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math.LO

math.LO: explore 9 source-linked works published from 2026 to 2026, with original documents and citations.

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Sources: arxiv. Collection updated 2026-09-15. Counts describe this index, not the complete source archives.

arXiv2026

Quantified propositional calculi and narrow implicit proofs

In the implicit version of a propositional proof system Q, we work with Q-proofs that are not written down directly, but are succinctly encoded by circuits. Thus implicit Q-proofs are potentially exponentially shorter than usual Q-proofs. We study narrow implicit proofs, a restricted version of this notion, in which lines in the encoded proof can only have polynomial size. We use a cut-elimination construction to show that G_{i+1} is equivalent to narrow implicit G_i, for i >= 1, where G_i is the extension of Frege allowing reasoning with Sigma^q_i quantified propositional formulas. We show that G_1 is equivalent to implicit resolution.

cs.LO↗

arXiv2026

Modalities in non-classical variations of $\mathsf{S4}$

A classical result in modal logic states that $\mathsf{S4}$ has $14$ modalities, that is, every sequence of negations, boxes, and diamonds is equivalent to one in a set of $14$ such sequences. We study analogous results for the non-classical analogues $\mathsf{CS4}$, $\mathsf{IS4}$, $\mathsf{GS4}$, and $\mathsf{GS4^c}$ of $\mathsf{S4}$. First, we show that, while all these logics have finitely many $\{\Box,\Diamond\}$- and $\{\neg,\Box\}$-modalities, the logic $\mathsf{CS4}$ has infinitely many $\{\neg,\Diamond\}$-modalities. Second, we show that $\mathsf{IS4}$ and $\mathsf{GS4}$ have finitely many $\{\neg,\Diamond\}$-modalities, but they have infinitely many $\{\neg,\Box,\Diamond\}$-modalities. At last, we show that $\mathsf{GS4^c}$ has finitely many $\{\neg,\Box,\Diamond\}$-modalities.

math.LO↗

arXiv2026

A categorical formulation of Kraus' paradox

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.

math.CT↗

arXiv2026

Proof-Carrying Analytic Approximation: Local-to-Global Evidence Transport at Encoding Cost

Under quasi-uniform refinement, bounded local-encoding hypotheses, and local $W^{r,2}$ approximation of order $r\ge 2$ in a rational piecewise-polynomial presentation of $W^{1,2}(0,1)$, carrying the complete proof genealogy up to the level required by an accuracy $\varepsilon$ costs the same asymptotic bit order as the finest-level conventional coefficient encoding. If $B_n=Θ(M_nβ_n)$ denotes that level-$n$ encoding size, our compiler transports supplied local approximation and overlap witnesses through exact partition-of-unity synthesis and geometric refinement to a represented limit with total certificate size $O(B_{m(\varepsilon)})$, where $m(\varepsilon)=O(\log(1/\varepsilon)/(r-1))$. The construction makes no oracle query to an independently supplied semantic target name ($Q_{\rm target}=0$). When $β_n=O(n+1)$, this becomes $O(\varepsilon^{-1/(r-1)}(1+\log(1/\varepsilon)))$. The surrounding framework is intentionally separated from this resource theorem. Every real computable Banach presentation admits a uniformly computable linear isometric embedding into standard computable $C([0,1])$, with computable inverse on its represented range. Complete metric evidence with rational strict slack collapses extensionally to the represented analytic metric once effective names are available, while chosen evidence transformations retain construction history and resource information. For the Lipschitz grammar used here, qualitative evidence-local lifting is canonical; the nontrivial question is therefore which evidence is retained and at what cost.

math.FA↗

arXiv2026

A Borel Concept Class of VC Dimension One with a Non-PAC Consistent Learner in ZFC

The fundamental theorem of statistical learning states that, under suitable measurability assumptions, finite Vapnik--Chervonenkis (VC) dimension guarantees that every proper consistent learning rule is probably approximately correct (PAC). Blumer, Ehrenfeucht, Haussler, and Warmuth showed, assuming the Continuum Hypothesis, that the "well-behavedness" condition of the concept class cannot be omitted: they constructed a concept class of Borel sets of VC dimension one admitting a consistent learning rule that is not PAC. We show that the Continuum Hypothesis is unnecessary. Working in Zermelo--Fraenkel set theory with the Axiom of Choice (ZFC) alone, we construct a concept class of Borel sets on $[0,1]$ of VC dimension one and a proper consistent learning rule that is not PAC. More precisely, for a suitable Borel probability measure and target concept, the rule has true risk one at every sample size on a set of samples of outer probability one. Consequently, finite VC dimension and Borel measurability of the individual concepts do not suffice to guarantee that every proper consistent learning rule is PAC. The result shows, with no need of extra set-theoretical assumptions, that the additional regularity assumption in the fundamental theorem cannot in general be omitted.

math.LO↗

arXiv2026

Exponential Gaps Between Intuitionistic Linear Extended Frege Systems

In this paper, we establish exponential separations between Extended Frege systems for a range of intuitionistic substructural and linear logics. More precisely, for any logic $L$ below the intuitionistic logic obtained by extending $\mathbf{ILL}$ with structural rules, and any logic $M$ not contained in $L$, we construct a family of $\mathsf{FL_e}$-provable formulas that have short proofs in $M$-Frege but require proofs of exponential size in $L$-Extended Frege. The same result holds in the $!$-free settings, using $\mathbf{IMALL}$ and $\mathbf{FL_e}$ in place of $\mathbf{ILL}$. The key ingredient in proving these separations is a variant of the feasible disjunction property for $L$-Frege, which may be of independent interest.

cs.LO↗

arXiv2026

Subgroup Accessibility in Group Order Logic

We investigate the expressive power of fixed-point logics (FP) and their extensions in defining generating sets for accessible subgroups of definable permutation groups. This operation, computable in polynomial time via the Schreier-Sims algorithm, plays a central role in the group-theoretic approach to Graph Isomorphism and Graph Canonisation. In particular, it underpins polynomial-time canonisation for bounded colour-class graphs--a class for which no natural logic capturing P is currently known. We first show that this operation cannot, in general, be expressed in any logic for P. This limitation arises from the fact that accessible subgroups need not admit symmetric generating sets of polynomial size. However, we prove that when the base group admits a definable ordered generating set, the accessible subgroup operation becomes definable in fixed-point logic with the group order operator (FP + ord). This is achieved by partially simulating the Schreier-Sims algorithm within FP + ord. As a corollary, we show that fixed-point logic with counting (FPC) can also define the operation when the base group is abelian. In particular, FPC can define the automorphism group of any graph with abelian colours--despite being unable to canonise such graphs.

cs.LO↗

arXiv2026

Bootstrapping Mutual Attestation with Kleene's Second Recursion Theorem

Mutual attestation among nodes with no central trusted operator requires each node to hold reference values (expected code measurements) for its peers. The naïve approach of mutually embedding these reference values in the nodes' code leads to an infinite regress. We call the problem of resolving this infinite regress the reference-value bootstrapping problem for mutual attestation. Existing solutions avoid this regress by relying on a trusted third party (TTP), externally supplied reference values, or architecture-specific measurement mechanisms. We instead express the bootstrapping problem as a system of mutual fixed-point equations and solve it by Kleene's second recursion theorem. The construction produces nodes that mutually reference one another's code and reconstruct every peer's exact source from built-in data alone. When a deployed source file is measured directly, as with a Python script, a node obtains the peer's reference value by applying the measurement function directly to the reconstructed source. When a built image is measured, as with AWS Nitro Enclaves, a node instead reproducibly rebuilds the peer's image from the reconstructed source and derives its reference measurement. For the first case, we develop PyReflect, a Python transpiler, and use it to implement a TPM mutual-attestation PoC. For the second, we develop NixReflect, a Nix transpiler, and use it in a PoC in which two Nitro Enclaves reproduce each other's reference PCRs from built-in data alone. Our solution is architecture-independent, requires neither a TTP nor externally supplied reference values, and works with existing attestation stacks unchanged.

cs.CR↗

arXiv2026

Unconditional $V^0_1$-independence of a certified hitting-set principle

We show that a certified formalization of the hitting-set-existence axiom of Atserias and Tzameret, instantiated on the parity-based Nisan-Wigderson compression class of Khaniki, is independent of the two-sorted theory $V^0_1$ of $\mathrm{AC}^0$-reasoning, unconditionally: $V^0_1$ proves neither it nor its negation. The same holds for the corresponding certified dual weak pigeonhole principle, whose refutation is witnessed by a single seed that certified-computes every string of the model simultaneously. The mechanism is a bounded-arithmetic transfer of Atserias-Tzameret's reduction from hitting sets to the dual weak pigeonhole principle: the amplification half of that reduction, the sole source of its NP-oracle, is unnecessary at the native stretch of the Nisan-Wigderson map, and the compression half becomes a $V^0_1$-provable implication once circuit evaluation is replaced by its certified $Σ^B_0$ unfolding. This is, to our knowledge, the first independence result for a derandomization-flavoured existence principle at the $\mathrm{AC}^0$-reasoning level, and it makes explicit the bridge between the Khaniki Nisan-Wigderson line and the Atserias-Tzameret reverse mathematics of hitting sets.

cs.CC↗

Compare source metadata on this page

Work	Published	Source identifier	Source
Quantified propositional calculi and narrow implicit proofs	2026-09-01	2606.10535	arxiv
Modalities in non-classical variations of $\mathsf{S4}$	2026-09-01	2609.00736	arxiv
A categorical formulation of Kraus' paradox	2026-08-31	2403.17961	arxiv
Proof-Carrying Analytic Approximation: Local-to-Global Evidence Transport at Encoding Cost	2026-08-31	2506.22693	arxiv
A Borel Concept Class of VC Dimension One with a Non-PAC Consistent Learner in ZFC	2026-08-31	2608.30246	arxiv
Exponential Gaps Between Intuitionistic Linear Extended Frege Systems	2026-08-31	2609.00422	arxiv
Subgroup Accessibility in Group Order Logic	2026-08-31	2609.00499	arxiv
Bootstrapping Mutual Attestation with Kleene's Second Recursion Theorem	2026-08-29	2608.20671	arxiv
Unconditional $V^0_1$-independence of a certified hitting-set principle	2026-08-28	2608.28813	arxiv

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