are introduced for proper tail calls. This alignment facilitates optimizations like β- and η-reductions, where tail simplify to direct jumps in the compiled code.

Delimited continuations

Delimited continuations represent a restricted variant of continuation-passing style where the captured is limited to the portion from the current execution point up to a dynamically specified boundary, or , rather than encompassing the entire remaining execution as in undelimited continuations. This delimitation allows for more precise control over flow, enabling the programmer to abstract and manipulate only a bounded segment of the . The originates from efforts to refine the expressive power of continuations while mitigating their potential for unintended global effects. In languages like Scheme, delimited continuations are typically implemented using the operators shift and reset. The reset operator establishes a delimiter, evaluating its body within a bounded context and ensuring that any captured continuations do not extend beyond this point; it effectively reifies the computation inside it as a value, discarding the outer context upon completion. The shift operator, in contrast, captures the delimited continuation up to the nearest enclosing reset and binds it to a parameter (often denoted as k), which is then passed to its body expression; this body can invoke k multiple times or compose it with other computations, but invocations respect the delimiter and do not affect the surrounding code. These operators work in tandem to provide first-class, composable representations of partial continuations that behave like ordinary functions. A illustrative example in is the expression (reset (+ 1 (shift k (k 2)))), which evaluates to 3. Here, shift captures the k corresponding to the of 1 within the reset boundary; invoking k with 2 applies this partial , yielding 3, while the prevents any further propagation of control outside the reset. This demonstrates how shift abstracts the local context for reuse without altering the broader program structure. Delimited continuations offer several advantages over full, undelimited continuations, including enhanced safety by restricting capture to avoid unintended inclusion or side effects, which can lead to non-termination or erroneous in larger programs. They facilitate modular reasoning about effects, as the bounded scope supports local handling and of control abstractions without pervasive impacts. Additionally, their design enables uniform of computational effects, such as monads, with efficient, low-overhead implementations derived directly from continuation-passing style transformations, promoting in paradigms.

Implementation

Code transformation

Continuation-passing style (CPS) code transformation is a mechanical, recursive process that converts direct-style functional programs into an equivalent form, making explicit by parameterizing functions with . This technique, foundational to compilers for languages like and , enables optimizations such as tail-call elimination and defunctionalization by representing all computations as tail calls to . The transformation applies to call-by-value λ-calculus and ensures that every value-producing expression passes its result to a continuation, eliminating implicit returns. The core transformation rules define how each syntactic construct is mapped to CPS. For a variable x, the CPS form is a function that applies the continuation k to x: 
[[x]] = λk. k(x)
This rule simply forwards the variable's value to the continuation. For function applications, the rule evaluates the operator and operands sequentially, then applies the operator to the operands and the continuation. For an application f(e₁, ..., eₙ), the CPS form is: 
[[f(e₁, ..., eₙ)]] = λk. [[f]](λv_f. [[e₁]](λv₁. ⋯ [[eₙ]](λvₙ. v_f(v₁, ..., vₙ, k)) ⋯ ))
Here, fresh variables v_f, v₁, ..., vₙ, and k are introduced to bind intermediate results, ensuring left-to-right evaluation order. Lambda abstractions are transformed by adding a parameter to the function body. For λx. e, the CPS form becomes: 
[[λx. e]] = λk. k(λx k'. [[e]] k')
The body e is recursively converted, and the resulting takes both the original parameter x and a fresh k'. This rule embeds the function as a value that expects a upon application. Let-bindings are handled by converting them into nested applications of the bound expression to a continuation that substitutes the value into the body. For let x = e in body, the CPS form is: 
[[let x = e in body]] = λk. [[e]](λv. let x = v in [[body]] k)
This evaluates e to a value v, binds it to x, and continues with the transformed body, preserving scoping and avoiding redundant computations. The overall algorithm for the transformation is a recursive traversal of the program's , typically implemented in a single pass for efficiency. It proceeds as follows: (1) For each expression, introduce a fresh variable k as the parameter to the enclosing λ-abstraction; (2) Recursively apply the rules to subexpressions, propagating k downward and binding intermediate values with fresh s; (3) Ensure every path through the expression concludes with a to k applied to the computed value, eliminating any implicit returns or branches that do not invoke the . This approach avoids generating unnecessary "administrative" redexes through careful fresh allocation and can incorporate optimizations like for linear-time execution. The transformation preserves the semantics of the original program: executing the CPS version with the identity continuation λx. x yields the same observable behavior as the direct-style code, including side effects and non-termination. This equivalence holds under and is proven using techniques such as logical relations, ensuring that the CPS form is a faithful suitable for further compilation stages. Additionally, the process preserves tail positions from the source, facilitating stack-safe in the output.

Handling non-local control

In continuation-passing style (CPS), non-local exits are managed by explicitly passing special continuations, such as abort handlers or error continuations, as additional arguments to functions, allowing to jump outside the normal path without relying on implicit stack unwinding. This approach models abrupt transfers, like gotos or early , by invoking the designated handler continuation directly with the relevant value or state. Exceptions in CPS are typically implemented using a double-barrelled style, where each function receives both a success continuation k (for normal returns) and an exception continuation h (for error cases), often written as f(x, k, h). For instance, a raise e operation evaluates e to a value v and invokes h(v), bypassing the success path, while a try e1 catch x => e2 installs a local handler by evaluating e1 with an updated h' that binds x to the raised value and proceeds to e2. This dual-continuation passing simulates try-catch semantics without altering the underlying CPS transformation, ensuring exceptions propagate to the appropriate handler. 
scheme
;; Example: Dual-continuation for [exception handling](/page/Exception_handling) in Scheme-like [CPS](/page/CPS)
(define (safe-div x y k h)
  (if (zero? y)
      (h "division by zero")  ; Invoke exception continuation
      (k (/ x y))))           ; Invoke success continuation

(define (try-div x y k_outer h_outer)
  (safe-div x y
            (lambda (result) (k_outer result))  ; Success: continue normally
            (lambda (err)    ;; Local catch handler: bind err (as x) and proceed to e2 (e.g., return 0)
                            (k_outer 0))))   ; Simulate catch returning 0 on error
Challenges in this approach include ensuring that all computational paths—normal and exceptional—correctly invoke the intended , which requires meticulous propagation of both k and h through recursive calls and compositions to avoid errors. In full undelimited , continuation leaks can occur if captured continuations are invoked multiple times or out of context, potentially leading to non-termination or incorrect state; delimited continuations mitigate this by bounding the scope of control transfers.

Applications

In functional programming

Continuation-passing style (CPS) plays a central role in the of languages, where it provides a mathematical framework for interpreting and related constructs by explicitly passing as arguments to . This approach, pioneered in early works on semantics, models as a series of transformations on continuations, enabling precise definitions of and without relying on operational details like stacks. For instance, in denotational semantics, the meaning of an expression is a function that takes a continuation and produces a result by applying it, as formalized in treatments of where computation types are defined as C_A = K_A \to R, with K_A as the continuation type for value type A and R as the response type. In interpreters for functional languages, CPS serves as an effective intermediate representation (IR) for evaluating lambda terms, facilitating optimizations and clear separation of value and control aspects. Compilers for functional languages, including historical experiments in the Glasgow Haskell Compiler (GHC), have employed CPS as an IR to transform direct-style code into a form where all is explicit, aiding in tasks like strictness analysis and for evaluation. This transformation ensures that function applications become tail calls in the CPS form, which is crucial for efficient recursive functional code. CPS forms the foundational basis for in , particularly in handling effects like I/O and through continuation-based structures. The monad in , defined as newtype Cont r a = Cont { runCont :: (a -> r) -> r }, directly embodies CPS by representing computations that take a continuation of type a -> r and yield a final result of type r, allowing monadic composition of effectful operations while preserving . This monad underpins implementations of and I/O monads, where the continuation captures the residual computation, enabling delimited control over effects in pure functional settings. In , CPS enables the implementation of non-deterministic constructs like the amb operator for search, such as in parsers or , by reifying to capture and resume alternative computation paths. The amb operator, which introduces ambiguous choice, can be realized using first-class to implement failure and : upon failure, the continuation is rewound to the last choice point, exploring alternatives in a depth-first manner without explicit search structures. This approach, rooted in early paradigms, integrates seamlessly with CPS interpreters for efficient evaluation of ambiguous expressions in functional contexts.

In concurrency and other domains

Continuation-passing style (CPS) has been extended to concurrency models by transforming threaded code into event-driven forms that unify native and cooperative threads. In Continuation-Passing C (CPC), a language extension for C, CPS converts imperative code to manage lightweight threads as heap-allocated continuations, enabling cooperative multitasking with yields that facilitate efficient context switches in the range of 16-46 nanoseconds on x86-64 architectures. This approach supports up to 50 million simultaneous threads with minimal overhead of about 82 bytes per continuation, as demonstrated in applications like the Hekate BitTorrent seeder. Delimited continuations further enhance safe concurrency in operating systems by capturing user process contexts during system calls, such as I/O operations, allowing suspension and resumption without race conditions through techniques like copy-on-write updates in structures like the Zipper File System. In , CPS models and choice points without relying on explicit , transforming clauses into binary forms where each subgoal receives a continuation for subsequent goals. The BinWAM, a simplified for binary , stores these on a copy managed by garbage collection, eliminating traditional environment instructions and reducing choice point overhead. This continuation-passing model, as in BinProlog, maps full to binary logic programs, enabling implicit via failure continuations while maintaining efficiency in search exploration. Beyond these areas, appears in for , where the Hakaru system transforms recursive probabilistic models into pure functions via CPS to enable and exact . In Hakaru's Mappl , CPS handles branching and calls by passing continuations that partition computations into dependent and independent parts, allowing summation over discrete variables with log-sum-exp operations to recover polynomial-time algorithms like the forward algorithm for hidden Markov models. In computer graphics, CPS supports tracing by lowering shaders to continuation form, pausing execution after ray queries and resuming via a stack-based that manages callbacks across shader stages in a big loop with switch statements. Modern applications include JavaScript's asynchronous programming, where callback-based patterns inherently follow CPS, passing continuations to handle non-blocking operations like data loading, with async/await providing over this style to simplify . In , recent semantics reconstruct core instructions in tail-recursive CPS to facilitate compositional analysis and optimizations, such as partial evaluation, by representing control structures like blocks and calls as a stack of continuations in a big-step interpreter.