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A Mathematical Theory of Pragmatic Information

Kai Niu, Ping Zhang

arXiv:2609.10986v1cs.ITcs.AIcs.ROeess.SY

TL;DR

The paper focuses on information used to guide decisions and actions, especially the optimal action class relevant to safe and efficient navigation. It develops pragmatic information measures and dualities, with cross-layer optimization and capacity results, and frames the framework for goal-directed systems extracting value under resource constraints.

  • Problem

    Information is evaluated by how it guides decisions and actions, with the optimal action class identified as key to safe and efficient navigation.

  • Method

    The framework recognizes only action-relevant distinctions, defines pragmatic value and cost, and establishes a Legendre–Fenchel duality for cross-layer analysis.

  • Results

    The simplified global Lagrangian is Lglobal(R) = Φp(R) − CoIp(R), while the Gaussian case gives R = 1 as free and R = 2 as costing 3 units.

  • Takeaways & Limitations

    The theory offers a unified language for intelligent systems that extract value from information under resource constraints rather than merely transmitting symbols.

  • Takeaways & Limitations

    The framework is scoped to intelligent systems operating under resource constraints.

Abstract

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We propose a pragmatic information theory unifying communication, control, and decision-making. Its core is the isoteleia mapping, formalizing equifinality: distinct semantic paths leading to the same optimal action are pragmatically equivalent. This induces a three-tier hierarchy of syntactic, semantic, and pragmatic information, each abstraction discarding task-irrelevant distinctions. We develop pragmatic entropy, up/down mutual information, channel capacity, and rate-distortion, and prove three coding theorems generalizing Shannon's classical results. We introduce pragmatic value (VoI) and cost (CoI) of information as decision-theoretic duals to rate-distortion and capacity, respectively, and formulate a Lagrangian dual framework for cross-layer optimization. The pragmatic efficiency bound $\mathcal{E}_p(λ)=\sup_R[Φ_p(R)-λ\,\mathrm{CoI}_p(R)]$ quantifies the maximum net utility any resource-constrained intelligent system can extract, thereby establishing a fundamental behavioral capacity limit---generalizing Shannon's symbol-level capacity to goal-directed action. Extensions to continuous messages yield closed-form Gaussian expressions, while dynamic settings are addressed via a Bellman equation for sequential decision-making. This framework provides a rigorous foundation for task-oriented communication, networked control, autonomous systems, and embodied AI, shifting focus from symbol fidelity to the effectiveness of information in guiding actions, and offers a unified mathematical language for next-generation intelligent systems.

I. INTRODUCTION

The paper introduces pragmatic information theory to unify communication, control, and decision-making around the effectiveness of information in guiding actions. It formalizes equifinality through a three-tier hierarchy and develops measures, coding theorems, and optimization principles that extend classical information theory toward goal-directed systems.

  • Motivation and framework: Pragmatic information links perception, communication, decision, and execution by focusing on how information affects a receiver’s decisions and actions.The framework shifts attention from signal transmission and meaning toward task-relevant action effectiveness.
  • Motivation and framework: The isoteleia mapping formalizes equifinality by treating distinct semantic paths that converge to the same optimal action as pragmatically equivalent.This mapping supports a hierarchy in which syntactic and semantic distinctions can be discarded when they are irrelevant to the task.
  • Core measures: The theory defines pragmatic entropy, mutual information, channel capacity, and rate-distortion, with hierarchies relating them to syntactic and semantic counterparts.The resulting measures characterize pragmatic information and quantify gains from pragmatic abstraction.
  • Decision-theoretic optimization: Pragmatic value and cost of information connect utility gains and resource expenditure to rate-distortion and channel-capacity dualities.The framework also supplies a Lagrangian approach for cross-layer resource allocation and task-oriented system design.
  • Coding theorems: Three pragmatic coding theorems establish limits for lossless compression, reliable transmission, and lossy compression, generalizing Shannon’s classical coding theorems.The minimum lossless rate is pragmatic entropy, the maximum reliable rate is pragmatic capacity, and the lossy rate is Rp(D).
  • Extensions and scope: The framework extends to Gaussian continuous-domain measures, dynamic Bellman optimization, and systems ranging from classical communication to embodied and social-cognitive agents.These extensions support a general model for goal-directed information processing under uncertainty.

II. PRAGMATIC INFORMATION SYSTEM AND ISOTELEIA MAPPING

The pragmatic information system unifies communication and control through a three-tier architecture that transforms observations and task goals into utility-maximizing actions. Its PCDE loop evaluates information by how effectively it supports terminal action rather than symbol reconstruction.

  • System Structure and Processing Flow: The framework integrates Shannon communication with cybernetic control in a perception-communication-decision-execution architecture aimed at maximizing task utility.The system includes physical-world sensing, task-intent generation, hierarchical source and destination modules, communication, demapping, and task execution.
  • System Structure and Processing Flow: The Source maps task intent through pragmatic, semantic, and syntactic levels, answering what action is best, why it is justified, and what signal should be transmitted.
  • System Structure and Processing Flow: The Reification Mapping composes isoteleia and synonymous mappings to enumerate syntactic realizations that fulfill a pragmatic purpose.
  • System Structure and Processing Flow: The channel is modeled by P(Yt|Xt) under power, bandwidth, and delay constraints, while decoding and demapping recover semantic meaning and pragmatic purpose.
  • Axiomatic Foundations: VoI and CoI evaluate the end-to-end system by balancing expected utility gain against the information resources required.
  • Axiomatic Foundations: Classical source coding, channel coding, and rate-distortion results emerge as special cases, extending information theory from symbol transmission to action effectiveness.

C. Design Principles of Pragmatic Information Systems

The design principles organize pragmatic information around utility-relevant action, semantic equivalence, and compatibility with established communication and control theories. Their mathematical basis is the isoteleia mapping, which groups distinct semantic paths that produce the same optimal action.

  • Principle I: Pragmatic Imperceptibility and Isoteleia: Pragmatic information is evaluated through utility gain and system performance, so design should prioritize resulting actions over symbol-reconstruction fidelity.Communication resources should be allocated according to the marginal Value of Information they provide.
  • Principle I: Pragmatic Imperceptibility and Isoteleia: Isoteleia makes semantic and syntactic realizations pragmatically equivalent when they lead to the same optimal terminal action under a utility function.The mapping partitions semantic meanings into classes sharing an optimal action.
  • Principle II: Semantic Imperceptibility and Synonymy: Synonymy treats different syntactic realizations as equivalent when they convey the same meaning, permitting syntax-level lossy compression when semantic content is preserved.The framework also calls for contextual disambiguation and task-relevant semantic feature transmission.
  • Principle III: Backward Compatibility and Unification: Backward compatibility requires the pragmatic framework to contain classical communication and control as special cases, collapsing to Shannon theory under symbol-fidelity utility.When communication is ignored, the framework collapses into classical control theory.
  • Structural Mappings: Isoteleia formalizes equifinality: distinct semantic paths can converge to the same ultimate goal while remaining pragmatically equivalent.
  • Structural Mappings: The three information layers form a hierarchy in which syntactic symbols group into semantic fibers and semantic fibers group into pragmatic action classes.

III. PRAGMATIC ENTROPY

Pragmatic entropy measures uncertainty about optimal terminal actions rather than symbols or meanings. The resulting entropy hierarchy formalizes how semantic and pragmatic abstraction remove distinctions irrelevant to decisions and coding.

  • Definition and Interpretation: Pragmatic entropy is the minimum average number of pragmatic bits required to specify the optimal terminal action.It represents irreducible decision uncertainty resolved through communication and control.
  • Fundamental Properties: Pragmatic entropy is nonnegative and equals zero exactly when there is only one possible optimal action regardless of the state.
  • Entropy Hierarchy: The entropy hierarchy removes distinctions successively: syntactic entropy retains symbol differences, semantic entropy removes meaningless variations, and pragmatic entropy removes decision-irrelevant meanings.Each coarsening reduces joint entropy and supports multilevel source coding and compression.
  • Illustrative Example: In the traffic-light example, entropy falls from 1.295 bits syntactically to 0.469 prabits pragmatically because the vehicle needs only distinguish Go from Stop.
  • Sequential and Joint Entropy: The binary chain-rule results bound pragmatic joint entropy between expressions involving marginal and conditional pragmatic entropies, formalizing layered uncertainty reduction.
  • Sequential and Joint Entropy: For the joint example, syntactic, semantic, and pragmatic entropies are approximately 3.6318 bits, 2.7565 sebits, and 1.7509 prabits, confirming Hp ≤ Hs ≤ H.

IV. PRAGMATIC RELATIVE ENTROPY AND PRAGMATIC MUTUAL INFORMATION

Pragmatic relative entropy aggregates probability distributions over equivalence classes defined by terminal actions, extending semantic relative entropy to decision-oriented comparisons. Its variants satisfy inequality, ordering, and convexity properties inherited through pragmatic coarse-graining.

  • Pragmatic Mutual Information: The framework extends these relative-entropy constructions to pragmatic mutual information evaluated at terminal actions.It introduces full, partial, and up/down pragmatic measures for dependencies after pragmatic aggregation.
  • Pragmatic Relative Entropy: The three pragmatic relative-entropy forms are full, partial Type I, and partial Type II, differing in which distribution is aggregated.Type I aggregates p against fine-grained q, whereas Type II keeps p fine-grained against aggregated q.
  • Pragmatic Relative Entropy: Pragmatic relative entropy satisfies a hierarchy of inequalities, with equality conditions determined by agreement within or across pragmatic classes.The equality cases distinguish matching pragmatic marginals, matching distributions within each class, and pointwise equality.
  • Pragmatic Relative Entropy: Pragmatic abstraction orders the relative entropies across pragmatic, semantic, and classical levels according to which argument is coarse-grained.Coarse-graining reduces divergence when the first argument is aggregated and increases it when the second argument is aggregated.
  • Pragmatic Relative Entropy: All three pragmatic relative entropies are jointly convex in the pair of distributions.The result follows because pragmatic aggregation is linear and the quantities decompose into classical KL divergences.
  • Pragmatic Relative Entropy: Pragmatic relative entropy compares distributions at the level of terminal actions rather than symbols or meanings.The full form compares pragmatic marginals, while partial forms mix pragmatic aggregation with fine-grained syntactic distributions.

B. Pragmatic Mutual Information

The paper defines up/down pragmatic mutual information and single-sided variants to quantify information after bilateral or asymmetric coarse-graining. These measures form ordered hierarchies and decompose into gains or losses from semantic and pragmatic abstraction.

  • Pragmatic Mutual Information: Up and down pragmatic mutual information capture reductions in decision uncertainty from complementary abstraction perspectives.Up mutual information measures extractable information about terminal actions, whereas down mutual information measures surviving pragmatic dependence after coarse-graining.
  • Pragmatic Mutual Information: The bilateral measures obey the hierarchy Ip(W; V ) ≤ Is(˜W; ˜V ) ≤ I(W; V ) ≤ Is(˜W; ˜V ) ≤ Ip(W; V ).The ordering follows from the entropy hierarchy across syntactic, semantic, and pragmatic representations.
  • Pragmatic Mutual Information: Up pragmatic mutual information equals up semantic mutual information plus a nonnegative pragmatic increment.The increment is Hs(˜W, ˜V ) − Hp(W, V ), representing additional reduction from pragmatic coarse-graining.
  • Pragmatic Mutual Information: Down pragmatic mutual information subtracts a nonnegative pragmatic loss from down semantic mutual information.The loss sums the marginal semantic-to-pragmatic entropy reductions for W and V.
  • Single-Sided Pragmatic Mutual Information: Four single-sided pragmatic mutual-information measures cover asymmetric perception, expression, and reconstruction settings.They distinguish whether one communication side is coarsened while the other remains syntactic, forming ordered perceptual and expressive chains.
  • Single-Sided Pragmatic Mutual Information: The measures form a complete lattice covering no, one-sided, and two-sided coarse-graining with syntactic or pragmatic marginal entropies.This provides finer-grained analysis of task-oriented information flow than bilateral measures alone.
  • Single-Sided Pragmatic Mutual Information: The single-sided measures form ordered chains around classical mutual information, while cross-comparisons between perceptual and expressive chains are generally incomparable.Each lower pragmatic-marginal measure is bounded above by each corresponding upper syntactic-marginal measure.

V. PRAGMATIC CHANNEL CAPACITY AND PRAGMATIC RATE DISTORTION

Pragmatic channel capacity and rate-distortion replace symbol-level fidelity with reliable terminal-action transmission and task distortion. Their hierarchies show how pragmatic abstraction can increase useful capacity and reduce the rate required for a given decision-relevant distortion.

  • Pragmatic Channel Capacity: Pragmatic channel capacity is the maximum reliable rate for transmitting pragmatic information, optimized over input distributions and joint reification mappings.It is defined using up pragmatic mutual information and supports single-sided extensions for asymmetric sensing and actuation.
  • Pragmatic Channel Capacity: The channel-capacity hierarchy places syntactic capacity below semantic capacity and semantic capacity below pragmatic capacity.The pragmatic capacity can exceed the syntactic rate because one pragmatic bit may represent multiple syntactic bits that do not alter meaning or optimal action.
  • Pragmatic Channel Capacity: Pragmatic abstraction can transmit more useful information per physical bit by allowing errors that preserve the optimal terminal action.The resulting capacity gain reflects relaxed reconstruction requirements rather than a change to the physical channel limit.
  • Pragmatic Rate Distortion: Pragmatic rate-distortion minimizes down pragmatic mutual information subject to a bound on task-utility loss.The optimization ranges over source and reconstruction reification mappings and test channels satisfying the pragmatic distortion constraint.
  • Pragmatic Rate Distortion: The rate-distortion hierarchy is Rp(D) ≤ Rs(D) ≤ R(D), so pragmatic distortion can require no more rate than semantic or syntactic distortion.The comparison assumes compatible distortion constraints, with pragmatic distortion defined directly on terminal actions.
  • Pragmatic Rate Distortion: Rp(D) can be significantly lower than classical R(D) when syntactic or semantic errors do not affect the terminal action.This supplies a theoretical basis for task-oriented compression by redefining fidelity around decision-relevant utility.
  • Single-Sided Extensions: Asymmetric systems receive four single-sided rate-distortion and achievable-rate measures for cases where only observations or action labels are pragmatically coarsened.These measures provide finer-grained analysis of task-oriented flows in perception, sensing, and actuation.

4) Prospective Expressive Pragmatic Achievable Rate:

The prospective expressive pragmatic achievable rate quantifies how effectively a low-dimensional pragmatic label conveys source information under resource constraints. It forms part of a broader asymmetric framework relating achievable rates, value, cost, and pragmatic abstraction.

  • Prospective expressive rate: The prospective expressive pragmatic achievable rate measures the maximum rate at which a pragmatic command or action label conveys source information under a resource constraint.The optimization ranges over conditional label distributions satisfying the resource constraint.
  • Applications: Expressive pragmatic communication models low-dimensional action labels encoding high-dimensional source information when designers seek the best performance under resource limits.The setting includes remote control, teleoperation, and related expressive systems.
  • Asymmetric abstraction: Single-sided pragmatic measures use data processing and mutual-information inequalities to show that one-sided coarsening preserves distinctions, requiring higher rates for equal distortion and lowering achievable rates.These measures support asymmetric perceptual, expressive, and edge-AI systems.
  • Unified framework: The four single-sided measures, together with bilateral counterparts, provide a pragmatic information-theoretic framework for task-oriented communication with asymmetric constraints.The framework includes perceptual and expressive paths and their associated value and cost relationships.
  • Pragmatic value: Pragmatic value is based on optimal action classes rather than raw states or observations, with isoteleia grouping distinct states or observations that induce the same optimal action.Both state and observation are coarsened before evaluating decision utility.
  • Properties and duality: The pragmatic value of information is nonnegative, monotone under observation refinement, and non-decreasing concave in communication rate.Its variational duality connects the value-rate function with the pragmatic rate-distortion function.

1) Source-Distortion-Value Lagrangian (Down Loop):

The Lagrangian framework couples decision-quality and physical-resource trade-offs through a common rate, selecting operating points where marginal value equals marginal cost. Its concavity and duality yield a global optimization principle and a behavioral-capacity measure.

  • Source-distortion-value loop: The down-loop Lagrangian balances decision value from reducing distortion against rate cost, with optimality requiring marginal value of rate to equal marginal rate cost.The down loop determines the rate needed to achieve a target utility or distortion level.
  • Channel-power-cost loop: The up-loop Lagrangian balances achievable rate against physical resource cost, with the shadow price equaling the marginal cost of increasing rate at optimality.The resource may be power or another physical cost represented through pragmatic cost.
  • Cross-layer coupling: The two loops are coupled through their common rate, and their dual multipliers coincide at the global optimum, linking decision-theoretic and physical-layer perspectives.The down-loop marginal rate cost equals the marginal value associated with pragmatic cost.
  • Global optimization: The global pragmatic Lagrangian is concave in rate when pragmatic value is concave and pragmatic cost is convex, so a differentiable stationary point is globally optimal.This establishes sufficient conditions for solving the cross-layer optimization by first-order conditions.
  • Behavioral capacity: The pragmatic efficiency functional is the supremum of net value minus resource-priced cost, representing the maximum net utility extractable by a resource-constrained goal-directed system.It is presented as the system’s behavioral capacity and master performance metric under resource pricing.
  • Duality structure: The framework unifies bilateral and single-sided value-cost dualities, including rate-distortion with value and capacity with cost, across symmetric and asymmetric systems.These relationships characterize trade-offs among rate, distortion, capacity, cost, and value.

VII. PRAGMATIC LOSSLESS SOURCE CODING

Pragmatic lossless source coding compresses sequences by preserving pragmatic equivalence classes rather than exact syntactic or semantic details. The three-tier AEP and typical-set hierarchy establishes pragmatic entropy as the exact threshold for vanishing-error coding and full pragmatic value.

  • Three-Tier Typical Sets: Pragmatic abstraction merges semantic distinctions that do not affect the optimal terminal action, reducing typical-set sizes and enabling more efficient lossless compression.The entropy hierarchy is Hp(W) ≤ Hs(W̃) ≤ H(W).
  • Three-Tier AEP: The three-tier AEP establishes asymptotic convergence for syntactic, semantic, and pragmatic sequences under synonymous, isoteleia, and reification mappings.The semantic and pragmatic convergences follow from aggregated probabilities over equivalence classes.
  • Three-Tier Typical Sets: Reified typical sets combine synonymous typical sets across isoteleia classes, with their additional freedom supplying the source of pragmatic compression gains.The nested partition first forms synonym classes and then groups them by pragmatic action.
  • Pragmatic Lossless Source Coding Theorem: Rates above pragmatic entropy are achievable with vanishing error, whereas rates below it cannot support pragmatic lossless source coding.The pragmatic lossless source coding theorem gives both achievability and converse results.
  • Pragmatic Lossless Source Coding Theorem: Full pragmatic Value of Information is attained if and only if the coding rate satisfies R ≥ Hp(W).Below this threshold, uncertainty about the optimal action class remains unresolved and achievable VoI is strictly lower.

VIII. PRAGMATIC CHANNEL CODING

Pragmatic channel coding groups syntactic input-output pairs by shared pragmatic purpose and decodes only the resulting pragmatic class. This abstraction increases the probability of pragmatic joint typicality and thereby supports a pragmatic capacity larger than the classical symbol-level capacity when action-irrelevant errors are tolerated.

  • Joint Reification Mapping: The joint reification mapping partitions syntactically jointly typical input-output sequences into disjoint classes indexed by pragmatic pairs.Each reified class is a union of jointly synonymous typical sets over the corresponding isoteleia typical set.
  • Pragmatic Abstraction: The resulting coding framework characterizes equivalence of input-output pairs according to their optimal terminal actions.Pragmatic abstraction merges semantically distinct but pragmatically equivalent pairs.
  • Pragmatic Joint AEP: The pragmatic joint AEP follows by applying the weak law of large numbers to aggregated pragmatic probabilities.This extends the joint asymptotic equipartition property to the pragmatic layer.
  • Jointly Reified Typical Set: The jointly reified typical set has approximate cardinality 2^n(H(X,Y)-Hp(X,Y)), exceeding a single synonymous typical set by a factor 2^n(Hs(X̃,Ỹ)-Hp(X,Y)).The factor measures the additional syntactic freedom retained when only pragmatic purpose must be preserved.
  • Jointly Reified Typical Set: Pragmatic joint typicality occurs with probability about 2^-nIp(X;Y), larger than the classical 2^-nI(X;Y), reflecting a pragmatic capacity gain.The jointly reified typical set aggregates syntactic pairs that share a pragmatic interpretation.

B. Pragmatic Channel Coding Theorem

The pragmatic channel coding theorem uses a two-level code that separates pragmatic indices from syntactic realizations. It establishes pragmatic channel capacity as the sharp threshold for reliable action-oriented transmission and connects that threshold to pragmatic information cost.

  • Two-Level Pragmatic Code: The encoder selects a pragmatic index for an optimal action class, chooses a reification index for its syntactic codeword, and the decoder recovers only the pragmatic index.Syntactic variations within each reified typical set are intentionally ignored.
  • Pragmatic Channel Coding Theorem: Reliable pragmatic transmission is achievable when total rate Rtot < Cp, while rates Rtot > Cp force error probability away from zero.The theorem proves both achievability and converse using random coding and jointly reified typical decoding.
  • Pragmatic Capacity: Pragmatic capacity is defined through up pragmatic mutual information, with Ip(X;Y) = H(X) + H(Y) − Hp(X,Y).The pragmatic index rate and reification rate together form the total transmission rate.
  • Pragmatic Cost of Information: The pragmatic cost of information is finite for R ≤ Cp and becomes unbounded beyond Cp, making capacity the resource threshold for reliable pragmatic communication.No finite resource can guarantee reliable transmission above this threshold.
  • Relation to Classical Coding: By tolerating errors that do not affect the optimal action, pragmatic coding can achieve strictly larger capacity than classical coding.The classical and semantic theorems are recovered when the corresponding mappings are trivial.

B. Pragmatic Rate-Distortion Coding Theorem

The pragmatic rate-distortion theorem characterizes lossy source coding when distortion is measured at the pragmatic level, and links the required rate to achievable pragmatic value. It generalizes classical and semantic rate-distortion results through reification-based abstraction.

  • Code construction: The code maps source sequences through pragmatic source and reconstruction classes, then transmits pragmatic and reification indices for decoding.The construction uses reification mappings, partitioned sourcebooks and codebooks, and deterministic encoding and decoding functions.
  • Theorem statement: Theorem 45 gives achievability when R > Rp(D) and a converse showing distortion exceeds D − ϵ when R < Rp(D).These bounds hold for sufficiently large blocklengths under an i.i.d. syntactic source and bounded pragmatic distortion.
  • VoI connection: Full pragmatic Value of Information is attainable if and only if R ≥ Rp(D); below this threshold, rate shortfall reduces achievable utility.The loss in pragmatic information produces a corresponding loss in utility.
  • Hierarchy and special cases: The pragmatic rate-distortion function satisfies Rp(D) ≤ Rs(D) ≤ R(D), recovering semantic or classical rate-distortion under the corresponding mappings.Trivial identity reification yields the classical function, while a nontrivial synonymous mapping yields the semantic function.
  • Hierarchy and special cases: Setting reification rates to zero achieves pure pragmatic compression, while increasing them preserves additional syntactic or semantic detail.The additional rates provide flexible trade-offs between abstraction and retained detail.

X. PRAGMATIC INFORMATION MEASURE OF CONTINUOUS MESSAGE

The continuous-message extension replaces discrete pragmatic classes with isoteleic volumes that measure signal-space regions leading to the same optimal action. It preserves the entropy hierarchy and derives Gaussian capacity, rate, and cost expressions with bilateral and single-sided abstractions.

  • Gaussian capacity and CoI: Gaussian channel extensions derive pragmatic capacity and CoI expressions, interpreting CoI as the minimum transmit or sensing power needed for a pragmatic rate.The formulas cover bilateral and single-sided cases, including perceptual and expressive costs.
  • Continuous pragmatic measures: The continuous entropy hierarchy remains Hp ≤ Hs ≤ H, with differential pragmatic entropy accounting for abstraction through average isoteleic volume.When the mapping is trivial, pragmatic entropy reduces to differential entropy.
  • Continuous pragmatic measures: Continuous isoteleic volumes quantify the average signal-space measure whose realizations map to the same optimal action.They arise from a quantization-and-limit construction for continuous alphabets.
  • Continuous pragmatic measures: Continuous pragmatic entropy is relative rather than absolute and requires a reference measure; nonnegative prabit entropy additionally assumes E[log Ω] ≤ h(W).This scope condition distinguishes differential pragmatic entropy from an absolute information measure.
  • Gaussian capacity and CoI: Bilateral abstraction requires less power than single-sided abstraction for the same pragmatic rate because it coarsens both source and observation sides.The resulting design guideline is to exploit abstraction on both sides whenever possible.

C. Pragmatic Rate Distortion of Gaussian Source and Pragmatic Value of Information

For Gaussian sources, pragmatic rate-distortion and Value of Information quantify how action-oriented abstraction changes the rate–utility trade-off. The broader source-channel theorem shows that reliable pragmatic communication requires pragmatic source demand not to exceed pragmatic capacity.

  • Gaussian rate-distortion: The Gaussian pragmatic rate-distortion function is derived by reducing the distinguishable source volume through isoteleia mappings and optimizing the test channel.The derivation assumes symmetric average isoteleic volumes and achieves equality with a Gaussian test channel and optimal partition.
  • Gaussian pragmatic VoI: At R = 1, bilateral coarsening yields 0.984 utils and distortion 0.0156, while single-sided coarsening yields 0.9375 utils and distortion 0.0625.The same rate therefore gives bilateral abstraction a fourfold reduction in distortion and higher utility gain.
  • Source-channel coding: Pragmatic source-channel coding is reliable when Hp(W) < Cp for lossless transmission or Rp(D) < Cp for distortion D; the converse rules out the corresponding guarantees above capacity.The lossy theorem guarantees E[dp(W, Ẇ)] ≤ D below capacity and denies such a sequence above it.
  • Value-cost characterization: Pragmatic Value of Information is non-decreasing in rate and saturates at full value once R reaches Hp(W) or Rp(D), while CoI is non-decreasing and convex.These properties support explicit resource–utility trade-offs.
  • Source-channel coding: The pragmatic condition can be weaker than the classical condition because errors that do not affect the optimal action may be tolerated.The paper states that pragmatic communication can operate reliably in regimes where classical communication would fail.

1) Separation Principle and the Imperative of Joint Optimization:

The pragmatic separation principle establishes asymptotic limits for source and channel coding, while finite resource, delay, memory, and complexity constraints require joint optimization of rate, abstraction, and communication resources.

  • Separation principle: Asymptotically, pragmatic source and channel codes can be optimized independently while the concatenated system achieves the fundamental limit when the source rate does not exceed channel capacity.The separation condition is expressed as Hp(W) ≤ Cp for lossless coding or Rp(D) ≤ Cp for lossy coding.
  • Joint optimization: Finite blocklength or computational, delay, and memory constraints make separated source and channel coding suboptimal for end-to-end task utility.The encoder must jointly consider source statistics, channel characteristics, and decision utility.
  • Joint optimization: The finite-resource problem is coupled through the rate-resource trade-off, because one rate controls both source-side abstraction and channel-side error protection.The Lagrangian framework jointly optimizes this trade-off to maximize net benefit.
  • Separation principle: Pragmatic separation generalizes classical and semantic separation results as special cases and supports task-oriented communication system design.The framework unifies prior results when the relevant mappings are trivial or only synonymous mappings are non-trivial.
  • Behavioral capacity: In the Gaussian setting, pragmatic abstraction amplifies effective signal-to-noise ratio and can substantially reduce the transmit power needed for a target utility.With Ω = 2 and zero transmit power, the reported value is approximately 0.996, exceeding the target 0.9.
  • Behavioral capacity: Pragmatic abstraction increases behavioral capacity, with diminishing returns toward perfect-information utility and greater robustness as resource costs rise.The capacity curve saturates at perfect-information utility, while larger abstraction reduces the decline caused by increasing resource shadow price.

2) Dynamic Joint Optimization: Sequential Decision Making:

The framework extends pragmatic information theory to sequential decision-making by coupling actions, communication rates, belief updates, and resource costs in a belief-state Bellman formulation. It establishes well-posed dynamic optimization and derives adaptive design principles for balancing information value against communication cost.

  • Dynamic Joint Optimization: The dynamic framework jointly optimizes actions and communication rates in a POMDP, maximizing discounted utility while accounting for pragmatic communication costs.The formulation uses controlled state transitions, an observation kernel affected by rate, and a Lagrangian resource constraint.
  • Dynamic Joint Optimization: The belief-state Bellman operator is a contraction when γ < 1, yielding a unique fixed point and a well-defined optimal value function.The proof applies the Banach fixed-point theorem after establishing the contraction bound.
  • Dynamic Joint Optimization: The dynamic isoteleia mapping is uniquely determined by the optimal value function and reduces to the static mapping when γ = 0.In steady state, the dynamic mapping collapses to the static mapping, preserving the hierarchical entropy bounds.
  • Dynamic Joint Optimization: The optimal rate adapts to belief uncertainty, allocating more resources when beliefs are diffuse or future rewards are especially sensitive to estimation accuracy.This produces a value-of-information-guided resource allocation policy.
  • Dynamic Joint Optimization: Action and rate decisions are coupled through belief updates, expressing a trade-off between control and sensing or communication.The framework balances immediate and future value through the marginal relationship between information benefit and communication cost.
  • Dynamic Joint Optimization: The resulting dynamic optimization provides a tractable method for co-designing communication and control while trading off information value, physical resource consumption, and utility.The framework is presented as a unified tool for task-oriented communication and control systems.

APPENDIX

The appendix proofs establish properties of pragmatic sequence constructions using definitions, probability bounds, independence, and the pragmatic entropy hierarchy. These arguments conclude the stated properties through direct inequalities and asymptotic reasoning.

  • APPENDIX: The appendix derives several sequence properties directly from the definitions of A(n), E(n), and B(n).The proof passages repeatedly invoke the defining properties of these constructions.
  • APPENDIX: Probability inequalities and division by positive sequence probabilities establish the required lower and upper bounds.The arguments use positivity conditions such as P(w^n) > 0 and P(x^n,y^n) > 0.
  • APPENDIX: The asymptotic arguments apply large-number reasoning to i.i.d. pragmatic sequence pairs and aggregate probabilities over reified equivalence classes.Independence is used for one of the properties, together with entropy-hierarchy bounds and sequence-size bounds.
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