Multi-Modal Logics of Bounded Density

• Multi-modal logics of bounded density are defined over Kripke frames with several accessibility relations that enforce density through intermediate refinement steps.
• They employ a window-based decision procedure using consistent classical saturations to mirror density axioms and achieve PSPACE and para-PSPACE complexity results.
• These techniques isolate modal depth as a critical resource, offering efficient and scalable methods for satisfiability checking in both fixed and parameterized modal settings.

Multi-modal logics of bounded density are modal systems interpreted over Kripke frames equipped with several accessibility relations whose interaction enforces a bounded form of density. In the formulation presented for grammar logics, one fixes $\pi \in \mathbb{N}$, uses modalities $[i]$ indexed by $\Pi=\{0,1,\dots,\pi\}$, and requires that every $R_i$-edge can be refined through an intermediate point by an $R_i$-step followed by an $R_{i+1}$-step. Their satisfiability problem admits a tableau-like decision procedure based on finite windows, is PSPACE-complete for fixed $\pi$, and in the monomodal setting yields a para-PSPACE analysis when modal depth is treated as a parameter. A related line of work studies $n$-dense modal logics characterized by axioms of the form $\Box^n p \rightarrow \Box p$ and extends the window technique to recursive windows, obtaining para-PSPACE upper bounds for fixed-$n$ and fixed-$[i]$0 multi-modal variants (Gasquet, 20 Jul 2025, Gasquet, 13 Apr 2026).

1. Formal setting: frames, language, and semantics

The bounded-density framework fixes $[i]$1 and sets

$[i]$2

A $[i]$3-frame is a pair

$[i]$4

where $[i]$5 and each $[i]$6. Such a frame is $[i]$7-dense, or of “density at most $[i]$8,” if for every $[i]$9 and all $\Pi=\{0,1,\dots,\pi\}$0,

$\Pi=\{0,1,\dots,\pi\}$1

The class of all such frames is written $\Pi=\{0,1,\dots,\pi\}$2 (Gasquet, 20 Jul 2025).

The corresponding multi-modal language is generated by

$\Pi=\{0,1,\dots,\pi\}$3

with $\Pi=\{0,1,\dots,\pi\}$4 and $\Pi=\{0,1,\dots,\pi\}$5. Standard abbreviations are used:

$\Pi=\{0,1,\dots,\pi\}$6

and the dual modality is defined by

$\Pi=\{0,1,\dots,\pi\}$7

The modal depth $\Pi=\{0,1,\dots,\pi\}$8 is the maximal nesting of box/diamond operators (Gasquet, 20 Jul 2025).

A model is

$\Pi=\{0,1,\dots,\pi\}$9

where the frame is $R_i$0-dense and $R_i$1 is a valuation. Truth is given by the usual clauses for atoms, Boolean connectives, and modal operators:

$R_i$2

$R_i$3

A formula is satisfiable if $R_i$4 in some $R_i$5-dense model (Gasquet, 20 Jul 2025).

2. Density axioms and related modal families

The bounded-density condition is mirrored syntactically by the axiom

$R_i$6

which is the condition explicitly enforced in the window-based decision procedure. In the tableau presentation, obligations produced by a formula at modality $R_i$7 are propagated along a finite $R_i$8-slice and recursively discharged one level higher via $R_i$9 (Gasquet, 20 Jul 2025).

The monomodal density logic appears as a specialization. When $R_i$0, the logic is described as

$R_i$1

and the well-studied monomodal density logic $R_i$2 arises as the special case in which modalities $R_i$3 and $R_i$4 coincide (Gasquet, 20 Jul 2025). This makes precise that “bounded density” is not merely a restatement of a single binary relation being dense; rather, the multi-modal presentation organizes density through a hierarchy of relations $R_i$5.

A related family is given by $R_i$6-dense modal logics. In the monomodal setting, $R_i$7 is the smallest normal modal system containing

$R_i$8

and

$R_i$9

and it is complete for frames satisfying

$R_{i+1}$0

The same idea extends to the multi-modal language with modalities $R_{i+1}$1, yielding systems $R_{i+1}$2 with axioms

$R_{i+1}$3

complete for frames satisfying $R_{i+1}$4 for each $R_{i+1}$5 (Gasquet, 13 Apr 2026). This related development does not identify the two families, but it shows that bounded-density techniques interact naturally with broader “reduction” or density axioms.

3. Consistent classical saturations and finite windows

The bounded-density decision procedure is organized around consistent classical saturations (CCS’s). Given a finite set of formulas $R_{i+1}$6, one considers the set $R_{i+1}$7 of all finite sets $R_{i+1}$8 such that $R_{i+1}$9, $\pi$0 is classically consistent, and $\pi$1 is saturated under the usual decomposition conditions for conjunction, negated conjunction, double negation, and modal formulas $\pi$2. Each $\pi$3 is called a CCS of $\pi$4 (Gasquet, 20 Jul 2025).

The central combinatorial object is a finite window. Fix a monotone “size-control” function $\pi$5 on CCS’s, for example $\pi$6 or $\pi$7. Let $\pi$8 be CCS’s, let $\pi$9, and let $n$0. A $n$1-window for $n$2 is a pair

$n$3

If $n$4, then $n$5 is the empty window. If $n$6, then:

• $n$7 is a sequence of CCS’s satisfying, for all $n$8,

$n$9

and at the end

$\Box^n p \rightarrow \Box p$0

• $\Box^n p \rightarrow \Box p$1 is a sequence where each $\Box^n p \rightarrow \Box p$2 is a $\Box^n p \rightarrow \Box p$3-window for the pair $\Box^n p \rightarrow \Box p$4.

Here

$\Box^n p \rightarrow \Box p$5

and for any set $\Box^n p \rightarrow \Box p$6 of formulas, $\Box^n p \rightarrow \Box p$7 is the collection of its CCS’s (Gasquet, 20 Jul 2025).

Intuitively, $\Box^n p \rightarrow \Box p$8 tracks a finite slice of an $\Box^n p \rightarrow \Box p$9-chain and $n$0 recurses at the next modality. The formal role of the condition

$n$1

is to mirror the density axiom

$n$2

so that obligations generated at level $n$3 are either carried directly from $n$4 or passed upward through $n$5 and an $n$6-chain (Gasquet, 20 Jul 2025).

4. Tableau-like decision procedures and recursive windows

The finite-window algorithm begins from an initial CCS $n$7 for a formula $n$8. It first verifies that $n$9. Then, for each diamond-formula $[i]$00, it guesses a CCS

$[i]$01

and a $[i]$02-window $[i]$03 for $[i]$04, with $[i]$05 chosen sufficiently large, and checks recursively that every $[i]$06 in $[i]$07 and every subwindow $[i]$08 in $[i]$09 is itself satisfiable. The procedure stores only one window in memory at a time and recurses on CCS’s of lower modal depth (Gasquet, 20 Jul 2025).

The related $[i]$10-dense analysis refines this method through recursive windows. There, a window is described as a self-contained finite piece of what would be a possibly infinite $[i]$11-tableau beneath a node labelled by a classical saturated set $[i]$12 of formulas. Windows enforce the density axiom by explicitly displaying all the required intermediate $[i]$13-successors. A $[i]$14-window for $[i]$15 is, when $[i]$16, a sequence of $[i]$17-saturated sets

$[i]$18

together with, between each successive pair, a sub-window of type $[i]$19 for enforcing the constraint $[i]$20; when $[i]$21 it is just empty. A continuation of a window is another window whose first $[i]$22 slices coincide pointwise with the last $[i]$23 slices of the first window, capturing the idea that a displayed pattern can be pumped indefinitely (Gasquet, 13 Apr 2026).

Five lemmas are identified as critical in the recursive-window analysis:

• Lemma 4.1: if $[i]$24 is a $[i]$25-continuation of $[i]$26, then they can be spliced to produce a $[i]$27-window for the same root pair.
• Lemma 4.2: from a sufficiently long $[i]$28-window one can use the pigeonhole principle to obtain a $[i]$29-window.
• Lemma 4.3: from an $[i]$30-dense model satisfying $[i]$31 at $[i]$32 and a direct successor $[i]$33 satisfying $[i]$34, one can read off an infinite $[i]$35-window for $[i]$36 with satisfiable slices.
• Lemma 5.1: if $[i]$37 is frame-satisfiable then the routine $[i]$38 returns “yes.”
• Lemma 5.2: if $[i]$39 returns “yes” then $[i]$40 is satisfiable in some $[i]$41-dense model, obtained by gluing together a tree of small model-pieces certified by the windows (Gasquet, 13 Apr 2026).

Taken together, these results show that finite windows are not merely a proof-search heuristic. They provide a compact representation of the chains required by density, and recursive windows make explicit how local witnesses can be extended, reused, and pumped.

5. Complexity landscape

For the multi-modal logic of bounded density with fixed $[i]$42, the satisfiability problem is PSPACE-complete (Gasquet, 20 Jul 2025). The upper bound is obtained from three facts stated in the proof sketch:

• each recursive call works with one CCS of size $[i]$43 and one window of size polynomial in $[i]$44;
• recursion depth is at most $[i]$45;
• at each stage one stores only the current CCS, the current window, and a counter $[i]$46.

Hence the nondeterministic search uses space

$[i]$47

and Savitch’s theorem yields a deterministic PSPACE algorithm. PSPACE-hardness follows because when $[i]$48 the logic is plain $[i]$49, which is PSPACE-hard via the usual reduction from QBF, and the bounded-density logic conservatively extends $[i]$50 (Gasquet, 20 Jul 2025).

The monomodal case has a more refined parameterized classification. When $[i]$51, the same window-based algorithm applies, but the recursion depth is bounded by $[i]$52 alone, and the number of windows of modal-degree $[i]$53 is at most $[i]$54. The search can therefore be done in space

$[i]$55

for some computable $[i]$56, that is, in para-PSPACE when modal depth is viewed as a fixed parameter (Gasquet, 20 Jul 2025).

For fixed $[i]$57, the satisfiability problem for $[i]$58, parameterized by the modal depth $[i]$59 of the input formula, lies in para-PSPACE. More concretely, there is an algorithm that on input $[i]$60 of modal depth $[i]$61 decides satisfiability in space

$[i]$62

Thus once $[i]$63 is fixed, the problem is in PSPACE. The same analysis extends to the fixed-$[i]$64 multi-modal case: nothing in the complexity analysis depends on having only one modality, so the final para-PSPACE upper bound holds equally for the fixed-$[i]$65 multi-modal case (Gasquet, 13 Apr 2026).

A common point of confusion is the relation between the PSPACE-completeness theorem and the para-PSPACE results. They concern different parameterizations, and in part different families: PSPACE-completeness is stated for the fixed-$[i]$66 bounded-density grammar logics, whereas para-PSPACE isolates the effect of bounded modal depth in the monomodal bounded-density case and, in related work, in fixed-$[i]$67 and fixed-$[i]$68 $[i]$69-dense logics.

6. Example, interpretation, and significance

A minimal example in the monomodal setting takes

$[i]$70

One starts with a CCS $[i]$71 of $[i]$72, for instance $[i]$73. The unique diamond-formula is $[i]$74 at $[i]$75, so the algorithm chooses

$[i]$76

hence $[i]$77. With $[i]$78, one takes

$[i]$79

with $[i]$80, so a valid CCS is $[i]$81. No subwindows are needed because at $[i]$82 the window is trivial, and one obtains

$[i]$83

Both $[i]$84 and $[i]$85 are satisfiable, so $[i]$86 is declared satisfiable. Model-theoretically, one can take

$[i]$87

so that $[i]$88 via $[i]$89 and $[i]$90-density is witnessed by the intermediate $[i]$91 (Gasquet, 20 Jul 2025).

The significance of the window method is stated in algorithmic as well as semantic terms. For bounded-density grammar logics, the finite-window method gives a uniform PSPACE decision procedure for all fixed $[i]$92 and provides a clear semantic and algorithmic framework for logics defined by “reduction” or “density” axioms (Gasquet, 20 Jul 2025). In the $[i]$93-dense setting, recursive windows explain directly how the bounded-density axiom $[i]$94 is enforced: every direct $[i]$95-edge must come equipped with a little $[i]$96-long chain sitting in the window, those chains never grow beyond exponentially many saturated nodes, and because one only ever looks “$[i]$97 steps” deep, only parameter-bounded polynomial space is required (Gasquet, 13 Apr 2026).

A plausible implication is that window-based proof search isolates modal depth as the structurally decisive resource in these bounded-density settings. The fixed-$[i]$98 PSPACE bound shows that the global satisfiability problem remains within classical polynomial space, while the para-PSPACE refinements show that once modal nesting is controlled, the remaining proof search can be organized within a parameterized space discipline.