Deterministic Horizon: Theory & Applications

• Deterministic Horizon is defined as a fixed evaluation interval used in control theory, chaos, and AI to set boundaries for reliable prediction and computation.
• In finite-horizon systems like MJLS-LQR, it enables transforming stochastic problems into deterministic formulations that yield reduced-order Riccati equations.
• Beyond control applications, it quantifies numerical predictability in chaos and establishes architecture-determined limits in transformer-based trustworthy AI.

“Deterministic Horizon” is used in current literature in several technically distinct ways. In finite-horizon control, it denotes a fixed, non-random time interval such as $[0,T]$, in contrast to random stopping times or asymptotic $t\to\infty$ formulations; in this sense it is central to the deterministic reformulation of finite-horizon LQR for Markov Jump Linear Systems (MJLS) (Narváez et al., 18 Nov 2025). In deterministic differential games and optimal control, the phrase also marks the difference between finite deterministic horizons and infinite deterministic horizons on $[0,\infty)$ (Asri, 2010, Asri et al., 2021). In chaos, it denotes the horizon of predictability $T_p$, namely the last time up to which a numerically computed orbit remains reliable under finite precision (Angelidis et al., 19 Oct 2025). Recent work in trustworthy AI uses “Deterministic Horizon” for an architecture-determined limit on transformer reasoning depth, denoted $d^*$ (Guo, 21 May 2026).

1. Core meanings across the literature

Taken together, these works use the expression in two main ways. The first is temporal and control-theoretic: a deterministic horizon is a fixed evaluation interval, either finite or infinite, on which dynamics, costs, and dynamic-programming equations are posed. The second is boundary-theoretic: a deterministic horizon is a computable limit on reliable prediction or reasoning, even when the underlying system is mathematically deterministic.

Domain	Meaning of “Deterministic Horizon”	Representative papers
MJLS-LQR	Fixed interval $[0,T]$ with exact deterministic reformulation	(Narváez et al., 18 Nov 2025)
Deterministic impulse control / differential games	Finite or infinite deterministic time domain	(Asri, 2010, Asri et al., 2021, Asri et al., 2021)
Deterministic optimal control via LP	Infinite discounted horizon in discrete or continuous time	(Gaitsgory et al., 2017, Gaitsgory et al., 2017, Kamoutsi et al., 2017)
Receding-horizon control on deterministic systems	Finite prediction horizon used repeatedly online	(Ding et al., 2012, Svoreňová et al., 2013)
Chaos	Horizon of predictability $T_p$ under finite precision	(Angelidis et al., 19 Oct 2025)
Trustworthy AI	Architecture-determined reasoning-depth limit $d^*$	(Guo, 21 May 2026)

In the most explicit control-theoretic usage, “finite horizon” means that performance is evaluated over a fixed deterministic interval $[0,T]$, with terminal time $T>0$ given and not random. This is stated directly in the MJLS-LQR formulation and contrasted there with infinite-horizon problems and stationary solutions (Narváez et al., 18 Nov 2025). The same contrast appears in deterministic finite-horizon minimax impulse control, where the horizon is a fixed $t\to\infty$0, and in deterministic income-consumption problems, where consumption is optimized up to a finite deterministic time horizon $t\to\infty$1 (Asri, 2010, Eisenberg, 2016).

2. Finite-horizon MJLS and deterministic reformulation

In "Finite-Horizon LQR for General Markov Jump Linear Systems: Deterministic Reformulation and Reduced-Order Solution" (Narváez et al., 18 Nov 2025), the deterministic horizon is the interval $t\to\infty$2 for the continuous-time MJLS

$t\to\infty$3

where $t\to\infty$4 is a homogeneous continuous-time Markov chain with finite state space $t\to\infty$5 and generator matrix $t\to\infty$6. The chain is fully general: no irreducibility or ergodicity is assumed, so transient, absorbing, and non-communicating states are allowed. The finite-horizon quadratic cost is

$t\to\infty$7

with $t\to\infty$8 and $t\to\infty$9 (Narváez et al., 18 Nov 2025).

The central move is to replace the stochastic state process by mode-indexed second moments

$[0,\infty)$0

These are collected into $[0,\infty)$1 in the Hilbert space

$[0,\infty)$2

equipped with inner product

$[0,\infty)$3

Because the definition uses unconditional expectations multiplied by indicators, it remains well-defined even when $[0,\infty)$4 (Narváez et al., 18 Nov 2025).

For a feedback law $[0,\infty)$5, the second moments satisfy a deterministic linear ODE on $[0,\infty)$6: $[0,\infty)$7 where

$[0,\infty)$8

The cost also becomes deterministic: $[0,\infty)$9 with $T_p$0 and $T_p$1. The stochastic finite-horizon LQR is therefore exactly equivalent to a deterministic optimal control problem on the same deterministic horizon $T_p$2 (Narváez et al., 18 Nov 2025).

This formulation is significant because the original system remains stochastic at the sample-path level, but the optimization problem no longer contains probability measures explicitly. In the paper’s synthesis, the horizon is deterministic, the reformulated problem is deterministic, and all stochasticity is encoded in the fixed coefficients of the deterministic equations (Narváez et al., 18 Nov 2025).

3. Visited states, reduced-order Riccati equations, and exact reduction

A distinctive feature of the general MJLS treatment is the projection onto the set of visited states

$T_p$3

If $T_p$4, then $T_p$5 for all $T_p$6, and consequently

$T_p$7

The second-moment trajectory therefore always lies in the subspace

$T_p$8

The orthogonal projection $T_p$9 that zeros out unvisited coordinates yields restricted operators $d^*$0 and $d^*$1, and the resulting reduction is exact rather than approximate (Narváez et al., 18 Nov 2025).

Dynamic programming on this reduced deterministic problem yields a reduced-order system of coupled Riccati differential equations. The value functional has the quadratic form

$d^*$2

and for each visited state $d^*$3,

$d^*$4

with terminal conditions

$d^*$5

Only $d^*$6 Riccati equations must be solved, rather than $d^*$7, so the deterministic horizon is paired with a mode-space reduction of dimension $d^*$8 instead of $d^*$9 (Narváez et al., 18 Nov 2025).

The optimal feedback law is the usual mode-dependent Riccati law restricted to visited states: $[0,T]$0 States outside $[0,T]$1 never occur on $[0,T]$2, so their gains are irrelevant. The framework naturally accommodates transient states, absorbing states, and multiple non-communicating recurrent classes; unreachable states become dead coordinates with $[0,T]$3 and $[0,T]$4 (Narváez et al., 18 Nov 2025).

The paper’s numerical examples support the exactness of this reduction. They report validation of the deterministic Riccati solution against Monte Carlo estimates with small relative error, approximately $[0,T]$5–$[0,T]$6, and include a satellite orbit control example with an absorbing failure mode (Narváez et al., 18 Nov 2025).

4. Hamilton–Jacobi structure and matrix-space foundations

The deterministic reformulation in (Narváez et al., 18 Nov 2025) is not merely computational. It is tied to a Hamilton–Jacobi–Bellman equation posed on the matrix Hilbert space $[0,T]$7: $[0,T]$8 with terminal condition

$[0,T]$9

The Fréchet gradient $T_p$0 is rigorously represented as an element of $T_p$1 via the Riesz–Fréchet representation theorem. This closes what the paper describes as an open conceptual gap in earlier work, where the quadratic value structure had been adopted as an ansatz without full derivation (Narváez et al., 18 Nov 2025).

This matrix-space HJB viewpoint clarifies what “deterministic horizon” means in the MJLS setting. The original control problem is stochastic because of Markovian jumps, but the horizon is fixed and deterministic, and the paper shows that over this interval the problem can be rewritten as deterministic dynamics plus deterministic integral and terminal costs in an augmented second-moment state (Narváez et al., 18 Nov 2025).

The result also situates the paper relative to classical continuous-time MJLS-LQR theory associated with Wonham, Mariton, and Costa–Fragoso. Classical treatments typically assume irreducibility or related communication hypotheses and sum over all modes. Here, the projection onto $T_p$2, the reduced-order Riccati system, and the Hilbert-space HJB formulation generalize that picture to arbitrary finite-state Markov chains (Narváez et al., 18 Nov 2025).

A common misunderstanding is therefore avoided by the deterministic reformulation: the phrase does not mean that the MJLS itself has ceased to jump. Rather, it means that on a fixed interval $T_p$3, optimal control can be carried out on a deterministic lifted system whose coefficients encode the original jump process (Narváez et al., 18 Nov 2025).

5. Deterministic horizons in broader control and game theory

Outside MJLS, the phrase retains its time-domain meaning but appears in several neighboring settings. In deterministic finite-horizon minimax impulse control, the horizon is the fixed interval $T_p$4, the value function is $T_p$5, and the problem is characterized by an Isaacs quasi-variational inequality with an explicit terminal condition at $T_p$6 (Asri, 2010). In deterministic income-consumption problems with stochastic interest-rate discounting, the optimization up to a finite deterministic time horizon $T_p$7 leads to a time-dependent HJB equation with terminal condition at $T_p$8, whereas the infinite-horizon versions are stationary and require different techniques (Eisenberg, 2016).

In deterministic receding-horizon control for finite transition systems under temporal-logic constraints, the horizon is a fixed prediction length $T_p$9. At each time step, the controller solves a finite-horizon optimization problem on a deterministic product automaton and applies only the first transition. Correctness with respect to an infinite LTL specification is enforced through terminal constraints derived from an energy function on the product automaton (Ding et al., 2012). A related framework for weighted deterministic transition systems with penalties and a specification of the form $d^*$0 combines an offline strategy minimizing expected average cumulative penalty per surveillance cycle with an online receding-horizon refinement over a finite horizon $d^*$1 (Svoreňová et al., 2013).

The infinite-horizon variant is equally prominent. Deterministic zero-sum impulse games with both players impulsive are posed on $d^*$2 with discount factor $d^*$3, and their value functions solve stationary HJBI quasi-variational inequalities (Asri et al., 2021). Deterministic differential games in which both players use continuous and impulse controls likewise produce stationary HJBI QVIs under Isaacs condition (Asri et al., 2021). In both cases, “infinite deterministic horizon” means that time is non-random and unbounded, so the PDE is stationary rather than backward in time (Asri et al., 2021, Asri et al., 2021).

A parallel strand formulates deterministic infinite-horizon optimal control through occupation measures and infinite-dimensional linear programming. Discrete-time discounted problems are rewritten as LPs over discounted occupational measures (Gaitsgory et al., 2017), while deterministic infinite-horizon discounted control in discrete time is linked to primal-dual LP formulations that yield necessary and sufficient optimality conditions and near-optimal controls (Gaitsgory et al., 2017). In continuous time, discounted deterministic optimal control with state constraints is relaxed to a measure LP equivalent to the original problem under mild assumptions, and polynomial data permit Lasserre-type moment/SOS relaxations (Kamoutsi et al., 2017). These papers use “deterministic horizon” in the sense of a fully deterministic time domain and deterministic system evolution, not in the sense of reduced-order finite-horizon reformulation.

The same terminology also appears in reinforcement learning and large-population games. "Deterministic Value-Policy Gradients" treats infinite-horizon discounted deterministic transitions and interprets the rollout length $d^*$4 in value-gradient estimation as a practical deterministic horizon for model-based backpropagation (Cai et al., 2019). "Truly Deterministic Policy Optimization" eliminates exploratory noise injection in essentially deterministic environments and reports strong performance on a robotic control task with a long horizon of $d^*$5 time steps (Saleh et al., 2022). In deterministic infinite-horizon LQ Nash games with many symmetric players, the $d^*$6 limit yields a mean-field-like deterministic aggregate, and the limiting stationary feedback approximates large finite-player equilibria (Papavassilopoulos, 2014).

6. Predictability, computable limits, and architecture-determined horizons

A different meaning appears in chaos theory. "How Far Can We Trust Chaos? Extending the Horizon of Predictability" formalizes the deterministic horizon as the horizon of predictability $d^*$7: the largest time or iteration up to which a numerically generated chaotic trajectory can be trusted, within a prescribed error tolerance, under finite-precision arithmetic (Angelidis et al., 19 Oct 2025). For local exponential divergence $d^*$8, the paper gives

$d^*$9

where $[0,T]$0 is initial numerical accuracy, $[0,T]$1 is acceptable error, and $[0,T]$2 is the maximal Lyapunov exponent. For the analytic computation of the Logistic map with mantissa $[0,T]$3 and acceptable precision $[0,T]$4, it derives

$[0,T]$5

In this usage, the system is mathematically deterministic, but the deterministic horizon is a numerical reliability boundary rather than a control interval (Angelidis et al., 19 Oct 2025).

The same paper shows that, by exploiting the conjugacy between the Logistic map and the Tent map and using exact rational arithmetic for the Tent map, one can construct chaotic time series whose horizon of predictability is effectively unlimited, with per-iteration error in the logistic coordinate bounded by

$[0,T]$6

This shifts the term from time-domain determinism to a computable reliability threshold for deterministic dynamics under numerical representation (Angelidis et al., 19 Oct 2025).

A further extension appears in trustworthy AI. "The Deterministic Horizon: Impossibility Results as Design Specifications for Trustworthy AI Systems" defines the Deterministic Horizon $[0,T]$7 for decoder-only transformers as the maximal effective reasoning depth up to which step-wise reasoning remains reliably better than random (Guo, 21 May 2026). If chain-of-thought uses $[0,T]$8 steps through a stack of $[0,T]$9 layers, the effective reasoning depth is

$T>0$0

The thesis defines

$T>0$1

It reports $T>0$2 across twelve transformer architectures, gives an architectural scaling law

$T>0$3

and fits the empirical form

$T>0$4

Past the horizon, the thesis states a super-exponential accuracy envelope

$T>0$5

and its fine-tuning impossibility theorem asserts that no architecture-preserving fine-tuning procedure can move the ceiling. The reported experiment on optimal-length traces yields only $T>0$6 percentage-point recovery at depth $T>0$7 (Guo, 21 May 2026).

These latter uses suggest a broader reinterpretation of the term. In control, the deterministic horizon is a prescribed time domain. In chaos and transformer reasoning, it becomes a computable boundary of reliable evolution imposed by numerical precision or architectural capacity. What remains constant across these usages is the role of the horizon as a sharp organizing boundary: before it, deterministic reasoning or control remains valid in the intended sense; beyond it, a different analytical regime begins (Angelidis et al., 19 Oct 2025, Guo, 21 May 2026).