Structured Abductive-Deductive-Inductive Reasoning for LLMs via Algebraic Invariants
Authors: Sankalp Gilda, Shlok Gilda
Organizations: DeepThought Solutions · Department of Computer Science · University of Florida
Abstract
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.
Chain-of-thought (CoT) prompting enables large language models (LLMs) to tackle multi-step reasoning tasks, yet the generated intermediate steps are not guaranteed to be logically sound. We present Reason Popper-ly, a neurosymbolic framework that uses inductive logic programming (ILP) to learn relation composition rules from reasoning traces and deploys them as an online verifier for step-level correction. Given an LLM-generated trace, the method checks each inferred step against the learned rule table, diagnoses the violation type, rewrites incorrect steps with symbolically derived repairs, and regenerates the remaining suffix so that the model can produce its final answer conditioned on a verified trace. We evaluate on CLUTRR, a multi-hop kinship reasoning benchmark, using five language models over reasoning chains of 2 to 10 hops. Across all models, Reason Popper-ly consistently improves terminal accuracy over standard CoT, with gains of up to 48 percentage points for small models and 15 points for frontier models on the longest chains. Compared with a fully exogenous symbolic pipeline, our method performs better on harder instances by preserving the model's successful grounding while correcting only verifiable reasoning failures. In addition, step-level ILP verification yields a fine-grained error taxonomy that provides diagnostic insight beyond final-answer accuracy.
While LLMs demonstrate impressive reasoning capabilities, they remain fragile in multi-step logical deduction, where a single transition error can propagate through the entire reasoning chain, leading to unstable performance. In this work, we identify logical connectives as primary points of this structural fragility. Through empirical analysis, we show that connective tokens function as high entropy forking points, at which models frequently struggle to determine the correct logical direction. Motivated by this observation, we hypothesize that intervening in logical connective selection can guide LLMs toward more correct logical direction, thereby improving the overall reasoning chain. To validate this hypothesis, we propose a multi-layered framework that intervenes specifically at these logic-critical junctions in the reasoning process. Our framework includes (1) Gradient-based Logical Steering to guide LLMs internal representations towards valid reasoning subspaces, (2) Localized Branching to resolve ambiguity via targeted look-ahead search, and (3) Targeted Transition Preference Optimization, a surgical reinforcement learning objective that selectively optimizes single-token preferences at logical pivots. Crucially, by concentrating intervention solely on logic-critical transitions, our framework achieves a favorable accuracy--efficiency trade-off compared to global inference time scaling methods like beam search and self-consistency.
Chain-of-thought (CoT) prompting can fail severely on constraint-dense logical reasoning tasks, where unverified errors accumulate silently across steps. We introduce SymStep: an LLM makes one atomic claim at a time (DEDUCE: Alice, pet, Cat), then a lightweight constraint propagator checks the claim for consistency with prior accepted deductions, rejects contradictions, and cascades implied facts automatically. SymStep+G additionally provides MRV guidance after each accepted step, directing the LLM toward the most constrained unresolved variable. On a 35-puzzle retained subset of ZebraLogicBench, a benchmark of 1,000 Einstein-style logic puzzles, Direct and CoT both achieve 0%, while SymStep+G reaches 97%. On AR-LSAT analytical reasoning problems, SymStep achieves 100% vs. CoT's 87%. On LGP-14, SymStep+G achieves 100% vs. 0% for CoT and Logic-LM, the strongest prior symbolic+LLM baseline we compare against. Ablation studies reveal that MRV guidance is a key mechanism for reducing directionless cycling, while consistency checking provides a safety net against explicit contradictions. Across six benchmarks spanning five task domains, SymStep variants match or exceed every baseline on constraint-dense and arithmetic tasks. Experiments on AQUA-RAT algebra confirm the advantage is constraint-density-specific.