Age of Incorrect Semantics (AoIS)

• AoIS is a semantics-aware metric that measures how long a system remains semantically incorrect by combining elapsed time with content-specific penalties.
• The framework generalizes Age of Incorrect Information by integrating semantic mismatch to guide scheduling, coding, and resource allocation in communication networks.
• Empirical studies show that AoIS-based policies can substantially improve system performance by aligning update strategies with semantic correctness.

Searching arXiv for papers on AoIS, AoII, and semantics-aware freshness metrics. Age of Incorrect Semantics (AoIS) is a semantics-aware freshness metric that quantifies how long a receiver, monitor, or decision-making system has remained semantically incorrect with respect to an underlying source or task state. In the literature surveyed here, AoIS is most naturally understood as a semantic generalization of Age of Incorrect Information (AoII): instead of penalizing only raw state mismatch or update staleness, it combines a notion of elapsed time since semantic correctness with a content-dependent penalty that measures semantic mismatch, task loss, or semantic utility degradation. Existing work uses this structure to bridge classical distortion, Age of Information (AoI), AoII, version-aware freshness, and broader goal-oriented semantic communication, and to derive transmission, scheduling, coding, and beamforming policies optimized for semantic correctness over time rather than for freshness alone (Luo et al., 14 Dec 2025, Han et al., 17 Aug 2025).

1. Conceptual definition and scope

AoIS is not uniformly named across the literature, but the underlying concept is explicit: it measures the age of being wrong at the semantic layer. In the most direct formalization, a semantic state $S_t$, a receiver-side semantic estimate $\hat{S}_t$, and a nonnegative semantic penalty $g_{\text{sem}}(S_t,\hat{S}_t)$ are used to define a “time since last semantic correctness” process and then an age-weighted semantic error. One canonical construction is

$$\Delta^{\text{sem}}_t \triangleq t - h^{\text{sem}}_t,\quad h^{\text{sem}}_t = \max\{ \tau \le t : g_{\text{sem}}(S_\tau,\hat{S}_\tau)=0\},$$

followed by

$$\delta^{\text{AoIS}}_t \triangleq \Delta^{\text{sem}}_t \cdot g_{\text{sem}}(S_t,\hat{S}_t).$$

This definition appears explicitly as a semantic extension of AoII in work on variable-length stop-feedback coding and semantic-age formulations (Bountrogiannis et al., 2024).

A closely related persistence-based definition treats AoIS as a semantic error-age process that increments while semantics remain incorrect and resets when semantics become correct. In the simplest form,

$$\Delta^{\text{IS}}_t = \begin{cases} \Delta^{\text{IS}}_{t-1}+1, & I_t=1,\ 0, & I_t=0, \end{cases}$$

where $I_t$ indicates semantic incorrectness. A more refined variant resets only when the semantic error disappears and restarts from $1$ when the error type changes, mirroring significance-aware consecutive-error metrics (Luo et al., 14 Dec 2025).

This places AoIS in a distinct position relative to AoI. AoI asks how old the latest received update is; AoIS asks for how long the receiver’s semantic interpretation has been wrong, and optionally how harmful that wrongness is. The literature repeatedly emphasizes that these are not equivalent objectives: information can be old but still semantically correct, or fresh yet semantically misleading (Chen et al., 2022, Luo et al., 14 Dec 2025).

2. Relation to AoII and neighboring freshness metrics

AoIS is best understood as a generalization of AoII. In AoII, the source state $X_t$ and receiver estimate $\hat{X}_t$ are compared through an information penalty $\hat{S}_t$0, often binary or distance-based, and weighted by a time component. One common AoII form is

$\hat{S}_t$1

with examples including linear and thresholded time penalties together with squared-error or thresholded content penalties (Dizdar et al., 2023). Another common discrete form tracks the time elapsed since the last instant of correctness, so that AoII is zero when the estimate is correct and positive only while the receiver is wrong (Bountrogiannis et al., 2024, Chen et al., 2023).

Distance-based AoII makes the semantic interpretation even more explicit. In pull-based monitoring of Markov sources, the expected AoII is expressed as an accumulated distance between the true source trajectory and the monitor’s estimate: $\hat{S}_t$2 which yields

$\hat{S}_t$3

under the paper’s birth-process model (Kriouile et al., 2022). This formulation already behaves like a semantics-aware metric whenever $\hat{S}_t$4 is interpreted as application-level harm rather than merely geometric distance.

Several neighboring metrics clarify AoIS by contrast. Version-aware measures such as Version Age of Information and Age of Incorrect Version track whether the receiver holds the wrong version rather than merely old data. In particular, AoIV evolves only when the receiver’s version is wrong, and increments when source versions change while the receiver remains incorrect (Salimnejad et al., 2024, Luo et al., 14 Dec 2025). Significance-aware error-persistence metrics such as Age of Missed Alarm, Age of False Alarm, and Age of Consecutive Error separate error types and assign error-specific age functions $\hat{S}_t$5, which is structurally very close to AoIS with semantic error classes (Luo et al., 14 Dec 2025).

A plausible implication is that AoIS is not a single fixed metric but a family of semantics-aware age constructions whose common principle is persistence of semantic wrongness. The literature supports this reading by treating AoII, AoIV, AoCE, and context-dependent semantic costs as members of a unified semantics-aware freshness taxonomy (Luo et al., 14 Dec 2025).

3. Formal models used for AoIS

Published AoIS formulations are currently concentrated in semantic communication settings for remote monitoring, especially video over wireless channels. In the most explicit instance, AoIS is introduced for MIMO semantic transmission of video frames, where $\hat{S}_t$6 is a semantic feature vector extracted at the base station and $\hat{S}_t$7 is the semantic feature recovered at the user. AoIS is defined as

$\hat{S}_t$8

where $\hat{S}_t$9 is a semantic mismatch term and $g_{\text{sem}}(S_t,\hat{S}_t)$0 is a non-decreasing time penalty (Han et al., 17 Aug 2025).

In that formulation, semantic mismatch is induced by a downstream task model $g_{\text{sem}}(S_t,\hat{S}_t)$1 and defined through cosine similarity of task outputs: $g_{\text{sem}}(S_t,\hat{S}_t)$2 Because the transmitter does not know the realized channel noise, the paper uses a Monte Carlo approximation of expected semantic mismatch based on sampled reconstructions (Han et al., 17 Aug 2025).

The time penalty is chosen as an exponential function of the time since the last received frame: $g_{\text{sem}}(S_t,\hat{S}_t)$3 where $g_{\text{sem}}(S_t,\hat{S}_t)$4 is a tunable constant and $g_{\text{sem}}(S_t,\hat{S}_t)$5 is the last frame reception time. Thus the operational AoIS is

$g_{\text{sem}}(S_t,\hat{S}_t)$6

This yields a semantics-aware age that grows with both semantic mismatch and elapsed time since meaningful refresh (Han et al., 17 Aug 2025).

The broader semantic-freshness literature supports more general formalizations. A semantic-age cost can be written as

$g_{\text{sem}}(S_t,\hat{S}_t)$7

where $g_{\text{sem}}(S_t,\hat{S}_t)$8 is a context-dependent semantic distortion and $g_{\text{sem}}(S_t,\hat{S}_t)$9 is an error-type-specific age function, potentially linear, logarithmic, or exponential (Luo et al., 14 Dec 2025). This suggests that existing AoIS models are special cases of a more general semantic cost-of-persistence framework.

4. Optimization frameworks and policy structure

AoIS inherits much of its optimization machinery from AoII. The dominant analytical tool is the Markov decision process, with states combining source dynamics, receiver estimate, and age or semantic-age variables. In AoII with random delay, the decision problem is formulated as an average-cost MDP, and under an easy-to-verify condition the optimal policy is to initiate a transmission whenever the channel is idle and AoII is not zero (Chen et al., 2022). The same threshold structure reappears under timeout-limited transmission and in bounded-delay channels, where threshold-$$\Delta^{\text{sem}}_t \triangleq t - h^{\text{sem}}_t,\quad h^{\text{sem}}_t = \max\{ \tau \le t : g_{\text{sem}}(S_\tau,\hat{S}_\tau)=0\},$$0 policies are shown optimal under explicit verifiable conditions (Chen et al., 2022, Chen et al., 2023).

Partially observed variants arise when the scheduler does not know the current source state. For Markov remote-source tracking, the problem is formulated as a partially observable MDP or restless multi-armed bandit. Belief states encode the probability that the receiver’s estimate is correct, and Whittle index policies emerge from Lagrangian relaxation and threshold optimality in the single-arm problem (Kriouile et al., 2021). Distance-based AoII for pull-based sensor scheduling yields a related Whittle-index structure with state-dependent thresholds that depend on source volatility $$\Delta^{\text{sem}}_t \triangleq t - h^{\text{sem}}_t,\quad h^{\text{sem}}_t = \max\{ \tau \le t : g_{\text{sem}}(S_\tau,\hat{S}_\tau)=0\},$$1, distance weight $$\Delta^{\text{sem}}_t \triangleq t - h^{\text{sem}}_t,\quad h^{\text{sem}}_t = \max\{ \tau \le t : g_{\text{sem}}(S_\tau,\hat{S}_\tau)=0\},$$2, and channel reliability $$\Delta^{\text{sem}}_t \triangleq t - h^{\text{sem}}_t,\quad h^{\text{sem}}_t = \max\{ \tau \le t : g_{\text{sem}}(S_\tau,\hat{S}_\tau)=0\},$$3 (Kriouile et al., 2022). This suggests that AoIS with semantic distortion weights should admit analogous belief-index formulations when semantic states are partially observed.

Under explicit power or transmission constraints, AoII optimization becomes a constrained MDP. For unreliable channels and multi-state Markov sources, the optimal policy is a mixture of two deterministic threshold policies, with the mixing coefficient chosen to meet the average transmission constraint exactly (Chen et al., 2021). For general discrete-time phase-type delays, an SMDP formulation has recently been used to optimize estimation-dependent multi-threshold policies, with thresholds depending on the current estimate value and costs given by estimate-specific AoII penalty functions $$\Delta^{\text{sem}}_t \triangleq t - h^{\text{sem}}_t,\quad h^{\text{sem}}_t = \max\{ \tau \le t : g_{\text{sem}}(S_\tau,\hat{S}_\tau)=0\},$$4 plus weighted transmission costs (Cosandal et al., 3 Dec 2025). A plausible implication is that AoIS with semantic-state-dependent penalty functions should fit naturally into the same SMDP template.

Lyapunov optimization and drift-plus-penalty methods provide an online alternative. In AoIS-aware semantic video transmission, virtual queues are introduced for average actuation-cost constraints, and the long-term stochastic optimization is transformed into a per-slot decision problem minimizing a drift-plus-AoIS bound (Han et al., 17 Aug 2025). Similar Lyapunov and token-based methods are identified in the broader semantics-aware literature as standard tools for semantic sampling and transmission control (Luo et al., 14 Dec 2025).

5. Coding, multiple access, and cross-layer realizations

AoIS and AoII are increasingly used beyond packet scheduling, particularly in coding and physical-layer design. For variable-length stop-feedback coding over Gaussian channels, AoII-optimal and delay-optimal feedback sequences are shown to differ, and lower average decoding delay does not necessarily imply lower average AoII (Bountrogiannis et al., 2024). The system uses a Markov source, VLSF coding, non-instantaneous feedback of duration $$\Delta^{\text{sem}}_t \triangleq t - h^{\text{sem}}_t,\quad h^{\text{sem}}_t = \max\{ \tau \le t : g_{\text{sem}}(S_\tau,\hat{S}_\tau)=0\},$$5, and an infinite-horizon average-cost MDP with state $$\Delta^{\text{sem}}_t \triangleq t - h^{\text{sem}}_t,\quad h^{\text{sem}}_t = \max\{ \tau \le t : g_{\text{sem}}(S_\tau,\hat{S}_\tau)=0\},$$6, where $$\Delta^{\text{sem}}_t \triangleq t - h^{\text{sem}}_t,\quad h^{\text{sem}}_t = \max\{ \tau \le t : g_{\text{sem}}(S_\tau,\hat{S}_\tau)=0\},$$7 is the elapsed time without successful correct decoding, $$\Delta^{\text{sem}}_t \triangleq t - h^{\text{sem}}_t,\quad h^{\text{sem}}_t = \max\{ \tau \le t : g_{\text{sem}}(S_\tau,\hat{S}_\tau)=0\},$$8 is the total number of symbols sent for the current sample, and $$\Delta^{\text{sem}}_t \triangleq t - h^{\text{sem}}_t,\quad h^{\text{sem}}_t = \max\{ \tau \le t : g_{\text{sem}}(S_\tau,\hat{S}_\tau)=0\},$$9 is the number of symbols sent since the last feedback. This establishes a coding-theoretic counterpart of the semantic-age principle: shorter delay is not automatically better if feedback placement harms correctness persistence (Bountrogiannis et al., 2024).

In downlink semantic-aware networks, AoII has been embedded into rate-splitting multiple access. There, one-slot-ahead AoII is minimized jointly over user scheduling, precoding, and common-rate allocation, with success determined by whether user $$\delta^{\text{AoIS}}_t \triangleq \Delta^{\text{sem}}_t \cdot g_{\text{sem}}(S_t,\hat{S}_t).$$0 receives an effective rate $$\delta^{\text{AoIS}}_t \triangleq \Delta^{\text{sem}}_t \cdot g_{\text{sem}}(S_t,\hat{S}_t).$$1 exceeding the required update rate $$\delta^{\text{AoIS}}_t \triangleq \Delta^{\text{sem}}_t \cdot g_{\text{sem}}(S_t,\hat{S}_t).$$2 (Dizdar et al., 2023). Big-$$\delta^{\text{AoIS}}_t \triangleq \Delta^{\text{sem}}_t \cdot g_{\text{sem}}(S_t,\hat{S}_t).$$3 reformulation and successive convex approximation convert the original non-convex conditional objective into an iterative convex program. Since the AoII formulation in that work already uses content penalty functions $$\delta^{\text{AoIS}}_t \triangleq \Delta^{\text{sem}}_t \cdot g_{\text{sem}}(S_t,\hat{S}_t).$$4, it serves as a direct bridge to AoIS by replacing raw numeric mismatch with semantic loss (Dizdar et al., 2023).

A related semantic-NOMA XR framework defines AoII as

$$\delta^{\text{AoIS}}_t \triangleq \Delta^{\text{sem}}_t \cdot g_{\text{sem}}(S_t,\hat{S}_t).$$5

where $$\delta^{\text{AoIS}}_t \triangleq \Delta^{\text{sem}}_t \cdot g_{\text{sem}}(S_t,\hat{S}_t).$$6 is packet AoI and $$\delta^{\text{AoIS}}_t \triangleq \Delta^{\text{sem}}_t \cdot g_{\text{sem}}(S_t,\hat{S}_t).$$7 is BERT-based semantic similarity under DeepSC, approximated through a generalized logistic function of user SINR (Chen et al., 2023). This is effectively an AoIS construction in which semantic error is measured through sentence embeddings and multiplied by age. The associated optimization decomposes into an AoI minimization over queueing service rates and a semantic-similarity maximization over transmit powers (Chen et al., 2023).

The explicit AoIS formulation for semantic video over MIMO channels pushes this cross-layer coupling further. The time-averaged AoIS minimization problem jointly optimizes the semantic actuation indicator $$\delta^{\text{AoIS}}_t \triangleq \Delta^{\text{sem}}_t \cdot g_{\text{sem}}(S_t,\hat{S}_t).$$8, semantic symbol length $$\delta^{\text{AoIS}}_t \triangleq \Delta^{\text{sem}}_t \cdot g_{\text{sem}}(S_t,\hat{S}_t).$$9, and transceiver beamformers $$\Delta^{\text{IS}}_t = \begin{cases} \Delta^{\text{IS}}_{t-1}+1, & I_t=1,\ 0, & I_t=0, \end{cases}$$0, subject to average actuation cost, transmit power, discrete symbol-length, and delay constraints (Han et al., 17 Aug 2025). The problem is handled by Lyapunov decomposition, exhaustive search over $$\Delta^{\text{IS}}_t = \begin{cases} \Delta^{\text{IS}}_{t-1}+1, & I_t=1,\ 0, & I_t=0, \end{cases}$$1 and $$\Delta^{\text{IS}}_t = \begin{cases} \Delta^{\text{IS}}_{t-1}+1, & I_t=1,\ 0, & I_t=0, \end{cases}$$2, and either SCA-based MIMO beamforming or a low-complexity zero-forcing design in the MISO case (Han et al., 17 Aug 2025).

6. Empirical behavior, design trade-offs, and open questions

The empirical literature consistently shows that minimizing freshness alone can be misaligned with semantic objectives. In VLSF-coded monitoring, “a lower average delay does not necessarily correspond to a lower average AoII,” because feedback timing changes the probability that delivered content is still correct when decoded (Bountrogiannis et al., 2024). In RSMA-based downlink semantic-aware networks, RSMA achieves lower AoII than SDMA because its interference management allows more users to satisfy correctness-driven rate requirements simultaneously (Dizdar et al., 2023). In semantic XR uplink NOMA, AoII captures trade-offs missed by pure AoI or pure similarity metrics: increasing transmit power improves semantic similarity and reduces AoII, while increasing the update generation rate can first reduce and then raise AoII as queues saturate (Chen et al., 2023).

For explicit AoIS in remote video monitoring, the reported gains are system-level. The proposed AoIS-aware schemes preserve more than $$\Delta^{\text{IS}}_t = \begin{cases} \Delta^{\text{IS}}_{t-1}+1, & I_t=1,\ 0, & I_t=0, \end{cases}$$3 of the original information under the same AoIS compared to constrained baselines, while also improving PSNR, MS-SSIM, and LPIPS in the reported experiments (Han et al., 17 Aug 2025). This indicates that AoIS can serve not only as an evaluation metric but also as a control objective that reshapes resource allocation toward semantically salient regions and updates.

Several common design principles recur across the literature. First, updates should be triggered by incorrectness or semantic mismatch rather than by elapsed time alone (Chen et al., 2022, Luo et al., 14 Dec 2025). Second, the appropriate threshold depends on source dynamics, channel reliability, and semantic cost asymmetry; high-volatility or high-importance semantics generally justify lower thresholds (Kriouile et al., 2021, Cosandal et al., 3 Dec 2025). Third, physical-layer latency, coding decisions, and feedback timing should be optimized jointly with semantic-age costs because delay-optimal and semantic-age-optimal designs need not coincide (Bountrogiannis et al., 2024, Han et al., 17 Aug 2025).

Open issues remain substantial. Existing explicit AoIS work is concentrated in monitoring-oriented semantic communication, especially vision and text-like semantic embeddings (Han et al., 17 Aug 2025, Chen et al., 2023). More general AoIS formulations will require semantic state models that are not merely Markov and low-dimensional, semantic penalties tied directly to downstream control or decision loss, and tractable approximations when semantic mismatch is continuous, multimodal, or history dependent. The unifying survey literature suggests that AoIS should be viewed as part of a larger family of history-dependent semantics-aware metrics rather than as a single canonical quantity (Luo et al., 14 Dec 2025).

In that sense, AoIS design is less about replacing AoI with one new scalar than about selecting a semantics-aware age functional appropriate to the task. The core invariant across formulations is clear: what matters is not only whether information is recent, but whether the system has been semantically wrong, how long that wrongness has persisted, and how costly that persistence is for the application.