Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algorithmically Expressive, Always-Terminating Model for Reversible Computation

Matteo Palazzo, Luca Roversi

TL;DR

This work introduces For_{\textsf{est}}, a terminating, reversible computational model that extends Matos' M-SRL with a FromTo-style iteration and formal inverse semantics. The authors prove termination and reversibility, and show that For_{\textsf{est}} is PRF-complete by embedding MSRL, thereby representing all PRF functions. They also demonstrate strict algorithmic superiority over MSRL by implementing a constant-time sign operation and a minimum function that runs in time $O(\min(|m|,|n|))$, which MSRL cannot achieve. These results establish For_{\textsf{est}} as a practical, expressive bridge between MSRL and Janus, with potential applications in reversible compression and encryption, while acknowledging future work to eliminate assertion-based exits and to relate the framework to APRA-like formalisms.

Abstract

Concerning classical computational models able to express all the Primitive Recursive Functions (PRF), there are interesting results regarding limits on their algorithmic expressiveness or, equivalently, efficiency, namely the ability to express algorithms with minimal computational cost. By introducing the reversible programming model Forest, at our knowledge, we provide a first study of analogous properties, adapted to the context of reversible computational models that can represent all the functions in PRF. Firstly, we show that Forest extends Matos' linear reversible computational model MSRL, the very extension being a guaranteed terminating iteration that can be halted by means of logical predicates. The consequence is that Forest is PRF complete, because MSRL is. Secondly, we show that Forest is strictly algorithmically more expressive than MSRL: it can encode a reversible algorithm for the minimum between two integers in optimal time, while MSRL cannot.

Algorithmically Expressive, Always-Terminating Model for Reversible Computation

TL;DR

This work introduces For_{\textsf{est}}, a terminating, reversible computational model that extends Matos' M-SRL with a FromTo-style iteration and formal inverse semantics. The authors prove termination and reversibility, and show that For_{\textsf{est}} is PRF-complete by embedding MSRL, thereby representing all PRF functions. They also demonstrate strict algorithmic superiority over MSRL by implementing a constant-time sign operation and a minimum function that runs in time , which MSRL cannot achieve. These results establish For_{\textsf{est}} as a practical, expressive bridge between MSRL and Janus, with potential applications in reversible compression and encryption, while acknowledging future work to eliminate assertion-based exits and to relate the framework to APRA-like formalisms.

Abstract

Concerning classical computational models able to express all the Primitive Recursive Functions (PRF), there are interesting results regarding limits on their algorithmic expressiveness or, equivalently, efficiency, namely the ability to express algorithms with minimal computational cost. By introducing the reversible programming model Forest, at our knowledge, we provide a first study of analogous properties, adapted to the context of reversible computational models that can represent all the functions in PRF. Firstly, we show that Forest extends Matos' linear reversible computational model MSRL, the very extension being a guaranteed terminating iteration that can be halted by means of logical predicates. The consequence is that Forest is PRF complete, because MSRL is. Secondly, we show that Forest is strictly algorithmically more expressive than MSRL: it can encode a reversible algorithm for the minimum between two integers in optimal time, while MSRL cannot.
Paper Structure (7 sections, 7 theorems, 16 equations, 4 figures)

This paper contains 7 sections, 7 theorems, 16 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The computational model For$_{\textsf{est}}$
  3. For$_{\textsf{est}}$ is terminating and reversible
  4. For$_{\textsf{est}}$ is PRF-complete because M-SRL is
  5. For$_{\textsf{est}}$ is algorithmically more expressive than M-SRL
  6. Conclusions and future work
  7. Future work.

Key Result

theorem 1

Let $P\in{\normalfont\textsf{For$_{\textsf{est}}$}}\xspace$, and $\bm{\sigma}\in\Sigma$. The interpretation of $P$ starting from $\bm{\sigma}$, either terminates in some $\bm{\tau}\in\Sigma$ or fails. Formally:

Figures (4)

  • Figure 1: The iteration of For$_{\textsf{est}}$ begins/halts depending on $e_u, e_v, e_{\operatorname{\textit{in}}}$, and $e_{\operatorname{\textit{out}}}$.
  • Figure 2: Interpretation of boolean expressions.
  • Figure 3: Operational Semantics on $\mathcal{F}_{\operatorname{\textit{ext}}}$.
  • Figure 4: Flow-chart of the premise of rule FromTo in Figure \ref{['fig:operationalSemantics']}.

Theorems & Definitions (24)

  • definition 1: Arithmetical and boolean expressions
  • definition 2: Set $\mathcal{F}$ of well-formed terms
  • definition 3: Inverse of a term
  • definition 4: States
  • definition 5: Evaluating arithmetic and boolean expressions
  • remark 1
  • definition 6: Extension $\mathcal{F}_{\operatorname{\textit{ext}}}$ of $\mathcal{F}$
  • definition 7: Operational semantics of $\mathcal{F}_{\operatorname{\textit{ext}}}$
  • definition 8: The computational model For$_{\textsf{est}}$
  • remark 2: For$_{\textsf{est}}$ syntax and operational semantics
  • ...and 14 more