Time, Space, and Energy in Reversible Computing Paul Vitan´ yi ¤ CWI University of Amsterdam National ICT of Australia ABSTRACT sipate energy by generating a corresponding amount of entropy for We survey results of a quarter century of work on computation by every bit of information that gets irreversibly erased; the logically reversible general-purpose computers (in this setting Turing ma- reversible operations can in principle be performed dissipation-free. chines), and general reversible simulation of irreversible computa- One should sharply distinguish between the issue of logical re- tions, with respect to energy-, time- and space requirements. versibility and the issue of energy dissipation freeness. If a com- Categories and Subject Descriptors: F.2 [Algorithms], F.1.3 puter operates in a logically reversible manner, then it still may dis- [Performance] sipate heat. For such a computer we know that the laws of physics General Terms: Algorithms, Performance do not preclude that one can invent a technology in which to im- plement a logically similar computer to operate physically in a dis- Keywords: Reversible computing, reversible simulation, adia- sipationless manner. Computers built from reversible circuits, or batic computing, low-energy computing, computational complex- the reversible Turing machine, [1, 2, 7], implemented with cur- ity, time complexity, space complexity, energy dissipation com- rent technology will presumably dissipate energy but may conceiv- plexity, tradeoffs. ably be implemented by future technology in an adiabatic fashion. But for logically irreversible computers adiabatic implementation 1. INTRODUCTION is widely considered impossible. Computer power has roughly doubled every 18 months for the Thought experiments can exhibit a computer that is both logi- last half-century (Moore’s law). This increase in power is due pri- cally and physically perfectly reversible and hence perfectly dissi- marily to the continuing miniaturization of the elements of which pationless. An example is the billiard ball computer, [7], and simi- computers are made, resulting in more and more elementary gates larly the possibility of a coherent quantum computer, [6, 19]. Our per unit area with higher and higher clock frequency, accompa- purpose is to determine the theoretical ultimate limits to which the nied by less and less energy dissipation per elementary computing irreversible actions in an otherwise reversible computation can be event. Roughly, a linear increase in clock speed is accompanied reduced. by a square increase in elements per unit area—so if all elements Currently, computations are commonly irreversible, even though compute all of the time, then the dissipated energy per time unit the physical devices that execute them are fundamentally reversible. rises cubicly (linear times square) in absence of energy decrease At the basic level, however, matter is governed by classical me- per elementary event. The continuing dramatic decrease in dissi- chanics and quantum mechanics, which are reversible. This con- pated energy per elementary event is what has made Moore’s law trast is only possible at the cost of efficiency loss by generating possible. But there is a foreseeable end to this: There is a minimum thermal entropy into the environment. With computational device quantum of energy dissipation associated with elementary events. technology rapidly approaching the elementary particle level it has This puts a fundamental limit on how far we can go with miniatur- been argued many times that this effect gains in significance to the ization, or does it? extent that efficient operation (or operation at all) of future comput- R. Landauer [11] has demonstrated that it is only the ‘logically ers requires them to be reversible, [11, 1, 2, 7, 10, 15, 8]. The mis- irreversible’ operations in a physical computer that necessarily dis- match of computing organization and reality will express itself in friction: computers will dissipate a lot of heat unless their mode of ¤Part of this work was done while the author was on sabbatical operation becomes reversible, possibly quantum mechanical. Since leave at National ICT of Australia, Sydney Laboratory at UNSW. 1940 the dissipated energy per bit operation in a computing device Supported in part by the EU Project RESQ IST-2001-37559, the has—with remarkable regularity—decreased at the inverse rate of ESF QiT Programmme, the EU NoE PASCAL, and the Netherlands BSIK/BRICKS project. Address: CWI, Kruislaan 413, 1098 SJ Moore’s law [10] (making Moore’s law possible). Extrapolation Amsterdam, The Netherlands. Email: [email protected] of current trends shows that the energy dissipation per binary logic operation needs to be reduced below kT (thermal noise) within 20 years. Here k is Boltzmann’s constant and T the absolute tem- perature in degrees Kelvin, so that kT 3 10 21 Joule at room ¼ £ ¡ Permission to make digital or hard copies of all or part of this work for temperature. Even at kT level, a future device containing 1 trillion personal or classroom use is granted without fee provided that copies are (1012) gates operating at 1 terahertz (1012) switching all gates all not made or distributed for profit or commercial advantage and that copies of the time dissipates about 3000 watts. Consequently, in contem- bear this notice and the full citation on the first page. To copy otherwise, to porary computer and chip architecture design the issue of power republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. consumption has moved from a background worry to a major prob- CF'05, May 4–6, 2005, Ischia, Italy. lem. Theoretical advances in reversible computing are scarce and Copyright 2005 ACM 1-59593-018-3/05/0005 ...$5.00. far between; many serious ones are listed in the references. For this Reversible Turing machines or other reversible computers will review we have drawn primarily on the results and summarized ma- require special reversible programs. One feature of such programs terial in [15, 17, 5]. It is a tell-tale sign of the difficulty of this area, is that they should be executable when read from bottom to top that no further advances have been made in this important topic as well as when read from top to bottom. Examples are the pro- since that time. grams in the later sections. In general, writing reversible programs will be difficult. However, given a general reversible simulation 2. REVERSIBLE TURING MACHINES of irreversible computation, one can simply write an oldfashioned irreversible program in an irreversible programming language, and There is a decisive difference between reversible circuits and re- subsequently simulate it reversibly. This leads to the following: versible special purpose computers [7] on the one hand, and re- versible universal computers on the other hand [1, 3]. While one DEFINITION 1. An irreversible-to-reversible compiler receives can design a special-purpose reversible version for every particular an irreversible program as input and compiles it to a reversible irreversible circuit using reversible universal gates, such a method program. does not yield an irreversible-to-reversible compiler that can exe- cute any irreversible program on a fixed universal reversible com- puter architecture as we are interested in here. 3. ADIABATIC COMPUTATION In the standard model of a Turing machine the elementary oper- All computations can be performed logically reversibly, [1], at ations are rules in quadruple format (p;s;a;q) meaning that if the the cost of eventually filling up the memory with unwanted garbage finite control is in state p and the machine scans tape symbol s, then information. This means that reversible computers with bounded the machine performs action a and subsequently the finite control memories require in the long run irreversible bit operations, for ex- enters state q. Such an action a consists of either printing a symbol ample, to erase records irreversibly to create free memory space. s0 in the tape square scanned, or moving the scanning head one tape The minimal possible number of irreversibly erased bits to do so is square left or right. believed to determine the ultimate limit of heat dissipation of the Quadruples are said to overlap in domain if they cause the ma- computation by Landauer’s principle, [11, 1, 2]. In reference [4] chine in the same state and scanning the same symbol to perform we and others developed a mathematical theory for the unavoid- different actions. A deterministic Turing machine is defined as a able number of irreversible bit operations in an otherwise reversible Turing machine with quadruples no two of which overlap in do- computation. main. Methods to implement (almost) reversible and dissipationless Now consider the special format (deterministic) Turing machines computation using conventional technologies are often designated using quadruples of two types: read/write quadruples and move by the catch phrase ‘adiabatic switching’. Many currently proposed quadruples. A read/write quadruple (p;a;b;q) causes the machine physical schemes implementing adiabatic computation reduce irre- in state p scanning tape symbol a to write symbol b and enter versibility by using longer switching times. This is done typically state q. A move quadruple (p; ;σ;q) causes the machine in state by switching over equal voltage gates after voltage has been equal- ¤ p to move its tape head by σ 1;+1 squares and enter state ized slowly. This type of switching does not dissipate energy, the 2 f¡ g q, oblivious to the particular symbol in the currently scanned tape only energy dissipation is incurred by pulling voltage up and down: square. (Here ‘ 1’ means ‘one square left’, and ‘+1’ means ‘one the slower it goes the less energy is dissipated. If the computation ¡ square right’.) Quadruples are said to overlap in range if they cause goes infinitely slow, zero energy is dissipated. Clearly, this coun- the machine to enter the same state and either both write the same teracts the purpose of low energy dissipation which is faster com- symbol or (at least) one of them moves the head.