Structured Abductive-Deductive-Inductive Reasoning for LLMs via Algebraic Invariants

Computer Science > Artificial Intelligence [Submitted on 17 Apr 2026] Structured Abductive-Deductive-Inductive Reasoning for LLMs via Algebraic Invariants Sankalp Gilda, Shlok Gilda Large language models exhibit systematic limitations in structured logical reasoning: they conflate hypothesis generation with verification, cannot distinguish conjecture from validated knowledge, and allow weak reasoning steps to propagate unchecked through inference chains. We present a symbolic reasoning scaffold that operationalizes Peirce's tripartite inference -- abduction, deduction, and induction -- as an explicit protocol for LLM-assisted reasoning. The framework enforces logical consistency through five algebraic invariants (the Gamma Quintet), the strongest of which -- the Weakest Link bound -- ensures that no conclusion in a reasoning chain can exceed the reliability of its least-supported premise. This principle, independently grounded as weakest link resolution in possibilistic logic and empirically validated for chain-of-thought reasoning, prevents logical inconsistencies from accumulating across multi-step inference. We verify all invariants through a property-based testing suite of 100 properties and 16 fuzz tests over 10^5+ generated cases, providing a verified reference implementation of the invariants suitable as a foundation for future reasoning benchmarks. Comments: 10 pages + 3 pages references. Accepted as a poster at the ICLR 2026 Workshop for LLM Reasoning Subjects: Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Logic in Computer Science (cs.LO) Cite as: arXiv:2604.15727 [cs.AI]   (or arXiv:2604.15727v1 [cs.AI] for this version)   https://doi.org/10.48550/arXiv.2604.15727 Focus to learn more Submission history From: Sankalp Gilda [view email] [v1] Fri, 17 Apr 2026 05:59:16 UTC (39 KB) Access Paper: HTML (experimental) view license Current browse context: cs.AI < prev   |   next > new | recent | 2026-04 Change to browse by: cs cs.LG cs.LO References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)