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A Theory of Semantic Communication

Yulin Shao, Qi Cao, Deniz Gunduz

arXiv:2212.01485v4cs.IT

TL;DR

The paper addresses the lack of a comprehensive framework for characterizing semantic communication by distinguishing language exploitation from language design. It places language design within joint source-channel coding and analyzes three language-exploitation strategies using semantic distortion-cost regions, characterizing the achievable region for each.

  • Problem

    Semantic communication lacks a comprehensive, widely accepted framework, particularly for distinguishing language exploitation from language design.

  • Method

    The paper defines semantic communication around a shared semantic language, formulates two fundamental problems, and analyzes semantic encoding, semantic decoding, and combined semantic encoding and decoding.

  • Results

    Achievable semantic distortion-cost regions are characterized for semantic encoding, semantic decoding, and combined semantic encoding and decoding.

  • Takeaways & Limitations

    Language design aligns with joint source-channel coding, whereas language exploitation addresses mismatches involving an agreed but undesignable semantic language.

Abstract

from arXiv · show

Semantic communication is an emerging research area that has gained a wide range of attention recently. Despite this growing interest, there remains a notable absence of a comprehensive and widely-accepted framework for characterizing semantic communication. This paper introduces a new conceptualization of semantic communication and formulates two fundamental problems, which we term language exploitation and language design. Our contention is that the challenge of language design can be effectively situated within the broader framework of joint source-channel coding theory, underpinned by a comprehensive end-to-end distortion metric. To tackle the language exploitation problem, we put forth three approaches: semantic encoding, semantic decoding, and a synergistic combination of both in the form of combined semantic encoding and decoding. Furthermore, we establish the semantic distortion-cost region as a critical framework for assessing the language exploitation problem. For each of the three proposed approaches, the achievable distortion-cost region is characterized. Overall, this paper aims to shed light on the intricate dynamics of semantic communication, paving the way for a deeper understanding of this evolving field.

I. INTRODUCTION

The paper frames semantic communication as conveying intended meaning through an agreed semantic language, distinct from technical symbol transmission and pragmatic task effectiveness. It introduces language exploitation and language design as two core problems, focusing on how mismatches between communicators can be mitigated.

  • A. What is semantic communication?: Semantic communication concerns how messages convey intended meanings, whereas technical communication reconstructs symbols and pragmatic communication generates meanings suited to the ultimate task.The framework distinguishes technical, semantic, and effectiveness problems as three levels of communication.
  • A. What is semantic communication?: The same transmitted message can fail semantically through ambiguity, while different messages can succeed when they preserve the intended meaning.The examples contrast “apple” as a fruit or phone with “does not like” and “dislikes” as equivalent expressions.
  • B. Fundamental problems of semantic communication: A semantic language is richer than a technical language because meanings may map to multiple messages and messages may admit multiple interpretations.This richness supports goals such as elegance and comprehensibility while requiring robustness to mismatched prior information.
  • B. Fundamental problems of semantic communication: Language exploitation assumes semantic and technical languages are fixed and designs available components to reduce transmitter–receiver misinterpretation.The paper identifies semantic encoding, semantic decoding, and their combination as relevant strategies, including prompt engineering as an example.
  • C. Contributions and roadmap: The paper presents semantic language as encompassing the semantic encoder, semantic decoder, and shared knowledge, and evaluates language exploitation through semantic distortion-cost regions.It characterizes achievable regions for semantic encoding, semantic decoding, and combined semantic encoding and decoding.
  • B. Fundamental problems of semantic communication: Language design instead allows the semantic and technical languages to be jointly crafted for efficient communication under a suitable distortion measure.The paper relates this problem to classical communication principles and joint source-channel coding.

III. PROBLEM FORMULATION

The paper formalizes semantic communication through semantic languages, channels, distortion, and cost, then frames language exploitation through encoding, decoding, and their combination. It introduces the semantic distortion-cost region as the central object for characterizing these approaches.

  • III. PROBLEM FORMULATION: A semantic language is modeled by words, syntax, expression, and interpretation, which together connect meanings and messages.Its message set S is determined by words and syntax, while expression and interpretation are represented by mappings P and Q.
  • III. PROBLEM FORMULATION: Meanings form a set W with intended-meaning probabilities, while messages may express multiple meanings and meanings may be expressed by multiple messages.These multiplicities capture expressive redundancy and interpretive ambiguity.
  • III. PROBLEM FORMULATION: The framework permits logically mismatched expression and interpretation, so q(w|s) need not satisfy logical self-consistency.This accommodates semantic communication in which transmitter and receiver mappings do not align perfectly.
  • III. PROBLEM FORMULATION: Human agents acquire distinct language systems from personal interaction with the world, yielding individual message sets and expression and interpretation mappings.The paper uses this observation to motivate language as agent-dependent rather than universally identical.
  • III. PROBLEM FORMULATION: A semantic channel combines technical message transmission with semantic expression and interpretation, and its quality depends partly on the technical communication implementation.The formulation uses transition probabilities for transmitted and received messages while abstracting coding and modulation into the semantic channel.
  • A. Semantic encoding: Language exploitation seeks to minimize receiver misinterpretation through semantic encoding, semantic decoding, or combined semantic encoding and decoding.The paper considers one-shot transmission and defines semantic encoding as choosing messages for intended meanings under channel, decoder, and cost constraints.
  • B. Semantic decoding and combined semantic encoding and decoding (CSED): Combined semantic encoding and decoding lets transmitter and receiver optimize their semantic operations simultaneously, and the paper characterizes its distortion-cost region alongside the encoding problem.The semantic distortion-cost region is defined through the lower envelope of achievable distortion-cost pairs.
  • A. Semantic distortion-cost region and function: Unlike Shannon rate-distortion theory, semantic distortion-cost behavior can include a boost segment where reducing distortion or cost improves the other quantity.The shape depends on the semantic language and the specific semantic cost and distortion definitions.

B. Characterizing the distortion-cost region

The paper characterizes the semantic encoding distortion-cost region by reducing attention to deterministic encoders and constructing its boundary from achievable distortion-cost pairs. Stochastic encoders are represented through time sharing, yielding a convex region and piecewise-linear boundary.

  • B. Characterizing the distortion-cost region: A stochastic semantic encoding scheme achieves a distortion-cost pair exactly when time sharing among deterministic schemes achieves that pair.This establishes deterministic encoders as sufficient for characterizing the full region.
  • B. Characterizing the distortion-cost region: The distortion-cost region Renc is convex, and its distortion-cost function D*U,Q(L) is convex.Consequently, the characterization can focus on deterministic semantic encoding schemes.
  • B. Characterizing the distortion-cost region: The boundary of Renc is a piecewise-linear connection of T+T+2 distortion-cost pairs generated by deterministic encoding schemes.Theorem 4.2 identifies these boundary vertices as the output of the constructive procedure in Algorithm 2.
  • B. Characterizing the distortion-cost region: Algorithm 2 constructs deterministic encoders by finding message subsets for each meaning and progressively changing selected messages to trace boundary points.Its inputs are the semantic language, message cost function, and distortion measure.
  • B. Characterizing the distortion-cost region: The lower envelope is the paper’s focus because it gives the minimum distortion attainable at each average message cost.The distortion-cost function is linear between boundary vertices, with non-vertex points achievable by stochastic encoding or time sharing.

V. SEMANTIC DECODING AND COMBINED SEMANTIC ENCODING AND DECODING

Semantic decoding optimizes receiver-side interpretation under potentially inaccurate prior information, while its distortion-cost region is constrained by a fixed transmitter-side semantic encoder. Under Hamming distortion, optimality depends on whether prior-induced probability points remain in the same partition regions, and inaccurate priors can make decoding worse than the original interpretation.

  • A. Semantic decoding: The paper characterizes the semantic-decoding distortion-cost region and studies conditions under which receiver-side decoding achieves the optimal distortion.The section introduces the achievable distortion-cost region and analyzes prior mismatch through Hamming distortion and a geometric partition criterion.
  • A. Semantic decoding: The semantic-decoding distortion-cost region is a vertical line because the fixed semantic encoder fixes the achievable message cost.Deterministic decoding maps each received message to a single meaning, while the encoder remains unchanged.
  • A. Semantic decoding: With an inaccurate prior q(w), the receiver selects the decoding mapping that minimizes its perceived distortion, but may lack the true optimal scheme.The receiver’s optimization is based on q(w), whereas the benchmark optimal decoder uses the true prior p(w).
  • A. Semantic decoding: Under Hamming distortion, semantic decoding is optimal if and only if αp(ˆs) and αq(ˆs) fall in the same probability-space region for every received message.For N = 3, the simplex is an equilateral triangle partitioned into three quadrilaterals; the relevant comparison is partition membership rather than L1 or KL distance.
  • A. Semantic decoding: An inaccurate prior can make semantic decoding worse than the original interpretation Q, whereas an accurate prior makes the optimized decoder strictly better than Q.The paper therefore proposes refining Q to construct a random decoding scheme with distortion no greater than the original interpretation.

B. Combined semantic encoding and decoding (CSED)

CSED combines semantic encoding at the transmitter with semantic decoding at the receiver, but this combination does not universally outperform either separate approach. The section defines CSED’s distortion-cost region and identifies conditions under which it can improve performance.

  • B. Combined semantic encoding and decoding (CSED): CSED combines semantic encoding and semantic decoding, allowing the transmitter and receiver to optimize their respective language operations simultaneously.The scheme is introduced because separate encoding and decoding are decoupled when negotiation is unavailable.
  • B. Combined semantic encoding and decoding (CSED): The CSED distortion-cost region Rcsed is defined as the convex hull of performance points generated by the selected deterministic encoding schemes and optimal decoding.Time sharing enables interpolation among the performance characteristics of different deterministic schemes.
  • B. Combined semantic encoding and decoding (CSED): CSED does not necessarily improve upon separate semantic encoding or decoding: in the nod-shake example, separate methods achieve Hamming distortion 0, whereas CSED yields distortion 1.The example uses equal-cost messages and opposite transmitter and receiver interpretations.
  • B. Combined semantic encoding and decoding (CSED): Under an error-free semantic channel, perfect receiver priors, logical self-consistency, and symmetric distortion, CSED is better than both semantic encoding and semantic decoding.The theorem provides a sufficient-condition result rather than a universal dominance claim.
  • B. Combined semantic encoding and decoding (CSED): Repeatedly predicting the other party’s updated encoding or decoding can make distortion alternate between 0 and 1, preventing stable communication improvements.The paper compares this iterative inference phenomenon with human communication between participants who continually revise their interpretations.

VI. SEMANTIC COMMUNICATION: AN EXAMPLE

The example instantiates the paper’s semantic communication framework in a grid world where a transmitter observes a bug’s trajectory and communicates its destination to a receiver. It specifies the semantic and technical languages, message costs, and error-free transmission assumptions.

  • VI. SEMANTIC COMMUNICATION: AN EXAMPLE: The grid-world example models a bug moving from (0, 0) toward destinations A: (1, 2) or B: (2, 2), with only up and right moves allowed.The transmitter observes the trajectory, while the receiver is interested in the bug’s destination.
  • VI. SEMANTIC COMMUNICATION: AN EXAMPLE: The transmitter and receiver agree on semantic and technical languages, with semantic words ‘U’ and ‘R’ representing the bug’s up and right actions.A message is a sequence of words describing the bug’s trajectory from (0, 0), such as ‘URR’.
  • VI. SEMANTIC COMMUNICATION: AN EXAMPLE: The receiver’s meaning prior is q(A) = 1/2 and q(B) = 1/2, while the example’s meaning distribution is p(A) = 1/3 and p(B) = 2/3.This creates a mismatch between the transmitter-side meaning distribution and the receiver’s prior information.
  • VI. SEMANTIC COMMUNICATION: AN EXAMPLE: The technical language maps ‘U’ to ‘0’ and ‘R’ to ‘10’, and the resulting message-to-channel-symbol mappings are one-to-one.The technical and semantic channels are assumed error-free, so transmitted messages are perfectly received.

2) Error-free technical communication:

Under error-free technical communication, the example defines message costs through channel-symbol lengths and constructs semantic encoding schemes to characterize the semantic encoding distortion-cost region. The resulting region is presented in Fig. 11(a).

  • 2) Error-free technical communication:: The example uses Hamming distortion and defines each message’s cost as the length of its associated bit sequence.This cost definition connects semantic performance to the technical-language representation.
  • 2) Error-free technical communication:: The receiver interpretation q(w|s) and transmitter expression p(s|w) are explicitly tabulated for the semantic language.Table I provides the complete interpretation and expression values for meanings and messages.
  • 2) Error-free technical communication:: The encoding analysis constructs subsets and deterministic encoding schemes, then uses the resulting schemes to characterize the semantic encoding region Renc in Fig. 11(a).The schemes and their achieved distortion and cost are listed in Table III.
  • 2) Error-free technical communication:: The example reports interpretation values such as q(A|U) = 3/5, q(B|U) = 2/5, q(A|R) = 3/4, and q(B|R) = 1/4.These values are among the tabulated expression and interpretation results for the semantic language.

5) Semantic decoding and CSED:

The paper characterizes semantic decoding and CSED distortion-cost regions, showing how decoding with an inaccurate prior and combined encoding-decoding behave in the example. It also identifies extensions and a selection-dependent limitation of CSED.

  • 5) Semantic decoding and CSED:: With an inaccurate prior, the derived optimal semantic decoding scheme achieves the optimal distortion in the example, as shown in Fig. 11(b).The original interpretation Q yields a separate distortion value, while the receiver’s optimized scheme is derived using Proposition 5.2.
  • 5) Semantic decoding and CSED:: The CSED distortion-cost region Rcsed is obtained from Theorem 5.5 and plotted in Fig. 11(c).CSED combines the semantic encoding and decoding procedures into a new achievable region.
  • 5) Semantic decoding and CSED:: Different equally optimal encoding choices can produce different CSED distortion-cost regions, even though they produce the same semantic encoding region.The dependence arises because CSED is determined by the particular encoding schemes selected during Algorithm 2.
  • 5) Semantic decoding and CSED:: The paper relates semantic communication to rate-distortion theory by treating meanings and messages analogously to messages and channel symbols, with semantic cost defined by channel-symbol length.This comparison frames language design through the classical joint source-channel coding perspective.
  • 5) Semantic decoding and CSED:: Future directions include multiple transmissions, feedback-aided interaction, learned languages, and semantic communication over networks.The paper presents these as extensions involving repeated channel use, interaction, language formation, and network settings.
  • 5) Semantic decoding and CSED:: The paper distinguishes language exploitation from classical information theory through an agreed but undesignable semantic language and studies encoding, decoding, and CSED using distortion-cost regions.Its conclusion frames these three approaches as mechanisms for reducing misinterpretation within shared semantic ground.

APPENDIX A PROOF OF PROPOSITION 4.1

The proof shows that stochastic semantic encoding schemes can be represented through deterministic schemes while preserving their distortion-cost performance.

  • Stochastic encoding schemes can be replaced by deterministic encoding schemes that achieve the same distortion-cost pair.
  • Therefore, deterministic schemes suffice to characterize the distortion-cost region of semantic encoding.
  • The proof establishes the converse by constructing a stochastic encoding scheme from a distribution over deterministic schemes.

APPENDIX B PROPERTIES OF THE SIX SUBSETS

This appendix develops structural properties of the six subsets used to construct deterministic semantic encoding schemes and characterize their distortion-cost region.

  • For each meaning, the relevant candidate messages are organized through the six subsets defined for constructing deterministic encoding schemes.The appendix states that these subsets support the characterization of the semantic-encoding distortion-cost region.
  • Four representative distortion-cost points are generated by deterministic schemes indexed through the selected message sets.
  • The selected message set excludes messages that are simultaneously less accurate and less cost-efficient than another message.
  • The propositions give minimum- and maximum-cost or accuracy characterizations for the associated subsets.
  • Any encoding scheme can be bounded by selected schemes whose distortion and cost satisfy the four inequalities in Lemma B.4.

APPENDIX C PROOF OF THEOREM 4.2

The proof establishes that Algorithm 2 constructs the exact semantic-encoding distortion-cost region by connecting a finite sequence of deterministic encoding schemes and proving convexity.

  • Algorithm 2 constructs a sequence of deterministic encoding schemes whose piecewise-linear connections define the candidate region.
  • The auxiliary G function is monotone across the sequence, supporting the ordered construction of successive encoding schemes.
  • The functions D(L) and D̄(L) are respectively convex and concave, and the resulting candidate region is convex.
  • The characterized region is exactly the semantic-encoding distortion-cost region Renc.
  • The converse shows that every achievable encoding distortion-cost pair lies within the constructed region.

APPENDIX D PROOF OF PROPOSITION 5.3

The appendix derives semantic decoding expressions under Hamming distortion and compares decoding under a prior distribution with decoding under the true distribution.

  • Under Hamming distortion, semantic decoding is expressed through the decoding probabilities and the associated semantic error.
  • For a chosen prior q(w), the decoder uses maximum a posteriori decoding for each received message.
  • The q(w)- and p(w)-based decoding distortions are equal if and only if condition (30) holds.

APPENDIX E PROOF OF THEOREM 5.6

The proof of Theorem 5.6 analyzes CSED mappings and establishes properties needed to characterize its distortion-cost function. It combines bijectivity arguments, induction, and deterministic encoding constructions.

  • The distortion-cost function D∗ is obtained using deterministic encoding schemes and linear combinations of two contiguous U(i) points, extending the result across D∗.
  • For CSED, meanings are mapped to encoded messages s and decoded messages ˆw, while the relevant message subset is defined explicitly.
  • The proof establishes that the mappings between W and [S|W], in both directions, are one-to-one under conditions 3) and 4).
  • The argument uses induction over w(n), with separate cases for the index β to verify the required construction at each step.
  • The proof shows β ≤ 1 and therefore β = 1 in the initial case, completing that part of the construction.
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