Breaking through the classical Shannon entropy limit: A new frontier through logical semantics (2501.00612v1)

Abstract: Information theory has provided foundations for the theories of several application areas critical for modern society, including communications, computer storage, and AI. A key aspect of Shannon's 1948 theory is a sharp lower bound on the number of bits needed to encode and communicate a string of symbols. When he introduced the theory, Shannon famously excluded any notion of semantics behind the symbols being communicated. This semantics-free notion went on to have massive impact on communication and computing technologies, even as multiple proposals for reintroducing semantics in a theory of information were being made, notably one where Carnap and Bar-Hillel used logic and reasoning to capture semantics. In this paper we present, for the first time, a Shannon-style analysis of a communication system equipped with a deductive reasoning capability, implemented using logical inference. We use some of the most important techniques developed in information theory to demonstrate significant and sometimes surprising gains in communication efficiency availed to us through such capability, demonstrated also through practical codes. We thus argue that proposals for a semantic information theory should include the power of deductive reasoning to magnify the value of transmitted bits as we strive to fully unlock the inherent potential of semantics.

• The paper introduces deductive reasoning into classical Shannon entropy models to enhance semantic communication efficiency and reduce transmission bits.
• It employs a probabilistic framework and novel logical semantic entropy to establish sharp bounds on coding costs in both symmetric and asymmetric knowledge scenarios.
• The study offers practical guidelines for designing advanced communication systems and managing misinformation through enhanced logical inference.

Exploring the Integration of Deductive Reasoning in Semantic Information Theory

The paper undertakes a rigorous exploration of integrating deductive reasoning in the classical framework of information theory, originally introduced by Claude Shannon in 1948. The primary focus is to analyze how deductive logic can transcend the traditional limits of Shannon entropy by enabling communication systems to encode semantic information more efficiently. It builds upon prior efforts by Carnap and Bar-Hillel, who proposed a formalization of semantics in logical terms, and extends this notion by incorporating deductive reasoning as a core mechanism for enhancing communication efficiency.

Key Contributions and Framework

The paper presents a theoretical model where communication systems use logical inference to improve the transmission efficiency of semantic information. It introduces deductive reasoning into classical communication system setups, allowing for significant gains in communicational efficiency as validated through both theoretical and empirical analyses.

Key components of the framework include:

• Logical Semantic Entropy: The authors introduce a novel metric called logical semantic entropy, denoted by $\Lambda$, which quantifies the expected number of bits needed for communication when deductive reasoning is utilized. The metric provides sharp bounds on communication costs, highlighting the efficiency gains obtainable by integrating deductive reasoning.
• Probabilistic Model: The paper discusses a probabilistic model that considers various logical statements' kernel sizes (representing information content) and explores scenarios of communication where Alice, the sender, possesses certain world knowledge, and Bob, the receiver, has a partial or complete logical model.
• Theoretical Results: Several results are established, offering upper and lower bounds on the expected number of communication bits necessary for successful semantic transmission. These bounds reveal that logical deduction allows for substantial reductions in communication costs, often outperforming traditional methods.

Technical Insights

The paper provides insights into several non-trivial scenarios in semantic communication. For instance, it describes a paradox akin to "Less is More," wherein more compact transmission can allow a receiver to prove more logical statements than explicitly communicated. The research examines both symmetric and asymmetric knowledge scenarios, as well as misinformation settings. The latter is modeled as a condition where Alice's and Bob's knowledge bases are logically inconsistent, thus requiring more significant resources to correct than mere ignorance.

Practical and Theoretical Implications

Practically, the incorporation of deductive reasoning into communication systems can enhance the efficiency of data exchange in several critical applications, including digital communication networks and AI systems. Theoretically, this approach opens new avenues for linking information theory with logic, inviting further exploration into semantic information representation beyond classical boundaries.

The development and testing of practical coding schemes based on logical semantics offer a promising direction for marrying logic and information theory. Additionally, the paper envisions the potential application in re-skilling and misinformation management, emphasizing the versatility and societal implications of this line of research.

Future Directions

The research suggests several future directions, including:

• Integrative Approaches: Combining deductive reasoning with existing semantic communication models could lead to enhanced systems that fully utilize the potential of semantic information theory. Leveraging ideas from coding theory and logical inference to extend the current framework might lead to new algorithmic solutions for more complex, real-world communication scenarios.
• Broadening Logic Frameworks: Extending the theoretical framework to encompass richer logics such as First-Order Logic with counting or Second-Order Logic could provide deeper insights and more robust applications.
• Cross-disciplinary Applications: Exploring connections between semantic information theory, Kolmogorov complexity, and machine learning models could recontextualize AI systems' learning and inference processes.

In conclusion, this paper presents a pioneering effort to redefine the role of semantics in information theory by introducing logical deduction mechanisms, offering a pathway to more efficient communication systems and inviting further exploration across multiple dimensions of information technology and its societal applications.