grounded-reasoning — Grounded, Guaranteed Reasoning for LLMs & Agents

TL;DR. LLMs hallucinate on multi-hop relational reasoning. This is a relation-algebra verifier an agent calls to check a claim before asserting it: zero model tokens, precision-guaranteed (accepts a claim iff a grounded proof path exists), language-agnostic, and provider-agnostic. Plugs in as a library, a function-calling tool, or an MCP server. Validated on real LLMs (DeepSeek et al.) and the public CLUTRR benchmark. See docs/integration.md.

📄 Full paper: PAPER.md · Integration guide: docs/integration.md · Try it in 30 seconds: quickstart notebook

Đọc bằng tiếng Việt: README.vi.md

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Why this exists

LLMs are solid on one-hop facts but collapse on composition — chaining several correct facts into a multi-step conclusion. On CLUTRR (kinship reasoning), DeepSeek's accuracy falls off with depth, while a grounded operator-composition solver holds ~100% flat — at zero tokens:

acc
100% ●─────●─────●─────●─────●─────●─────●   ● Grounded solver (algebra, 0 tokens)
 90% |
 80% ○
 70% |  ╲
 60% |   ╲
 50% |    ╲
 40% |     ○           ○                     ○ DeepSeek (LLM)
 30% |      ╲         ╱ ╲
 20% |       ○─────○     ╲
 10% |                    ○─────○
  0% +──┴─────┴─────┴─────┴─────┴─────┴─────┴─
      hop 2    3     4     5     6     7     8   (composition steps)

     hop:      2     3     4     5     6     7     8
     DeepSeek: 83%   42%   25%   25%   42%   17%   8%
     Solver:   100%  100%  100%  100%  100%  100%  100%

(CLUTRR/v1 gen_train234_test2to10, clean-chain, n=12/hop; full test set n=635: solver covers 99.5%, accuracy 99.2%. grounded_reasoning/experiments/clutrr_eval.py.)

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What it is / is NOT (honestly)

Is: a guaranteed reasoning-verification layer built on relation operator algebra.

• Precision = 1.0, guaranteed (Theorem G) — accepts a claim only if a grounded proof path exists.
• Zero extra tokens — local matrix multiplication, no LLM call. Compare to "have the LLM self-verify," which costs +110% tokens for 34% precision.
• Two-sided guarantee (Theorem I) — precision and recall both have tight bounds.
• No external KB required (SGDC) — uses the LLM's own internal consistency. Precision=1.0 is conditional on the LLM's own atomic facts being sound; that assumption can be measured too — calibrate_transitivity doesn't care whether facts came from an external KB or the model's own assertions, so it already calibrates SGDC's real output precision with zero new code (see self_grounded_calibration_eval.py, PAPER.md §6's remark).

Is not: an "unprecedented breakthrough." The Katz index, the Neumann series, graph reachability, and neuro-symbolic grounding are all classical math and technique. The contribution here is unification, a measured guarantee, and benchmark numbers — not a new primitive. The guard needs a relation graph (supplied, or extracted from LLM facts); flexibility is bounded (see PAPER §5).

Two sharp edges the algebra itself can't see (and how to guard them)

Raised in review, reproduced, and fixed with an opt-in guard each — not swept under the rug:

• Entity identity is exact-string by default. If an LLM extraction is inconsistent about one entity's surface form ("Bob" vs "bob"), the graph treats them as two nodes and a real path silently breaks — the guard then (correctly, per its own contract) rejects a claim that is actually true. Fix (binary): GroundedReasoner(normalize=lambda s: s.strip().casefold()) folds surface-form variants together before they become graph keys; proofs still display each entity's original first-seen spelling. Theorem N characterizes exactly when this is safe: precision stays exactly 1.0 as long as normalize never merges two genuinely distinct entities — that's the only way it can go wrong, so it's exactly what gr.calibrate_normalization(labeled_pairs) measures from held-out evidence, reusing the same Clopper-Pearson machinery as Theorem M.
• Theorem G doesn't know if via is transitive in reality. It guarantees "a path exists under the closure of via," not "via actually composes in the world." Compose a relation that's only partially/conditionally transitive ("trusts": A trusts B, B trusts C, does not imply A trusts C) and you get a confident, mathematically correct grounded=True that answers a different question than the one you meant to ask. Fix (binary): GroundedReasoner(transitive_relations={"parent", "is_a", ...}) makes the guard raise ValueError for any undeclared relation, turning a silent modeling assumption into an explicit, checked one. Fix (measured — Theorem M): gr.calibrate_transitivity(rel, labeled_pairs) replaces the binary declare-or-reject with an actual number — a Clopper-Pearson lower confidence bound on "a graph-grounded claim for rel is really true," computed from held-out labeled pairs. Where the binary guard can only guess or block outright, the calibrated bound tells you how much to trust it.

Both opt-in guards are off by default (identical behavior to previous releases). Reproductions: tests/test_agent.py::TestEntityNormalization, ::TestTransitiveRelationsGuard, ::TestTransitivityCalibration, ::TestNormalizationCalibration; the A/B comparisons: transitivity_calibration_eval.py, normalization_calibration_eval.py.

Heterogeneous relation chains. verify(via=rel) composes ONE relation with itself; gr.verify_path(subject, obj, via=["parent","employer"]) composes an exact sequence of different relations (e.g. a derived "financially dependent on" claim) — not new math (OperatorRelationAlgebra.follow already composes mixed-relation chains exactly per Theorem G, this just exposes it at the facade with proof-path reconstruction) — and gr.calibrate_path(via, labeled_pairs) calibrates that fixed pattern with the same Clopper-Pearson engine as calibrate_transitivity (see PAPER.md §5.3.4). Checked against independent ground-truth BFS across 8,000 triples with zero mismatches: tests/test_agent.py::TestHeterogeneousPathVerification, heterogeneous_path_calibration_eval.py.

How this differs from the usual fixes

Approach	Extra tokens	Guarantee	Needs an external KB
LLM self-verification (2nd call)	+110%	none (measured 34% precision)	no
Self-consistency / majority vote	multiplies with sample count	none, statistical only	no
RAG / external KG grounding	varies	only as good as retrieval	yes
This guard	+0	precision = 1.0 (Theorem G)	no
This guard, self-grounded (SGDC)	+0	precision = 1.0 given sound atomic facts (Theorem I)	no
This guard, conformal	+0	coverage ≥ 1−α, distribution-free (Theorem K)	no

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Three theorems, one operator (F = G = H)

The reasoning core rests on a single unification (numerically verified, zero error):

View	Theorem	Content
Fuzzy diffusion inference	F	conf(a→b) = Σ αᵏ(Pᵏ)[a,b], calibrated + grounded
Relation operator algebra	G	composition = operator product, transitive closure = Σ powers
Spectral analysis (Katz)	H	engine.infer = resolvent (I−αP)⁻¹−I (matches 0.0 error)

⟹ fuzzy inference is spectral analysis of the relation operator. grounded_reasoning/reasoning/.

Six further theorems extend this core: I (two-sided precision/recall guarantee for a self-grounded, no-external-KB variant), J (closure-learning completeness, validated on CLUTRR), K (conformal reasoning — distribution-free coverage under a noisy relation graph, including one extracted by an LLM from raw text), L (Horn forward-chaining, generalizing transitive closure to conjunctive rules), M (empirical transitivity calibration — a Clopper-Pearson confidence bound replacing a blind transitivity assumption with a measured one), and N (normalization precision isolation — precision=1.0 breaks only via an over-merge, and only that is what needs calibrating). All nine are stated, proved, and numerically verified in PAPER.md.

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Evidence on real LLMs (DeepSeek)

Experiment	Result
Hallucination guard (kinship)	precision 33% → 100%, catches 92/92 (two seeds), 0 false rejects
Hallucination guard, harder stress test (48-person tree, sibling/spouse distractor facts, shuffled prose, T=0.7, guaranteed-empty trap questions)	raw DeepSeek precision 4.6% (2124 fabricated names, 86/90 trap questions answered with a fabrication); guarded precision 100%, 0 leaked, 0 correct answers dropped — guard_llm_stress_eval.py
Guard token cost	+0 tokens (vs. LLM self-verify: +110% tokens, 34% precision)
SGDC (self-grounded, no external KB)	precision 78% → 100% from internal consistency alone
Dense, anti-commonsense ontology	precision 31% → 100%, catches 106/106, 0 false rejects — nl_ontology_eval.run_dense
CLUTRR (public benchmark)	solver ~100% at every hop vs. DeepSeek 83%→8%
Hard passage (9-step chain, 8 questions)	DeepSeek fabricates 1/8 (wrong direction); grounded system 8/8, with proofs — examples/hallucination_demo.py

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Guaranteed reasoning over a graph an LLM extracted from raw text

The guard/solver needs a clean graph. But if you let an LLM extract relations from natural-language text, the graph is noisy (missing/spurious edges). Conformal Reasoning (Theorem K) fixes exactly that: use operator confidence as a score, calibrate a threshold ⟹ distribution-free coverage ≥ 1−α, even on a noisy graph.

End-to-end demo: DeepSeek extracts an "is a" graph from text → conformal runs on that extracted graph (ground truth is used only for scoring):

Text	LLM extraction (P / R)	Coverage (target ≥90%)	Efficiency (FPR)
Easy	100% / 99.7%	91.3%	0.0
Hard (nested clauses + near-miss distractors)	99.5% / 68.5%	93.0%	0.77

The LLM's extraction drops 31% of the edges (a genuinely noisy graph) → the coverage guarantee still holds (93% ≥ 90%), only efficiency degrades. Validity always holds; efficiency scales with graph quality.

⟹ A path to guaranteed reasoning over natural-language relations — where the hard guard can't reach. grounded_reasoning/experiments/conformal_llm_eval.py.

Efficiency can be pushed further under dropout-dominant noise, at no cost to validity. ConformalReasoner.calibrate(..., group_fn=...) calibrates a separate threshold per group instead of one global one (Mondrian conformal — classical, not new); redundancy_group groups a pair by whether it has more than one walk in the extracted graph, computable with no ground truth. A different grouping tried first (hop-distance) was numerically falsified before shipping — it made efficiency worse, not better, and was discarded. Redundancy grouping cuts FPR from 98.7% → 80.8% when dropped edges dominate the noise (matching this system's real LLM-extraction noise mode) while coverage still holds ≥90% — and honestly gives ~no benefit when spurious added edges dominate instead. redundancy_conformal_eval.py, PAPER.md §7.1's remark.

A different, orthogonal weakness — the noise level DRIFTING over time, not being heterogeneous — needs a different classical tool. Split-conformal (and its Mondrian extension above) assumes calibration and test data share a distribution; that breaks if extraction quality changes between document batches. AdaptiveConformalReasoner (Adaptive Conformal Inference — Gibbs & Candès, 2021, classical, not new) updates its threshold from a stream of confirmed-true examples instead of freezing it after one calibration. When noise shifts partway through a stream (p_drop 0.05 → 0.45), a frozen threshold's coverage collapses from 88.6% to 47.6% — well below the 90% target, silently — while ACI recovers to 89.6%, in 15/15 trials tested. drift_conformal_eval.py, PAPER.md §7.1's remark.

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Self-verification with NO external knowledge base (SGDC)

The guard above needs some relation graph handed to it. Self-Grounded Deductive Consistency (Theorem I) removes even that: it exploits the fact that LLMs are reliably accurate on atomic (1-hop) facts but hallucinate on composition. Take the model's own confident 1-hop facts, build the operator closure from those, then reject any of the model's own multi-hop conclusions that fall outside its own closure — self-contradiction is the hallucination signal, not disagreement with an external source.

from grounded_reasoning import GroundedReasoner

# the LLM's OWN atomic facts (no external KB) -- taken at face value
gr = GroundedReasoner()
gr.add_facts([("sparrow", "is_a", "bird"), ("bird", "is_a", "animal")])

# the LLM's OWN multi-hop conclusion, self-verified against ITS OWN facts above
gr.verify("sparrow", "animal", via="is_a")   # grounded=True: self-consistent
gr.verify("sparrow", "plant",  via="is_a")   # grounded=False: self-contradiction, blocked

	precision	recall
Raw multi-hop (LLM)	78%	87%
SGDC (self-grounded, zero external knowledge)	100%	72%
Ceiling: filtering with an external graph	100%	87%

The honest cost is recall (72% vs. 87%): self-closure is conservative. And Theorem I's precision=1.0 is conditional — it holds if the model's own atomic facts are sound; in a counter-prior domain (e.g. "a whale is a fish"), atomic precision itself can drop, and recall suffers with it (PAPER.md §6 records this honestly rather than hiding it).

That assumption can be measured too, with zero new code. gr.calibrate_transitivity(rel, labeled_pairs) (Theorem M) doesn't care whether gr's facts came from an external KB or the model's own atomic self-assertions — so calling it on a reasoner built purely from an LLM's own facts calibrates SGDC's actual output precision directly, from held-out evidence, instead of assuming atomic soundness. In a synthetic domain with 15% of the atomic facts deliberately wrong, SGDC's real precision fell to ~74% (not the naively-expected ~85% — a single wrong atomic edge composes into several downstream claims, amplifying its damage), and the calibrated bound correctly stayed below that in 98.3% of trials — self_grounded_calibration_eval.py, PAPER.md §6's remark.

Runnable: examples/self_grounded_demo.py (offline) · live on DeepSeek: grounded_reasoning/experiments/self_grounded_eval.py.

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Quickstart

pip install grounded-reasoning

# or, for development (tests + lint):
git clone https://github.com/ALEXaquarius/grounded-reasoning
cd grounded-reasoning && pip install -e ".[dev]"
pytest tests/                       # every theorem + offline-locked logic, no network needed

# Use it right now (no LLM/network needed):
python -c "from grounded_reasoning import GroundedReasoner as G; r=G(); r.add_facts([('a','p','b'),('b','p','c')]); print(r.verify('a','c',via='p'))"

# Real-LLM experiments (need a key — read from an env var, NEVER hardcoded):
export DEEPSEEK_API_KEY=sk-...        # bring your own; .env is gitignored
python -m grounded_reasoning.experiments.guard_llm_eval        # hallucination guard
python -m grounded_reasoning.experiments.guard_llm_stress_eval # harder: distractors + traps + high temperature
python -m grounded_reasoning.experiments.self_grounded_eval    # SGDC
python -m grounded_reasoning.experiments.clutrr_eval           # public CLUTRR benchmark
python -m grounded_reasoning.experiments.conformal_llm_eval    # end-to-end conformal (LLM-extracted graph)
python -m grounded_reasoning.experiments.guard_cost_eval       # token cost: guard vs. LLM self-verify
python -m grounded_reasoning.experiments.nl_ontology_eval      # dense anti-commonsense ontology (add run_dense() for the 106/106 result)

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Integrating with an Agent / LLM (grounded_reasoning/agent/)

A relation-reasoning verifier for agents: check a multi-hop claim before asserting it — zero model tokens, precision guaranteed (accepts iff a grounded proof path exists).

from grounded_reasoning import GroundedReasoner
gr = GroundedReasoner()
gr.add_facts([("alice","parent","bob"),("bob","parent","carol")])
gr.verify("alice","carol", via="parent")   # Verdict(grounded=True, proof=['alice','bob','carol'], confidence=0.36, relation='parent')
gr.verify("alice","zed",   via="parent")   # Verdict(grounded=False, proof=None, confidence=0.0, relation='parent')  ← hallucination blocked

Three integration paths (details: docs/integration.md):

• Library: GroundedReasoner.verify / filter_claims / contradictions.
• Function-calling: TOOL_SPEC (Anthropic) / openai_tool_spec() (OpenAI) + run_tool — a stateless verify_relation tool.
• MCP server: python -m grounded_reasoning.agent.mcp_server — plugs into Claude or any MCP-compatible agent.

Multi-provider (not just DeepSeek): LLMClient(provider=...) for DeepSeek / OpenAI / Groq / OpenRouter / Together / Mistral / Ollama (local) — all OpenAI-compatible, switch providers without changing code. Multilingual: entities/relations are opaque Unicode strings ⟹ works with any language (cha, 父, والد…) with zero configuration.

A real function-calling demo (agent verifies itself, blocks hallucination): python -m grounded_reasoning.experiments.agent_demo. When the graph is noisy (relations extracted by an LLM from text), use ConformalReasoner for a coverage ≥1−α guarantee instead of hard precision.

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Source map

Path	Content
grounded_reasoning/	Public package — GroundedReasoner, verify_relation, TOOL_SPEC, ConformalReasoner, AdaptiveConformalReasoner, LLMClient
grounded_reasoning/agent/{verifier,tool,mcp_server}.py	Public API implementation — HallucinationGuard, function-calling tool, MCP server
grounded_reasoning/reasoning/abstract_inference.py	FuzzyInferenceEngine, TypedInferenceEngine, HallucinationGuard (Theorem F)
grounded_reasoning/reasoning/operator_algebra.py	Relation operator algebra (Theorem G)
grounded_reasoning/reasoning/relation_spectrum.py	Spectrum, nilpotency, Katz resolvent (Theorem H)
grounded_reasoning/reasoning/conformal_reasoning.py	Conformal — coverage guarantee under noise (Theorem K)
grounded_reasoning/reasoning/composition_algebra.py	Composition-table learning, validated on CLUTRR (Theorem J)
grounded_reasoning/reasoning/horn.py	Horn forward-chaining, least-model semantics (Theorem L)
grounded_reasoning/reasoning/transitivity_calibration.py	Clopper-Pearson calibration — reused for both the transitivity assumption (Theorem M) and the normalization over-merge risk (Theorem N)
grounded_reasoning/reasoning/llm_client.py	Provider-agnostic LLM client (key read from an env var)
grounded_reasoning/theory/theorems.py	Nine theorems (F–N) with numerical verification
grounded_reasoning/experiments/{guard_llm,guard_llm_stress,self_grounded,self_grounded_calibration,nl_ontology,guard_cost,clutrr,conformal_llm,redundancy_conformal,drift_conformal,inference,transitivity_calibration,normalization_calibration,heterogeneous_path_calibration}_eval.py	Real-LLM and benchmark experiments backing every claim above
examples/hallucination_demo.py	End-to-end function-calling demo (real LLM, needs a key)
examples/self_grounded_demo.py	SGDC (Theorem I): self-verify a model's own multi-hop claim with NO external KB (offline)
examples/rag_pipeline_demo.py	filter_claims as a RAG/agent post-processing guard, heterogeneous claims (offline)
examples/calibration_demo.py	Theorem M + N side by side: measuring transitivity and normalization trust instead of assuming it (offline)
examples/conformal_demo.py	Coverage guarantee vs. noise tradeoff, clean vs. noisy graph side by side (offline)
examples/quickstart.ipynb	Runnable tour of the library (offline, Colab-ready)

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Origin story

This project began as an attempt to invent an embedding-free retrieval algorithm that could compete with dense/RAG retrieval. That research question reached a rigorous, fully honest negative conclusion (ties BM25, loses significantly to dense embeddings — with a proof of why). The same mathematical toolkit — operator algebra, spectral analysis — turned out to have real, measurable value on a different problem: guaranteeing multi-hop relational reasoning. This repository ships only that validated, tested reasoning system; the full retrieval research trail (including every failed attempt, honestly recorded) lives in a separate research repository and is not part of this package. See PAPER.md §1 for the full framing.

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Contributing & Community

• How to contribute + research principles: CONTRIBUTING.md
• Code of conduct: CODE_OF_CONDUCT.md · Security: SECURITY.md
• Version history: CHANGELOG.md · Citation: CITATION.cff
• License: MIT (LICENSE)

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Principle: proof before code, formal definitions, falsifiability, and honest reporting of negative results — see CONTRIBUTING.md.