boolean-algebra-engine

Coding agents: machine-readable docs are in llms.txt (summary) and llms-full.txt (full API reference).

Your AI agent's decision rules might contradict each other. LLMs can't reliably catch that. This engine can — provably, in under 10ms.

AI agents with decision logic — loan approval, compliance checks, access control, policy enforcement — run on boolean rules written by humans. Nobody verifies those rules don't conflict before the agent acts on them. This engine does.

The benchmark shows why you need it: even a 70B model gets ~20% of boolean logic questions wrong. You can't ask an LLM if your rules conflict and trust the answer. You need a deterministic layer that computes it.

90 tests passing · <10ms evaluation · zero dependencies · exhaustive enumeration, not sampling

Quick start

pip install boolean-algebra-engine

from boolean_algebra_engine import evaluate, synthesize

# Does a contradiction exist?
table, _ = evaluate("A.!A")
print(table.satisfiable)   # False — always a contradiction

# Can two rules both be true simultaneously?
table, _ = evaluate("(A.B).(!A)")
print(table.satisfiable)   # False — A and !A can't both hold

# Full truth table
table, _ = evaluate("A.(B+C)")
print(table.variables)     # ['A', 'B', 'C']
print(table.minterms)      # [5, 6, 7]
print(table.satisfiable)   # True

# Simplify to minimal form
minimal, _ = synthesize(table)
print(minimal)             # A.C+A.B

One-liner check:

python -c "from boolean_algebra_engine import evaluate; t,_ = evaluate('A.!A'); print(t.satisfiable)"
# False

pip install "boolean-algebra-engine[cli]"          # CLI / REPL
pip install "boolean-algebra-engine[mcp]"          # MCP server (Claude Desktop)
pip install "boolean-algebra-engine[api]"          # REST API
pip install "boolean-algebra-engine[nl-anthropic]" # NL layer via Anthropic
pip install "boolean-algebra-engine[nl-openai]"    # NL layer via OpenAI

The problem

Six rules. Three variables. Written by four people over six months.

A fintech AI agent auto-approves or rejects loan applications based on these rules — nobody ever verified them together. The engine checks all input combinations for every rule, in every combination:

# pip install boolean-algebra-engine[mcp]
from boolean_algebra_engine.mcp.server import check_prompt_logic

result = check_prompt_logic([
    "A.B",  # approve: good credit AND income verified
    "!A",   # reject:  bad credit
    "C",    # approve: collateral exists
    "!C",   # reject:  no collateral
])

print(result["summary"])
# {'total': 4, 'contradictions': 0, 'tautologies': 0,
#  'equivalent_pairs': 0, 'conflicting_pairs': 2}

print([(p["rule1"], p["rule2"]) for p in result["pairwise"] if p["always_conflict"]])
# [('A.B', '!A'), ('C', '!C')]

What it found:

• A.B and !A conflict — good credit approval and bad credit rejection fire simultaneously when A=1. The agent picks a winner arbitrarily.
• C and !C conflict — collateral approval and no-collateral rejection are mutually exclusive by definition. Both rules can never apply at the same time.

Nobody caught these by reading the rules. The engine caught them by checking every combination.

The benchmark

Every model tested hallucinates on boolean logic — but in different ways depending on size and architecture.

The small models aren't reasoning at all — they picked a default and stuck to it. The larger models reason but still hallucinate. llama-3.3-70b scores 20% but makes only one type of error: it assumes rules conflict when they don't (0% missed conflicts, 40% missed compatibles).

Variable curve — does complexity make it worse?

qwen3-32b was run across variable counts from 3 to 10 (8 to 1,024 truth table rows), 100 cases each. The hallucination rate stayed flat at 16–19% throughout. Complexity doesn't degrade it — the errors are a consistent baseline, not caused by harder logic.

python3 benchmark.py --provider ollama --model tinyllama --cases 20
python3 benchmark.py --provider ollama --model llama3.2:3b --cases 20
python3 benchmark.py --provider ollama --model gemma3:4b --cases 20

correct: 10/20  ·  hallucination rate: 50.0%
missed conflicts: 10/10 (100%)  ·  missed compatibles: 0/10 (0%)
pattern: always outputs "yes" — no case-by-case reasoning

correct: 10/20  ·  hallucination rate: 50.0%
missed conflicts: 0/10 (0%)  ·  missed compatibles: 10/10 (100%)
pattern: always outputs "no" — opposite default, same reasoning failure

correct: 13/20  ·  hallucination rate: 35.0%
missed conflicts: 4/10 (40%)  ·  missed compatibles: 3/10 (30%)
pattern: engages with each case individually — but wrong 1 in 3

correct: ~83/100  ·  hallucination rate: ~17%
flat across 3–10 variables — errors are a consistent baseline, not complexity-driven

correct: ~80/100  ·  hallucination rate: ~20%
missed conflicts: 0%  ·  missed compatibles: ~40%
pattern: over-cautious — never flags a conflict that isn't there, but misses 2 in 5 compatible pairs

Core API

from boolean_algebra_engine import evaluate, synthesize

# Forward: expression → truth table
table, _ = evaluate("A.(B+C)")
print(table.variables)    # ['A', 'B', 'C']
print(table.minterms)     # [5, 6, 7]
print(table.satisfiable)  # True

# Inverse: truth table → minimal expression
minimal, _ = synthesize(table)
print(minimal)            # A.C+A.B

# Equivalence and satisfiability (via MCP server functions — no HTTP, direct call)
# pip install boolean-algebra-engine[mcp]
from boolean_algebra_engine.mcp.server import equivalent, satisfiable

print(equivalent("A.(B+C)", "A.B+A.C")["equivalent"])  # True — distributive law
print(satisfiable("A.!A")["satisfiable"])               # False — contradiction

The core module has zero external dependencies. Import it into any Python project.

MCP — Claude calls the engine

Wire the engine into Claude Desktop and Claude stops predicting boolean logic. It computes it.

{
  "mcpServers": {
    "boolean-algebra-engine": {
      "command": "python",
      "args": ["-m", "boolean_algebra_engine.mcp.server"]
    }
  }
}

Five tools Claude can call mid-conversation:

• evaluate — expression → truth table
• simplify — expression → minimal form
• equivalent — are two expressions identical?
• satisfiable — does any input make this true?
• check_prompt_logic — audit a full rule set for contradictions, tautologies, conflicts, duplicates

NL Layer

The LLM translates. The engine decides. The logic is always exact.

Most tools that accept plain English rules pass them to an LLM and trust whatever comes back. That works for summarisation. It doesn't work for logic — the benchmark in this repo proves it, with hallucination rates from 17% to 50% across every model tested.

The NL layer takes a different approach. It uses the LLM for the one thing it's actually good at: mapping human language to symbols. Once the sentence becomes an expression, the LLM is done. The engine takes over, evaluates every possible input combination, and returns a result that is provably correct.

How it works

Take this sentence:

"access granted if the user is an admin, or if they're verified and not suspended"

"access granted if admin, or verified and not suspended"
        │
        ▼  LLM — maps words to variables, returns an expression
   A+(V.!S)
        │
        ▼  core engine — evaluates every input combination
   truth table: 8 rows · 6 rows where output = 1
   minimal form: A+V.!S
   satisfiable: yes  ·  tautology: no  ·  contradiction: no
        │
        ▼  LLM — turns the result back into plain English
   "Access is granted in 6 of 8 cases. A suspended verified
    user is always denied, regardless of admin status."

The LLM is involved twice — both times doing something fuzzy (language), never something exact (logic).

Try it in 5 minutes

The fastest path is Ollama — runs locally, no API key, free.

# 1. Install Ollama
curl -fsSL https://ollama.com/install.sh | sh

# 2. Pull a model
ollama pull deepseek-r1:latest   # 5.2 GB, best quality
# ollama pull deepseek-r1:1.5b  # 1.1 GB, for low-memory machines

# 3. Install the package
pip install boolean-algebra-engine

from boolean_algebra_engine.nl.nl import ask

result = ask("alarm on if door open or window open, but not if system disabled")

print(result.expression)   # (D+W).!S
print(result.minimal)      # D.!S+W.!S
print(result.satisfiable)  # True
print(result.variables)    # {'D': 'door is open', 'W': 'window is open', 'S': 'system is disabled'}
print(result.explanation)  # plain English summary

The provider is auto-detected: if Ollama is running, it is used automatically. No configuration needed.

Or via the CLI:

boolcalc ask "alarm on if door open or window open, but not if system disabled"
boolcalc check-rules "access if admin" "access if verified" "no access if suspended"

Choosing a provider

pip install "boolean-algebra-engine[nl-anthropic]"
export ANTHROPIC_API_KEY=sk-ant-...

from boolean_algebra_engine.nl.nl import ask, AnthropicProvider, OpenAIProvider, OpenAICompatProvider, OllamaProvider

# Anthropic
result = ask("...", provider=AnthropicProvider())
result = ask("...", provider=AnthropicProvider(model="claude-opus-4-7"))

# OpenAI
result = ask("...", provider=OpenAIProvider())
result = ask("...", provider=OpenAIProvider(model="gpt-4-turbo"))

# OpenAI-compatible (Groq, Together, LM Studio, vLLM)
result = ask("...", provider=OpenAICompatProvider(
    api_key="your-key",
    base_url="https://api.groq.com/openai/v1",
    model="llama3-8b-8192",
))

# Ollama with specific model or remote host
result = ask("...", provider=OllamaProvider(model="deepseek-r1:1.5b"))
result = ask("...", provider=OllamaProvider(base_url="http://192.168.1.10:11434"))

Bring your own model

Any model that can receive a system prompt and a user message can be plugged in:

from boolean_algebra_engine.nl.nl import ask, Provider

class MyProvider(Provider):
    def complete(self, system: str, user: str, max_tokens: int = 512) -> str:
        response = my_llm.generate(system_prompt=system, user_message=user, max_tokens=max_tokens)
        return response.text

result = ask("access granted if admin or verified user", provider=MyProvider())

complete() is called twice per ask() — once to parse the sentence (expects JSON back), once to explain the result (expects plain text).

What you get back

Every ask() call returns an NLResult:

@dataclass
class NLResult:
    input_sentence: str
    expression:     str          # boolean expression the LLM parsed from your sentence
    variables:      dict[str, str]  # {'A': 'user is admin', ...}
    minimal:        str          # Quine-McCluskey simplified form
    satisfiable:    bool
    tautology:      bool
    contradiction:  bool
    minterms:       list[int]    # row indices where output = 1
    maxterms:       list[int]    # row indices where output = 0
    explanation:    str          # plain English summary from the LLM
    rows:           list[dict]   # full truth table

Operators

Variables: uppercase A–Z. Parentheses override precedence. Up to 26 variables, arbitrary nesting.

Common mistake: &&, ||, ~ are not valid — use ., +, !.

Interfaces

vs SymPy and boolean.py

SymPy (sympy.logic) is more powerful for pure boolean mathematics — its DPLL-based satisfiable() scales better beyond 15 variables, and simplify_logic() covers similar minimization ground. If you're doing symbolic mathematics, use SymPy.

boolean.py handles expression parsing and symbolic simplification cleanly. If you need to manipulate boolean expressions as objects, it's the right tool.

This engine is different in three ways:

1. Zero-dependency core. SymPy pulls in numpy, mpmath, and the full symbolic stack. The core module is plain Python — no install side-effects, embeds anywhere.

2. Built for AI pipelines, not mathematics. check_prompt_logic audits a set of rules for pairwise conflicts — the kind of check you run on a system prompt or a business rule engine before an agent acts on it. Neither SymPy nor boolean.py has this concept.

3. The integration layer. MCP server for Claude Desktop, NL layer for plain English input, REST API, benchmark against LLMs — none of this exists in math-focused libraries because it's not a math problem. It's an AI reliability problem.

If you want to do boolean algebra, SymPy is the answer. If you want to verify that your AI agent's rules don't contradict each other, this is built for that.

Credibility

The engine does not sample, approximate, or predict. It evaluates every possible input combination:

• Satisfiable — an actual row where output = 1 was found
• Contradiction — every row was checked, all were 0
• Equivalent — output columns compared row-by-row across the full truth table
• Conflict — conjunction of both rules evaluated for every input, always returned 0

No black box, no model weights, no probability — just arithmetic. This is a stronger correctness claim than any probabilistic tool can make.

90 tests across unit, integration, edge cases, and round-trips. All passing.

For bugs and feature requests: open an issue.