Compositional Cognitive Tasks

Updated 24 October 2025

Compositional cognitive tasks are defined as tasks whose structure supports the systematic combination, reuse, and decomposition of subtasks to build complex behaviors.
They leverage formal operations such as weighted mixtures and function compositions to precisely model task difficulty and assess generalization in both AI and cognitive science.
The framework integrates computational metrics with modular decomposition principles to enhance skill transfer, robust policy discovery, and scalable cognitive architectures.

Compositional cognitive tasks are tasks whose structure supports, requires, or benefits from the systematic combination, reuse, and decomposition of subtasks or concept constituents. In cognitive sciences and artificial intelligence, compositionality underpins the capacity to construct complex behaviors or meanings from simpler, reusable components. This notion is central to evaluating and engineering systems able to generalize far beyond the regimes covered by their training or evolutionary experience. The paper of compositional cognitive tasks therefore provides a rigorous basis for analyzing the structural organization, difficulty, computational mechanisms, and generalization properties of artificial and natural intelligent systems.

1. Formalization of Tasks and Compositional Structure

A cognitive task, in this context, is modeled as an interactive, possibly episodic process between an agent and an environment, characterized by a set of episodes (trials), observations, actions, and measured responses. The innate structure of a task—distinct from any learning or training process—involves how subtasks or concept primitives are organized, whether in sequence, as a weighted mixture, or via hierarchical or recursive composition.

Compositionality is made explicit via operations such as:

Task mixture: $p = \alpha p_1 + (1-\alpha) p_2$ , where, on each episode, the agent faces either $p_1$ or $p_2$ chosen by probabilistic routing.
Function composition or recursive application: Complex operations or meanings are constructed from successive functional application of task subroutines.
Algebraic formulations: In categorical and neuro-symbolic frameworks, meanings or transformations are composed according to categorical semantics, e.g., via compact closed categories and associated linear maps (Bolt et al., 2017, Al-Mehairi et al., 2016).

Formally, the meaning or solution for a composite task can often be expressed as:

$M(x_1, x_2, ..., x_n) = f(M(x_1), M(x_2), ..., M(x_n)),$

where atomic units $x_i$ are composed via $f$ reflecting a grammar, rule, or mechanism of composition (Sinha et al., 13 Jun 2024).

2. Measuring Task Difficulty and Complexity

Task difficulty is tied not only to the complexity of the problem but to the minimal computational effort required to construct an $\varepsilon$ -acceptable policy—a program or mechanism producing responses above a defined threshold.

Key quantitative measures include:

Kolmogorov-style task complexity:

$K_{\varepsilon, v}(p) = \min_{\pi \in A[\varepsilon]} L(\pi)$

where $L(\pi)$ is the description length (i.e., encoding size) of policy $\pi$ , and $A[\varepsilon]$ denotes the set of policies achieving performance within an $\varepsilon$ margin (Hernandez-Orallo, 2015).

Levin-style time complexity:

$K_t(\varepsilon, v)(p) = \min_{\pi \in A[\varepsilon]} \left\{ L(\pi) + \log S_{\rightarrow v}(\pi, p) \right\}$

where $S_{\rightarrow v}(\pi, p)$ is the expected number of computational steps required for $v$ trials under policy $\pi$ . The logarithmic metric reflects "bits of effort" expended in finding and validating a policy.

When tasks are composed (e.g., $p = \alpha p_1 + (1-\alpha) p_2$ ), the difficulty function $h$ is often expected to satisfy:

$h(\alpha p_1 + (1-\alpha) p_2) \approx \alpha h(p_1) + (1-\alpha) h(p_2)$

when optimal policies share or reuse components (Hernandez-Orallo, 2015).

3. Task Composition and Decomposition Principles

Compositional cognitive tasks are governed by:

Mixture and assembly: Composition is often formalized as a weighted mixture or function composition, naturally inducing a structure where subtasks are assembled to solve more complex tasks.
Decomposition: The reverse operation enables understanding how a complex task can be split into constituent subtasks or primitive component problems. This is critical for analyzing skill transfer, modularity, and curriculum learning.
Complexity distance: Task similarity or distance can be quantified via differences in difficulty, e.g.,

$d(p_1, p_2) = 2h(0.5p_1 + 0.5p_2) - h(p_1) - h(p_2),$

which measures divergence in resource requirements for policies that generalize across both $p_1$ and $p_2$ (Hernandez-Orallo, 2015).

The set of operations defining how tasks are composed and decomposed supports modularity—a central principle in building scalable cognitive and artificial systems.

4. Episodic, Asynchronous, and Stochastic Task Models

Unlike standard Markov Decision Processes (MDPs), compositional cognitive tasks often require a more general account of temporal and reward structure:

Episodes and responses: Each episode is modeled as a trial where observations, actions, and intermediate rewards may occur, but performance is often determined by an external, final response aggregated at the end of the episode.
Asynchronous-time stochastic processes: Agents and environments interact on a continuous or discrete clock, with actions and observations occurring at arbitrary, non-even intervals. Waiting (as in reaction time tasks) can be modeled with dormant phases (e.g., a sleep(t) instruction) that do not incur computational cost.
Stochastic verification: Task difficulty must account for the need to verify that policies succeed under randomness; robustness and generalizability require successful performance over several trials, not just one (Hernandez-Orallo, 2015).

This modeling flexibility is necessary to accommodate the full range of cognitive tasks encountered in both human and AI evaluation that are not easily reduced to simple MDPs.

5. Computational and Information-Theoretic Foundations

The computational approach to compositional cognitive tasks uses algorithmic information theory and resource-bounded search:

Resource-bounded search: The process of constructing policies is itself subject to complexity constraints, integrating exploration, program synthesis, and policy evaluation within a single framework.
Trade-offs: The interplay between description length (compression of the policy or knowledge structure) and execution cost (policy runtime verification) is crucial; optimal compositional structure is found where shared components achieve maximal reuse and minimal redundancy.
Program induction: The search for acceptable policies aligns with Levin’s universal search paradigm, where the logarithmic sum of program length and verification time sets a lower bound for task difficulty.

Task composition and decomposition are, therefore, not only matters of symbolic recombination but are bound by the cost of generating and verifying effective solutions.

6. Implications for Evaluation and Cognitive Architecture

The formalization of compositional cognitive tasks as outlined above has several theoretical and practical implications:

Unified metrics of performance: By focusing on the minimal computational effort for policy discovery and execution, difficulty metrics become comparable across modalities, species, or systems.
Compositionality in evaluation: Formal mixtures and decompositions allow for constructing benchmarks and curricula that probe not just rote memorization but systematic recombination and abstraction.
Structural transfer and adaptability: The compositional approach enables principled evaluation of transfer learning—measuring how knowledge from one subtask (module) accelerates performance on complex or novel compositions.
Design of learning systems: Modular, compositional architectures are naturally favored, with empirical and theoretical evidence that such systems are better equipped for generalization, robustness, and efficiency.

7. Mathematical Representations and Key Formulas

Several key mathematical expressions from the literature encapsulate the above principles:

Formula	Purpose	Reference
$K_{\varepsilon,v}(p) = \min_{\pi \in A[\varepsilon]} L(\pi)$	Minimal policy complexity for acceptable performance	(Hernandez-Orallo, 2015)
$K_t(\varepsilon,v)(p) = \min_{\pi \in A[\varepsilon]} \{ L(\pi) + \log S_{\to v}(\pi,p)\}$	Policy complexity plus execution effort	(Hernandez-Orallo, 2015)
$p = \alpha p_1 + (1-\alpha) p_2$	Task composition by mixture	(Hernandez-Orallo, 2015)
$h(\alpha p_1 + (1-\alpha) p_2) \approx \alpha h(p_1) + (1-\alpha) h(p_2)$	Additive difficulty for composite task	(Hernandez-Orallo, 2015)
$d(p_1, p_2) = 2 h(0.5 p_1 + 0.5 p_2) - h(p_1) - h(p_2)$	Complexity-based task “distance”	(Hernandez-Orallo, 2015)

These formal tools provide a foundation for building and analyzing systems capable of robust, efficient, and systematic compositional cognition.

Compositional cognitive tasks, as rigorously formalized in the literature, provide a unifying perspective for understanding the structure, complexity, and generalizability of intelligent systems, supporting principled benchmarks and design strategies for both artificial intelligence and computational cognitive science (Hernandez-Orallo, 2015).