Dynamical Semantics: Processes and Models

• Dynamical semantics is a framework that models meaning as dynamic state transitions, emphasizing processes over static truth conditions.
• It employs various formalisms—such as Kripke models, topological systems, and game semantics—to update information and capture context changes in logic, computation, and language.
• The approach highlights trade-offs between expressivity and tractability, unifying categorical, dynamic, and static models for analyzing epistemic, computational, and linguistic phenomena.

Dynamical Semantics, also referred to as dynamic semantics, encompasses a spectrum of formal frameworks that interpret semantic meaning not as a static attribute but as a process or transformation—typically in the form of state-space evolution, model update, or context change. These approaches are unified by the core idea that the meaning of linguistic expressions, programs, logics, or logical theories is characterized by their effect on some semantic “state,” often interpreted as temporal, epistemic, informational, or computationally “dynamic” in nature. This entry surveys foundational theories, technical principles, representative models, and key results in dynamical semantics, spanning logic, theoretical computer science, linguistics, and systems theory.

1. Foundations and Scope of Dynamical Semantics

The central notion in dynamical semantics is to model the evolution of some abstract information state under semantic or syntactic transformations. This state can range from Kripke models (as in dynamic epistemic logic, DEL), topological dynamical systems, game-based contexts (game semantics), context vectors (as in distributional semantics), or combinatorial state spaces (as in the semantics of logic programs or automata). In contrast to static semantics, where the meaning of an expression is typically a truth value, denotation, or extension in a fixed structure, dynamical semantics focuses on processes—sequences, trajectories, or updates—on a space of states.

This dynamical perspective has been fruitfully realized in diverse areas, notably:

• Dynamic and Epistemic Logics: DEL, public announcement logic, transition logic, and topological dynamical logics interpret meaning through model updates reflecting knowledge or informational changes (Klein et al., 2017, Kishida, 2017, Lerouvillois et al., 28 Oct 2025, Galliani, 2012, Fernández-Duque, 2016, Fernández-Duque et al., 2021, Montacute, 2023, Fernández-Duque, 2016).
• Programming Semantics and Game Theory: Game semantics, geometry of interaction, and interaction graphs model computation as interactive processes, tracking fine-grained steps or communication flows (Yamada et al., 2016, Seiller, 2016).
• Natural Language Semantics: Dynamic vector semantics and event semantics treat linguistic meaning as context change, with entailment encoded by (in)variance under context updates (Sadrzadeh et al., 2018, Qian et al., 2011).
• Systems and Numeration Theory: Some approaches directly treat semantic numeration or logic programs as explicit discrete-time dynamical systems (Chunikhin, 28 Jul 2025, Trinh et al., 7 Jan 2026).

2. Logical and Categorical Models of Dynamical Semantics

2.1. Dynamic Epistemic Logic and Topological Semantics

DEL models epistemic transitions by updating Kripke models via action models, with each update representing an epistemic or ontic event (Klein et al., 2017, Kishida, 2017). A central insight is that the set of pointed Kripke models modulo logical equivalence (i.e., the modal state space) can be topologized (Stone topology), turning action-model updates into continuous maps and making DEL update processes a subclass of discrete-time topological dynamical systems (Klein et al., 2017). Propositionally, every consistent formula can be made true in any information state via a suitable event [0610093].

Dynamic topological logic (DTL) and its intuitionistic variants further merge modal-temporal and topological modalities, defining semantics with respect to tuples $(X, \tau, f, V)$, where $f$ is a continuous transition map (Fernández-Duque, 2016, Fernández-Duque, 2016). Dynamic derivative logics, incorporating the Cantor derivative operator $d$, interpret modalities as limit-point or “derivative” dynamics, yielding completeness and finite model properties for classes of topological dynamical systems, especially over TD- or scattered spaces (Fernández-Duque et al., 2021, Montacute, 2023).

2.2. Game Semantics, Category Theory, and Process Models

Dynamic game semantics refines traditional (static) game semantics by distinguishing internal vs. external moves and introducing a hiding operator $H^d$ that mirrors small-step computation (e.g., $\beta$-reduction in the $\lambda$-calculus) at the semantic level (Yamada et al., 2016). This perspective gives rise to a cartesian closed bicategory (CCBoC) of games and strategies, capturing intensional and operational distinctions absent in classical denotational models. The dynamic semantics internalizes the reduction (normalization) process as part of the semantic structure, rather than external to it.

From a categorical viewpoint, DEL and related logics can be recast in terms of categories of relations, Kripke frames, and bounded morphisms, with action model updates formalized as colimit and pullback constructions (Kishida, 2017). Modalities correspond to adjoint functors, while product updates correspond to pushouts or colimits, and dynamic modalities (e.g., $[A,e]$, $\langle A,e\rangle$) are interpreted as universal or existential quantification along relations. This categorical infrastructure naturally generalizes to first-order logic, sheaf models, and quantifies over families or bundles of individuals.

3. Representative Formalisms and Technical Principles

3.1. Dynamic Modal and Topological Logics

Dynamic logics of dynamical systems extend modal logic with program constructs or system evolution, often for hybrid or continuous systems, using first-order languages and temporal/modal operators for reasoning about reachability, invariance, controllability, and liveness (Platzer, 2012). Key results concern compositionality of program constructs, well-foundedness of solutions, soundness and completeness of proof calculi, and fixpoint characterizations of loop constructs. Topological semantics for modal logic via the Cantor derivative $d$ provides modal logics (e.g., wK4C, K4C, GLC) sound and complete for classes of (respectively) arbitrary, TD, and scattered topological dynamic systems, each exhibiting the finite model property and explicit axiom systems (Fernández-Duque et al., 2021, Montacute, 2023).

3.2. Update, Trap, and Transition Semantics

For logic programs, the trap space (dynamical) semantics interprets programs as discrete-time state-transition systems on the space of interpretations, analyzing behavior in terms of fixed points, cycles (oscillations), and minimal trap sets (attractors) (Trinh et al., 7 Jan 2026). Each update operator (e.g., stable-model $F_P$ and supported-model $T_P$ updates) induces a transition graph whose structure explicates the relationship between stable, supported, regular, and $L$-stable models. Minimal trap spaces correspond to classical model-theoretic constructs, showing that the full spectrum of logic programming semantics can be unified within a coherent dynamical picture.

Transition Logic (TL) and its variants reformulate dependence and game logics via reachability in transition systems, making the dynamic aspect of semantics—particularly in imperfect information or team semantics—explicit syntactically (Galliani, 2012).

4. Application to Linguistics, Computation, and Systems

4.1. Context-Change and Vector Semantics

In natural language semantics, dynamic approaches (e.g., context change potentials (CCPs)) model the meaning of sentences as context updates rather than truth conditions. In dynamic vector semantics, contexts are encoded as real-valued matrices or higher-order tensors; the application of a sentence is a context function $$C : \text{Context} \times \text{Sentence} \to \text{Context}$$ (Sadrzadeh et al., 2018). Admittance (entailment) is characterized by context invariance under an update. Similarly, event-based compositional dynamic semantics employs continuation-passing style $\lambda$-calculus with explicit event variables and dynamic glue to model discourse structure, anaphora, and rhetorical relations (Qian et al., 2011).

4.2. Semantic Numeration and Discrete-Time Systems

Semantic numeration systems recast numeration and symbolic computation as concrete discrete-time dynamical systems, where the state is a “cardinal semantic multeity” (a vector of entity counts), and updates are determined by “cardinal semantic operators” represented in a configuration matrix, encoding the system topology, parameters, and update structure (Chunikhin, 28 Jul 2025). The resulting systems are subject to standard systems-theoretic analyses (stability, controllability, observability), bridging semantic theory with control and systems engineering.

5. Complexity, Decidability, and Model-Theoretic Properties

A recurring theme in dynamical semantics is the trade-off between expressivity and tractability. For example, in intuitionistic temporal logic of dynamical systems (${ITL}^c$), validity is decidable via a finite quasimodel construction (at a super-exponential bound), allowing for the expression of deep dynamical properties like minimality and Poincaré recurrence (Fernández-Duque, 2016). In dynamic topological and derivative logics, completeness and finite model properties are achieved for restricted classes (notably scattered spaces), giving finite, explicit axiomatizations for nontrivial dynamical phenomena (Fernández-Duque et al., 2021, Montacute, 2023).

For dynamic epistemic logic, the general decision problem for properties such as existence of limit cycles or nontrivial recurrence is undecidable when updates are sufficiently expressive (static-non-Boolean or Boolean-non-static), often directly related to Turing-completeness within the update mechanisms (Klein et al., 2017). Finite-model and filtration techniques are used to transfer results from Kripke to topological semantics.

6. Structural and Methodological Innovations

The integration of dynamical and static semantics via categorical collapse (e.g., triskell → weighted relation → quantitative coherence space) establishes a principled correspondence between operationally “dynamic” models (process-like, stepwise, path-sensitive) and denotationally “static” models (matrix transformations, coherence spaces), preserving computational invariants and enabling cross-fertilization between the approaches (Seiller, 2016).

Dynamic proof theory, as in the calculus of dynamic hypersequents for public announcement logic (PAL), realizes the dynamic nature of semantic model updates at the level of syntactic proof, endowing sequent calculi with explicit mechanisms for tracking model transitions, while maintaining classical properties (admissibility, invertibility, cut-elimination) (Lerouvillois et al., 28 Oct 2025).

7. Open Questions and Ongoing Directions

Despite major advances in the modeling of dynamical phenomena, several open questions persist:

• The search for natural, finite, or recursively enumerable axiomatizations for expressively rich dynamical logics (e.g., full ${ITL}^c$) remains incomplete (Fernández-Duque, 2016).
• The complexity landscape for dynamic model-checking (especially over infinite or continuous state spaces) is only partly characterized, with the frontier often lying along the boundary of Turing-completeness and decidability (Klein et al., 2017).
• Bridging modal/epistemic update logics with continuous or hybrid system theory, as in dynamic logics of dynamical systems, is an area of ongoing research, especially for the verification of cyber-physical and safety-critical systems (Platzer, 2012).
• Dynamical semantics continues to inspire new categorical, algebraic, and computational models, where the interplay of operational reduction, interaction, and context change is central.

Dynamical semantics thus forms a rich, interdisciplinary landscape, providing deep unification and sharp analytical tools for the study of meaning, information, computation, and evolution within formal systems.