Consensus & MAS Mappings¶

Overview¶

Consensus is the fundamental problem in distributed systems: how do multiple agents agree on a single value despite failures, delays, and disagreements? AgentiCraft Foundation formalizes the four classical consensus properties, extends them with quality-weighted quorums for LLM-based agents, and provides bidirectional mappings to established multi-agent systems (MAS) theories -- BDI, Joint Intentions, SharedPlans, and Contract Net.

Consensus Properties¶

The four classical consensus properties, originally formalized for distributed processes, apply directly to multi-agent coordination.

Agreement¶

No two correct processes decide differently.

\[\forall \, p_i, p_j \in \text{Correct} : \text{decision}(p_i) = \text{decision}(p_j)\]

In an agent mesh, this means all non-faulty agents converge on the same answer, plan, or action.

Validity¶

The decided value was proposed by some correct process.

\[\text{decision}(p_i) \in \{\text{proposal}(p_j) \mid p_j \in \text{Correct}\}\]

This prevents the consensus mechanism from inventing values -- the outcome must originate from an actual agent's proposal.

Integrity¶

Every correct process decides at most once.

\[\forall \, p_i \in \text{Correct} : |\{\text{decision}(p_i)\}| \leq 1\]

Once an agent commits to a decision, it does not change its mind.

Termination¶

Every correct process eventually decides.

\[\forall \, p_i \in \text{Correct} : \exists \, t : \text{decided}(p_i, t) = \text{true}\]

The protocol does not run forever -- all correct agents reach a decision in bounded time (in synchronous systems) or with probability 1 (in randomized asynchronous systems).

Weighted Consensus¶

In LLM-based multi-agent systems, not all agents are equally reliable. An agent with a history of accurate responses should carry more weight than one prone to hallucination. Weighted consensus assigns quality weights \(w_i \in [0, 1]\) to each agent \(i\) based on historical reliability metrics, and requires weighted quorums rather than simple majorities.

Let \(W = \sum_{i} w_i\) be the total weight of all agents.

Weighted Agreement¶

No two correct weighted quorums decide differently.

\[\forall \, Q_1, Q_2 \subseteq \text{Correct} : \left(\sum_{i \in Q_1} w_i > \frac{2W}{3} \wedge \sum_{j \in Q_2} w_j > \frac{2W}{3}\right) \implies \text{decision}(Q_1) = \text{decision}(Q_2)\]

Weighted Validity¶

Decided values must come from proposals with sufficient aggregate weight.

\[\text{decision} \in \left\{\text{proposal}(p_j) \;\middle|\; \sum_{p_k \in \text{supporters}(p_j)} w_k > \frac{W}{3}\right\}\]

Weighted Quorum Intersection¶

Any two quorums share an honest-majority overlap, ensuring conflicting decisions are impossible.

\[\forall \, Q_1, Q_2 : \sum_{i \in Q_1 \cap Q_2 \cap \text{Correct}} w_i > \frac{W}{3}\]

This property is the weighted generalization of the classical requirement \(n \geq 3f + 1\). It ensures that even if some high-weight agents are Byzantine, the overlap between any two quorums contains enough honest weight to prevent disagreement.

MAS Theory Mappings¶

AgentiCraft Foundation provides bidirectional mappings between formal consensus primitives and four classical multi-agent systems theories. Each mapping preserves the formal properties of both sides -- consensus properties map to MAS invariants, and MAS constructs map to verifiable consensus configurations.

Theory	Mapping	Formal Preservation
BDI (Belief-Desire-Intention)	Beliefs \(\leftrightarrow\) context state; Desires \(\leftrightarrow\) task objectives; Intentions \(\leftrightarrow\) active assignments	Intention persistence \(\leftrightarrow\) consensus integrity
Joint Intentions (Cohen & Levesque)	Mutual belief \(\leftrightarrow\) consensus state; Persistent goal \(\leftrightarrow\) task completion condition	Joint commitment \(\leftrightarrow\) agreement property
SharedPlans (Grosz & Kraus)	Recipe \(\leftrightarrow\) task decomposition DAG; Subgroup plans \(\leftrightarrow\) agent cluster assignments	Plan completeness \(\leftrightarrow\) termination property
Contract Net (Smith 1980)	Manager broadcasts CFP \(\leftrightarrow\) proposal phase; Bidders respond \(\leftrightarrow\) vote phase; Manager awards \(\leftrightarrow\) decision phase; Execution reports \(\leftrightarrow\) commitment	Contract binding \(\leftrightarrow\) validity property

BDI Mapping¶

The Belief-Desire-Intention architecture maps naturally to consensus:

Beliefs are the agent's local state -- what it knows about the world and other agents. In consensus, this corresponds to the agent's view of proposed values and received messages.
Desires are the agent's objectives. In a consensus context, the desire is to reach agreement on a value that satisfies the task.
Intentions are the agent's committed plans. Once consensus is reached, the decided value becomes an intention that persists (integrity property).

Joint Intentions Mapping¶

Cohen and Levesque's Joint Intentions theory models how agents form and maintain shared commitments:

Mutual belief corresponds to the consensus state -- all agents believe the same decided value.
Persistent goal corresponds to the task completion condition -- agents maintain their commitment until the goal is achieved or known to be impossible.

SharedPlans Mapping¶

Grosz and Kraus's SharedPlans theory models collaborative activity through hierarchical plan structures:

Recipe (the plan structure) maps to the task decomposition DAG in a workflow.
Subgroup plans map to agent cluster assignments -- which subset of agents is responsible for which subtask.

Contract Net Mapping¶

Smith's Contract Net Protocol maps directly to a single round of consensus:

Manager broadcasts a Call For Proposals (CFP) -- analogous to the proposal phase
Bidders respond with bids -- analogous to the voting phase
Manager selects and awards -- analogous to the decision phase
Execution and reporting -- analogous to the commitment phase

Verification¶

Each mapping is verified programmatically through verify_mapping_preservation(), which checks that the formal properties of the source theory are maintained through the mapping and back.

How It Maps to Code¶

from agenticraft_foundation.specifications import (
    ConsensusSpecification, ConsensusState,
    Agreement, Validity, Integrity, Termination,
    WeightedConsensusState, WeightedAgreement, WeightedValidity,
    BDIMapping, JointIntentionMapping,
    SharedPlanMapping, ContractNetMapping,
    verify_mapping_preservation,
)

# Create a consensus specification and verify properties
spec = ConsensusSpecification()
state = ConsensusState(
    proposals={"agent-0": "plan-A", "agent-1": "plan-A", "agent-2": "plan-B"},
    decisions={"agent-0": "plan-A", "agent-1": "plan-A", "agent-2": "plan-A"},
)

# Check the four classical properties
agreement = Agreement()
validity = Validity()
assert agreement.check(state).status.name == "SATISFIED"
assert validity.check(state).status.name == "SATISFIED"

# Weighted consensus with quality-based agent weights
weighted_state = WeightedConsensusState(
    proposals={"agent-0": "plan-A", "agent-1": "plan-A", "agent-2": "plan-B"},
    decisions={"agent-0": "plan-A", "agent-1": "plan-A", "agent-2": "plan-A"},
    weights={"agent-0": 0.9, "agent-1": 0.7, "agent-2": 0.5},
)
w_agreement = WeightedAgreement()
assert w_agreement.check(weighted_state).status.name == "SATISFIED"

# Map to BDI theory and verify preservation
bdi = BDIMapping()
assert verify_mapping_preservation(bdi)