Efficient Communication of Logic Statements: Bridging Information Theory and Propositional Logic

מושגי ליבה

The core message of this article is to develop a theory of communication that allows a sender (Alice) to efficiently convey logic statements to a receiver (Bob) with the goal of enabling Bob to deduce specific logic propositions, even when their knowledge about the underlying logic system differs.

תקציר

The article introduces a theory of communication that covers the following scenario: Alice knows more than Bob about a certain set of logic propositions, and Alice and Bob wish to communicate as efficiently as possible with the shared goal that, following their communication, Bob should be able to deduce a particular logic proposition that Alice knows to be true.

The authors make the following key contributions:

1. They introduce a function Λ that appears to be fundamental to problems involving the communication of logic information. This function arises naturally in their analysis and provides sharp upper and lower bounds on the communication cost.

2. They provide a series of theorems that study various assumptions on what Alice and Bob know and what their goal is. These theorems reveal several interesting insights, such as:

   • When the goal is to communicate only a portion of Alice's knowledge, the optimal communication cost can be lower than expected, yet it may allow Bob to prove even more than originally intended.
   • In some scenarios, Alice need not know the logic statements that Bob already has in order to achieve asymptotically the same communication efficiency as if she knew them.
   • The communication cost of correcting misinformation (when Bob's knowledge excludes Alice's) grows unboundedly compared to correcting ignorance (when Bob's knowledge is a subset of Alice's).

3. The authors introduce practical coding schemes, comprised of a combination of linear codes and enumerative source codes, that are asymptotically optimal for some scenarios.

4. The work establishes connections to existing literature in information theory, semantic information theory, and logic, building on concepts such as Shannon's information theory, Carnap and Bar-Hillel's work on semantic information, and ideas from rate-distortion theory and Slepian-Wolf/Wyner-Ziv coding.

סטטיסטיקה

The article does not contain any explicit numerical data or statistics. It focuses on developing a theoretical framework and deriving analytical results.

ציטוטים

"Relative to the broad subject of communication, there seem to be problems at three levels. Thus it seems reasonable to ask serially: LEVEL A. How accurately can the symbols of communication be transmitted? (The technical problem) LEVEL B. How precisely do the transmitted symbols convey the desired meaning? (The semantic problem.) LEVEL C. How effectively does the received meaning affect conduct in the desired way? (The effectiveness problem.)" - Warren Weaver

תובנות מפתח מזוקקות מ:

Towards a Unification of Logic and Information Theory

by Luis A. Last... ב- arxiv.org 04-17-2024

https://arxiv.org/pdf/2301.10414.pdf

שאלות מעמיקות

How can the insights from this work on efficient communication of logic statements be extended to more expressive logics beyond propositional logic, such as first-order logic or modal logic

The insights gained from the work on efficient communication of logic statements in propositional logic can be extended to more expressive logics, such as first-order logic or modal logic, by considering the structure and complexity of these logics.
In first-order logic, where quantifiers and predicates are used to express relationships between objects, the communication of logical statements would involve transmitting not just simple propositions but also complex structures involving variables, functions, and quantifiers. The challenge would be to encode and transmit these structures efficiently while ensuring that the receiver can deduce the intended logical conclusions.
One approach to extending the insights to first-order logic could involve developing encoding schemes that capture the quantified nature of the logic, allowing for the transmission of quantified statements and logical relationships. This may involve encoding not just the truth values of propositions but also the quantifiers and variables involved in the logic statements.
Similarly, in modal logic, which deals with modalities such as necessity and possibility, the communication of logical statements would involve conveying not just factual information but also modal relationships and constraints. Extending the insights to modal logic would require encoding and transmitting modal operators and their associated semantics effectively.
Overall, the extension to more expressive logics would involve adapting the encoding and communication strategies developed for propositional logic to capture the additional complexity and structure present in first-order logic and modal logic.

What are the implications of this work for the design of communication protocols in applications where the semantics of the transmitted information is crucial, such as in safety-critical systems or legal/regulatory domains

The implications of this work for the design of communication protocols in applications where the semantics of the transmitted information is crucial, such as in safety-critical systems or legal/regulatory domains, are significant.
In safety-critical systems, where the correct interpretation of communicated information can have life-or-death consequences, the efficient communication of logic statements can ensure that critical information is transmitted accurately and reliably. By optimizing the communication process based on the insights from this work, safety-critical systems can enhance their ability to convey essential information in a clear and concise manner.
In legal and regulatory domains, where precise semantics and logical reasoning are paramount, the design of communication protocols based on efficient logic transmission can improve the clarity and accuracy of legal documents, contracts, and regulatory guidelines. By incorporating strategies for efficient communication of logic statements, legal and regulatory frameworks can reduce ambiguity, improve compliance, and enhance the overall effectiveness of communication in these domains.
Overall, the design of communication protocols based on the insights from this work can lead to more robust and reliable communication systems in safety-critical systems and legal/regulatory domains, ensuring that the semantics of transmitted information are preserved and accurately conveyed.

Can the techniques developed in this work be adapted to address challenges in other areas of computer science that involve reasoning about logical knowledge, such as in knowledge representation, automated reasoning, or multi-agent systems

The techniques developed in this work can be adapted to address challenges in other areas of computer science that involve reasoning about logical knowledge, such as in knowledge representation, automated reasoning, or multi-agent systems.
In knowledge representation, where the goal is to represent knowledge in a formal and structured way, the efficient communication of logic statements can improve the sharing and exchange of knowledge between different systems or agents. By applying the communication strategies developed in this work, knowledge representation systems can enhance their ability to transmit logical information effectively and accurately.
In automated reasoning, where systems are designed to automatically infer logical conclusions from given information, the optimization of communication protocols based on efficient logic transmission can improve the efficiency and accuracy of automated reasoning processes. By streamlining the communication of logical statements, automated reasoning systems can enhance their ability to derive logical conclusions in a timely and reliable manner.
In multi-agent systems, where multiple autonomous agents interact to achieve common goals, the design of communication protocols informed by efficient logic transmission can facilitate effective communication and coordination among agents. By incorporating strategies for transmitting logic statements efficiently, multi-agent systems can improve their ability to share information, make decisions, and collaborate towards shared objectives.
Overall, the techniques developed in this work have broad applications in various areas of computer science that involve reasoning about logical knowledge, offering opportunities to enhance communication, reasoning, and collaboration in complex systems and domains.