[Paper Review] A Survey on Analog Models of Computation

Olivier Bournez, Amaury Pouly|arXiv (Cornell University)|May 14, 2018

Computability, Logic, AI Algorithms4 citations

TL;DR

This survey presents a comprehensive analysis of analog models of computation, focusing on continuous-time dynamical systems defined by polynomial ordinary differential equations (ODEs). It establishes foundational characterizations of classical complexity classes—such as P—using ODEs, showing that polynomial-time computable functions can be precisely captured by the solution trajectories of such systems, with convergence precision tied to the length of the solution curve.

ABSTRACT

We present a survey on analog models of computations. Analog can be understood both as computing by analogy, or as working on the continuum. We consider both approaches, often intertwined, with a point of view mostly oriented by computation theory.

Motivation & Objective

• To unify and survey analog models of computation that operate over continuous spaces and time, emphasizing their theoretical foundations in computation theory.
• To clarify the dual meaning of 'analog'—as computation by analogy or as continuous computation—highlighting their interplay in historical and modern models.
• To investigate the computational power of continuous dynamical systems, particularly those governed by polynomial ODEs, in capturing classical complexity classes.
• To provide a rigorous characterization of polynomial-time computable functions (P) using continuous-time ODEs with length-based convergence criteria.
• To bridge classical computation theory with analysis by showing that concepts like computability and complexity can be defined purely through differential equations.

Proposed method

• Model analog computation as dynamical systems defined by flows φ: T × X → X, where T is a subgroup of ℝ (continuous or discrete time), and X is a space (typically ℝⁿ or ℝⁿ×ℕᵏ).
• Represent continuous-time systems via initial value problems (IVPs) y′ = f(y), y(0) = x₀, where f is a vector of polynomials with rational coefficients.
• Use the length of the solution curve ℓ(t) = ∫₀ᵗ ||y′(s)|| ds as a key computational resource, linking it to precision and runtime in analog computation.
• Encode inputs (e.g., real numbers or binary words) into initial conditions, using rational encodings like ψ(w) = Σ wᵢ2⁻ⁱ for binary strings.
• Define acceptance via a designated output component y₁(t): y₁ ≥ 1 for acceptance, y₁ ≤ -1 for rejection, with stability ensured by the invariance condition.
• Introduce a lower bound on ||y′(t)|| ≥ 1 to prevent pathological slow convergence and ensure that sufficient length implies required precision.

Experimental results

Research questions

• RQ1 Can classical complexity classes like P be characterized using continuous-time dynamical systems defined by polynomial ODEs?
• RQ2 How does the geometric length of a solution curve relate to computational precision and runtime in analog models?
• RQ3 To what extent can analog computation over the reals simulate classical digital computation, and vice versa?
• RQ4 What role do polynomial ODEs with rational coefficients play in capturing computable functions and polynomial-time decidability?
• RQ5 How do the dual interpretations of 'analog'—as computation by analogy or as continuous computation—interact in the design and analysis of computational models?

Key findings

• A language L ⊆ {0,1}* is in P if and only if there exists a system of polynomial ODEs with rational coefficients such that the solution trajectory's length ℓ(t) ≥ L(|w|) implies y₁(t) ≥ 1 (accept) or y₁(t) ≤ -1 (reject), with the output stable after acceptance.
• The precision of the output y₁(t) improves with the length of the solution curve: if ℓ(0,t) ≥ L(μ), then |f(x) - y₁(t)| ≤ e⁻ᵘ, showing that longer curves yield higher accuracy.
• The condition ||y′(t)|| ≥ 1 ensures that the system does not stall, preventing degenerate cases where long curve length does not correspond to meaningful computation.
• The characterization of P via ODEs is robust: it extends to multivariate functions and unbounded domains by incorporating the norm of the input into the length function L.
• The framework allows for a purely analytical definition of classical computational concepts—such as polynomial-time decidability—using only differential equations and geometric measures like curve length.
• The results demonstrate that polynomial-time computation can be captured without discrete steps, using only continuous evolution governed by rational polynomial ODEs.