arXiv subjects

Søren Brinck Knudstorp

Publications and source records attributed to Søren Brinck Knudstorp.

9 recordsLinked to original sources

arXiv2026

Convex Team Logics

We prove expressive completeness results for convex propositional and modal team logics, where a logic is convex if, for each formula, if it is true in two teams $t$ and $u$ and $t\subseteq s\subseteq u$, then it is also true in $s$. We introduce propositional/modal team logics which are expressively complete for the class of all convex propositional/modal team properties. We also answer an open question concerning the expressive power of classical propositional logic extended with the nonemptiness atom NE: we show that it is expressively complete for the class of all convex and union-closed properties. A modal analogue of this result additionally yields an expressive completeness theorem for Aloni's (2022) Bilateral State-Based Modal Logic. There is a specific sense in which one of the propositional convex logics we introduce extends propositional dependence logic; this sense corresponds to the existence of a compositional translation between the two logics. Another of the convex logics extends propositional inquisitive logic in the same manner. We generalize the notion of uniform definability, as considered in the team semantics literature, to formalize this notion of compositional extension.

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arXiv2026

Teams, Truthmakers, Translations

We present a faithful translation from Aloni's logic BSL into a classical modal logic over semilattices combining ideas from earlier modal translations of truthmaker semantics with algebraic representation theory. We show how our method can also be made to work for other current hyperintensional logics. Finally, we place our results in a broader setting of translating hyperintensional logics into classical modal ones, and discuss applications and new challenges that arise for both areas of philosophical logic connected in this way.

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arXiv2026

Diamonds and Dominoes: Impossibility Results for Associative Modal Logics

We show that for any class of Boolean algebras with an associative operator, if it contains the complex algebra of (P(N), U), its equational theory is undecidable. Equivalently, any associative normal modal logic valid over the frame (P(N), U) is undecidable. This settles a long-open question on the decidability of hyperboolean modal logic (Goranko and Vakarelov, 1999), and addresses several related problems.

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arXiv2026

Knocking Down Boxes: The FMP for $\mathbf{K} \oplus \Box^{m+k} p \to \Box^m p$

It is a long-standing open problem whether modal logics of the form $\mathbf{K} \oplus \Box^n p \to \Box^m p$ for $n>m>1$ have the finite model property (FMP). We solve this by showing that any modal logic axiomatized by formulas of the form $\Box^n p \to \Box^m p$ where $n>m>1$ has the FMP.

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arXiv2026

Medvedev logic is undecidable

We show that Medvedev's logic of finite problems, a well-known superintuitionistic logic, is undecidable. The key method is a reduction from the periodic tiling problem to non-theoremhood in Medvedev's logic. This settles a longstanding open problem. Using similar techniques, but reducing instead to the ordinary tiling problem, we likewise obtain undecidability of Skvortsov's logic of infinite problems, and the fact that the two logics are distinct -- in fact, they are separated by any aperiodic tiling of the plane. Due to the fact that Medvedev's logic figures in so many different areas, these results have implications for several fields -- for example, the study of schematic fragments of logics such as propositional dependence logic, or the study of internal logics of toposes. The core idea and technical work of the undecidability proof were obtained using ChatGPT Sol 5.6, and formally verified in Lean by Claude Opus 5. A detailed methodology section outlines how such results were obtained.

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arXiv2026

Possibly Relevant Translations

We develop translations from relevant logics into normal modal logics, and use them to clarify structural connections between relevant and modal logic, obtain a few corollary results, and raise questions for future work.

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arXiv2026

Undecidability in Relevant Logic

We prove undecidability for every positive relevant logic extending the system axiomatized by hypothetical syllogism, prefixing, and suffixing and contained in the logic of the semilattice frame $(P_{\mathrm{fin}}(\mathbb{N}), \cup, \varnothing)$. This settles the longstanding decision problem for the semilattice relevant logic S in the negative, contrary to prevailing expectations of decidability. It also provides a new proof of Urquhart's (1984) undecidability theorem for R, E, and T, now by reduction from the Wang tiling problem for arbitrarily large finite isosceles right triangular regions of the plane.

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arXiv2026

The logic of bunched implications is undecidable

The logic of bunched implications (BI), introduced by O'Hearn and Pym (1999), has attracted significant attention due to its elegant proof calculus, varied semantics, and close connections to the propositional fragment of separation logic. We show here that provability in BI is undecidable by encoding Wang tilings into its ternary relational semantics. Equivalently, this yields the undecidability of the equational theory of BI-algebras. Our result is much more general, applying to the {and, or, not, --*}-fragment of stronger and weaker logics: the negation simply needs to be disjointive, and the multiplicative conjunction need not be commutative (then --* splits into two divisions \, /). Consequently, our result covers an interval that includes BI, the non-commutative logic GBI, and Boolean BI (BBI), the latter already known to be undecidable. This result contrasts with a long-standing expectation that BI might be decidable. We also identify the gaps in the publications claiming decidability.

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arXiv2024

Tho Modal Logic of Minimal Upper Bounds

To formalize patterns of information increase and decrease, Van Benthem (1996) proposed modal information logic (MIL), a modal logic over partial orders. In MIL, points are interpreted as information states and least upper bounds, when existent, as informational sums. A natural counterpart to this logic is the modal logic of minimal upper bounds (MIN), interpreting minimal, rather than least, upper bounds as informational sums. This paper presents the logic MIN, and in the main result, it is shown that the modal language cannot distinguish the two interpretations: a formula is valid in MIN if and only if it is valid in MIL. Leveraging the work of [11], as corollaries, an axiomatization of MIN and a proof of decidability are obtained.

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