Modal Refinement (MoRe)

• Modal Refinement (MoRe) is a set of closely related semantic and proof-theoretic notions that enable controlled structural changes while preserving essential modal properties.
• It applies across various domains including Kripke models, refinement modal logic, and specification theories, unifying behavioral equivalence via weakened bisimulation.
• Practical insights include advanced model checking techniques, tableau-based complexity analysis, and abstraction refinement methods useful in verification frameworks.

Modal Refinement (MoRe) denotes a cluster of closely related notions in modal logic, specification theory, verification, and proof theory, all centered on controlled change of system structure while preserving designated modal information. In the literature considered here, it appears as a relation between Kripke models and transition systems, as a quantificational device in Refinement Modal Logic (RML), as a specification-theoretic preorder for may/must formalisms, as a notion of structural or hierarchical refinement, and as a proof-theoretic transformation from labelled calculi to refined or nested systems (Bozzelli et al., 2012). More recent work also uses modal operators to reason directly about abstraction refinement, interpreting $\lozenge$ as “there is a refinement, in which …” and $\Box$ as “in all refinements, …” (Piribauer et al., 9 Jan 2026).

1. Kripke-style refinement and its generalizations

In the Kripke-model setting, refinement is presented as a weakening of bisimulation. Formally, model $M'$ is a refinement of $M$ if there exists a non-empty relation $\mathcal{R} \subseteq W \times W'$, with $M = (W, R, V), M' = (W', R', V')$, such that: for all $(s, s') \in \mathcal{R}$ one has $V(s) = V'(s')$; and for all $s \in W, s', t' \in W'$ with $(s, s') \in \mathcal{R}$ and $\Box$0, there is $\Box$1 such that $\Box$2 and $\Box$3. This preserves propositional facts and the “back” condition, but omits the “forth” condition. Informally, a refined model can lose possible transitions, but not propositional information (Achilleos et al., 2013).

This perspective is made explicit in RML, where a refinement is “like a bisimulation, except that from the three relational requirements only atoms' andback' need to be satisfied.” The same family of ideas also admits dual and mixed variants. In the 2025 treatment of simulation and refinement, simulation keeps atoms and forth, while refinement keeps atoms and back, and both are interpreted as quantification over information change in multi-agent systems (Ditmarsch et al., 27 Nov 2025). In the 2022 covariant-contravariant setting, CC-refinement generalizes bisimulation, simulation, and refinement by partitioning actions into covariant actions $\Box$4, contravariant actions $\Box$5, and bivariant actions $\Box$6, with forth required for $\Box$7 and back required for $\Box$8 (Xing, 2022).

A common source of confusion is to treat “modal refinement” as a single invariant definition across all subfields. The surveyed literature instead uses the term for a family of structurally related notions: all are refinement-like weakenings or reorganizations of behavioural equivalence, but they differ in whether they quantify over models, compare specifications, constrain action polarities, or transform proof systems. This suggests that “MoRe” is best understood as a semantic and proof-theoretic pattern rather than a single formalism.

2. Refinement Modal Logic and quantified reasoning about change

Refinement Modal Logic extends standard modal logic with refinement quantifiers. One presentation uses

$\Box$9

where $M'$0 means that there exists a refinement of the current model where $M'$1 holds, and $M'$2 quantifies over all refinements (Achilleos et al., 2013). In the earlier multi-agent presentation, the operator $M'$3 acts as a universal quantifier over all $M'$4-refinements of a pointed model, and the logic is given a sound and complete axiomatization (Bozzelli et al., 2012).

The semantics are direct. For a pointed model $M'$5,

$M'$6

The universal operator is defined analogously. These quantifiers allow reasoning about the truth of formulas under possible structural changes of the underlying transition system (Achilleos et al., 2013).

RML also supports reduction-style axiomatizations. In the 2012 account, propositional formulas are invariant under refinement, with axioms such as $M'$7 and $M'$8, together with interaction axioms for modal operators and cover modalities. The same paper presents an extension to the modal $M'$9-calculus and an axiomatization for the single-agent version of this logic (Bozzelli et al., 2012). In the 2022 CC-refinement modal $M$0-calculus, the language $M$1 extends ordinary modal $M$2-calculus with $M$3, and the resulting system is sound, complete, and decidable. A key technical point is that every CCRML$M$4 formula is provably equivalent to a modal $M$5-calculus formula without CC-refinement quantifiers (Xing, 2022).

The same quantificational pattern also appears in logics combining refinement, simulation, and mutual factual ignorance. There, reduction-based, modular axiomatizations eliminate the dynamic modalities back to base modal logic, while the origin modality refers to a model of mutual factual ignorance in which no agent knows any factual information and this ignorance is common knowledge (Ditmarsch et al., 27 Nov 2025).

3. Complexity, satisfiability, model checking, and succinctness

The computational profile of refinement logics is highly sensitive to the fragment under study. A central result is that the satisfiability problem for the existential fragment $M$6 is PSPACE-complete. The 2013 paper establishes a tight PSPACE upper bound by introducing a new tableau system, thereby settling the open problem posed by Bozzelli, van Ditmarsch and Pinchinat (Achilleos et al., 2013).

That tableau method uses labels of the form $M$7, where $M$8 encodes the model and $M$9 encodes the state within that model. The key adaptation is that the tableau must handle not only the evolution of states within a model but also of entire models related by refinement. The paper proves soundness and completeness, and shows that in any branch the model-prefix length is at most the nesting depth of $\mathcal{R} \subseteq W \times W'$0, while the state-prefix length is at most the nesting depth of modal operators. This yields a non-deterministic polynomial-space algorithm (Achilleos et al., 2013).

The same work also shows that model checking for RML is PSPACE-complete, even for formulas with a single quantifier, and that the PSPACE algorithm for $\mathcal{R} \subseteq W \times W'$1 closes the previous complexity gaps for logic fragments with a fixed number of quantifier alternations. By contrast, the 2012 treatment reports that RML in the single-agent case is EXPSPACE-complete, while refinement $\mathcal{R} \subseteq W \times W'$2-calculus is non-elementary because it can encode quantified propositional temporal logic. It also states that RML is doubly exponentially more succinct than basic modal logic, and refinement $\mathcal{R} \subseteq W \times W'$3-calculus is doubly exponentially more succinct than modal $\mathcal{R} \subseteq W \times W'$4-calculus (Bozzelli et al., 2012).

These results collectively show that refinement quantification can preserve strong axiomatic structure while sharply increasing decision complexity. A plausible implication is that the expressive payoff of quantifying over structural change is accompanied by different algorithmic regimes for existential, alternation-bounded, and fixed-point extensions.

4. Modal refinement in specification theories

In specification theory, modal refinement usually formalizes when a more concrete system correctly realizes a more abstract specification by preserving must behaviour and restricting may behaviour. For Modal Transition Systems, modal refinement represents a step of the design process in which some optional behaviour is discarded while other optional behaviour becomes necessary. Yet modal refinement is not complete with respect to implementation sets: it can happen that $\mathcal{R} \subseteq W \times W'$5 even though $\mathcal{R} \subseteq W \times W'$6 (Basile, 2023).

This incompleteness motivates richer specification models and alternative refinement relations. For Boolean MTS and parametric MTS, modal refinement is reduced to QBF, with BMTS modal refinement being $\mathcal{R} \subseteq W \times W'$7-complete and PMTS modal refinement being $\mathcal{R} \subseteq W \times W'$8-complete. The same work shows that modal refinement implies thorough refinement, but not conversely in general, and that for deterministic BMTS the two coincide. It also gives improved upper bounds for thorough refinement: NEXPTIME for BMTS and 2-EXPTIME for PMTS (Křetínský et al., 2013).

A related line develops Non-reducible Modal Transition Systems (NMTS), a subset of MTS in which states reachable by the same sequence cannot disagree on the modality of equally labelled outgoing transitions. A novel refinement relation $\mathcal{R} \subseteq W \times W'$9 is proposed for NMTS, with the aim of aligning refinement more closely with implementation-set inclusion. The same source notes, however, that soundness issues remain and further adjustments are a subject of ongoing research (Basile, 2023).

For interface theories, IR-MIA introduces modal refinement that distinguishes between obligatory and allowed output behaviours, as well as between implicitly underspecified and explicitly forbidden input behaviours. In that setting, modal refinement is a preorder over weak may-input-enabled IR-MIA, is preserved under demonic input completion, and under suitable restrictions supports compositionality of modal-irioco with respect to parallel composition and decompositionality via quotient (Luthmann et al., 2016).

For quantitative systems, Abstract Probabilistic Timed Automata define a modal specification language for combined probabilistic timed systems. The paper distinguishes semantic thorough refinement from syntactic strong and weak refinement, establishes the hierarchy

$M = (W, R, V), M' = (W', R', V')$0

and shows that conjunction is a greatest lower bound with respect to weak refinement, while weak refinement is a precongruence for parallel composition (Han et al., 2013).

5. Structural, hierarchical, and proof-theoretic forms of MoRe

A distinct but related use of refinement appears in logical specification formalisms. For the modal $M = (W, R, V), M' = (W', R', V')$1-calculus, a new notion of structural refinement is introduced as a sound abstraction of logical implication. Using translations between the modal $M = (W, R, V), M' = (W', R', V')$2-calculus and disjunctive modal transition systems, the paper shows that these formalisms are structurally equivalent and transfers composition and quotient from DMTS to the modal $M = (W, R, V), M' = (W', R', V')$3-calculus. With these operations, modal $M = (W, R, V), M' = (W', R', V')$4-calculus expressions modulo mutual refinement form a commutative, residuated lattice (Fahrenberg et al., 2014).

Hierarchical and layered refinement arise in $M = (W, R, V), M' = (W', R', V')$5-layered transition systems. The 2016 hybrid-modal framework defines layered refinement as the existence of a total $M = (W, R, V), M' = (W', R', V')$6-layered simulation and hierarchical refinement as the corresponding notion for hierarchical models. In this setting, positive formulas are preserved under refinement, and a modal invariance theorem states that bisimilar tuples satisfy the same strict $M = (W, R, V), M' = (W', R', V')$7-layered formulas (Madeira et al., 2016).

The term “Modal Refinement (MoRe)” is also used explicitly for a proof-theoretic transformation. The 2021 thesis defines MoRe as the transformation from a labelled system with structural rules encoding frame constraints to a refined system, labelled or nested, where those constraints are enforced by propagation side-conditions on logical rules. The method systematically replaces structural rules by propagation or reachability rules, preserves soundness, completeness, cut-admissibility, and invertibility, and yields systems with stronger adherence to the subformula property and improved suitability for automated reasoning (Lyon, 2021).

These proof-theoretic uses differ from Kripke-style or specification-style modal refinement, but the shared theme is again controlled structural change under preservation of modal consequences. This suggests a broad methodological unity: refinement is used either to compare models or to compress the proof theory that represents them.

6. Abstraction refinement as a modal frame

A recent development treats abstraction refinement itself as the accessibility relation of a modal logic. In this setting, worlds are transition systems, accessibility is the refinement relation $M = (W, R, V), M' = (W', R', V')$8, and atomic propositions stand for CTL-definable system properties. The modal operator $M = (W, R, V), M' = (W', R', V')$9 is interpreted as “There exists a refinement in which $(s, s') \in \mathcal{R}$0 holds,” and $(s, s') \in \mathcal{R}$1 as “In all refinements, $(s, s') \in \mathcal{R}$2 holds” (Piribauer et al., 9 Jan 2026).

The paper investigates three scenarios. For all finite abstractions of a fixed system, $(s, s') \in \mathcal{R}$3 is always valid, and there exists at least one $(s, s') \in \mathcal{R}$4 for which $(s, s') \in \mathcal{R}$5. For all abstractions of a fixed system, $(s, s') \in \mathcal{R}$6 is always valid, and there exists at least one $(s, s') \in \mathcal{R}$7 for which $(s, s') \in \mathcal{R}$8. For the class of all transition systems, the lower bound is $(s, s') \in \mathcal{R}$9 and the upper bound is $V(s) = V'(s')$0 (Piribauer et al., 9 Jan 2026).

Not every modal principle is valid in this setting. The axiom $V(s) = V'(s')$1 is always valid because every system refines itself, but $V(s) = V'(s')$2 is not always valid. The paper also develops control statements—pure buttons, switches, pure weak buttons, restricted switches, and decisions—to obtain upper bounds. These statements are used to simulate frame classes such as pre-Boolean algebras, lollipop frames, and finite partial functions posets (Piribauer et al., 9 Jan 2026).

This line has direct implications for verification. The same source states that universal properties (ACTL) are preserved under refinement, whereas existential and mixed properties may not be preserved, and relates this to why typical abstraction refinement verification frameworks are restricted to universally quantified properties. A common misconception is therefore that refinement uniformly preserves arbitrary temporal properties; the modal-logic bounds show that preservation is constrained by the structure of the refinement frame (Piribauer et al., 9 Jan 2026).