Balanced- Ternary Logic for Improved and Advanced Computing

Logical Systems are the essence of our computing machines. They determine the design, understanding and most importantly the performance of the compuuting machines. Contemporary and traditional binary logic is way too old and limited in application and scope. This paper is meant to present balanced ternary logic as the most suitable logical system for our modern computing machines in terms of performance, simplicity, cost and the future prospects that it can bestow upon our modern computing machines. The paper also deals with the fundamental logical gates and operations in balance ternary logic. INTRODUCTION: For decades we have been using computing machines which work on the concept of fundamental switches having only two possible states often represented by 0 and 1, or ON and OFF.[1] We have witnessed innumerable advancements and improvements in the realm of these devices but none to have so strong an impact so as to question this fundamental property of our “modern” computers itself. Consequently, we have been working and improving upon the same binary logic based systems which has obvious limits. This paper, however, presents another logical system, balanced ternary logic for our computing machines and its advantages over any other proposed logic. Modern computing devices based on binary logic implement Boolean logic in which the fundamental components of the internal circuitry are required to have only two differentiable states φ1 φ2 [2]from where it gets the title as base-2 as well. Binary logic became popular due to the presence of simple and readily available two state switches and also extensive works on the same logic and machines. Based on two state-logic, modern computers lack various aspects expected from a good logical system which we could recover in our machines by implementing balanced ternary logic. Any logical system can have an associated power set PL to hold elements {φ1, φ2, φ3,..., φn} where n(PL) gives the base bL of that particular logic. Likewise, (balanced) ternary logic, also called “flip-flap-flop”[4] may be associated with a power set defined as PBT to hold elements as PBT = {φ1, φ2, φ3} and has n(PT)=3 which is ultimately bBT. This one extra state φ and the fact that bBT =3, make balanced ternary logic, the supreme logic. This paper is essentially a work on balanced ternary logic; we will see how it has an essential advantage over other possible logics including binary logic as well. The paper also defines the basic logical gates, the basic logical setups for this logic and also simple arithmetical operations. Balanced ternary logic has the least hardware complexity [5] and thus economical [6]. Unlike binary representation of information, there is no difference between “signed” and “unsigned” numbers when signed numbers are represented in this logic. To further add to its beauty, the amount of conditional instructions, using this logic, decrease twice which necessarily improves the performance and simplifies things. Special proofs and arguments in the paper shall further embark upon the efficiency, simplicity, versatility and overall supremacy of ternary logic over any other proposed logic. What if the fundamentals of our machines has better accordance with the Nature and informal human thinking? We need not worry; ternary logic has such advantages as well. Using three different states, it can, on the fundamental level itself have states referring to ‘True’ ‘False’ and interestingly ‘Unknown’[7] which is very relevant to informal human thinking. Moreover, in terms of versatility, balanced ternary logic, with a clever designing, can also provide an economical space for the conventional binary logic as well [12]. BALANCED TERNARY LOGIC: Balanced ternary logic is a non-binary, multi-valued logic and special case of ternary logic in which the PT is represented as PBT where PBT ={1,0,-1}, though, for shorthand, we use PBT ={+, 0, -}It is noteworthy that (φ) are simple states and that their representation as symbols and integral values carries no physical significance. Here, we define ‘Trit’ as the name given to the basic unit of information in a balanced ternary logic based machine. We also define ‘tryte’ for our purpose as a collection of six trits capable of holding 729 unique values. I Complexity: Besides it other numerous advantages, ternary logic turns out to be the least complex logic for computing. A proposed method for measuring complexity, CL of a logic would be by examining the maximum number of information represented by fixed n number of φ for any particular logic. Clearly, the logical system representing the least number of information using n φ would be least economical and would be the most complex. Conversely, the logical system that would represent the highest number of information for n number of φ would win and will be the most simple (least complex) and most economical [6] of all the proposed logics and will have a base equal to b. Let N number of numerical information be represented by any logical system with base b. We can define a function f(b) to find N to give the maximum possible number of information that could be represented using n φ. f (b) = b ^ (n /b ) Clearly, f(b), at the highest value of N, would correspond to the base b of most efficient logic. It is seen that with every Shamshad Ahmad et al, / (IJCSIT) International Journal of Computer Science and Information Technologies, Vol. 5 (4) , 2014, 5157-5160