Euclid-MCP

MCP server for logical reasoning — turns facts into formal proofs.

Euclid-MCP is a hybrid cognitive architecture: a lightweight LLM describes the world in facts, and a deterministic engine performs the actual deduction. The LLM never needs to reason — it only needs to describe.

With Euclid-MCP, an 8B model can solve reasoning tasks that stump even 400B+ cloud models — because the engine handles deduction deterministically. Every answer comes with a proof tree, so you can trace why a conclusion holds, not just what it is. Use it to enforce RBAC policies, audit cloud compliance, validate loan eligibility rules, or reason over any domain where answers must be explainable and verifiable.

Euclid-MCP is written in Python and uses Euclid-IR, a human-readable intermediate language designed for both AI agents and humans. It currently uses SWI-Prolog as its inference engine and can be consumed in multiple ways: via MCP by AI agents (OpenCode, Claude, Cursor), via HTTP by tools and automation platforms (n8n, Zapier, Make), and via Python API for direct integration. Euclid-IR rules can also be used to augment RAG pipelines with deterministic policy enforcement.

How it works

┌──────────────┐     ┌──────────────────┐     ┌──────────────┐     ┌──────────────┐
│  LLM/Agent   │────▶│  Euclid-MCP      │────▶│  Translator  │────▶│  SWI-Prolog  │
│  (MCP Client)│◀────│  (FastMCP)       │◀────│  + Meta-IP   │◀────│ (subprocess) │
└──────────────┘     └──────────────────┘     └──────────────┘     └──────────────┘

1. Receive facts, rules, and a query in a simple intermediate language

2. Translate into Prolog with a meta-interpreter for proof tree capture

3. Execute via SWI-Prolog subprocess

4. Return solutions + proof trees as structured JSON

Additional tools (diagnose, what_if, check_kb) extend this core flow with analysis, scenario testing, and validation.

LLMs describe. Euclid MCP proves.

Knowledge Base

In the current implementation, facts and rules are provided with each request.

The long-term architecture also supports persistent Knowledge Bases, where stable business rules are loaded once into the inference engine, while agents provide only the session-specific facts required for the current query.

This minimizes token usage, improves performance, and allows small LLMs to reason over large rule sets without reconstructing the entire knowledge base for every request.

Related MCP server: Pyke MCP Server

Intermediate Language

Even if currently Euclid-MCP uses a Prolog Engine, no Prolog syntax is required.
Euclid-IR (Intermediate Representation) is a declarative intermediate representation for logical inference. Variables use $name, implication is IF, conjunction is AND.

Text format:

human(socrates)
mortal($x) IF human($x)

? mortal($who)

YAML format:

facts:
  - parent(tom, bob)
  - parent(bob, ann)
  - parent(tom, liz)
rules:
  - ancestor($x, $y) IF parent($x, $y)
  - ancestor($x, $y) IF parent($x, $z) AND ancestor($z, $y)

query: ancestor(tom, $who)

Full language reference: docs/EUCLID_IR.md

Euclid-IR Syntax Reference

Arithmetic Comparisons

Rules support arithmetic comparisons that are evaluated during deduction:

# Stale access: users who haven't logged in for 90+ days
stale_access($user) IF
    user($user) AND last_login_days($user, $days) AND $days > 90

# Excessive permissions: more than 15 direct permissions
excessive_permissions($user, $count) IF
    user($user) AND permission_count($user, $count) AND $count > 15

# Clearance check: user clearance >= resource classification
can_access($user, $resource) IF
    user($user) AND resource($resource, _, _, _, _, $cls) AND
    classification($cls, $cls_level, _) AND
    user_clearance($user, $user_level) AND $user_level >= $cls_level

Supported operators: >, >=, <, <=, =:=, is, !=

Multi-line Rules

Rules can span multiple lines for readability:

can_deploy($user, $env) IF
    user($user) AND
    has_role($user, $role) AND
    deploy_requires_level($env, $min) AND
    deploy_role_level($role, $level) AND
    $level >= $min AND
    user_has_permission($user, deploy_code)

Conjunctions in Queries

Queries can combine multiple predicates:

? can_access_resource($who, $res) AND resource($res, _, _, _, _, secret)

This returns solutions where both conditions are satisfied simultaneously.

Why External Inference?

The external inference gives several advantages:

• deterministic

• explainable

• verifiable

• inexpensive

• replaceable backend

In the current implementation Euclid-MCP uses Prolog.
Prolog is a 50-year-old battle-tested logic engine. Using it as a "deduction coprocessor" lets small LLMs perform complex multi-step reasoning without needing larger, more expensive models. The intermediate language strips away Prolog's syntax quirks while keeping its logical core.

Some internal benchmarks demonstrate the difference: with 1 000+ facts, LLMs alone score 2/5 while Euclid-MCP scores 5/5 — and runs 7× faster while outputting 14× fewer tokens.

Tools

Euclid-MCP exposes 4 tools, each with a specific purpose:

reason

Main tool for verifiable deterministic reasoning.

Returns ReasonResult with solutions[] — each containing variable bindings and a proof tree.

diagnose

Query analysis — understand why a query succeeds or fails.

Modes:

• why — explain why a query holds (or that it doesn't)

• why_not — explain why a query fails (missing facts/rules)

• what_needs — suggest what would make a false query true

Returns DiagnosisResult with holds, findings[], conclusion, and optionally proof.

what_if

Scenario analysis — apply modifications to a knowledge base and compare results.

Returns WhatIfResult with before_count, after_count, delta, solutions_before, solutions_after, conclusion.

check_kb

Knowledge base validator — check for consistency before running deduction.

Returns KBCheckResult with valid, errors[], warnings[], facts_count, rules_count, predicates_count.

Installation

pip

# Prerequisites: Python ≥ 3.10, SWI-Prolog
brew install swi-prolog

# Install
pip install euclid-mcp

From source

git clone https://github.com/meob/Euclid-MCP
cd Euclid-MCP
python3 -m venv .venv && source .venv/bin/activate
pip install -e .

Docker

No local SWI-Prolog installation needed — the image bundles everything.

# Build
docker build -t euclid-mcp .

# MCP stdio mode (for local MCP clients)
docker compose run --rm euclid-mcp

# HTTP API mode (for n8n, Zapier, remote access)
docker compose up euclid-api
# API available at http://localhost:8080

See Docker in Integrations for full details.

Usage

Via MCP (OpenCode, Claude, etc.)

{
  "mcpServers": {
    "euclid-mcp": {
      "command": "python3",
      "args": ["-m", "euclid_mcp"],
      "cwd": "/path/to/euclid-mcp"
    }
  }
}

Via Python

from euclid_mcp.server import reason, diagnose, what_if, check_kb

# Reasoning
result = reason(knowledge="""
    human(socrates)
    mortal($x) IF human($x)
    ? mortal($who)
""")
for sol in result.solutions:
    print(sol.substitutions, sol.proof.type)

# Diagnosis — why does a query fail?
diag = diagnose(
    knowledge="human(socrates)\nmortal($x) IF human($x)",
    query="mortal(plato)",
    mode="why_not"
)
print(diag.conclusion)

# What-if — how does adding a fact change results?
scenario = what_if(
    base_knowledge="human(socrates)\nmortal($x) IF human($x)",
    modifications="+ human(plato)",
    query="mortal($who)"
)
print(f"Before: {scenario.before_count}, After: {scenario.after_count}")

# KB validation
check = check_kb(knowledge="human(socrates)\nmortal($x) IF human($x)")
print(f"Valid: {check.valid}, Errors: {check.errors}")

Example output

{
  "query": "ancestor(tom, $who)",
  "solutions": [
    {
      "substitutions": {"who": "bob"},
      "proof": {
        "type": "rule",
        "goal": "ancestor(tom, bob)",
        "body": "parent(tom, bob)",
        "subproof": {"type": "fact", "goal": "parent(tom, bob)"}
      }
    },
    {
      "substitutions": {"who": "ann"},
      "proof": {
        "type": "rule",
        "goal": "ancestor(tom, ann)",
        "body": "parent(tom, bob), ancestor(bob, ann)",
        "subproof": {
          "type": "and",
          "left": {"type": "fact", "goal": "parent(tom, bob)"},
          "right": {
            "type": "rule",
            "goal": "ancestor(bob, ann)",
            "body": "parent(bob, ann)",
            "subproof": {"type": "fact", "goal": "parent(bob, ann)"}
          }
        }
      }
    }
  ]
}

Diagnose output

{
  "holds": false,
  "findings": [
    "Fact 'mortal(plato)' not found in knowledge base",
    "No rule with head 'mortal(plato)' found"
  ],
  "conclusion": "Query fails: no derivation path for mortal(plato)"
}

What-if output

{
  "before_count": 1,
  "after_count": 2,
  "delta": 1,
  "solutions_before": [{"substitutions": {"who": "socrates"}}],
  "solutions_after": [
    {"substitutions": {"who": "socrates"}},
    {"substitutions": {"who": "plato"}}
  ],
  "conclusion": "Adding 'human(plato)' adds 1 new solution"
}

Use cases

• Small LLM reasoning: Offload deduction from LLMs (3-8B) to a deterministic engine

• Explainable decisions: Every answer comes with a proof tree which allows explanation, reasoning trace, and justification

• Business rules: Validate logic chains (permissions, workflows, compliance)

• Dependency analysis: Circular dependency detection, topological ordering

• Education: Interactive logic tutoring with visible proof chains

• Knowledge preload: Complex business rules can be loaded in Euclid instead of using a vector database

• Query diagnosis: Understand why queries fail and what facts/rules are missing

• Scenario analysis: Test "what-if" modifications before applying them to production

• KB validation: Check knowledge bases for consistency before reasoning

Real-world examples

After installing (via pip or from source with an active virtualenv):

# Genealogy — recursive family tree reasoning
python examples/01_genealogy.py

# RBAC — Role-Based Access Control
python examples/02_rbac.py

# Classification — biological taxonomy
python examples/03_classification.py

# Business rules — loan eligibility
python examples/04_loan_eligibility.py

# Compliance auditor — cloud resource policy enforcement
python examples/05_compliance_auditor/auditor.py

# Loan officer — CSV-driven eligibility with detailed breakdown
python examples/06_loan_eligibility/loan_officer.py

# IT Security & Compliance — multi-layer policy reasoning
python examples/07_it_security_compliance/demo.py --small

Each example runs a complete reasoning session and prints solutions with proof trees — no LLM required.
Use them as templates for integrating Euclid-MCP into your own agents.

The two newer examples (05, 06) demonstrate a data-driven agent workflow:

• Read external data (JSON, CSV) that simulates API/CRM exports

• Convert structured data to Euclid facts in Python

• Load policy rules from .euclid files (separated from data)

• Call reason() for deduction

• Format results into human-readable reports with proof chains

This mirrors how a real agent would work: collect data, describe it as facts, let Euclid reason, and present the results.

Example 07: IT Security & Compliance

The most advanced example demonstrating:

• 3-layer architecture: Standards (CIS, AWS IAM) → Company Policies → Data Facts

• Arithmetic comparisons: $days > 90 for stale access detection

• Multi-line rules: Complex policies split across lines

• Conjunction queries: Combining multiple predicates

• Negative tests: Verifying empty results for invalid access patterns

# Quick test (30 users, 50 resources, ~578 facts)
python3 examples/07_it_security_compliance/demo.py --small

# Full dataset (200 users, 300 resources, ~3,872 facts)
python3 examples/07_it_security_compliance/demo.py

Integrations

OpenCode

Euclid-MCP includes a pre-configured agent in .opencode.json:

{
  "mcpServers": {
    "euclid-mcp": {
      "command": "python3",
      "args": ["-m", "euclid_mcp"],
      "cwd": "."
    }
  },
  "agents": {
    "reasoning-engine": {
      "description": "Deterministic logic engine",
      "instructions": "Write facts in Euclid IR, use the reason tool...",
      "mcpServers": ["euclid-mcp"]
    }
  }
}

n8n / Zapier / Make

Run the HTTP API:

python3 integrations/euclid_api.py --port 8080

# Reasoning
curl -X POST http://localhost:8080/reason \
  -H "Content-Type: application/json" \
  -d '{"knowledge": "human(socrates)\nmortal($x) IF human($x)\n? mortal($who)"}'

# Diagnosis
curl -X POST http://localhost:8080/diagnose \
  -H "Content-Type: application/json" \
  -d '{"knowledge": "human(socrates)\nmortal($x) IF human($x)", "query": "mortal(plato)", "mode": "why_not"}'

# What-if
curl -X POST http://localhost:8080/what-if \
  -H "Content-Type: application/json" \
  -d '{"base_knowledge": "human(socrates)\nmortal($x) IF human($x)", "modifications": "+ human(plato)", "query": "mortal($who)"}'

# KB validation
curl -X POST http://localhost:8080/check-kb \
  -H "Content-Type: application/json" \
  -d '{"knowledge": "human(socrates)\nmortal($x) IF human($x)"}'

Docker

The Docker image bundles SWI-Prolog + Python, so no local prerequisites are needed. Base image: swipl:stable (Debian Bookworm).

Two modes via docker-compose:

# MCP stdio — pipe to a local MCP client
docker compose run --rm euclid-mcp

# HTTP API — expose REST endpoints on port 8080
docker compose up euclid-api

Standalone usage:

# Build
docker build -t euclid-mcp .

# Run HTTP API
docker run --rm -p 8080:8080 euclid-mcp \
  python3 integrations/euclid_api.py --port 8080

# Run MCP stdio (interactive)
docker run --rm -i euclid-mcp

# Quick test — reason directly from CLI
docker run --rm euclid-mcp python3 -c "
from euclid_mcp.server import reason
r = reason(knowledge='human(socrates)\nmortal(\$x) IF human(\$x)\n? mortal(\$who)')
print(r.solutions[0].substitutions)
"

Docker image size: ~370 MB (SWI-Prolog + Python 3.11 + dependencies).

CLI pipeline

echo '{"knowledge": "red(apple)\\n? red($x)"}' | python3 integrations/euclid_cli.py

See integrations/README.md for full details.

How is Euclid?

Euclid was an ancient Greek mathematician. Living and teaching in Alexandria, he built the foundations of geometry and number theory using rigorous logical proofs.

Euclid-MCP is not:

• an LLM

• a knowledge base

• a vector database

• an agent framework

• a planner

Euclid-MCP is a deterministic inference engine that can be used by any of them.
Euclid-MCP allows deterministic and explainable replies from small LLMs on Edge hardware too.

License

Apache 2.0