Algebraic Telic Problems in Real Dynamics

• Algebraic telic problems are decision problems in the BSS model that determine whether an orbit of a computable map reaches a target interval in a finite number of steps using bounded-precision inputs.
• They bridge dynamical systems theory and real computational complexity by framing finite-time reachability as an NP_R problem and highlighting intrinsic circuit complexity barriers.
• Canonical examples like the expanding and tent maps demonstrate how dynamical invariants influence the feasibility of mapping reductions and establish arithmetic circuit lower bounds.

An algebraic telic problem is a decision problem in the Blum–Shub–Smale (BSS) model of computation over the real numbers, formulated to express finite-time reachability in one-dimensional dynamical systems. Algebraic telic problems formalize, within an explicit $\textsf{NP}_\mathbb{R}$ structure, whether an orbit of a computable map can reach a designated target interval in a specified number of steps, given bounded-precision initial data. This framework captures the computational complexity of verifying reachability in algebraically defined systems and provides a bridge between dynamical systems theory and real computational complexity, particularly in the study of reductions and circuit lower bounds for real decision problems (Everett, 15 Jan 2026).

1. Formal Definition and Model

Let %%%%1%%%% denote the state space and let $F: I \to I$ be a BSS-computable map, specified as a rational function evaluated by a real machine. A homeomorphism $g : I \to I$ is also BSS-computable (e.g. piecewise-linear).

For each input length $n$, an instance consists of $$(a, b, 1, 1, \ldots, 1) \in I \times I \times \{1\}^n$$ with $a \leq b$ in $[0,1]$. The certificate is a mesh point $$y \in I_{n^2} := \{ k / 2^{n^2} : 0 \leq k \leq 2^{n^2} \}$$. The decision form is:

• Does there exist $y \in I_{n^2}$ such that $F^{n}(g(y)) \in [a,b]$?

The associated structured decision problem is

$$\begin{aligned} B &= \bigcup_{n \geq 1} ([0,1] \times [0,1] \times \{1\}^n) \ A^{(n)} &= \{ (a, b, 1^n) : a \leq b \text{ and } \exists y \in I_{n^2},\, F^n(g(y)) \in [a,b] \} \ R_{(I, F)}(g) &= (B, A) \end{aligned}$$

Deciding membership in $A$ is in $\textsf{NP}_\mathbb{R}$ via nondeterministic guessing of $y$ and explicit evaluation of $F^n(g(y))$ in $O(n)$ steps (Everett, 15 Jan 2026).

2. Canonical One-Dimensional Examples

Two explicit systems illuminate the structure of algebraic telic problems:

• Expanding Map: $F(x) = E_2(x) = 2x$ mod $1$. The telic decision is: given $(a, b, 1^n)$, determine whether there exists $y \in I_{n^2}$ such that $E_2^n(y) \in [a,b]$.
• Tent Map: $F(x) = 2 \min\{x, 1 - x\}$. The analogous decision: for $(a, b, 1^n)$, does there exist $y \in I_{n^2}$ with $(2\ \min)^n(y) \in [a,b]$?

These encode finite-time reachability, with orbits after $n$ steps given by $f^{n}(x) = f \circ \dots \circ f(x)$. For $E_2$, the equation reduces to $2^n y \bmod 1 \in [a,b]$ for some dyadic rational $y$ with $n^2$ bits of precision (Everett, 15 Jan 2026).

3. Structure of Reductions Between Telic Problems

Natural mapping reductions between algebraic telic problems are highly constrained by the dynamical invariants of the underlying systems.

• Natural Reductions: Given two systems $(I, F)$ and $(I, T)$ (possibly with positive and zero topological entropy, respectively), and homeomorphisms $g, \tilde{g}$, a "level 1" reduction requires a sequence of maps $\eta_n$ such that $\eta_n(F^n(x)) = T^n(x)$ for all $x \in I_{n^2}$. Higher "levels" permit more freedom but must still mirror combinatorial and topological structure.
• Impossibility Theorems: There exist $(I, F)$ with positive entropy and $(I, T)$ with zero entropy, such that $R_{(I, F)}(g)$ is not reducible to $R_{(I, T)}(\tilde{g})$ via any of the first three levels of mapping reductions. Existence of such a reduction implies the systems share major dynamical features: periodic-orbit structure, entropy, and invariant measures (Everett, 15 Jan 2026).
• Obstruction via Fiber-Counting: For $F$ 2-to-1 (e.g. the doubling map) and $T$ 1-to-1, no $\eta_n$ as required by the reduction definition can exist (the preimage counts mismatch).

This demonstrates that complexity-theoretic reducibility among telic problems mirrors dynamical-systems conjugacy at the level of finite encodings.

4. Circuit Complexity Barriers

Algebraic telic problems derived from chaotic or expanding systems display intrinsic barriers to small arithmetic circuit representations.

• Arithmetic Circuit Lower Bound: For the map $E_2$, any uniform family of arithmetic circuits over $\mathbb{R}$ using only $+$ and $\times$ gates, which on input a rational interval of length $2^{-n^2}$ outputs a witness $y \in I_{n^2}$ for $E_2^n(y) \in [a,b]$, must have depth at least $\Omega(n^2)$.
• Sensitivity and Nonuniformity: For suitable choices of the homeomorphism $g = \alpha$, the map $s \mapsto E_2^n(\alpha(s))$ is so sensitive that perturbing $[a,b]$ by $2^{-n^2/2}$ shifts the answer for $y$ by $O(1)$, necessitating high-degree polynomial interpolation. Thus, naive circuit approaches are unfeasible in this regime (Everett, 15 Jan 2026).

These circuit-theoretic barriers distinguish algebraic telic problems from analogous discrete $\textsf{NP}$ problems and show that real reachability can require superpolynomial resources in certain dynamical settings.

5. Proof Strategies and Algorithmic Techniques

Central proof strategies for complexity separation and depth bounds utilize dynamical properties and combinatorial arguments:

• Reduction-Impossibility: By counting fiber-preimages or analyzing pullbacks under proposed $\eta_n$, one shows that imposed mappings cannot exist unless strong conjugacy or factor relations hold between maps.
• Degree Lower Bounds: Constructing perturbations $c_1, c_2$ of the target so that the solutions $s_1, s_2$ are far apart enforces that any interpolating polynomial used in an arithmetic circuit must have exponentially large degree, translating immediately to depth lower bounds through established real-circuit complexity techniques (Everett, 15 Jan 2026).

All methods emphasize the interplay between combinatorial orbit structure and algebraic encoding of the reachability decision.

6. Broader Impact, Open Problems, and Future Directions

Algebraic telic problems enable a precise analysis of the interplay between dynamical systems and real computation.

• Reflecting Dynamical Invariants: The existence and nonexistence of "natural" reductions among telic problems encapsulate invariants such as entropy and periodic structure. Computational separations map directly to qualitative properties of dynamical systems.
• Extensions and Open Problems:
  • Higher-dimensional analogues: the computational complexity of reachability for polynomial ODEs on $\mathbb{R}^d$ is open and would generalize the current theory.
  • Level-4 reductions: the possibility of ruling out even more flexible reduction schemas remains open.
  • Separation of decision and search in $\textsf{NP}_\mathbb{R}$: concrete dynamical constructions that witness a genuine complexity-theoretic gap are yet to be established.
  • Impact of richer gates (e.g., logical or sign gates) on computational lower bounds for telic problems is under investigation (Everett, 15 Jan 2026).

A plausible implication is that the algebraic telic problem framework offers a new computational lens for interpreting dynamical-system regularity and algorithmic hardness, particularly for problems where symbolic reachability or orbit-finding is essentially "dynamically protected" by entropy or topological mixing.

7. Connections to Telescoping Problems and Related Algebraic Frameworks

While the algebraic telic problem formalizes reachability in real dynamical systems, there is a structural parallel to "telescopers" in symbolic computation and creative telescoping problems.

• In symbolic computation, telescopers for rational/algebraic functions in multiple variables are differential or recurrence operators certifying the existence of certificates (as in the Zeilberger paradigm) (Chen et al., 2012, Chen et al., 2019).
• Algebraic telic problems correspondingly address whether certain polynomial-time verifiable orbits exist, analogously to detecting the existence of a certificate annihilating a function up to a differential or difference (Everett, 15 Jan 2026).

This suggests a fertile ground for further research at the interface of algorithmic dynamical systems, algebraic complexity, and the theory of integrability and summation in multiple variables.

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References:

• "Correspondences in computational and dynamical complexity II: forcing complex reductions" (Everett, 15 Jan 2026)
• "On the Existence of Telescopers for Rational Functions in Three Variables" (Chen et al., 2019)
• "Telescopers for Rational and Algebraic Functions via Residues" (Chen et al., 2012)