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Group Theoretical

Aspects of

Reversible Logic

Gates

Abstract

• Logic gates with three input bits and three output bits have a

privileged position within fundamental computer science. They are a

sufficient building block for constructing arbitrary reversible boolean

networks and therefore are the key to reversible digital computers.

• Such computers can, in principle, operate without heat production.
• There exist as many as 8! = 40,320 different 3-bit reversible truth

tables

• The question : which ones to choose as building blocks.
• Because these gates form a group with respect to the operation

`cascading', we can apply group theoretical tools, in order to make

such a choice.

• We will study permutations

1 Introduction

• Landauer's principle: logic computations that are not reversible, necessarily generate heat: - i.e. kT log(2), for every bit of information that is lost. where k is Boltzmann's constant and T the temperature.
• For T equal room temperature, this package of heat is small, i.e. 2.9 x 10 -21^ joule, but non- negligible.
• In order to produce zero heat, a computer is only allowed to perform reversible computations.
• Such a logically reversible computation can be undone': the value of the output suffices to recover what the value of the inputhas been'.
• The hardware of a reversible computer cannot be constructed from the conventional gates
• On the contrary, it consists exclusively of logically reversible building blocks.
• The number of output bits of a reversible logic gate necessarily equals its number of input bits.
• We will call this number the `logic width' of the gate.

Gates of logic width 3

• Fredkin and Toffoli:
  • demonstrated that three inputs and three outputs is necessary and

sufficient in order to construct a reversible implementation of an

arbitrary boolean function of a finite number of logic variables.

• Thus, from the fundamental point of view, reversible logic

gates with a width equal to three have a privileged position.

remark

• OR gate is not universal but very useful and

cheap, so this assumption is not very

realistic in general

• but it is good for reversible logic, as based

on my examples.

• think why?

2 Calculation with a single bit

• There exist only four different truth tables with one bit input and one bit output.
• Two of them are logically irreversible: the resetter ( P = 0) and the setter ( P = 1).
• The two others are reversible: the follower ( P = A ) and the inverter ( P = NOT A ).
  • If, for example, we have `forgotten' the value of A , knowledge of the value of the inverter's output P suffices to recover it.
• Note that among the 1-bit reversible gates, the NOT gate is a `generator'.
  • This means we can make any reversible gate of width 1 by combining a finite number of this particular gate.
• Indeed, a follower can be fabricated by the sequence of two inverters.
• The opposite is not true: one cannot fabricate an inverter by cascading followers.

3 Calculation with two bits

• On the contrary, truth Table 3b is an example of a 2-bit reversible table that cannot be reduced to two separate 1-bit reversible tables, and therefore is called a true two-bit reversible gate.
• Among the 24 reversible 2-bit tables, only 16 are true 2-bit tables.

Table 3: Feynman's truth tables: (a) NOT, (b) CONTROLLED NOT, (c) CONTROLLED CONTROLLED NOT. Docsity.com

CONTROLLED NOT by Feynman

• All reversible true 2-bit gates can be fabricated from the same building block, combined with an inverter before and/or an inverter after.
• Indeed, Table 3b together with the inverter (Table 3a) forms a set of two building blocks with which we can synthetize an arbitrary reversible 2-bit gate.
• Truth Table 3b is called the CONTROLLED NOT by Feynman
• Its logic operation looks like this:

P = A Q = A XOR B , where XOR is the abbreviation of the EXCLUSIVE OR function.

• The gate is the reversible form of the classical (irreversible) XOR gate.

Table 3: Feynman's truth tables: (a) NOT, (b) CONTROLLED NOT, (c) CONTROLLED CONTROLLED NOT. Docsity.com

Controlled NOT gate

• These three boolean expressions are identical, but lead to different physical realizations.
• We note, however, that these implementations not only make use of the 1-bit NOT function and the 2-bit CONTROLLED NOT function, but also of the 2-bit exchanger, i.e. the gate that interchanges two logic variables (realizing P = B as well as Q = A ).
• This is an example of a general property: the EXCHANGER, the NOT, and the CONTROLLED NOT form a natural `generating set' for the twenty-four 2-bit reversible gates.
• More precisely, each reversible 2-bit gate can be synthesized by taking one or zero CONTROLLED NOTs and adding one or zero EXCHANGERs and one or zero NOTs to the left and to the right of it.

Table 4: The three generators of the 2- bit reversible gates: (a) EXCHANGER, (b) NOT, (c) CONTROLLED NOT.

• Note that:
  • neither the EXCHANGE R nor the NOT is a true 2-bit gate,
  • but the CONTROLL ED NOT is one.
• See Table 4

4 Calculation with three bits

• There exist 8 8 = 16,777,216 different truth tables

with 3 inputs and 3 outputs.

• Among them, only 8! = 40,320 are reversible.
• However, 48 of these truth tables fall apart into three

separate 1-bit reversible tables and another 288 fall

apart into one 1-bit and one (true) 2-bit reversible

gate.

• Thus, among the 40,320 reversible 3-bit gates, only

39,984 are true 3-bit gates.

4 Calculation with three bits

• Both have a particular property: each is a universal

primitive.

• This means that any Boolean function of any finite number

of logic input variables can be implemented by combining a finite number of such building blocks.

• The proof consists of two steps:
  • one first proves that the building block suffices to implement the

NAND function (Table 1b),

• then one refers to the fact that the NAND function is a universal

primitive.

4 Calculation with three bits

• The latter step is a well-known theorem.
• The former step is demonstrated by introducing a so-called

preset : we keep one or two inputs fixed and look how the three outputs are function of the remaining input(s).

• Among the 39,984 reversible true 3-bit gates, many have

the universality property.

• It is clear, however, that the number 39,984 is too large to

allow `manual' inspection.

• We have to recur to computer-algebra software specially

dedicated to group theory, such as GAP and Magma

• In the present study, we have chosen the GAP approach, because

of GAP's built-in commands DoubleCoset and DoubleCosets.

Table 5: Truth tables: (a) Fredkin's conservative gate, (b) a `pseudo-

inverting' gate.

• Cascade

composition of two

Fredkin Gate is

identity circuit

• Cascade

composition of two

pseudo-inverting

gates is the identity

5 Groups and subgroups

• Because of the universality property of some of the 3-bit

reversible gates, we now continue the w = 3 case in more

detail.

• When a reversible 3-bit gate x is cascaded by a reversible

3-bit gate y (i.e. when the P output of gate x is connected

to the A input of y , etc.), then a new reversible 3-bit gate

is formed, denoted xy.

• The 40,320 reversible truth tables of width 3 therefore

form a group , say R , which is isomorphic to the

symmetric group S 8.