Topological Logics of Path-Reachability

Abstract: The topological semantics of modal logic has been an active area of research ever since their introduction in the 1940s, with attention shifting in recent years from standard unimodal logic to more expressive frameworks. In particular, an Until-like path-reachability modality has recently been studied in Bezhanishvili et al. (2024) in polyhedral semantics; this paper investigates its topological counterpart. Focusing on the language combining said modality with the classical Cantor derivative modality, we exhibit an axiomatic system sound and complete both for the class of T1 topologies and for the class of all metric spaces, and establish its decidability. We also axiomatize the logic of all topological models in a weaker language obtained by substituting the closure modality for the Cantor derivative. To prove our results, we introduce an equivalent neighborhood-like semantics allowing for the finite model property.

• The paper axiomatizes and shows the decidability of path-reachability in topological spaces using modalities through topical calculus TRL and TLR_,c
• The method demonstrates completeness and decidability outcomes for T1 topologies and metric spaces, bridging canonical flanked Kripke models
• Under Neumann path-reachability neighborhood semantics enables $T_1$ topological space $d
• The applicable axiomation provided reifies completeness of differentiability
• The disaggregated modular axionization shows completeness of interpolative analytic topologies.

Overview

This paper studies modal logics interpreted over topological spaces through the lens of a binary path-reachability modality $\gamma(\varphi,\psi)$, which holds at a point $x$ when there exists a continuous path starting at $x$, remaining within the region defined by $\varphi$ at all intermediate points, and ending at a point satisfying $\psi$. The modality was previously studied in polyhedral and Alexandroff semantics, where its axiomatizations were known. The contribution here is to extend this program to general topologies: the authors axiomatize the logic of all $T_1$ topological spaces — and simultaneously that of all metric spaces — in the language combining $\gamma$ with the Cantor derivative ($d$-semantics), establish decidability via the finite model property, and axiomatize the logic of all topological spaces for the weaker language where the derivative is replaced by closure ($c$-semantics). A key methodological device is an equivalent neighborhood-like ("flanked Kripke") semantics in which the intermediate points of a path are treated as a "neighborhood" of the path's origin.

The central results can be stated compactly:

• The calculus TLR (K4 plus eight axioms (A0)–(A8) and a monotonicity rule) is sound and complete both for $T_1$ topologies and for metric spaces, and is decidable.
• The calculus TLR_c (S4 plus (A0)–(A7) and Mon) is sound for all topologies, complete for metric spaces, and decidable.
• Consequently, the $x$0 logic coincides with the logic of Hausdorff, regular Hausdorff, and metric spaces alike — a collapse analogous to McKinsey–Tarski's classical result that S4 is the logic of any dense-in-itself metric space.

Defining T1 by path-reachability

A technically notable early result is a modal definition of the separation axiom $x$1 in the $x$2-language: a topology is $x$3 if and only if the formula $x$4 is valid over it. Here $x$5 defines a discrete set, so a continuous path through it must be constant on the open interval; validity then forces every such path's start to equal its end, exactly excluding pairs of points that cannot be separated by an open set. This formula becomes axiom (A8) of TLR, and its role is precisely to cut the class of all topologies down to $x$6. Notably, the authors show (via a three-point countermodel) that merely dropping (A8) yields a system sound but not complete even for $x$7 spaces, because an additional valid formula involving the interaction of $x$8 with the derivative escapes the remaining axioms. This indicates that the behavior of $x$9 in non-$x$0 spaces involves genuinely new phenomena not captured by the present framework.

Soundness and the flanked Kripke semantics

Each axiom of TLR corresponds to a natural property of paths: (A1) encodes concatenation of paths, (A2) the existence of constant paths, (A3) non-emptiness of the path image, (A4) reversibility of paths, (A5) the fact that interior points of a path also reach the endpoint, (A6) a decomposition along the first exit from a closed region, and (A7) a subtle covering argument using suprema of preimages of closed sets. All of these except (A8) are valid in arbitrary topologies.

For completeness, the authors introduce flanked Kripke frames: triples $x$1 with $x$2 transitive (interpreting the derivative as usual) and a monotone ternary relation $x$3 interpreting $x$4, motivated by the observation that in a topological model one may take $x$5 to be the set of triples witnessed by actual paths. Finite frames satisfying eight frame conditions (F1)–(F8), called suitable frames, mirror axioms (A1)–(A8). Conditions such as (F6) and (F7) involve $x$6-upsets, which form an Alexandroff topology; (F7) requires existence of certain "sequences" decomposing paths across covers by upsets, and corresponds to the most intricate axiom (A7).

Finite model property

Completeness for suitable frames proceeds by a canonical flanked Kripke model construction, followed by a filtration of size at most $x$7 for any refuted formula $x$8. The filtration is built as the smallest extension of the projected relation satisfying (F1) — i.e., closed under concatenation — mirroring the classical proof of S4's finite model property via transitive closures of filtrations. The main technical work verifies that the minimal filtration already satisfies (F2)–(F8) and that closing under (F1) preserves them; each preservation argument recombines sequences or witnesses from the two concatenated pieces. The result is a finite countermodel of exponential size, yielding decidability directly.

From suitable frames to metric spaces

The bridge back to topology is a notion of path-morphism $x$9 from a metric space $\varphi$0 to a flanked frame, with forth/back conditions for both $\varphi$1 and $\varphi$2; path-morphisms preserve satisfaction of all formulas. Given a point $\varphi$3 in a suitable frame, the authors construct a metrizable tree-like space $\varphi$4 — similar to a real tree, but with edges isomorphic either to $\varphi$5 or to the convergent sequence $\varphi$6 — and define a path-morphism onto the frame. Two lemmas carry the weight:

• Every triple in $\varphi$7 is matched either trivially or by a good path, i.e., a map $\varphi$8 satisfying ($\varphi$9-forth) and condition (G). The proof uses a well-founded strict partial order on $\psi$0 with five cases (C1)–(C5); notably, case (C5) employs the standard Cantor-set construction, weaving good paths into complementary intervals of $\psi$1 with a dense enumeration, echoing earlier techniques from the polyhedral setting.
• The tree construction transfers ($\psi$2-forth) and ($\psi$3-forth) from individual edges to the whole map, the latter requiring a case analysis on the relative order of a path's endpoints in the tree partial order, using connectedness of $\psi$4.

Combining the chain of equivalences — $\psi$5 validity, metric validity, refutation on suitable frames of bounded size, and derivability in TLR — gives the main theorem: these four properties coincide for every formula, and the logic is decidable.

The c-semantical fragment

For the language without explicit occurrences of $\psi$6, the calculus TLR_c replaces K4 by S4. A translation substituting each $\psi$7 with $\psi$8 maps TLR-theorems in this fragment to TLR_c-theorems; the instance (A8) of the translation is handled via S4 reasoning plus (A3). It follows that TLR_c is sound for all topological spaces, complete for metric spaces, and decidable. An interesting asymmetry emerges: while in polyhedral and Alexandroff models $\psi$9 coincides with closure, in arbitrary topologies (e.g., totally path-disconnected ones) it does not, so c-semantics and $T_1$0-semantics genuinely diverge outside those restricted classes.

Limitations and open problems

The paper is explicit about what remains unresolved. Completeness for the full derivative language over the class of all topologies is left open; the countermodel showing incompleteness of TLR minus (A8) for $T_1$1 spaces suggests that new axioms or an enriched tree construction — the authors propose adding "jump" edges isomorphic to the two-point Sierpiński space — would be needed. Second, the problem of axiomatizing the $T_1$2-logic of the Euclidean plane, already raised in prior work, remains open; since the present completeness proof targets metric spaces in general, it does not specialize to planar geometry. Finally, the filtration argument gives only a naive exponential bound; no tight complexity analysis of the satisfiability problem is provided.

Conclusion

The paper extends the axiomatization theory of the path-reachability modality from polyhedral and Alexandroff models to general topology, delivering complete, decidable calculi for $T_1$3 spaces and metric spaces under d-semantics and for all topologies under c-semantics. The introduction of flanked Kripke semantics, together with the Cantor-set-based realization of abstract path witnesses inside genuine metric trees, provides reusable machinery for spatial modal logics. The outstanding questions — completeness for all spaces in the derivative language and the Euclidean plane — mark the natural next targets for this line of work.