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Simon Santschi, Niels C. Vooijs
We complete Maksimova's classification of Craig and deductive interpolation in normal extensions of S4. In particular, we prove Craig interpolation for the six extensions of S4 for which the status of the property was still open. The proof strategy builds upon the ideas of Smoryński, but employs a novel approach using Fine's frame formulas for splitting clusters in transitive models.
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Tommaso Flaminio, Sara Ugolini
We provide a new foundational approach to the generalization of terms up to equational theories. We interpret generalization problems in a universal-algebraic setting making a key use of projective and exact algebras in the variety associated to the considered equational theory. We prove that the generality poset of a problem and its type (i.e., the cardinality of a complete set of least general solutions) can be studied in this algebraic setting. Moreover, we identify a class of varieties where the study of the generality poset can be fully reduced to the study of the congruence lattice of the 1-generated free algebra. We apply our results to varieties of algebras and to (algebraizable) logics. In particular we obtain several examples of unitary type: abelian groups; commutative monoids and commutative semigroups; all varieties whose 1-generated free algebra is trivial, e.g., lattices, semilattices, varieties without constants whose operations are idempotent; Boolean algebras, Kleene algebras, and Gödel algebras, which are the equivalent algebraic semantics of, respectively, classical, 3-valued Kleene, and Gödel-Dummett logic. Finally, we prove that the variety of MV-algebras, the equivalent algebraic semantics of Lukasiewicz logic, has nullary type.
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Rob Arthan, Paulo Oliva
We investigate four well-known negative translations of classical logic into intuitionistic logic within a substructural setting. We find that in \emph{affine logic} the translation schemes due to Kolmogorov and Gödel both satisfy Troelstra's criteria for a negative translation. On the other hand, the schemes of Glivenko and Gentzen both fail for affine logic, but for different reasons: one can extend affine logic to make Glivenko work and Gentzen fail and {\it vice versa}. By contrast, in the setting of \emph{Łukasiewicz logic}, we can prove a general result asserting that a wide class of formula translations including those of Kolmogorov, Gödel, Gentzen and Glivenko not only satisfy Troelstra's criteria with respect to a natural intuitionistic fragment of Łukasiewicz logic but are all equivalent.
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Kirill Osipov
For a strictly increasing $G:\mathbb{N}\to\mathbb{N}$ with $G(0)\geq 1$, we study minimal-basis function algebras generated by recursion along generalized inverses of its iterates. Bounded recursion requires an earlier pointwise majorant; unbounded recursion has no such requirement. If the orbit $G^{[t]}(0)$ has no polynomial upper bound, we obtain exact fixed-function membership spectra, reverse-divisibility inclusion, and indexed intersections that need not be single-stride algebras. Every finite divisibility-downset is realized in both regimes, and mixed strides admit fixed functions expressing any finite monotone Boolean condition on their additive stride monoid. For every generator, the realized bounded spectra form a bounded meet-semilattice: a construction sharing a bounding input coordinate realizes finite intersections without an unrestricted selector. Polynomial orbit growth is equivalent to definability of each of ordinary predecessor, unary doubling, and addition in any fixed unbounded stride algebra. In this case the algebra is ordinary primitive recursion relative to its inverse map, giving full collapse for primitive-recursive $G$. Ordinary predecessor belongs to a bounded stride algebra exactly when the orbit is polynomially bounded and $G$ has bounded successive differences; equivalently, ordinary bounded recursion embeds. A locality obstruction explains the additional bounded requirement, and two primitive-recursive generators with quadratic orbits have opposite bounded collapse behavior.
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Sofia Brenner, Jan Hubička, Matěj Konečný
We prove that finite two-graphs have coherent EPPA and that neither graphs modulo global complementation nor two-graphs modulo global complementation have EPPA. This completes the classification of EPPA and coherent EPPA for the reducts of the random graph, answering a question and confirming a conjecture of Evans, Hubička, Konečný and Nešetřil.
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Kirill Osipov
We study finite-terminating, address-only step recursion on the nonnegative reals, from zero, unit successor and projections. For reduced positive rational strides $s=a/q$ and $t=b/v$, we prove $\mathcal{C}s\subseteq\mathcal{C}t$ if and only if $q$ divides $v$, for both the bounded and unbounded algebras $\mathcal{C}\in{\mathcal{B},\mathcal{U}}$. The bounded compiler preserves earlier global majorants on every auxiliary input. Mixed rational stride sets are classified by least common denominators and supernatural denominators. Primitive-recursive translated-coordinate grid codes characterize the unbounded algebras and yield $\mathcal{U}{1/q}\cap\mathcal{U}{1/v}=\mathcal{U}{1/\gcd(q,v)}$. For bounded intersections, we prove an exact reduction: all corresponding identities hold if and only if the single identity $\mathcal{B}{1/2}\cap\mathcal{B}_{1/3}=\mathcal{B}_1$ holds, equivalently if the Boolean parts of the integer successor and addition algebras under scalar bounded recursion coincide. A Boolean reconstruction reduces general functions to predicates; a clipped fine-grid simulation with a frozen global resource bound identifies the union of predicate precision classes with the addition level. These reductions do not decide the resulting predicate equality, so literal bounded intersections remain unresolved. We also prove that unbounded-program equality is $Π^0_1$-complete and extensional bounded-algebra membership is $Σ^0_2$-complete. The model has no free real arithmetic or Zeno limits; semantic majorant validity is distinguished from syntax transformation.
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George Barmpalias, Lu Liu
Muchnik's paradox says that enumerable betting strategies are not always reducible to enumerable strategies whose bets are restricted to either even rounds or odd rounds. In other words, there are outcome sequences x where an effectively enumerable strategy succeeds, but no such parity-restricted effectively enumerable strategy does. We characterize the effective Hausdorff dimension of such $x$, showing that it can be as low as 1/2 but not less. We also show that such reals that are random with respect to parity-restricted effectively enumerable strategies with packing dimension as low as $\log\sqrt3$. Finally we exhibit Muchnik's paradox in the case of computable integer-valued strategies.
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Michael Benedikt, Chia-Hsuan Lu, Alex Perry, Tony Tan
We look at a two-variable logic over finite graphs that allows Presburger quantifiers: linear arithmetic inequalities between the number of elements that satisfy subformulas. The prior literature shows that closing two-variable logic under Presburger quantification leads to undecidability, while closing under \emph{local quantification} -- the counts are always over neighbors of a given node -- preserves decidability. Here we show decidability of satisfiability for an intermediate logic that allows both local counts and global counts, but where the linear constraints do not mix the two. The proof requires a significant extension of the prior machinery developed for two-variable logic with counting. As a corollary we obtain decidability of satisfiability for an important family of Graph Neural Networks (GNNs). GNNs are a popular graph learning formalism. For example, in a variation of GNNs considered in the past literature, local aggregation is used in each layer to create modified values at each vertex of a graph, and then global aggregation is used in a final layer to obtain a single numerical value for the graph as a whole. Our logic is motivated by the fact that mixing of local counting and global aggregation in GNNs has been shown to lead to undecidability. Our results here show that if global and local aggregation are separated, one regains decidability. Decidability of other static analysis problems, like equivalence of GNNs, follows from the decidability of satisfiability.
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Victor Barroso-Nascimento, Joaquim T. Waddington, Elaine Pimentel
We investigate the relationship between base-extension semantics, Beth semantics, and Prawitz's ecumenical logic. We first present a base-extension semantics for the propositional fragment of Prawitz's system and prove that it is sound and complete. The semantics interprets intuitionistic and classical operators directly within a common proof-theoretic framework, without reducing the classical operators to their intuitionistic counterparts by means of a translation. We then analyse the effect of incorporating Beth-style clauses into base-extension semantics. When arbitrary bases are admitted and $\bot$ is governed by atomic explosion, these clauses trivialise. By contrast, when $\bot$ is treated as a basic sentence and attention is restricted to consistent bases, the Beth-style clauses for atoms and disjunction characterise, respectively, classical atoms and classical disjunction in Prawitz's system. We also obtain equivalent, path-independent formulations of the classical clauses for atoms, disjunction, and implication. These results reveal that the consistency constraint is not merely a technical restriction: within base-extension semantics, it transforms clauses originally associated with intuitionistic semantics into semantic characterisations of classical operators.
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Takuma Imamura
We study algebras for the $\mathbf{BM}$-, $\mathbf{BI}$-, and $\mathbf{BMI}$-fragments of combinatory logic, where $\mathbf{B} xyz \to_{w} x\left(yz\right)$, $\mathbf{M} x \to_{w} xx$ and $\mathbf{I} x \to_{w} x$. We prove that every finite $\mathbf{BM}$-algebra $\mathcal{A}$ contains an element $Y$ satisfying $Yx = x\left(Yx\right)$ for all $x\in A$. Moreover, a single closed $\mathbf{BM}$-term represents such an element in every finite $\mathbf{BM}$-algebra. On the other hand, we show that there exists an infinite $\mathbf{BMI}$-algebra with no such element. Consequently, the variety of $\mathbf{BM}$-algebras is not generated by finite $\mathbf{BM}$-algebras. We also characterise finite $\mathbf{BI}$-algebras as finite semigroups with two distinguished left units $B$ and $I$, and finite $\mathbf{BMI}$-algebras as finite idempotent semigroups with three distinguished left units $B$, $M$ and $I$.
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Oualid Merzouga
The fibrational framework of Licata, Shulman, and Riley gives a uniform account of substructural and modal logics for single-conclusion judgments. This thesis extends it to a multiple-output setting in which cartesian resources may be used freely, while linear resources occur exactly once on either side of the turnstile. Judgments are indexed by descriptors: directed acyclic graphs generalizing the tree-shaped expressions of the original framework. Descriptor $2$-cells encode structural transformations, and explicit gates provide weakening, contraction, coweakening, and cocontraction at linear modes. We define a sequent calculus generated by two connectives, $F$ and $U$. We prove identity, invertibility, and cut admissibility. The mode and type theories present linear-non-linear $2$-polycategories, and the projection is a local discrete fibration and a fibration, with opcartesian lifts for one-output base morphisms. Two adequacy results validate the framework: its cartesian fragment recovers the Licata--Shulman--Riley calculus, and its purely linear fragment presents classical multiplicative linear logic. Finally, we relate indexed and fibrational semantics. After developing representability correspondences for multicategories and polycategories, we establish an equivalence between multi-indexed categories over a fixed representable base and multicategorical opfibrations over it, together with its biclosed bifibrational version. We also give a one-way construction from planar poly-indexed data to a planar polycategory.
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Xu Huang
This note documents the specification of normal forms in cubical type theory. The definition is already present in the proof of normalization for cubical type theory, but we present it in a more traditional style explicitly for reference.
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Christoph Benzmüller
Gödel's ontological argument is modal by design. Possibility and necessity are among its central concerns, and connected interpretation decisions -- extensions or intensions for the positive properties, constant or varying quantifier domains, the choice of modal logic, etc. -- subtly shape the argument and, in particular, whether the modal collapse is implied or not. It is tempting to abstract the modality away and to read the positive properties as merely forming an ultrafilter -- which, however, may replace Gödel's argument with a perhaps more manageable alternative that may have little to do with Gödel's original intention. In this spirit, Odifreddi and Gomes argue that modal collapse is an intrinsic structural feature of any theory that characterises positive properties as an ultrafilter and identifies God as its principal generator. This comment shows that the structural claim is false. There is an explicit S5 model, with constant domains, in which positivity is at every world a modal ultrafilter principally generated by a unique and necessarily existing Godlike individual, and in which the modal collapse nevertheless fails -- already for an atomic formula. What drives the collapse is not the ultrafilter structure but the rigidity of positivity -- itself a thoroughly modal condition: given only that positivity is a modal ultrafilter, modal collapse and rigidity are equivalent. Rigidity is thus exactly what the collapse adds to the ultrafilter structure, and the disagreement is whether it is to be counted part of that structure at all. Two further claims of Odifreddi and Gomes are clarified: one on the derivability of Gödel's Theorem IV, one on the extensionality of positivity. All results are machine-checked in Isabelle/HOL, which distinguishes kernel-checked proofs from model-finder output.
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Yoshiki Nakamura
The validity problem for first-order logic is a well-known undecidable problem. The undecidability also holds even for $\mathsf{FO}3$ and (equational formulas of) the calculus of relations. In this paper we tighten these undecidability results to the following: (1) $\mathsf{FO}3$ with just one binary relation is undecidable even without equality; and (2) the calculus of relations with just one character and with only composition, union, and complement is undecidable. Additionally we prove that the finite validity problem is also undecidable for the above two classes.
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Alfredo Burrieza, Fernando Soler-Toscano, Antonio Yuste-Ginel
We propose a novel modal study of bisimulations. Our contribution is threefold. First, we extend the basic modal language with a new modality $\nbi$, whose intended meaning is universal quantification over all states that are bisimilar to the current one under a given bisimulation. We show that bisimulations are definable in this object language via frame validity (for the forth and back conditions) and model validity (for the atomic harmony condition). Additionally, some other related notions (refinements, simulations, bisimulations for a restricted set of variables, and $n$-bisimulations) are also shown to be definable. Second, we provide a sound and complete axiomatisation of the class of all pairs of Kripke models that are bisimulation-related. Third, we show that the satisfiability problem of this logic is decidable and PSPACE-complete via a translation to standard modal logic $K$ under a simple frame condition. All our results are encoded and verified by Isabelle/HOL.
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Andrew W. Swan
We give a categorical formulation of Kraus' "magic trick" for recovering information from truncated types. Rather than type theory, we work in Van den Berg-Moerdijk path categories with a univalent universe, and rather than propositional truncation we work with arbitrary cofibrations, which includes truncation as a special case. We show, using Kraus' argument that any cofibration with homogeneous domain is a monomorphism. We give some simple concrete examples in groupoids to illustrate the interaction between homogeneous types, cofibrations and univalent fibrations.
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Bryan Ford
What does a recursive definition mean when the logic is grounded, so a sentence is true only if a terminating process makes it so? Kripke's answer for the special case of truth was the least fixed point. This paper takes that answer as a rule. Reflective Grounded Arithmetic (RGA) already writes any recursive definition as one self-referential term, paradoxes included. Inductive Grounded Arithmetic (IGA) adds the elimination rule for every positive one, so that accessibility, the winning positions of a game, and grounded truth itself each come with their own induction principle. The rule is sound for the omega model and the system is consistent, both machine-checked in Isabelle/HOL. With the rule, IGA proves Goodstein's theorem outright, in a logic without implication at an undecided hypothesis, and the Kirby-Paris hydra game for primitive recursive strategies, both with every step of the arithmetic a grounded derivation. It proves the Cantor normal forms well-founded, from which the well-foundedness of the standard $\varepsilon_0$ notation system in HOL follows, and it certifies the Veblen notation as an ordinal below every $\varphi(n,0)$. It defines Kripke truth for RGA's own language as one fixed point that agrees with the omega model exactly. Two equivalences locate what the classical world has and the grounded one does not. IGA proves the whole Veblen notation well-founded exactly when it admits bivalence of accessibility for that order, and bivalence of accessibility for an order is sound exactly when the order is well-founded; so $Γ_0$ lies at most one instance of excluded middle away, at the accessibility predicate of the notation, and that instance, the well-foundedness sentence and one membership claim in a primitive-recursive form of Kleene's $\mathcal O$ are interderivable over IGA.
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