k3c 0.4.1

Kulkarni Calculus (K3) — Design causality, and the system emerges.

k3c — Kulkarni Calculus (K3) Python SDK

Design causality, and the system emerges.

pip install k3c

Python 3.12+ | Zero dependencies | MIT License

---

What is K3?

K3 is a minimal causal calculus for discrete systems. A system is fully determined by specifying:

• What events are permitted (guards)
• How state transforms (transitions)
• What must always be true (invariants)
• What must eventually become true (liveness)

The spec IS the implementation. Drift between specification and implementation is structurally impossible because the spec compiles into the system's own guards, transitions, and invariants.

Quick Start

from k3c import (
    Spec, Permit, Maintain, Universe, Ok, Impossible, Violated,
    Always, Compare, CmpOp, Field, Var, EventField, LInt,
)

# 1. Define the spec (declarative, frozen, serializable)
spec = Spec(
    name="bank",
    state0={"balance": 100},
    permits=(
        Permit(
            name="has_funds",
            on="Withdraw",
            when=Compare(CmpOp.GE, Field(Var("state"), "balance"), EventField("amount")),
        ),
    ),
    maintains=(
        Maintain(
            name="non_negative",
            expr=Always(Compare(CmpOp.GE, Field(Var("state"), "balance"), LInt(0))),
        ),
    ),
)

# 2. Write a pure transition function
def bank_transition(state: dict, event: dict) -> dict:
    match event.get("type"):
        case "Deposit":
            return {**state, "balance": state["balance"] + event["amount"]}
        case "Withdraw":
            return {**state, "balance": state["balance"] - event["amount"]}
        case _:
            return state

# 3. Create a Universe and apply events
u = Universe(spec=spec, transition=bank_transition)

match u.apply({"type": "Deposit", "amount": 50}):
    case Ok(state=s):       print(s["balance"])      # 150
    case Impossible(why=w): print(w.message)
    case Violated(why=w):   w.raise_()

match u.apply({"type": "Withdraw", "amount": 200}):
    case Ok(state=s):       print(s["balance"])
    case Impossible(why=w): print(w.rule)            # "has_funds"

The Three Results

apply() is total — it never raises. It always returns one of:

Result	Meaning	Bug?
Ok(state, ctx, step_hash)	Guards passed, invariants held	No
Impossible(why)	Guard rejected event. State unchanged	No
Violated(why)	Invariant broken. Implementation diverged	Yes

The 9-Tuple

A system is a 9-tuple K3d = (S, S0, E, G, T, N, P, Ctx, L):

Element	Role	Defined by
S, S0	State space & initial state	state0 dict
E	Event space	Dicts with "type" key
G	Guards (admissibility)	Permit clauses
T	Transition function	Pure (state, event) -> state
N	Safety invariants	Maintain(expr=Always(...))
P	Projections (derived views)	Projection clauses
Ctx	Spec context (witness)	Require clauses
L	Liveness obligations	Maintain(expr=Always(Eventually(...)))

Universe Algebra

Compose (parallel)

def router(event):
    return "right" if event.get("type", "").startswith("Pay") else "left"

composed = order_u.compose(payment_u, router)
composed.apply({"type": "PlaceOrder", "amount": 100})  # routes to order_u
composed.apply({"type": "PayInvoice", "amount": 100})  # routes to payment_u

Bridge (cross-universe)

from k3c import BridgeMode

audited = order_u.bridge(
    audit_u,
    mapper=lambda s, e, ns: {"type": "AuditEntry", "action": e.get("type")},
    mode=BridgeMode.SYNCHRONOUS,
)

Bridge modes: SYNCHRONOUS (atomic), ASYNC (source first), BEST_EFFORT (fire and forget).

Fuzz Testing (KC-5)

report = u.fuzz(sequences=1000, steps=100, seed=42)

if not report.passed:
    v = report.violations[0]
    print(f"Violation: {v.violated.why.rule}")
    print(f"Minimal reproducer: {v.shrunk_sequence}")

Explain (Debugging)

trace = u.explain({"type": "Withdraw", "amount": 500})
print(trace.summary())

for entry in trace.trace:
    print(f"{entry.phase}: {entry.clause} -> {entry.verdict}")

Simulate + Replay (KC-3)

run = u.simulate(events)
for t in range(run.processed):
    print(f"Step {t}: {run.state_at(t)}")

# Replay for determinism verification
u2 = Universe(spec=spec, transition=bank_transition)
replay = u2.replay(events, expected_hashes=run.step_hashes)
assert replay.passed

Parallel Processing

from k3c import parallel_reduce

specs = [spec.slice(from_state=checkpoint, events=["RT3"]) for checkpoint in checkpoints]
result = parallel_reduce(transition=fn, specs=specs, chunks=chunks, workers=8)
assert result.passed

Hash Chain

Every step produces a tamper-evident chained hash:

step_hash = SHA-256(state, event, prev_step_hash)

Pluggable: sha256 (default), blake2b (2x faster), blake3 (4-6x, pip install blake3).

KC Compliance Levels

Level	Name	What it adds
KC-1	Core Semantics	S, S0, E, G, T, N
KC-2	Observable	+ Projections (P)
KC-3	Traceable	+ Replay (Samsara)
KC-4	Compositional	+ Compose, Bridge
KC-5	Verified	+ Fuzz, Verify
KC-6	Certified	+ Signed step_hash chain

Architecture

k3c.ir       — K3l expression IR (43 nodes), eval(), serde, type system
k3c.spec     — Declarative Spec model, compile, extractors
k3c.engine   — Apply pipeline, SpecCtx, results, explain
k3c.runtime  — Universe, compose, bridge, isolate, parallel, samsara
k3c.emit     — Code emission (Python, TypeScript, SQL)
k3c.testing  — Fuzz testing with automatic shrinking

---

k3c.dev | Python 3.12+ | Zero dependencies | MIT License