On Ternary Coding and Three-Valued Logic (1807.06419v1)

Abstract: Mathematically, ternary coding is more efficient than binary coding. It is little used in computation because technology for binary processing is already established and the implementation of ternary coding is more complicated, but remains relevant in algorithms that use decision trees and in communications. In this paper we present a new comparison of binary and ternary coding and their relative efficiencies are computed both for number representation and decision trees. The implications of our inability to use optimal representation through mathematics or logic are examined. Apart from considerations of representation efficiency, ternary coding appears preferable to binary coding in classification of many real-world problems of AI and medicine. We examine the problem of identifying appropriate three classes for domain-specific applications.

• The paper demonstrates that ternary coding achieves nearly optimal efficiency (0.366 nats) with only a 0.5% deviation from the optimal performance, outperforming binary coding by 5.7%.
• It reveals that ternary number representation and decision trees offer shorter average path lengths and enhanced information capacity, useful in Huffman coding scenarios.
• It illustrates how ternary classification in AI and medicine provides more nuanced decision-making, aligning well with complex systems like drug response and quantum indeterminacy.

Comparative Analysis of Binary and Ternary Coding Systems

In the paper "On Ternary Coding and Three-Valued Logic," Subhash Kak explores the relative efficiencies of binary and ternary coding systems, extending the discussion to the broader implications in AI, medicine, and theoretical aspects of logic and mathematics. This essay aims to provide a succinct analytical summary of the key findings, numerical results, and theoretical implications presented in the paper.

Efficiency Metrics

The efficiency of different coding bases is analyzed through the lens of logarithmic measures and information theory. The efficiency per symbol, $E_b$, is given by $\frac{ \ln b }{ b }$. Using this metric, Kak shows that the optimal base is $e$ with $E(e) = 0.368$ nats or 0.531 bits. Ternary coding ($b=3$) achieves an efficiency of 0.366 nats, which is nearly optimal, with a deviation of only 0.5%. In contrast, binary coding ($b=2$) falls short by 5.7%.

Number Representation

Kak provides a thorough comparison between binary and ternary number representation, emphasizing that while binary coding is widely established in technology, ternary coding exhibits theoretical advantages. The carrying capacity of binary coding is considerably lower than that of ternary coding: $\frac{ e }{ \ln 2 } = 0.693$, whereas $\frac{ e }{ \ln 3 } = 1.099$. This demonstrates that ternary coding can convey information more efficiently.

Decision Trees

The paper also discusses the implications of using ternary coding in decision trees, which are fundamental structures in AI and machine learning. Binary trees, the prevalent choice, are contrasted with ternary trees. Kak demonstrates that for certain symbol distributions, ternary trees provide shorter average path lengths and greater efficiency. This finding is particularly relevant for Huffman coding scenarios, where probabilities are not perfect powers of the inverse of the coding base.

Ternary Classification in AI and Medicine

One of the most compelling practical implications of the paper is the notion that ternary classification outperforms binary classification in various real-world applications. Problems in medicine, such as drug response classification, can benefit significantly from a ternary approach. For example, classifying biomedical activities into agonistic, antagonistic, or neutral categories aligns naturally with ternary logic.

Philosophical and Theoretical Issues

Kak extends the discussion to philosophical questions, suggesting that ternary logic may offer a more nuanced model of reality, particularly when interpreting quantum mechanics. The indeterminate third value ("maybe") parallels quantum indeterminacy, proposing a potential deep-seated ontological basis for ternary logic.

Future Implications and Research Directions

The paper suggests that future research should focus on identifying optimal three-class systems for specific domains in AI and medicine. Such research might leverage unsupervised learning algorithms—like hierarchical clustering or k-means clustering—to uncover inherent ternary structures. Additionally, the application of ternary logic in database management and other technology sectors offers exciting avenues for exploration.

Conclusion

Subhash Kak’s comparative paper on ternary and binary coding not only highlights the theoretical efficiencies of ternary systems but also posits significant practical benefits. The paper’s insights into number representation, decision trees, and classification problems challenge the conventional reliance on binary systems, advocating for broader consideration of ternary logic in both AI applications and philosophical discourse. Future research has the potential to further validate and expand upon these findings, possibly leading to advancements in computational and cognitive frameworks.