entropy Article Generalization of the Landauer Principle for Computing Devices Based on Many-Valued Logic Edward Bormashenko Department of Chemical Engineering, Engineering Sciences Faculty, Ariel University, Ariel 407000, Israel; [email protected]; Tel.: +972-074-729-68-63 Received: 7 October 2019; Accepted: 22 November 2019; Published: 25 November 2019 Abstract: The Landauer principle asserts that “the information is physical”. In its strict meaning, Landauer’s principle states that there is a minimum possible amount of energy required to erase one bit of information, known as the Landauer bound W = kBTln2, where T is the temperature of a thermal reservoir used in the process and kB is Boltzmann’s constant. Modern computers use the binary system in which a number is expressed in the base-2 numeral system. We demonstrate that the Landauer principle remains valid for the physical computing device based on the ternary, and more generally, N-based logic. The energy necessary for erasure of one bit of information (the Landauer bound) W = kBTln2 remains untouched for the computing devices exploiting a many-valued logic. Keywords: Landauer principle; binary logic; ternary logic; Landauer bound; trit 1. Introduction Modern computers use the binary system, whereby a number is expressed in the base-2 numeral system. The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit. The base-2 is ubiquitous in computing devices because of its straightforward implementation in digital electronic circuitry using binary logic gates. However, one of the first computing machines was based on the ternary logic. In 1840, Thomas Fowler, a self-taught English mathematician and inventor, created a unique ternary mechanical calculating machine, completely manufactured of wood [1]. Ternary logic based computers based in the “trit” unit of information were successfully developed in Soviet Union by Nicolay Brousentsov [2]. The Setun computer, based on the ideas of ternary logic, ternary symmetrical number system and ternary memory element (“flip-flap-flop”) was designed in 1958 in Moscow University [2,3]. In principle, computer may be based on a many-valued logics, exposed in recent years to a growing interest due to the fundamental aspects and numerous applications [4,5]. Ternary computer TERNAC was reported in 1973 by Frieder et al. in Reference [6]. The present paper does not come into the mathematical details of the ternary (or another) many-valued logics, but extends the Landauer principle to the erasing of the information by the computing machine, based on the many-valued logics. Informational theory is usually supplied in a form that is independent of any physical realization. In contrast, Rolf Landauer, in his papers, argued that “information is physical” and has an energy equivalent [7–9]. It may be stored in physical systems, such as books and memory chips, and it is transmitted by physical devices exploiting electrical or optical signals [6–8]. Therefore, he concluded, it must obey the laws of physics, and first and foremost, the laws of thermodynamics. The Landauer principle [7–9] established the energy equivalent of information and remains a focus of investigations in the last decade [10–18]. In its strictest, tightest, and simplest meaning, the Landauer principle states that the erasure of one bit of information requires a minimum energy cost equal to kBTln2, where T is the temperature of a thermal reservoir used in the process and kB is Boltzmann’s constant [7–14]. The Landauer principle is usually demonstrated with Entropy 2019, 21, 1150; doi:10.3390/e21121150 www.mdpi.com/journal/entropy Entropy 2019, 21, 1150 2 of 6 Entropy 2019, 21, 1150 2 of 6 usedEntropy in2019 the, 21 ,process 1150 and is Boltzmann’s constant [7–14]. The Landauer principle is usually2 of 6 demonstratedused in the process with the and computers, is Boltzmann’s based on constantthe binary [7–14]. logic. The We Landauer demonstrate principle how itis mayusually be extendeddemonstrated to devices with thatthe exploitcomputers, a many-valued based on logics.the binary logic. We demonstrate how it may be the computers, based on the binary logic. We demonstrate how it may be extended to devices that extended to devices that exploit a many-valued logics. 2.exploit Discussion a many-valued logics. 2. Discussion 2. DiscussionConsider the computing device exploiting a particle enclosed within a chamber (cylinder) dividedConsider by half theby acomputing partition, as device shown exploiting in Figure 1.a Findingparticle the enclosed particle within M in the a certainchamber (left (cylinder) or right) Consider the computing device exploiting a particle enclosed within a chamber (cylinder) divided halfdivided of the by halfchamber by a partition, corresponds as shown to the in recording Figure 1. Finding of 1 bit the of particleinformation. M in the When certain the (left partition or right) is by half by a partition, as shown in Figure1. Finding the particle M in the certain (left or right) half removed,half of the the chamber location corresponds of the particle to theis uncertain, recording andof 1 this bit correspondsof information. to theWhen erasure the partitionof 1 bit ofis of the chamber corresponds to the recording of 1 bit of information. When the partition is removed, information.removed, the The location location of theof a particleparticle ison uncertain, a certain halfand ofthis the corresponds chamber corresponds to the erasure to “1”, of and1 bit the of the location of the particle is uncertain, and this corresponds to the erasure of 1 bit of information. uncertaininformation. location The location of the particle of a particle corresponds on a certai to “0n”, half thus of our the particle-based chamber corresponds computer to works “1”, and on the The location of a particle on a certain half of the chamber corresponds to “1”, and the uncertain location binaryuncertain logical location system. of the The particle work ofcorresponds this computer to “0 may”, thus be exemplifiedour particle-based by the computersingle-particle works thermal on the of the particle corresponds to “0”, thus our particle-based computer works on the binary logical system. engine,binary logical suggested system. by TheLeo workSzilard of thisin 1929 computer [19] anmayd depicted be exemplified in Figure by the 2. single-particleThe smallest possiblethermal The work of this computer may be exemplified by the single-particle thermal engine, suggested by thermodynamicengine, suggested machine by Leo consists Szilard of ain single 1929 particle[19] an ofd massdepicted m in ina closed Figure cylinder, 2. The whichsmallest has possible contact Leo Szilard in 1929 [19] and depicted in Figure2. The smallest possible thermodynamic machine withthermodynamic a thermal reservoirs. machine consists Consider of the a single “evergreen” particle Carnotof mass cycle, m in a performed closed cylinder, by a minimal which has engine, contact is consists of a single particle of mass m in a closed cylinder, which has contact with a thermal reservoirs. depictedwith a thermal in Figure reservoirs. 1 from Consider an informational the “evergreen” point of Carnot view cycle,[9,20,21]. performed At the firstby a minimalstage, the engine, particle is Consider the “evergreen” Carnot cycle, performed by a minimal engine, is depicted in Figure1 from contactsdepicted thein Figurethermal 1 reservoirfrom an informational(bath) T1 and pointundergoes of view a reversible[9,20,21]. Atisothermal the first stage,expansion, the particle which an informational point of view [9,20,21]. At the first stage, the particle contacts the thermal reservoir doublescontacts itsthe available thermal volume reservoir [21]. (bath) Note, T that1 and the undergoes particle initially a reversible occupies isothermal the left side expansion, of the cylinder. which (bath) T1 and undergoes a reversible isothermal expansion, which doubles its available volume [21]. Heatdoubles kBT 1its ln2 available is drawn volume from the [21]. bath Note, and that work the particle2 initiallyis extracted. occupies This processthe left side is equivalent of the cylinder. to the Note, that the particle initially occupies the left side of the cylinder. Heat kBT1 ln2 is drawn from the removalHeat kBT 1of ln2 the is partitiondrawn from at the the midpoint bath and ofwork the cylinder,2 is thus, extracted. one bit This of information process is equivalent is erased, ifto one the bath and work kBT ln2 is extracted. This process is equivalent to the removal of the partition at the bitremoval finds particleof the partition m 1at a certain at the sidemidpoint (left in of our the case) cylinder, of the thus, cylinder, one bit as of shown information in Figure is erased2 [10]., ifThus, one midpoint of the cylinder, thus, one bit of information is erased, if one bit finds particle m at a certain side heatbit finds kBT1 particleln2 spent m by at thea certain thermal side bath (left was in our exploite case)d of for the erasure cylinder, of 1 as bit shown of information. in Figure 2 [10]. Thus, (left in our case) of the cylinder, as shown in Figure2[ 10]. Thus, heat kBT1 ln2 spent by the thermal heat kBT1 ln2 spent by the thermal bath was exploited for erasure of 1 bit of information. bath was exploited for erasure of 1 bit of information. Figure 1. Finding of the particle M in the certain (left or right) half of the chamber corresponds to the Figure 1. Finding of the particle M in the certain (left or right) half of the chamber corresponds to the recordingFigure 1. Finding of 1 bit ofof information.the particle M in the certain (left or right) half of the chamber corresponds to the recording of 1 bit of information. recording of 1 bit of information. Figure 2. SketchSketch of of the minimal single-particle thermal machine is depicted.