Propositional Logic-Based Action Models

Updated 5 November 2025

Propositional logic-based action models are formal systems that combine Boolean logic with dynamic update mechanisms to represent discrete actions and evolving knowledge.
They integrate algebraic, modal, and categorical methods to define action preconditions, postconditions, and epistemic transitions with rigorous logical precision.
These models enable practical applications in planning, ontology integration, and learning, offering tools for automated deduction and advanced formal verification.

Propositional logic-based action models provide an abstract, algebraically and semantically precise machinery for modeling, representing, and reasoning about discrete actions, their effects, dynamic updates, and agent knowledge within a purely propositional (Boolean or finite-valued) setting. Drawing from contemporary research—including algebraic, categorical, epistemic, and dynamic frameworks—this article presents a comprehensive overview of the core methodologies and theoretical properties underpinning these models, with explicit reference to modern results in logic, dynamic epistemic logic, knowledge representation, and categorical semantics.

1. Algebraic and Logical Foundations

Propositional logic-based action models arise from the union of propositional logic and formalisms for describing actions, transitions, or updates. At their core, such models employ:

Propositional representations for states: Each propositional assignment (valuation) encodes a state.
Explicit action structures: Actions are modeled as transitions/transformations between propositional states, determined via logical, algebraic, or categorical mechanisms.
Product Update or Transition Systems: Many frameworks use product update (DEL), interpreted transitions (PDL), or algebraic updates (polynomial derivatives) as the underlying state-action mapping.

Algebraic Methods

A logic-algebraic approach interprets propositional formulas as polynomials in $\mathbb{F}_2[x_1, ..., x_n]$ , enabling fine-grained manipulation of knowledge bases and action pre/postconditions via algebraic operations. For example, the independence rule (derived from Boolean derivatives) provides a sound and refutationally complete inference mechanism for variable elimination and modular reasoning within propositional action models (Alonso-Jiménez et al., 2018). This algebraic approach links logical entailment and variable relevance to the vanishing of Boolean derivatives, thus supporting tools for modularity, reduction, and automated deduction.

Classical Propositional Dynamic Logic (PDL) and its many-valued generalizations encode structured actions (atomic, sequential, nondeterministic, iterative) as program expressions interpreted over Kripke models (Teheux, 2014, Sedlár, 2020). The transition relation for each action $a$ is specified, and modal formulas $[a]\phi$ describe universal properties over possible outcomes. Finitely-valued or FL-algebraic models generalize this approach, assigning degrees, weights, or costs to state transitions and permitting graded reasoning about structured actions.

2. Epistemic and Dynamic Action Models

In dynamic epistemic logic (DEL), action models are formal structures encoding the epistemic and ontic effects of actions:

Propositional action models: For a finite proposition set $P$ $P$ , an action model $A = (E, Q, pre, post)$ $A = (E, Q, p re, p os t)$ comprises:
- A finite set of events $E$
- An equivalence relation $Q$ (indistinguishability, for epistemic relevance)
- Precondition and postcondition maps assigning a propositional formula to each event
Product update: The execution of $A$ in state $s \subseteq P$ yields all $s'$ where $pre(e)$ holds in $s$ and $post(e)$ describes $s'$ (Bolander et al., 2015).

DEL frameworks analyze knowledge evolution via such action models, supporting public and private announcements, epistemic program specifications, and learning from observations. Importantly, deterministic finite propositional action models are finitely identifiable from observable state transitions, enabling theoretically grounded learning algorithms (Bolander et al., 2015).

3. Categorical, Coalgebraic, and Higher-Order Semantics

Modern trends have pursued the categorification of propositional logic-based action models:

Order-enriched and bicategorical models: Propositions, proofs, and rewrites are organized hierarchically as objects (formulas), 1-cells (proofs/actions), and 2-cells (reductions/rewrites), as realized in bicategories such as $\mathbf{Rel}$ , $\mathbf{Span}$ , and $\mathbf{Prof}$ . These models support symmetry (irrelevance of reduction order) and non-degeneracy (distinguishability at all dimensions), thus encoding not only that an action transforms a state, but also how such transformations can relate at higher (meta-)levels (Yamamoto, 2023).
Coalgebraic foundations: In stochastic, probabilistic, or game-theoretic settings, actions and programs correspond to morphisms in (co)algebraic categories. Each PDL program is canonically represented by an irreducible tree via rewrite rules, and mapped to a predicate lifting (natural transformation) describing the effect on state sets or distributions. Predicate lifting semantics, built around the Giry monad and Kleisli composition, underpins a robust framework for dynamic and stochastic transitions (Doberkat, 2011, Doberkat, 2014).

4. Expressiveness, Variations, and Phenomena

Propositional logic-based action models are highly expressive:

Structural expressiveness: Finitely-valued and FL-algebraic extensions encode not only possibility but degree—such as cost, weight, or probability—of action realization (Sedlár, 2020, Teheux, 2014).
Update expressivity: The update capabilities of action models (DEL) are formally incomparable to other update mechanisms, such as communication patterns, particularly on epistemic models—revealing limitations in simulating distributed information flows using traditional action model logic. However, action models subsume such patterns in interpreted systems equipped with history variables (Castañeda et al., 2023).
Forgetting and variable elimination: Uniform, non-uniform, or context-sensitive "forgetting" operators, modeled as dedicated action models or Kripke modifications, allow precise control over the agent’s knowledge and state description. Uniform forgetting can be axiomatized in action model logic, while non-uniform (dependent) variants exceed this expressivity (Fernández-Duque et al., 2015).

Table: Update Mechanism Expressivity

Update Mechanism	Update Expressivity on Epistemic Models	Update Expressivity on Interpreted Systems
Action Models (DEL)	Arbitrary; not subsumed by communication patterns	Strictly more expressive (with history)
Communication Patterns	Not subsumed by action models	Simulable by action models (with history)

5. Action Theories, Revision, and Nonmonotonic Reasoning

Propositional logic-based action theories are subject to evolution, revision, and contraction, driven by the discovery of new laws or exceptions:

Minimal change operators for contraction and revision are defined via a distance between Kripke models, delivering robust, postulate-compliant belief change mechanisms at the propositional action theory level (0811.1878).
Argumentation-theoretic frameworks explicitly represent assumptions (e.g., frame, qualification), minimize abnormality, and define plausible (minimal, conflict-free) sets, thus elegantly resolving frame, qualification, and ramification problems (Foo et al., 2011).
Reactive/proactive models: Model-theoretic and operational semantics are formally linked; operational semantics restricted to reactive rules only generate reactive models, i.e., all and only the models where every action is strictly motivated by rule antecedents satisfied by the current state (Kowalski et al., 2016).

6. Applications: Planning, Ontology Integration, and Learning

Concrete applications reflect the generality of these models:

Action languages: Formal action languages such as $\mathcal{BC}^+$ are based on propositional stable model semantics, directly encoding transition systems, supporting modern ASP constructs (choice, aggregates, constraints), and mapping propositional action descriptions to plans/paths via answer set computation (Babb et al., 22 Jun 2025).
Ontological action theories: Temporal answer set-based approaches encode domain knowledge with description logic (EL $^\bot$ ) constraints, ensuring action consistency via polynomial reductions and integrating non-deterministic actions and causal repair rules (Giordano et al., 2021).
Learning action models: DEL-inspired update learning procedures iteratively restrict initial action hypotheses using observed state transitions, achieving finite identifiability for deterministic propositional action models, and characterizing the limits of learnability in nondeterministic settings (Bolander et al., 2015).

7. Future Directions and Impact

Ongoing research in propositional logic-based action models extends into higher-dimensional categorical semantics, quantitative and stochastic models, logic-based learning, automated synthesis, and variable forgetting. The field increasingly bridges discrete action formalism, epistemic and dynamic phenomena, modular and distributed systems, and computational logic, ensuring continued theoretical as well as practical relevance in AI, multi-agent systems, formal verification, and computational knowledge representation.