arXiv subjects

Anggha Nugraha

Publications and source records attributed to Anggha Nugraha.

4 recordsLinked to original sources

arXiv2026

A Constructive Field of Infinitesimals: Chunk and Permeate Approach

While intuitive, naïve infinitesimal reasoning is classically inconsistent, and rigorous nonstandard analysis relies on non-constructive machinery. We resolve this tension by constructing an explicit, totally ordered field $\mathbb{R}^{\mathbb{Z}_{<}}$ using only real sequences and Cauchy convolution. We model the combined real and hyperreal axioms via the Chunk and Permeate strategy, a paraconsistent technique that isolates contradictions without global collapse. Equipping $\mathbb{R}^{\mathbb{Z}_{<}}$ with a two-tier topology, we develop a calculus where infinitesimal derivatives and integrals permeate cleanly to their classical counterparts. We further introduce a $(k,n)$-continuity hierarchy capturing infinitesimal smoothness invisible to standard or transfer-based models. Finally, $\mathbb{R}^{\mathbb{Z}_{<}}$ yields a direct algebraic consistency proof for Sergeyev's Grossone arithmetic, and we establish strict computability bounds on field operations. By guaranteeing infinitesimal contradictions never reach the classical chunk, this work bridges paraconsistent logic, constructive mathematics, and nonstandard analysis into a transparent, computationally tractable framework for infinitesimal reasoning.

math.LO↗

arXiv2026

Paraconsistent Dominated Convergence

The Levi-Civita field $\mathcal{R}$ of formal Laurent series is a constructive, non-Archimedean ordered field that supports a full Lebesgue measure and integration theory, including a Dominated Convergence Theorem. This paper embeds that integration theory into the paraconsistent Chunk and Permeate framework, extending it from elementary calculus to genuine measure theory. The source chunk is modelled by $\mathcal{R}$ with its measure and integral, while the target chunk is the classical real line $\mathbb{R}$. A permeability relation exports the standard part of the internal integral, and it is shown that the Dominated Convergence Theorem permeates from the source chunk to the target chunk, yielding the classical Lebesgue Dominated Convergence Theorem without any choice principles and without the ultrafilters required by nonstandard measure theory. The construction is entirely explicit and demonstrates that paraconsistent logic can provide a rigorous foundation for deep analytical tools while keeping inconsistencies safely confined.

math.LO↗

arXiv2026

Inductive Satisfiability Certification for Universal Quantifiers and Uninterpreted Function Symbols

The combination of uninterpreted function symbols and universal quantification occurs in many applications of automated reasoning, for example, due to their ability to reason about arrays. Yet the satisfiability of such formulas is, in general, undecidable. In practice, SMT solvers are often successful in the unsatisfiable case, using heuristics. However, in the satisfiable case, they rely on explicit model construction, which fails for formulas whose smallest model is not small enough. We introduce an alternative approach that certifies satisfiability using induction arguments, and apply it to the case of linear integer arithmetic. The resulting algorithm is able to prove satisfiability of formulas that are out of reach for current SMT solvers.

cs.LO↗

arXiv2020

Naïve Infinitesimal Analysis: Its Construction and Its Properties

This paper aims to build a new understanding of the nonstandard mathematical analysis. The main contribution of this paper is the construction of a new set of numbers, $\mathbb{R}^{\mathbb{Z}_< }$, which includes infinities and infinitesimals. The construction of this new set is done naïvely in the sense that it does not require any heavy mathematical machinery, and so it will be much less problematic in a long term. Despite its naïvety character, the set $\mathbb{R}^{\mathbb{Z}_< }$ is still a robust and rewarding set to work in. We further develop some analysis and topological properties of it, where not only we recover most of the basic theories that we have classically, but we also introduce some new enthralling notions in them. The computability issue of this set is also explored. The works presented here can be seen as a contribution to bridge constructive analysis and nonstandard analysis, which has been extensively (and intensively) discussed in the past few years.

math.LO↗