Reversible Computation in Term Rewriting

Essentially, in a reversible programming language, for each forward computation from state $S$ to state $S'$, there exists a constructive method to go backwards from state $S'$ to state $S$. Besides its theoretical interest, reversible computation is a fundamental concept which is relevant in many different areas like cellular automata, bidirectional program transformation, or quantum computing, to name a few. In this work, we focus on term rewriting, a computation model that underlies most rule-based programming languages. In general, term rewriting is not reversible, even for injective functions; namely, given a rewrite step $t_1 \rightarrow t_2$, we do not always have a decidable method to get $t_1$ from $t_2$. Here, we introduce a conservative extension of term rewriting that becomes reversible. Furthermore, we also define two transformations, injectivization and inversion, to make a rewrite system reversible using standard term rewriting. We illustrate the usefulness of our transformations in the context of bidirectional program transformation.Comment: To appear in the Journal of Logical and Algebraic Methods in Programmin

Reversible Term Rewriting

Essentially, in a reversible programming language, for each forward computation step from state S to state S', there exists a constructive and deterministic method to go backwards from state S' to state S. Besides its theoretical interest, reversible computation is a fundamental concept which is relevant in many different areas like cellular automata, bidirectional program transformation, or quantum computing, to name a few. In this paper, we focus on term rewriting, a computation model that underlies most rule-based programming languages. In general, term rewriting is not reversible, even for injective functions; namely, given a rewrite step t1 -> t2, we do not always have a decidable and deterministic method to get t1 from t2. Here, we introduce a conservative extension of term rewriting that becomes reversible. Furthermore, we also define a transformation to make a rewrite system reversible using standard term rewriting.This work has been partially supported by the EU (FEDER) and the Spanish Ministerio de Economía y Competitividad (MINECO) under grant TIN2013-44742-C4-1-R, by the Generalitat Valenciana under grant PROMETEO-II/2015/013 (SmartLogic) and by the COST Action IC 1405 on Reversible Computation. A. Palacios was partially supported by the the EU (FEDER) and the Spanish Ayudas para contratos predoctorales para la formación de doctores de la Sec. Estado de Investigación, Desarrollo e Innovación del MINECO under FPI grant BES-2014-069749. Part of this research was done while the second and third authors were visiting Nagoya University; they gratefully acknowledge their hospitality.Nishida, N.; Palacios Corella, A.; Vidal Oriola, GF.; Nishida (2016). Reversible Term Rewriting. Schloss Dagstuhl-Leibniz-Zentrum für Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.FSCD.2016.28

Techniques for the Synthesis of Reversible Toffoli Networks

This paper presents novel techniques for the synthesis of reversible networks of Toffoli gates, as well as improvements to previous methods. Gate count and technology oriented cost metrics are used. Our synthesis techniques are independent of the cost metrics. Two new iterative synthesis procedure employing Reed-Muller spectra are introduced and shown to complement earlier synthesis approaches. The template simplification suggested in earlier work is enhanced through introduction of a faster and more efficient template application algorithm, updated (shorter) classification of the templates, and presentation of the new templates of sizes 7 and 9. A novel ``resynthesis'' approach is introduced wherein a sequence of gates is chosen from a network, and the reversible specification it realizes is resynthesized as an independent problem in hopes of reducing the network cost. Empirical results are presented to show that the methods are effective both in terms of the realization of all 3x3 reversible functions and larger reversible benchmark specifications.Comment: 20 pages, 5 figure

A Reduction-Preserving Completion for Proving Confluence of Non-Terminating Term Rewriting Systems

We give a method to prove confluence of term rewriting systems that contain non-terminating rewrite rules such as commutativity and associativity. Usually, confluence of term rewriting systems containing such rules is proved by treating them as equational term rewriting systems and considering E-critical pairs and/or termination modulo E. In contrast, our method is based solely on usual critical pairs and it also (partially) works even if the system is not terminating modulo E. We first present confluence criteria for term rewriting systems whose rewrite rules can be partitioned into a terminating part and a possibly non-terminating part. We then give a reduction-preserving completion procedure so that the applicability of the criteria is enhanced. In contrast to the well-known Knuth-Bendix completion procedure which preserves the equivalence relation of the system, our completion procedure preserves the reduction relation of the system, by which confluence of the original system is inferred from that of the completed system

Optimized Compilation of Aggregated Instructions for Realistic Quantum Computers

Recent developments in engineering and algorithms have made real-world applications in quantum computing possible in the near future. Existing quantum programming languages and compilers use a quantum assembly language composed of 1- and 2-qubit (quantum bit) gates. Quantum compiler frameworks translate this quantum assembly to electric signals (called control pulses) that implement the specified computation on specific physical devices. However, there is a mismatch between the operations defined by the 1- and 2-qubit logical ISA and their underlying physical implementation, so the current practice of directly translating logical instructions into control pulses results in inefficient, high-latency programs. To address this inefficiency, we propose a universal quantum compilation methodology that aggregates multiple logical operations into larger units that manipulate up to 10 qubits at a time. Our methodology then optimizes these aggregates by (1) finding commutative intermediate operations that result in more efficient schedules and (2) creating custom control pulses optimized for the aggregate (instead of individual 1- and 2-qubit operations). Compared to the standard gate-based compilation, the proposed approach realizes a deeper vertical integration of high-level quantum software and low-level, physical quantum hardware. We evaluate our approach on important near-term quantum applications on simulations of superconducting quantum architectures. Our proposed approach provides a mean speedup of $5\times$, with a maximum of $10\times$. Because latency directly affects the feasibility of quantum computation, our results not only improve performance but also have the potential to enable quantum computation sooner than otherwise possible.Comment: 13 pages, to apper in ASPLO

Generating reversible circuits from higher-order functional programs

Boolean reversible circuits are boolean circuits made of reversible elementary gates. Despite their constrained form, they can simulate any boolean function. The synthesis and validation of a reversible circuit simulating a given function is a difficult problem. In 1973, Bennett proposed to generate reversible circuits from traces of execution of Turing machines. In this paper, we propose a novel presentation of this approach, adapted to higher-order programs. Starting with a PCF-like language, we use a monadic representation of the trace of execution to turn a regular boolean program into a circuit-generating code. We show that a circuit traced out of a program computes the same boolean function as the original program. This technique has been successfully applied to generate large oracles with the quantum programming language Quipper.Comment: 21 pages. A shorter preprint has been accepted for publication in the Proceedings of Reversible Computation 2016. The final publication is available at http://link.springer.co