Datasets:

11NOel11
/

ChaosBench-Logic

This document defines the formal ontology used in ChaosBench-Logic: a 27-predicate logical ontology (15 core predicates plus structural extensions) and the First-Order Logic (FOL) axioms that govern their relationships.

Core Predicate Set (15)

Each dynamical system in ChaosBench-Logic is characterized by a predicate inventory that includes a core 15-predicate set plus additional structural predicates used for extended reasoning. The original 11 predicates (1-11) are required for system eligibility; the 4 extension predicates (12-15) were added in v2 for deeper reasoning chains. The full 27-predicate inventory is defined in chaosbench/logic/ontology.py.

1. Chaotic

Definition: The system exhibits deterministic chaos — long-term unpredictable behavior arising from sensitive dependence on initial conditions, despite being governed by deterministic equations.

Formal properties:

Has positive Lyapunov exponent
Exhibits sensitive dependence on initial conditions
Deterministic (not stochastic)
Aperiodic dynamics

Example: Lorenz-63 system, Hénon map, logistic map at r=4

2. Deterministic

Definition: The system's future state is uniquely determined by its current state and the governing equations, with no random or stochastic components.

Formal properties:

Evolution described by deterministic equations (ODEs, maps, PDEs)
No noise terms or random variables
Same initial conditions always produce the same trajectory

Example: All non-stochastic ODEs, discrete maps without noise

Counter-example: Ornstein-Uhlenbeck process (has stochastic term)

3. PosLyap (Positive Lyapunov Exponent)

Definition: The system has at least one positive Lyapunov exponent, meaning nearby trajectories diverge exponentially on average.

Formal properties:

Largest Lyapunov exponent λ₁ > 0
Quantifies exponential divergence rate: d(t) ≈ d₀ e^(λ₁t)

Example: Lorenz-63 (λ₁ ≈ 0.9), Hénon map (λ₁ ≈ 0.42)

Note: Positive Lyapunov exponent is necessary but not sufficient for chaos (needs bounded dynamics).

4. Sensitive (Sensitive Dependence on Initial Conditions)

Definition: Arbitrarily small differences in initial conditions lead to large differences in trajectories after sufficient time.

Formal properties:

Related to positive Lyapunov exponent
Makes long-term prediction impossible in practice
Butterfly effect

Example: Weather systems, double pendulum

5. StrangeAttr (Strange Attractor)

Definition: The system's long-term behavior is confined to an attractor with fractal structure and non-integer dimension.

Formal properties:

Fractal dimension (not integer)
Self-similar structure at different scales
Bounded in phase space
Attracts nearby trajectories

Example: Lorenz attractor (dimension ≈ 2.06), Hénon attractor

Note: Strange attractor is sufficient but not necessary for chaos. Some chaotic systems lack attractors (e.g., Arnold cat map is area-preserving).

6. PointUnpredictable (Pointwise Unpredictable)

Definition: Precise point prediction of the system state at a specific future time is impossible beyond a finite horizon, even with arbitrarily accurate initial conditions.

Formal properties:

Due to sensitive dependence and finite-precision measurements
Lyapunov time: τ = 1/λ₁ (time scale for losing one bit of information)
Individual trajectories cannot be predicted long-term

Example: Weather (can't predict specific temperature two weeks ahead)

7. StatPredictable (Statistically Predictable)

Definition: While individual trajectories are unpredictable, statistical properties (ensemble averages, distributions, moments) remain predictable over long times.

Formal properties:

Ergodicity: time averages equal ensemble averages
Invariant measure and attractor statistics are stable
Climate vs. weather distinction

Example: Can predict average summer temperature (climate) but not specific day's weather

Note: Chaotic systems are pointwise unpredictable but statistically predictable.

8. QuasiPeriodic

Definition: The system exhibits motion that is almost periodic but never exactly repeats, typically characterized by multiple incommensurate frequencies.

Formal properties:

Dynamics on a torus in phase space
Multiple independent frequencies: ω₁, ω₂, ..., ωₙ
Ratios ωᵢ/ωⱼ are irrational (incommensurate)
All Lyapunov exponents are zero or negative

Example: Circle map (quasiperiodic regime), coupled oscillators with incommensurate frequencies

Counter-example: Chaotic systems (not quasiperiodic)

9. Random (Stochastic)

Definition: The system includes intrinsic randomness or noise in its governing equations.

Formal properties:

Contains stochastic terms (Wiener process, random variables)
Described by stochastic differential equations (SDEs)
Multiple realizations from same initial conditions differ

Example: Ornstein-Uhlenbeck process, Brownian motion, systems with thermal noise

Counter-example: All deterministic systems

10. FixedPointAttr (Fixed-Point Attractor)

Definition: The system's long-term behavior converges to a single equilibrium point in phase space.

Formal properties:

Attractor is a 0-dimensional point
All trajectories approach the fixed point asymptotically
All Lyapunov exponents are negative
Stable equilibrium

Example: Damped harmonic oscillator, systems below bifurcation threshold

Counter-example: Chaotic, periodic, or quasiperiodic systems

11. Periodic

Definition: The system exhibits perfectly repeating behavior with a fixed period.

Formal properties:

Attractor is a closed orbit (limit cycle)
State returns exactly to starting point after period T
One Lyapunov exponent is zero, others negative
All Lyapunov exponents ≤ 0

Example: Limit cycles, periodic orbits in maps, undamped harmonic oscillator

Counter-example: Chaotic or quasiperiodic systems

12. Dissipative

Definition: The system's phase space volume contracts over time (div(v) < 0).

Formal properties:

Volume contraction → trajectories converge to lower-dimensional attractor
Distinguishes dissipative chaos from Hamiltonian (conservative) chaos

Example: Lorenz system (dissipative), Rössler system, Hénon map (area-contracting)

Counter-example: Standard map (conservative/area-preserving), Arnold cat map

13. Bounded

Definition: All trajectories remain in a bounded region of phase space.

Formal properties:

Attractors are bounded by definition
Statistical predictability requires bounded ensemble
Required for physical realizability of chaotic dynamics

Example: Lorenz system (bounded attractor), logistic map (bounded on [0,1])

14. Mixing

Definition: The system has the strong mixing property: correlation functions decay to zero.

Formal properties:

Mixing is stronger than ergodicity (mixing ⊂ ergodic)
Essential for statistical mechanics interpretation of chaos
Hyperbolic chaotic systems typically exhibit mixing

Example: Lorenz system, Arnold cat map (hyperbolic mixing)

Counter-example: Periodic orbits, quasi-periodic tori

15. Ergodic

Definition: Time averages along typical trajectories equal ensemble averages (Birkhoff ergodic theorem).

Formal properties:

Foundation of statistical mechanics
Weaker than mixing but essential for statistical predictability
Enables long-time statistical forecasting

Example: Lorenz system (ergodic on attractor), quasi-periodic torus

Counter-example: Periodic orbits, multi-attractor systems

First-Order Logic (FOL) Axioms

The predicates are not independent — they obey logical constraints formalized as FOL axioms.

Axiom Structure

Each axiom has two components:

Requires: If predicate A is true, these predicates must also be true
Excludes: If predicate A is true, these predicates must be false

Axiom 1: Chaotic Systems

Chaotic(s) → Deterministic(s) ∧ PosLyap(s) ∧ Sensitive(s) ∧ PointUnpredictable(s) ∧ StatPredictable(s)
Chaotic(s) → ¬Random(s) ∧ ¬Periodic(s) ∧ ¬QuasiPeriodic(s) ∧ ¬FixedPointAttr(s)

Interpretation: Chaos requires determinism, positive Lyapunov exponent, sensitivity, pointwise unpredictability, and statistical predictability. Chaos excludes randomness, periodicity, quasiperiodicity, and fixed-point attractors.

Design choice: StrangeAttr is not in the requires list because strange attractors are sufficient but not necessary for chaos (e.g., Arnold cat map is chaotic without an attractor).

Axiom 2: Random Systems

Random(s) → ¬Deterministic(s) ∧ ¬Chaotic(s) ∧ ¬QuasiPeriodic(s) ∧ ¬Periodic(s)

Interpretation: Stochastic systems are not deterministic and cannot be chaotic, quasiperiodic, or periodic (which require determinism).

Axiom 3: QuasiPeriodic Systems

QuasiPeriodic(s) → Deterministic(s)
QuasiPeriodic(s) → ¬Chaotic(s) ∧ ¬Random(s) ∧ ¬Periodic(s) ∧ ¬FixedPointAttr(s)

Interpretation: Quasiperiodicity requires determinism and excludes chaos, randomness, periodicity, and fixed-point attractors.

Axiom 4: Periodic Systems

Periodic(s) → Deterministic(s)
Periodic(s) → ¬Chaotic(s) ∧ ¬Random(s) ∧ ¬QuasiPeriodic(s) ∧ ¬StrangeAttr(s)

Interpretation: Periodicity requires determinism and excludes chaos, randomness, quasiperiodicity, and strange attractors.

Axiom 5: Fixed-Point Attractors

FixedPointAttr(s) → Deterministic(s)
FixedPointAttr(s) → ¬Chaotic(s) ∧ ¬Random(s) ∧ ¬QuasiPeriodic(s) ∧ ¬Periodic(s) ∧ ¬StrangeAttr(s)

Interpretation: Fixed-point attractors require determinism and exclude chaos, randomness, quasiperiodicity, periodicity, and strange attractors.

Axiom 6: Deterministic Systems

Deterministic(s) → ¬Random(s)

Interpretation: Determinism and randomness are mutually exclusive.

Design Choices & Rationale

1. Unidirectional Implications

Axioms are one-way implications. For example:

Chaotic → Deterministic (chaos implies determinism)
But NOT: Deterministic → Chaotic (determinism doesn't imply chaos)

Rationale: Many deterministic systems are not chaotic (e.g., simple harmonic motion).

2. StrangeAttr Not Required for Chaos

StrangeAttr is not in the requires list for Chaotic.

Rationale:

Strange attractors are sufficient for chaos (if you have a strange attractor, the system is chaotic)
But they are not necessary (some chaotic systems lack attractors)
Example: Arnold cat map is chaotic but area-preserving (no attractor)

This design allows the ontology to handle both dissipative and conservative chaotic systems.

3. Symmetric Exclusions

If A excludes B, then B excludes A. For example:

Chaotic → ¬Random
Random → ¬Chaotic

Rationale: Exclusion is inherently symmetric. This ensures logical consistency.

4. Incomplete Specification

The axioms specify necessary conditions and exclusions but do not fully constrain all relationships.

Example: Deterministic has no requires list (only excludes Random). A deterministic system could be:

Chaotic (Lorenz)
Periodic (limit cycle)
Quasiperiodic (torus)
Fixed-point (damped oscillator)

Rationale: This reflects the mathematical reality — determinism alone doesn't determine the type of dynamics.

Validation & Consistency

Logical Consistency Checks

The evaluation pipeline (eval_chaosbench.py) checks model predictions against these axioms to detect FOL violations.

Example violation:

Model predicts: Chaotic=YES, Deterministic=NO
Violation: Chaotic → Deterministic

This is counted as a logical inconsistency even if the model doesn't contradict itself (didn't give both YES and NO to the same question).

Ground Truth Assignment

Each system in systems/*.json has a truth_assignment field with boolean values for the predicates (11 core required, 4 extension optional):

{
  "system_id": "lorenz63",
  "truth_assignment": {
    "Chaotic": true,
    "Deterministic": true,
    "PosLyap": true,
    "Sensitive": true,
    "StrangeAttr": true,
    "PointUnpredictable": true,
    "StatPredictable": true,
    "QuasiPeriodic": false,
    "Random": false,
    "FixedPointAttr": false,
    "Periodic": false
  }
}

All truth assignments are verified to satisfy the FOL axioms.

Predicate Extraction from Questions

The evaluation pipeline maps natural language questions to predicates using keyword matching:

Keywords	Predicate
"chaotic", "chaos"	`Chaotic`
"deterministic"	`Deterministic`
"positive lyapunov", "lyapunov exponent"	`PosLyap`
"sensitive dependence", "sensitivity"	`Sensitive`
"strange attractor"	`StrangeAttr`
"pointwise prediction", "point-wise predictable"	`PointUnpredictable`
"statistically predictable", "statistical prediction"	`StatPredictable`
"quasi-periodic", "quasiperiodic"	`QuasiPeriodic`
"random", "randomness", "stochastic"	`Random`
"fixed point", "fixedpoint"	`FixedPointAttr`
"periodic"	`Periodic`
"dissipative", "volume-contracting"	`Dissipative`
"bounded", "bounded attractor"	`Bounded`
"mixing", "topological mixing"	`Mixing`
"ergodic", "ergodicity"	`Ergodic`

See chaosbench/logic/ontology.py:KEYWORD_MAP for implementation.

Example System Definitions

Chaotic System: Lorenz-63

{
  "system_id": "lorenz63",
  "name": "Lorenz-63 system",
  "category": "chaotic",
  "equations": "dx/dt = σ (y - x); dy/dt = x (ρ - z) - y; dz/dt = x y - β z",
  "parameters": {
    "sigma": 10.0,
    "rho": 28.0,
    "beta": 2.6666666667
  },
  "truth_assignment": {
    "Chaotic": true,
    "Deterministic": true,
    "PosLyap": true,
    "Sensitive": true,
    "StrangeAttr": true,
    "PointUnpredictable": true,
    "StatPredictable": true,
    "QuasiPeriodic": false,
    "Random": false,
    "FixedPointAttr": false,
    "Periodic": false
  }
}

Satisfies: All Chaotic axioms — deterministic, positive Lyapunov, sensitive, etc.

Stochastic System: Ornstein-Uhlenbeck Process

{
  "system_id": "stochastic_ou",
  "name": "Ornstein-Uhlenbeck process",
  "category": "stochastic",
  "equations": "dX = θ(μ - X)dt + σ dW",
  "truth_assignment": {
    "Random": true,
    "Deterministic": false,
    "Chaotic": false,
    "PosLyap": false,
    "Sensitive": false,
    "StrangeAttr": false,
    "PointUnpredictable": false,
    "StatPredictable": false,
    "QuasiPeriodic": false,
    "FixedPointAttr": false,
    "Periodic": false
  }
}

Satisfies: Random axioms — not deterministic, not chaotic, etc.

QuasiPeriodic System: Circle Map

{
  "system_id": "circle_map_quasiperiodic",
  "name": "Circle map (quasiperiodic regime)",
  "category": "quasiperiodic",
  "equations": "θₙ₊₁ = θₙ + Ω - (K/2π) sin(2π θₙ) mod 1",
  "truth_assignment": {
    "QuasiPeriodic": true,
    "Deterministic": true,
    "Chaotic": false,
    "Random": false,
    "PosLyap": false,
    "Sensitive": false,
    "StrangeAttr": false,
    "PointUnpredictable": false,
    "StatPredictable": false,
    "FixedPointAttr": false,
    "Periodic": false
  }
}

Satisfies: QuasiPeriodic axioms — deterministic but not chaotic, periodic, or random.

References

Chaos Theory: Strogatz, S. H. (2015). Nonlinear Dynamics and Chaos. Westview Press.
Lyapunov Exponents: Wolf, A., et al. (1985). "Determining Lyapunov exponents from a time series." Physica D.
Strange Attractors: Ott, E. (2002). Chaos in Dynamical Systems. Cambridge University Press.
Ergodic Theory: Walters, P. (1982). An Introduction to Ergodic Theory. Springer.

Citation

If you use this ontology in your research, please cite:

@software{thomas2025chaosbench,
  author = {Thomas, Noel},
  title = {ChaosBench-Logic: A Benchmark for Evaluating Large Language Models on Complex Reasoning about Dynamical Systems},
  year = {2025},
  url = {https://github.com/11NOel11/ChaosBench-Logic}
}

Contact

For questions about the ontology or to report errors:

Open an issue on GitHub
Contact: Noel Thomas (MBZUAI)