Euclid-MCP

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 uses SWI-Prolog as its primary inference engine — and, where SWI-Prolog is not available (e.g. minimal containers), a pure-Python native engine that interprets Euclid-IR directly (see docs/NATIVE_ENGINE.md). It 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)│◀────│ (MCPServer) │◀────│ + Meta-IP │◀────│ (persistent) │ └──────────────┘ └──────────────────┘ └──────────────┘ └─────────────────┘

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 a persistent SWI-Prolog engine process (JSON-lines protocol on stdin/stdout; the workspace is reloaded per call, no process spawn overhead)
4. Return solutions + proof trees as structured JSON

Additional tools (explain, diagnose, what_if, check_kb) extend this core flow with natural-language explanations, analysis, scenario testing, and validation.

LLMs describe. Euclid MCP proves.

Knowledge Base

For small knowledge bases, facts and rules can be provided with each request.

A knowledge base can be loaded at server startup and reused across calls, so agents only pass the session-specific facts 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.

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.

Disjunction (OR)

Rule bodies can express alternatives with OR. The parser expands each alternative into its own Horn clause at load time — the solver only ever sees pure Horn clauses:

# A user can deploy when they are an admin, OR a dev on an approved env

can_deploy($u, $env) IF is_admin($u) OR (has_role($u, dev) AND env_approved($u, $env))

AND binds tighter than OR; parenthesized groups expand distributively. NOT applies to single goals only. A # RULE: is carried by every expanded branch, so proofs stay auditable against the source rule.

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.

A specific benchmark 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 8 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.

explain

Deterministic proof-tree → natural-language reasoning steps. No LLM involved: it walks the proof tree of each solution and renders every step in plain language, citing the rule ID (# RULE: ) when a rule has one. Use it to turn a proof into an auditable, human-readable explanation.

Returns ExplanationResult with explanations[] — each containing variable bindings, an ordered list of natural-language steps, and language-independent structured_steps (typed kind/goal/rule_id/body, ready for localized rendering in a UI).

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.

| delta_knowledge | string? | — | Session-specific facts appended to the `