Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.10008

Modal Extensions of Generalised Nelson Logics

Igor Sedlár·Pietro Vigiani
View original source ↗Read source document ↗Export metadata ↓

Abstract

Motivated by relevant epistemic logic, we extend Dunn's simple Kripke-style binary relational semantics for the semi-relevant logic RM to fit a modal extension of RM. It is shown that the modal axioms and rules that need to be added to RM to obtain a sound and complete axiomatisation of its modal extension give an axiomatisation of modal extensions of all logics in the family of generalised Nelson logics, also studied by Dunn.

Explore related subjects

cs.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Igor Sedlár, Pietro Vigiani. 2026-10-07. Modal Extensions of Generalised Nelson Logics. https://arxiv.org/abs/2610.10008

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Logspace-Constructive Proof of L=SL

We formalize the proof of Reingold's Theorem that SL=L [Rei05] in the theory of bounded arithmetic VL, which corresponds to ``logspace reasoning''. As a consequence, we get that VL=VSL, where VSL is the theory of bounded arithmetic for ``symmetric-logspace reasoning''. This resolves in the affirmative an old open question from Kolokolova [Kol05] (see also Cook-Nguyen [NC10]). Our proof relies on the Rozenman-Vadhan alternative proof of Reingold's Theorem ([RV05]). To formalize this proof in VL, we need to avoid reasoning about eigenvalues and eigenvectors (common in both original proofs of SL=L). We achieve this by using some results from Buss-Kabanets-Kolokolova-Koucký [Bus+20] that allow VL to reason about graph expansion in combinatorial terms.

cs.LO↗
arXiv2026

A new method for proving confluence on abstract reduction systems --- Confluence of non-E-overlapping weakly-shallow TRSs ---

This paper proposes a new method for proving the confluence of an abstract reduction system (ARS) by clarifying the sufficient conditions, called compatibility and edge commutativity, for expanding a given finite sub-ARS into a confluent one by adding rewrite edges. This method can be regarded as an extension of our earlier work, which showed that a weakly non-overlapping, shallow, and non-collapsing term rewriting system (TRS) is confluent. Furthermore, we apply our method to demonstrate that a non-$E$-overlapping and weakly shallow TRS is confluent. Here, a term is weakly shallow if each defined function symbol occurs either at the root or in the ground subterms, and a TRS is weakly shallow if both sides of all its rewrite rules are weakly shallow. This drops the non-collapsing condition assumed in our previous work on weakly shallow TRSs. Moreover, since a weakly shallow TRS is non-$E$-overlapping whenever it is non-$ω$-overlapping, and the latter property is decidable, we also obtain a decidable sufficient condition for confluence: non-$ω$-overlapping and weakly shallow TRSs are confluent.

cs.LO↗
arXiv2026

Two-Variable Logic with Arbitrarily Many Successor Relations

We study two-variable first-order logic with a finite, input-dependent number of distinguished successor relations and arbitrary additional unary and binary predicates. Each distinguished relation must be the exact immediate-successor relation of some linear order on the common domain; the orders themselves are not available in the language and may have arbitrary order types. We prove that satisfiability belongs to 2NEXPTIME, uniformly in the number of successors. The proof gives finite certificates for possibly infinite models. A closure operation separates local neighbourhood descriptions, called star types, into those of bounded multiplicity and those that can be realized infinitely often. The bounded part is represented explicitly. Outside it, integer height vectors allow overlapping successor requirements to be matched without creating cycles or unintended identifications. Ordering the resulting path components densely then recovers exact successor relations. Consistent assignments of the additional binary predicates handle witnesses involving the bounded part. The same certificates decide infinite satisfiability in 2NEXPTIME and give a doubly exponential threshold above which a finite model guarantees an infinite model. A doubly exponentially large finite core can be forced with only two successors and unary predicates. For finite models, we prove that satisfiability subject to a bound on the total number of reference adjacencies lost by the other orders is NEXPTIME-complete, even when the budget is encoded in binary and the number of successors is part of the input.

cs.LO↗