Related papers: Reflection calculus and conservativity spectra
For any partial combinatory algebra (PCA for short) A, the class of A-representable partial functions from N to A quotiented by the filter of cofinite sets of N, is a PCA such that the representable partial functions are exactly the…
We introduce a type and effect system, for an imperative object calculus, which infers "sharing" possibly introduced by the evaluation of an expression, represented as an equivalence relation among its free variables. This direct…
Non-classical negations may fail to be contradictory-forming operators in more than one way, and they often fail also to respect fundamental meta-logical properties such as the replacement property. Such drawbacks are witnessed by intricate…
Gradual dependent types can help with the incremental adoption of dependently typed code by providing a principled semantics for imprecise types and proofs, where some parts have been omitted. Current theories of gradual dependent types,…
Walsh [MR4525964, Zbl 1569.03151] has shown that comparing proof-theoretic ordinals is equivalent to comparing $\Pi^1_1$-consequence comparison and $\Pi^1_1$-reflection comparison, all modulo true $\Sigma^1_1$-sentences. In this paper, we…
Nakano's later modality can be used to specify and define recursive functions which are causal or synchronous; in concert with a notion of clock variable, it is possible to also capture the broader class of productive (co)programs. Until…
Large Reasoning Models (LRMs) demonstrate strong performance in complex tasks but often face the challenge of overthinking, leading to substantially high inference costs. Existing approaches synthesize shorter reasoning responses for LRMs…
We study the logical complexity of proofs in cyclic arithmetic ($\mathsf{CA}$), as introduced in Simpson '17, in terms of quantifier alternations of formulae occurring. Writing $C\Sigma_n$ for (the logical consequences of) cyclic proofs…
In combinatorics on words, the well-studied factor complexity function $\rho_{\infw{x}}$ of a sequence $\infw{x}$ over a finite alphabet counts, for every nonnegative integer $n$, the number of distinct length-$n$ factors of $\infw{x}$. In…
The purpose of this paper is to introduce the concept of reflecting numbers to the realm of number theory and to classify reflecting numbers of certain types. For us, reflecting numbers are coming from congruent numbers, above congruent…
Inductive and coinductive specifications are widely used in formalizing computational systems. Such specifications have a natural rendition in logics that support fixed-point definitions. Another useful formalization device is that of…
We investigate the complexity of satisfiability for finite-variable fragments of propositional dynamic logics. We consider three formalisms belonging to three representative complexity classes, broadly understood,---regular PDL, which is…
A fully algebraic approach to reconstructing one-dimensional reflectionless potentials is described. A simple and easily applicable general formula is derived, using the methods of the theory of determinants. In particular, useful…
Algebraic characterizations of the computational aspects of functions defined over the real numbers provide very effective tool to understand what computability and complexity over the reals, and generally over continuous spaces, mean. This…
Large language models (LLMs) have achieved strong performance on complex reasoning tasks using techniques such as chain-of-thought and self-consistency. However, ensemble-based approaches, especially self-consistency which relies on…
In two recent papers (math.LO/0003164 and math.LO/0003165) we answered a question raised in the book by Eklof and Mekler (p. 455, Problem 12) under the set theoretical hypothesis of diamondsuit_{aleph_1} which holds in many models of set…
Let $G$ be a connected reductive algebraic group $G$ over an algebraically closed field $k$ of prime characteristic $p$, and $\ggg=\Lie(G)$. In this paper, we study modular representations of the reductive Lie algebra $\ggg$ with…
Multimodal large language models (MLLMs) have shown promising capabilities in reasoning tasks, yet still struggle with complex problems requiring explicit self-reflection and self-correction, especially compared to their unimodal text-based…
We introduce a novel decidable fragment of first-order logic. The fragment is one-dimensional in the sense that quantification is limited to applications of blocks of existential (universal) quantifiers such that at most one variable…
We systematically investigate the complexity of model checking the existential positive fragment of first-order logic. In particular, for a set of existential positive sentences, we consider model checking where the sentence is restricted…