The information-theoretic complexity of differentiable functions
Pith reviewed 2026-05-20 01:35 UTC · model grok-4.3
The pith
V-complexity quantifies the complexity of differentiable functions using piecewise constant approximations.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
V-complexity is introduced as a measure that takes into account the quality of the approximation and the number of intervals in the approximating function. It is shown to formalize some intuitions about the simplicity or complexity of f(x). The V-complexity is hypothesized to be equivalent to the compression measure for the Run Length Encoding and the Lempel Ziv 77 algorithms. This allows the effective complexity of a complex system to be defined as the V-complexity of the differentiable function describing its perceived regularities, as demonstrated in the diffusion of cream in a cup of coffee where it starts at zero, increases to a maximum, and decreases back to zero.
What carries the argument
V-complexity, a measure combining the error in approximating a differentiable function by a piecewise constant function with the number of intervals used in that approximation.
If this is right
- V-complexity can be used to define the effective complexity of complex systems when their regularities are given by a differentiable function.
- In the cream diffusion model, V-complexity starts at zero, rises quickly to a maximum, then falls back to zero at equilibrium.
- The same rise and fall behavior is obtained using a cellular automaton and the concept of apparent complexity.
- V-complexity is equivalent to the compression measure using run length encoding and Lempel Ziv 77.
Where Pith is reading between the lines
- This measure provides a way to track the evolution of complexity in physical systems over time without relying on specific simulation methods.
- It suggests that many systems may exhibit maximum complexity during their transition to equilibrium rather than at the start or end.
- The hypothesized equivalence to compression could allow using V-complexity as a faster proxy for compressibility in function data.
Load-bearing premise
The premise that approximations by piecewise constant functions, weighted by both their accuracy and the number of pieces, provide a general and meaningful information-theoretic complexity measure for differentiable functions.
What would settle it
A specific differentiable function for which V-complexity gives a value that contradicts the compressibility under run length encoding or Lempel-Ziv 77, or that does not align with standard intuitions about its complexity.
Figures
read the original abstract
A measure for the complexity of a differentiable function f(x) on an interval is introduced. It is based on approximations of the function by piecewise constant functions. The measure takes into account the quality of the approximation and the number of intervals in the approximating function. This measure, called the V-complexity of f(x), is shown to formalize some intuitions about the simplicity or complexity of f(x). The V-complexity is then compared to another measure of complexity, namely how compressible an approximation of f(x) is. It is hypothesized that V-complexity is equivalent to the compression measure, in the case of the Run Length Encoding and the Lempel Ziv 77 algorithms. V-complexity can be used as an ingredient in the definition of the Effective Complexity (EC) of a Complex System. When the perceived regularities of such a system are described by a differentiable function on an interval, the EC can be defined as the V-complexity of that function. EC is applied to the model of diffusion of cream in a cup of coffee. The perceived regularity of this model is given by the diffusion equation. The V-complexity of the solution of the equation starts at zero, quickly increases to a maximum and then decreases back to zero as the liquid reaches its equilibrium state. It is shown that this is also the result when a cellular automaton approach and the concept of Apparent Complexity is used.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript introduces V-complexity, a measure for the complexity of a differentiable function f(x) on an interval. It is defined via the quality of approximation by piecewise-constant functions together with the number of intervals required. The paper claims that this measure formalizes intuitions about simplicity and complexity of functions. It hypothesizes equivalence between V-complexity and compressibility under Run-Length Encoding and Lempel-Ziv 77. V-complexity is then proposed as the basis for Effective Complexity (EC) of a complex system whose perceived regularities are captured by such a function; the diffusion equation solution for cream mixing in coffee is used as an example, with V-complexity shown to rise from zero to a peak and return to zero, matching the behavior obtained from a cellular-automaton model and Apparent Complexity.
Significance. If the hypothesized link to compression algorithms can be made rigorous and the measure shown to be independent of the specific approximation scheme, V-complexity would supply a concrete, approximation-based route to information-theoretic complexity for differentiable functions and a practical ingredient for Effective Complexity. The diffusion example, together with its consistency check against cellular automata, illustrates how the measure can track the emergence and disappearance of regularities in a physical model; this is a concrete strength that could be developed further.
major comments (2)
- [Abstract / hypothesis paragraph] Abstract (paragraph introducing the hypothesis): the claim that V-complexity is equivalent to the compression length under RLE and LZ77 is stated as a hypothesis, yet no explicit construction, choice of approximation rule (uniform grid, adaptive, or minimax), error metric, or side-by-side numerical comparison is supplied that would show the resulting V-value equals or ranks identically with the compressed bit length of the same discretized function. Because this equivalence is the stated bridge to information theory, its absence leaves the central interpretive claim unanchored.
- [Definition of V-complexity] Definition of V-complexity (section following the abstract): the quality of piecewise-constant approximation and the interval count are asserted to yield a meaningful complexity measure that captures perceived regularities independently of the concrete approximation algorithm or error metric. No robustness check or invariance argument is given to support this independence, which is load-bearing for the claim that V-complexity is a general information-theoretic quantity.
minor comments (1)
- [Diffusion application] The diffusion example would benefit from an explicit statement of the discretization used to compute V-complexity from the analytic solution of the diffusion equation.
Simulated Author's Rebuttal
We thank the referee for the constructive comments. We agree that the hypothesis linking V-complexity to compression requires concrete support and that independence from approximation details needs explicit justification. Below we respond point by point and indicate the planned revisions.
read point-by-point responses
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Referee: [Abstract / hypothesis paragraph] Abstract (paragraph introducing the hypothesis): the claim that V-complexity is equivalent to the compression length under RLE and LZ77 is stated as a hypothesis, yet no explicit construction, choice of approximation rule (uniform grid, adaptive, or minimax), error metric, or side-by-side numerical comparison is supplied that would show the resulting V-value equals or ranks identically with the compressed bit length of the same discretized function. Because this equivalence is the stated bridge to information theory, its absence leaves the central interpretive claim unanchored.
Authors: We accept the observation. The equivalence is presented as a hypothesis without numerical validation or specification of the approximation procedure. In the revision we will add a new subsection that fixes the approximation rule to uniform partitioning with supremum-norm error, supplies explicit constructions for several test functions, and reports side-by-side comparisons of the resulting V-values against the compressed lengths produced by RLE and LZ77 on the same discretizations. This will provide the missing anchor for the information-theoretic interpretation. revision: yes
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Referee: [Definition of V-complexity] Definition of V-complexity (section following the abstract): the quality of piecewise-constant approximation and the interval count are asserted to yield a meaningful complexity measure that captures perceived regularities independently of the concrete approximation algorithm or error metric. No robustness check or invariance argument is given to support this independence, which is load-bearing for the claim that V-complexity is a general information-theoretic quantity.
Authors: We agree that an invariance argument or robustness demonstration is required. We will revise the definition section to include a short asymptotic argument showing that, for fine enough partitions, the V-complexity ordering is independent of the choice between uniform and adaptive grids and between L2 and L-infinity error metrics. We will also add a brief numerical check on a small collection of example functions confirming that relative complexities remain stable across these variants. revision: yes
Circularity Check
V-complexity defined from first principles via piecewise-constant approximation; no reduction to inputs or self-citation chain.
full rationale
The paper introduces V-complexity directly from the quality of approximation by piecewise constant functions together with the number of intervals; this construction is independent of any compression length or fitted parameter. The subsequent hypothesis of equivalence to RLE and LZ77 is stated as a conjecture without any equation that equates the two measures by definition or by construction. The application to effective complexity of the diffusion equation and the matching result with cellular automata is presented as an external consistency check rather than a self-referential derivation. No self-citation load-bearing step, ansatz smuggling, or renaming of a known result appears in the derivation chain. The measure therefore remains self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Differentiable functions on an interval admit useful approximations by piecewise constant functions whose quality and piece count can serve as a complexity measure.
invented entities (1)
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V-complexity
no independent evidence
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
V(f) = 1/4 (∫_a^b |f'(x)|^{1/2} dx)^2 ... equidistribution of errors ... N(ε) = V(f) ε^{-1}
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IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
V-complexity ... hypothesized equivalent to ... Run Length Encoding and ... Lempel Ziv 77
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
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discussion (0)
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