LevelEnv Abstraction Framework

• LevelEnv Abstraction is a formal and algorithmic framework that maps low-level details to higher-level, interpretable representations.
• It leverages techniques in process mining, reinforcement learning, and description logics to control granularity and boost efficiency.
• Applications include unsupervised event abstraction, state compression, and modular design in virtual environments and knowledge systems.

LevelEnv Abstraction denotes a family of formal and algorithmic frameworks for structuring, compressing, and manipulating complex environments—whether event logs, state spaces, knowledge bases, or virtual worlds—by introducing and utilizing explicit abstraction levels. Its core objective is to lift low-level observations, activities, or definitions to coarser high-level abstractions that are more interpretable, tractable, or behaviorally meaningful for downstream analysis, process discovery, learning, or reasoning. Across domains such as process mining, reinforcement learning, description logics, and environment engineering, LevelEnv abstractions serve as a means to control granularity, improve efficiency, enable generalization, and align system structure with human conceptualization.

1. Formal Definitions and Theoretical Foundations

LevelEnv abstraction formalizes the mapping from a dense, low-level domain (states, events, concepts) to a higher-level, coarser domain using explicit surjective mappings or refinement/abstraction operators. In process mining, let $\Sigma$ denote the universe of low-level events, with event logs $L \in \mathbb{N}^{\Sigma^*}$; the aim is to construct a high-level event log $L^H \in \mathbb{N}^{H^*}$, for $H$ a set of abstract activities ($|H| \ll |\Sigma|$), via a mapping $\psi: \Sigma \to H \cup \{\bot\}$ where $\psi(e) = \bot$ denotes unmapped events (Mannhardt et al., 2017). In RL, state abstraction is encoded by $\phi: \mathcal{S} \to \mathcal{X}_\phi$ with regularity constraints ensuring model-irrelevant or value-irrelevant grouping (Kamalaruban et al., 2020).

In knowledge representation, LevelEnv constructs are made first-class within logic. For description logics, abstraction levels $L, L', L_i$ are indexed explicitly, and bridging operators such as $\Ref_{L \to L'}(q, C)$ (refinement) and $\Abs_{L \to L'}(C, q)$ (abstraction) map between concepts at different levels via conjunctive queries over the finer level (Lutz et al., 2023). These operators guarantee correspondence between interpretations at coarse and fine levels under a tree of levels, preserving semantic integrity through partial refinement functions.

2. Unsupervised Abstraction Pipeline in Process Mining

The LevelEnv abstraction in process mining leverages unsupervised techniques to detect recurring high-level patterns from activity-rich event logs without domain-supplied activities (Mannhardt et al., 2017):

1. Local Process Model (LPM) Discovery: Candidate LPMs, formalized as accepting Petri nets $N = (P,T,F,\ell, M_0, M_f)$, are mined from the event log, each covering a localized, frequent pattern over a small activity subset. Support is quantified by maximal fitting segments in traces via segmentation functions.
2. Diversity Filtering: To enforce distinctiveness among high-level activities, candidate LPMs are filtered by a diversity threshold $t_{\text{div}}$, measuring overlap of activity sets.
3. Pattern-Based Abstraction: The resulting Petri nets are composed into a single abstraction net (interleaving or parallel), and trace alignment segments the raw trace into intervals attributable to specific LPMs. The corresponding mapping $\psi$ lifts these to high-level events.
4. Model Discovery: The lifted high-level event log is then used with standard process discovery algorithms, yielding compact models with superior trade-offs in fitness and precision.

This abstraction reduces overgeneralization and overfitting; empirical evaluation demonstrates up to 20-point F-score improvements, dramatic rises in precision ($\approx 0.50 \to >0.85$), and balanced model quality when using 1–5 diverse LPMs (Mannhardt et al., 2017).

3. Abstraction in Reinforcement Learning: State and Environment Shaping

LevelEnv abstraction in RL compresses large or noisy state spaces for sample-efficient policy learning. The key steps follow (Kamalaruban et al., 2020, Patil et al., 2024):

• State Abstraction: A surjective map $\phi: \mathcal{S} \to \mathcal{X}_\phi$ creates abstract states whose model-irrelevance is formally bounded: for all $s_1, s_2$ in a class, $|R(s_1,a)-R(s_2,a)|\leq \epsilon_R$ and abstracted transition kernels are close in total variation.
• Environment Construction: An abstract MDP $M_\phi$ is built, with parameters (rewards, transitions, initial state) induced from projected statistics of the raw MDP. Shaping rewards may be added for uniqueness of the abstract optimal policy.
• Lifting: The abstract policy is lifted back to the original state space via the LevelEnv, ensuring performance close to optimal (with provable value-loss bounds not depending on $|\mathcal{S}|$).

Contrastive abstraction extends this via self-supervised representation learning, using contrastive loss (InfoNCE) for local smoothness, followed by modern Hopfield network clustering that defines the “level” granularity via the number of memory attractors. This pipeline is reward-free and allows rapid meta-policy or graph-planning atop the learned abstraction; the abstraction granularity is controlled through the Hopfield temperature parameter $\beta$ (Patil et al., 2024).

4. Knowledge Representation: Abstraction and Refinement Operators

In DLs, LevelEnv abstraction extends the logic by making abstraction levels and bridging explicit. Syntax introduces labeled inclusions, refinement/abstraction of concepts and roles using full conjunctive queries. Semantically, an A-interpretation is a tree of interpretations at different levels with a refinement function $\rho$ managing entity correspondences. Decidability is maintained under certain syntactic constraints (tree of levels, full CQs) with complexity of 2ExpTime for the general logic, ExpTime when only refinement is used. Undecidability arises with DAG level structures or non-full queries (Lutz et al., 2023). This supports rigorous ontology management over multiple granularities.

5. Abstraction Layers in Virtual Environment and System Design

LevelEnv abstraction is also realized in modular frameworks for virtual environment engineering, as a multi-tiered stack from high-level logic to hardware:

• Application Logic & AI: Scripting and rule-sets (e.g., via Python), decoupled from rendering/physics, operate at the highest abstraction layer; policies and AI rely on navigation mesh structures.
• Content Authoring & Scene Representation: Designers work in authoring tools (e.g., Blender), expressing objects via scene graphs and exporting physics/graphics metadata in extended XML formats.
• Scene Importer & Core Engine: Parses abstract representations, instantiates objects and physics via native engines (OGRE, PhysX), manages transformations.
• Subsystem Integration: Exposes low-level rendering, physics, audio, input, and networking APIs via wrappers, ensuring clean up/down-layer separation.
• Hardware & OS: The lowest abstraction exposes device drivers and interfaces to higher levels.

Each level is modular, allowing targeted replacement without affecting upper/lower layers. Empirical results demonstrate substantial reductions in prototyping time and code duplication, validating the LevelEnv/inspired design for both flexibility and rapid iteration (Catanese et al., 2011).

6. LevelEnv as a Framework for Environment Generation and Generalization

Environment generators, such as AutoEnv, instantiate LevelEnv abstraction by treating environments as factorizable distributions over transitions, observations, and rewards, governed by a small set of YAML-encoded parameters (Zhang et al., 24 Nov 2025). Each environment is:

• Defined as $E = (\mathcal{M}, T, \Omega, R, \tau)$ ($\mathcal{M} = (\mathcal{S}, \mathcal{A})$), with sampling and validation involving LLM-driven synthesis, DSL configuration, code generation, and rigorous reliability and consistency checks.
• Parameterized such that transitions, observations, and rewards vary independently, enabling controlled heterogeneity in generated environments.
• Used to study cross-environment generalization, as agents trained across factorizable families face a sharply diminishing return for fixed adaptation strategies; adaptive selection helps, but scalability remains an open challenge.

Generation is cost-efficient and reliable (mean \$4.12 per validated environment), enabling scale and heterogeneity in experimental design; limitations currently include the text-only format and the bounded space of learning adaptation methods (Zhang et al., 24 Nov 2025).

7. Trade-offs, Evaluation Metrics, and Practical Considerations

Across all domains, LevelEnv abstraction introduces fundamental trade-offs:

• Granularity vs. Fidelity: Coarse abstractions accelerate learning, model discovery, or reasoning but increase value loss or semantic loss; finer abstractions preserve detail but reduce efficiency.
• Evaluation: Fitness (coverage of target instances), precision (avoidance of spurious behaviors), and simplicity (model size) are the primary metrics in process mining; in RL, bounds on near-optimality and empirical sample efficiency are central.
• Design Choices: Abstraction mappings may be domain-informed (expert-driven), unsupervised (e.g., clustering, contrastive learning), or logic-based (CQ-defined), with error bounds or complexity guarantees deriving from their theoretical properties.

Optimal configuration of abstraction level, regularity of mapping, and system modularity depends on the target application and desired trade-offs.

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References:

• Unsupervised event abstraction and LevelEnv in process mining: (Mannhardt et al., 2017)
• Modular abstraction in virtual environments: (Catanese et al., 2011)
• State abstraction and environment shaping in RL: (Kamalaruban et al., 2020)
• Contrastive abstraction in RL: (Patil et al., 2024)
• Factorizable environment generators and cross-environment analysis: (Zhang et al., 24 Nov 2025)
• Multi-level abstraction and refinement in description logics: (Lutz et al., 2023)