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Continuation-passing style (often abbreviated as CPS) is a programming paradigm and transformation technique used in functional programming. It involves rewriting a function so that its continuation—the sequence of operations to be performed after the function completes—is passed explicitly as an argument. This approach shifts the control flow of the program, allowing for more flexible and composable code structures.

In CPS, a function does not return a value directly but instead invokes its continuation with the result. This technique is particularly useful for handling asynchronous operations, error handling, and resource management, as it enables explicit control over the program's flow. Many functional programming languages, such as Haskell and Erlang, support or encourage CPS-style programming.

One of the key benefits of CPS is its ability to simplify complex control structures, such as loops and conditionals, by transforming them into recursive calls with continuations. This can make code easier to reason about and optimize, as it avoids implicit control flow. Additionally, CPS can be used to implement coroutines, generators, and other advanced concurrency patterns.

The transformation to CPS involves systematically replacing each function call with an explicit continuation. For example, a simple function `f x` might be rewritten as `f x k`, where `k` is the continuation representing the operations to follow after `f` completes. This process can be automated using compiler techniques or libraries in some languages.

While CPS can improve modularity and control flow, it may introduce additional complexity in some cases, particularly for developers unfamiliar with the paradigm. However, its advantages in certain scenarios—such as handling side effects or asynchronous operations—make it a valuable tool in functional programming. Many modern languages and frameworks incorporate CPS-like patterns to manage non-deterministic or resource-intensive operations effectively.