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Reversible Turing Machines (RTMs)

The Reversible Turing Machine (RTM) is a foundational model in reversible computing. An RTM is defined as a Turing machine whose transition function is invertible, ensuring that each machine configuration (state and tape content) has at most one predecessor configuration. This guarantees backward determinism, allowing the computation history to be traced uniquely.

Formal definitions of RTMs have evolved over the last decades. While early definitions focused on invertible transition functions, more general formulations allow for bounded head movement and cell modification per step. This generalization ensures that the set of RTMs is closed under composition (executing one RTM after another results in another RTM) and inversion (the inverse of an RTM is also an RTM), forming a group structure for reversible computations. This contrasts with some classical TM definitions where composition might not yield a machine of the same class. The dynamics of an RTM can be described by a global transition function that maps configurations based on a local rule.

Yves Lecerf proposed a reversible Turing machine in a 1963 paper, but apparently unaware of Landauer's principle, did not pursue the subject further, devoting most of the rest of his career to ethnolinguistics.

A landmark result by Charles H. Bennett in 1973 demonstrated that any standard Turing machine can be simulated by a reversible one. Bennett's construction involves augmenting the TM with an auxiliary "history tape". The simulation proceeds in three stages:

1. Compute: The original TM's computation is simulated, and a record of every transition rule applied is written onto the history tape.
2. Copy Output: The final result on the work tape is copied to a separate, initially blank output tape. This copy operation itself must be done reversibly (e.g., using CNOT gates).
3. Uncompute: The simulation runs in reverse, using the history tape to undo each step of the forward computation. This process erases the work tape and the history tape, returning them to their initial blank state, leaving only the original input (preserved on its tape) and the final output on the output tape.

This construction proves that RTMs are computationally equivalent to standard TMs in terms of the functions they can compute, establishing that reversibility does not limit computational power in this regard. However, this standard simulation technique comes at a cost. The history tape can grow linearly with the computation time, leading to a potentially large space overhead, often expressed as S'(n) = O(S(n)T(n)) where S and T are the space and time of the original computation. Furthermore, history-based approaches face challenges with local compositionality; combining two independently reversibilized computations using this method is not straightforward. This indicates that while theoretically powerful, Bennett's original construction is not necessarily the most practical or efficient way to achieve reversible computation, motivating the search for methods that avoid accumulating large amounts of "garbage" history.

RTMs compute precisely the set of injective (one-to-one) computable functions. They are not strictly universal in the classical sense because they cannot directly compute non-injective functions (which inherently lose information). However, they possess a form of universality termed "RTM-universality" and are capable of self-interpretation.