Dynamic Reasoning Systems

Updated 4 December 2025

Dynamic Reasoning Systems are formal architectures that combine static structural constraints with dynamic adaptations for evolving reasoning processes.
They integrate logical, probabilistic, algorithmic, and learning-based methods into a unified framework to support rigorous runtime updates and analysis.
DRS employs metrics such as coherence, soundness, and completeness alongside iterative update rules to maintain system integrity and adaptability.

A Dynamic Reasoning System (DRS) is a formal architecture for modeling reasoning as a temporally evolving process, characterized by both static structural constraints and dynamic adaptation in response to new data, failures, or shifting objectives. DRS paradigms synthesize logical, probabilistic, algorithmic, and learning-based reasoning under a unified structural schema, facilitating rigorous analysis, principled runtime updates, and systematic comparison of inference architectures across diverse application domains (Nikooroo et al., 3 Aug 2025).

1. Formal Structural Definition

A general DRS instance is formally specified as a 5-tuple: $\mathcal R = (\Phi, \mathcal{E}, I, G, \mathcal{P})$ where:

$\Phi$ is the phenomena space (inputs, observations, or problem instances),
$\mathcal{E}$ is the explanation space (candidate solutions, hypotheses, outputs),
$I: \Phi \rightharpoonup \mathcal{E}$ is the inference map (encoder/solver/deduction),
$G: \mathcal{E} \rightharpoonup \Phi$ is the generation map (decoder/verifier/predictor),
$\mathcal{P} = \{\pi_1, \pi_2, \ldots, \pi_k\}$ is the principle base (constraints, axioms, admissibility rules).

Principles $\pi_j$ are domain-dependent predicates on $(\phi, e)$ or single arguments, specifying admissibility, consistency, or other structural requirements. This schema abstracts over logic-based, algorithmic, and learning-based processes, allowing broad unification.

2. Criteria for Internal Correctness

DRS analysis rests on three central formal criteria:

2.1 Coherence

A DRS is coherent if reconstructed input matches the original: $G(I(\phi)) = \phi \quad \forall \phi \in \mathrm{dom}(I)$ More generally, for a metric $d$ , it is $\varepsilon$ -coherent if $d(G(I(\phi)), \phi) \leq \varepsilon$ . Coherence diagnoses information loss or distortion through inference/generation cycles.

2.2 Soundness

Soundness requires all produced explanations respect prescribed principles: $I(\phi) \models \mathcal{P} \quad \forall \phi \in \Phi$ Formally, for every $\pi \in \mathcal{P}$ , $\pi(I(\phi))$ holds. Violation yields contradiction.

2.3 Completeness

Completeness is the guarantee that all inputs admit a valid solution: $\forall \phi \in \Phi \; \exists e \in \mathcal{E} \; \text{s.t.} \; I(\phi) = e \wedge e \models \mathcal{P}$ Absence of such $e$ signals incompleteness.

Fixed-point cases, rare in practice, occur when $G(I(\phi)) = \phi$ and $I(G(e)) = e$ everywhere.

3. Dynamic Update Rules and Evolution

DRS instances evolve over discrete time or iteration index $t$ : $\mathcal{R}_t = (\Phi, \mathcal{E}, I_t, G_t, \mathcal{P}_t)$ Key dynamic mechanisms include:

Repeatedly compose inference and generation: $e_{n+1} := I_t(G_t(e_n))$ or equivalently $F_t = I_t \circ G_t$ ; iteration $e_{n+1} = F_t(e_n)$ seeks stable fixed points $e^*$ .

3.2 Error-Driven Update

After each iteration, compute reconstruction error: $\delta(\phi) = \phi - G_t(I_t(\phi))$ Update $I_{t+1} = U_I(I_t, \{\delta(\phi)\})$ , $G_{t+1} = U_G(G_t, \{\delta(\phi)\})$ —e.g., via gradient/descent or rule revision—to minimize $\|\delta\|$ .

3.3 Principle Evolution

Edit the principle base in response to contradiction or performance feedback: $\mathcal{P}_{t+1} = \mu(\mathcal{P}_t, \text{feedback}_t)$ This may involve addition/removal of principles, creating epistemic drift or adaptation.

Illustrative Pseudocode

initialize I0, G0, P0
for t = 0,1,2,...
    for each φ in Φ:
        e0 = I_t(φ)
        repeat n times:
            e_{n+1} = I_t(G_t(e_n))
    compute errors δ(φ) = φ - G_t(e_n)
    update I_{t+1}, G_{t+1} via U_I, U_G on {δ(φ)}
    adjust P_{t+1} = μ(P_t, performance_metrics)
end

4. Failure Modes and Diagnosis

The framework supports precise identification of pathological states:

Failure Mode	Formal Condition	Systemic Consequence
Contradiction	$\exists \phi: I(\phi) \not\models \mathcal{P}$	Principle violation
Incompleteness	$\exists \phi$ no $e = I(\phi)$ with $e \models \mathcal{P}$	Input non-coverage
Non-Convergence	$\lim_{n\to\infty} F^n(e_0)$ does not exist	Iteration cycles/divergence
Deadlock/Inertia	$\forall \phi: I(\phi)=e^$ , $e^ \models \mathcal{P}$	Trivial output

These conditions enable resilience analysis and principled system comparison.

5. Comparative Framework and Design Relevance

Viewing any DRS as the $\mathcal{R} = (\Phi, \mathcal{E}, I, G, \mathcal{P})$ quintuple allows direct comparison of logic engines, optimization procedures, neural architectures, and hybrid reasoning pipelines under uniform criteria:

Coherence, soundness, and completeness become diagnostic metrics subject to continuous monitoring and targeted adjustment, rather than ad hoc, architecture-specific checks.
Dynamic update rules ( $I_{t+1}$ , $G_{t+1}$ , $\mathcal{P}_{t+1}$ ) enable iterative adaptation—spanning meta-reasoning, principle revision, and runtime learning—within the same language as static system analysis.
Failure diagnosis is systematic; recovery protocols such as principle revision or local inference map updates are supported by the formalism.
Facilitates the engineering of adaptive reasoning architectures that balance global consistency against local flexibility, support runtime failure recovery, and maintain cross-domain comparability.

6. Integration with Broader DRS Research

This abstraction encompasses the rich spectrum of DRSs, including:

Nonmonotonic predicate logics with explicit belief revision controllers (Schwartz, 2014, Schwartz, 2013),
Dynamic knowledge representation substrates with context-sensitive inheritance and traceable reasoning chains (Varey et al., 7 May 2025),
Probabilistic and stochastic network-based DRSs, supporting dynamic topology and sequential adaptation (Kjærulff, 2013, Provan, 2013),
Dynamic multimodal and chain-of-thought frameworks that tightly couple evolving state spaces to inference and generation modules (Ou et al., 22 May 2025),
Situation calculus-based probabilistic systems employing goal regression to reduce dynamic world belief updating to static analysis of initial states (Belle et al., 2013).

The formal $\mathcal{R}$ model both systematizes DRS anatomy and provides technical levers (criteria, update rules, failure diagnosis) for principled design, runtime monitoring, and theoretical investigation into reasoning architectures subject to dynamic change, constraint, and adaptation (Nikooroo et al., 3 Aug 2025).