Ternary Logic

Ternary Logic

         It is well known that the numeral system used in most if not all modern computer systems is the binary numerical system, as again it is known that binary logic, also known as Boolean algebra, also brings his significant contribution to the building of the same computer systems which I will name them more briefly as  computers. In other words we distinguish here, on the one hand between a binary arithmetic in which we do    basic calculations such as addition, subtraction, multiplication, division in a manner as possible like we do  arithmetic calculations with numbers written in the decimal numerical system and on the other hand we call into question this time an algebra, which has also two digits that are the truth values of sentences and I mean by   this the propositional calculus of the above mentioned Boole algebra. 

         Why does the binary numeral system and binary logic have so much significance in the construction of computers? The answer is at hand if we think about how easy it is to build electronic circuits with two states and I mention here the known bistable circuit.  In the same manner we can easily distinguish between a closed electrical circuit and an open one, and going further, between a forward biased LED and a switched off one, as again, one is a magnetized portion and a different thing is a non-magnetized portion of the same magnetic tape. In the spiritual plane we discover the dualist philosophies, among which the two religious concepts of Yin and Yang with roots in Chinese philosophy and metaphysics, is prominently situated, and also Manichaeism which supports radical ontological  dualism between the two eternal principles, Good and Evil, which oppose each other in the course of history, in an endless confrontation. A key element of Manichean doctrine is the non-omnipotence of the power of God, denying the infinite perfection of divinity that has they say a dual nature, consisting of two equal but opposite sides (Good-Bad.

 
Ternary digits

         The two figures of the binary numeral system are 0 and 1, each bearing the name of Binary digIT, hence the name of BIT with its plural bits. The figures of the ternary numeral system are 0, 1 and 2. As each of these figures would be known in English as a Ternary digIT we get the word TIT (with the plural TITs) that I propose to use from now on, when we refer to the figures of the ternary numeral system.

Counting and converting
         Let’s count in the ternary numeral system: 0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, etc. Let's convert into the base-10 number system, the number 21023 written in base-3 number system: 21023 = 2.33 + 1.32 + 0.31 + 2.30 = 54 + 9 + 0 + 2 = 6510. The resources of the Word Editor does not allow me to convert in base-3 a number written in base-10, but the procedure is similar to that used in the passage in base-2 of the numbers written in base-10, namely we do the integer division of the number with three and then successively of the obtained quotients with the same number three until the moment the quotient is zero, then we write the obtained remainders one after another from the last and reaching up to the first.

Numbers one to twenty-seven in standard ternary

Ternary	1	2	10	11	12	20	21	22	100
Binary	1	10	11	100	101	110	111	1000	1001
Decimal	1	2	3	4	5	6	7	8	9
Ternary	101	102	110	111	112	120	121	122	200
Binary	1010	1011	1100	1101	1110	1111	10000	10001	10010
Decimal	10	11	12	13	14	15	16	17	18
Ternary	201	202	210	211	212	220	221	222	1000
Binary	10011	10100	10101	10110	10111	11000	11001	11010	11011
Decimal	19	20	21	22	23	24	25	26	27

Practical usage

         A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a single hand for counting prayers (as alternative for the Misbaha). The benefit—apart from allowing a single hand to count up to 99 or to 100—is that counting doesn't distract the mind too much since the counter needs only to divide Tasbihs into groups of three.

         A rare "ternary point" is used to denote fractional parts of an inning in baseball. Since each inning consists of three outs, each out is considered one third of an inning and is denoted as .1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game , meaning 3⅔. In this usage, only the fractional part of the number is written in ternary form. Ternary numbers can be used to convey self-similar structures like a Sierpinski Triangle or a Cantor set conveniently. Additionally, it turns out that the ternary representation is useful for defining the Cantor Set and related point sets, because of the way the Cantor set is constructed. The Cantor set consists of the points from 0 to 1 whose ternary expression does not contain any instance of the digit 1.Ternary is the integer base with the highest radix economy, followed closely by binary and quaternary. It has been used for some computing systems because of this efficiency. It is also used to represent 3 option trees, such as phone menu systems, which allow a simple path to any branch.

Ternary computer

         Setun was a balanced ternary computer developed in 1958 at Moscow State University. The device was built under the lead of Sergei Sobolev and Nikolay Brusentsov. It was the only modern ternary computer, using three-valued ternary logic instead of two-valued binary logic prevalent in computers before and after Setun's conception. The computer was built to fulfill the needs of the Moscow State University and was manufactured at the Kazan Mathematical plant. Fifty computers were built and production was then halted in 1965. In the period between 1965 and 1970, a regular binary computer was then used at Moscow State University to replace it. However, although this replacement binary computer performed equally well, the device was still 2.5 times as expensive as the Setun. In 1970, a new ternary computer, the Setun-70, was designed. The computer was named after the Setun River, which ends near Moscow University.

 Referance

1.  Mohsen Soltanifar, On A sequence of cantor Fractals, Rose Hulman Undergraduate Mathematics Journal, Vol 7, No 1, paper 9, 2006.  
2. Mohsen Soltanifar, A Different Description of A Family of Middle-a Cantor Sets, American Journal of Undergraduate Research, Vol 5, No 2, pp 9–12, 2006.  
3. Brousentsov, N. P.; Maslov, S. P.; Ramil Alvarez, J.; Zhogolev, E.A.. "Development of ternary computers at Moscow State University". http://www.computer-museum.ru/english/setun.htm. Retrieved 20 January 2010. 
4. Hayes, Brian (2001), "Third base", American Scientist 89 (6): 490–494, 
5. Klimenko, Stanislav V.: Computer science in Russia: A personal view. IEEE Annals of the history of computing, v 21, n 3, 1999
6. Malinovski, B. N.: Istorija vychislitel’noj tekhniki v licakh. Kiev, 1995, (in Russian)
7. Žogolev, Y. A.: The order code and an interpretative system for the Setun computer. USSR Comp. Math. And Math. Physics (3), 1962, Oxford, Pergamon Press, p 563-578 (in English)
8. G. Trogemann, A. Y. Nitussov, W. Ernst (Hg.), Computing in Russia: The History of Computer Devices and Information Technology revealed. Vieweg Verlag, July 2001 (in English)
9. Hunger, Francis: SETUN. An Inquiry into the Soviet Ternary Computer. Institut für Buchkunst Leipzig, 2008, ISBN 3-932865-48-0 (Englisch, German)