# Bellatrix -- The Beacon Chain

## Table of contents

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- [Introduction](#introduction)
- [Custom types](#custom-types)
- [Preset](#preset)
  - [Execution](#execution)
  - [Updated penalty values](#updated-penalty-values)
- [Configuration](#configuration)
  - [Transition settings](#transition-settings)
- [Containers](#containers)
  - [Extended containers](#extended-containers)
    - [`BeaconBlockBody`](#beaconblockbody)
    - [`BeaconState`](#beaconstate)
  - [New containers](#new-containers)
    - [`ExecutionPayload`](#executionpayload)
    - [`ExecutionPayloadHeader`](#executionpayloadheader)
- [Helper functions](#helper-functions)
  - [Predicates](#predicates)
    - [`is_merge_transition_complete`](#is_merge_transition_complete)
    - [`is_merge_transition_block`](#is_merge_transition_block)
    - [`is_execution_enabled`](#is_execution_enabled)
  - [Misc](#misc)
    - [`compute_timestamp_at_slot`](#compute_timestamp_at_slot)
  - [Beacon state accessors](#beacon-state-accessors)
    - [Modified `get_inactivity_penalty_deltas`](#modified-get_inactivity_penalty_deltas)
  - [Beacon state mutators](#beacon-state-mutators)
    - [Modified `slash_validator`](#modified-slash_validator)
- [Beacon chain state transition function](#beacon-chain-state-transition-function)
  - [Execution engine](#execution-engine)
    - [`notify_new_payload`](#notify_new_payload)
  - [Block processing](#block-processing)
    - [Execution payload](#execution-payload)
      - [`process_execution_payload`](#process_execution_payload)
  - [Epoch processing](#epoch-processing)
    - [Slashings](#slashings)
- [Testing](#testing)

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## Introduction

This upgrade adds transaction execution to the beacon chain as part of Bellatrix upgrade.

Additionally, this upgrade introduces the following minor changes:
* Penalty parameter updates to their planned maximally punitive values

## Custom types

*Note*: The `Transaction` type is a stub which is not final.

| Name | SSZ equivalent | Description |
| - | - | - |
| `Transaction` | `ByteList[MAX_BYTES_PER_TRANSACTION]` | either a [typed transaction envelope](https://eips.ethereum.org/EIPS/eip-2718#opaque-byte-array-rather-than-an-rlp-array) or a legacy transaction|
| `ExecutionAddress` | `Bytes20` | Address of account on the execution layer |

## Preset

### Execution

| Name | Value |
| - | - |
| `MAX_BYTES_PER_TRANSACTION` | `uint64(2**30)` (= 1,073,741,824) |
| `MAX_TRANSACTIONS_PER_PAYLOAD` | `uint64(2**20)` (= 1,048,576) |
| `BYTES_PER_LOGS_BLOOM` | `uint64(2**8)` (= 256) |
| `MAX_EXTRA_DATA_BYTES` | `2**5` (= 32) |

### Updated penalty values

Bellatrix updates a few configuration values to move penalty parameters to their final, maximum security values.

*Note*: The spec does *not* override previous configuration values but instead creates new values and replaces usage throughout.

| Name | Value |
| - | - |
| `INACTIVITY_PENALTY_QUOTIENT_BELLATRIX` | `uint64(2**24)` (= 16,777,216) |
| `MIN_SLASHING_PENALTY_QUOTIENT_BELLATRIX` | `uint64(2**5)` (= 32) |
| `PROPORTIONAL_SLASHING_MULTIPLIER_BELLATRIX` | `uint64(3)` |

## Configuration

### Transition settings

| Name | Value |
| - | - |
| `TERMINAL_TOTAL_DIFFICULTY` | `58750000000000000000000` (Estimated: Sept 15, 2022)|
| `TERMINAL_BLOCK_HASH` | `Hash32()` |
| `TERMINAL_BLOCK_HASH_ACTIVATION_EPOCH` | `FAR_FUTURE_EPOCH` |

## Containers

### Extended containers

#### `BeaconBlockBody`

```python
class BeaconBlockBody(Container):
    randao_reveal: BLSSignature
    eth1_data: Eth1Data  # Eth1 data vote
    graffiti: Bytes32  # Arbitrary data
    # Operations
    proposer_slashings: List[ProposerSlashing, MAX_PROPOSER_SLASHINGS]
    attester_slashings: List[AttesterSlashing, MAX_ATTESTER_SLASHINGS]
    attestations: List[Attestation, MAX_ATTESTATIONS]
    deposits: List[Deposit, MAX_DEPOSITS]
    voluntary_exits: List[SignedVoluntaryExit, MAX_VOLUNTARY_EXITS]
    sync_aggregate: SyncAggregate
    # Execution
    execution_payload: ExecutionPayload  # [New in Bellatrix]
```

#### `BeaconState`

```python
class BeaconState(Container):
    # Versioning
    genesis_time: uint64
    genesis_validators_root: Root
    slot: Slot
    fork: Fork
    # History
    latest_block_header: BeaconBlockHeader
    block_roots: Vector[Root, SLOTS_PER_HISTORICAL_ROOT]
    state_roots: Vector[Root, SLOTS_PER_HISTORICAL_ROOT]
    historical_roots: List[Root, HISTORICAL_ROOTS_LIMIT]
    # Eth1
    eth1_data: Eth1Data
    eth1_data_votes: List[Eth1Data, EPOCHS_PER_ETH1_VOTING_PERIOD * SLOTS_PER_EPOCH]
    eth1_deposit_index: uint64
    # Registry
    validators: List[Validator, VALIDATOR_REGISTRY_LIMIT]
    balances: List[Gwei, VALIDATOR_REGISTRY_LIMIT]
    # Randomness
    randao_mixes: Vector[Bytes32, EPOCHS_PER_HISTORICAL_VECTOR]
    # Slashings
    slashings: Vector[Gwei, EPOCHS_PER_SLASHINGS_VECTOR]  # Per-epoch sums of slashed effective balances
    # Participation
    previous_epoch_participation: List[ParticipationFlags, VALIDATOR_REGISTRY_LIMIT]
    current_epoch_participation: List[ParticipationFlags, VALIDATOR_REGISTRY_LIMIT]
    # Finality
    justification_bits: Bitvector[JUSTIFICATION_BITS_LENGTH]  # Bit set for every recent justified epoch
    previous_justified_checkpoint: Checkpoint
    current_justified_checkpoint: Checkpoint
    finalized_checkpoint: Checkpoint
    # Inactivity
    inactivity_scores: List[uint64, VALIDATOR_REGISTRY_LIMIT]
    # Sync
    current_sync_committee: SyncCommittee
    next_sync_committee: SyncCommittee
    # Execution
    latest_execution_payload_header: ExecutionPayloadHeader  # [New in Bellatrix]
```

### New containers

#### `ExecutionPayload`

```python
class ExecutionPayload(Container):
    # Execution block header fields
    parent_hash: Hash32
    fee_recipient: ExecutionAddress  # 'beneficiary' in the yellow paper
    state_root: Bytes32
    receipts_root: Bytes32
    logs_bloom: ByteVector[BYTES_PER_LOGS_BLOOM]
    prev_randao: Bytes32  # 'difficulty' in the yellow paper
    block_number: uint64  # 'number' in the yellow paper
    gas_limit: uint64
    gas_used: uint64
    timestamp: uint64
    extra_data: ByteList[MAX_EXTRA_DATA_BYTES]
    base_fee_per_gas: uint256
    # Extra payload fields
    block_hash: Hash32  # Hash of execution block
    transactions: List[Transaction, MAX_TRANSACTIONS_PER_PAYLOAD]
```

#### `ExecutionPayloadHeader`

```python
class ExecutionPayloadHeader(Container):
    # Execution block header fields
    parent_hash: Hash32
    fee_recipient: ExecutionAddress
    state_root: Bytes32
    receipts_root: Bytes32
    logs_bloom: ByteVector[BYTES_PER_LOGS_BLOOM]
    prev_randao: Bytes32
    block_number: uint64
    gas_limit: uint64
    gas_used: uint64
    timestamp: uint64
    extra_data: ByteList[MAX_EXTRA_DATA_BYTES]
    base_fee_per_gas: uint256
    # Extra payload fields
    block_hash: Hash32  # Hash of execution block
    transactions_root: Root
```

## Helper functions

### Predicates

#### `is_merge_transition_complete`

```python
def is_merge_transition_complete(state: BeaconState) -> bool:
    return state.latest_execution_payload_header != ExecutionPayloadHeader()
```

#### `is_merge_transition_block`

```python
def is_merge_transition_block(state: BeaconState, body: BeaconBlockBody) -> bool:
    return not is_merge_transition_complete(state) and body.execution_payload != ExecutionPayload()
```

#### `is_execution_enabled`

```python
def is_execution_enabled(state: BeaconState, body: BeaconBlockBody) -> bool:
    return is_merge_transition_block(state, body) or is_merge_transition_complete(state)
```

### Misc

#### `compute_timestamp_at_slot`

*Note*: This function is unsafe with respect to overflows and underflows.

```python
def compute_timestamp_at_slot(state: BeaconState, slot: Slot) -> uint64:
    slots_since_genesis = slot - GENESIS_SLOT
    return uint64(state.genesis_time + slots_since_genesis * SECONDS_PER_SLOT)
```

### Beacon state accessors

#### Modified `get_inactivity_penalty_deltas`

*Note*: The function `get_inactivity_penalty_deltas` is modified to use `INACTIVITY_PENALTY_QUOTIENT_BELLATRIX`.

```python
def get_inactivity_penalty_deltas(state: BeaconState) -> Tuple[Sequence[Gwei], Sequence[Gwei]]:
    """
    Return the inactivity penalty deltas by considering timely target participation flags and inactivity scores.
    """
    rewards = [Gwei(0) for _ in range(len(state.validators))]
    penalties = [Gwei(0) for _ in range(len(state.validators))]
    previous_epoch = get_previous_epoch(state)
    matching_target_indices = get_unslashed_participating_indices(state, TIMELY_TARGET_FLAG_INDEX, previous_epoch)
    for index in get_eligible_validator_indices(state):
        if index not in matching_target_indices:
            penalty_numerator = state.validators[index].effective_balance * state.inactivity_scores[index]
            # [Modified in Bellatrix]
            penalty_denominator = INACTIVITY_SCORE_BIAS * INACTIVITY_PENALTY_QUOTIENT_BELLATRIX
            penalties[index] += Gwei(penalty_numerator // penalty_denominator)
    return rewards, penalties
```

### Beacon state mutators

#### Modified `slash_validator`

*Note*: The function `slash_validator` is modified to use `MIN_SLASHING_PENALTY_QUOTIENT_BELLATRIX`.

```python
def slash_validator(state: BeaconState,
                    slashed_index: ValidatorIndex,
                    whistleblower_index: ValidatorIndex=None) -> None:
    """
    Slash the validator with index ``slashed_index``.
    """
    epoch = get_current_epoch(state)
    initiate_validator_exit(state, slashed_index)
    validator = state.validators[slashed_index]
    validator.slashed = True
    validator.withdrawable_epoch = max(validator.withdrawable_epoch, Epoch(epoch + EPOCHS_PER_SLASHINGS_VECTOR))
    state.slashings[epoch % EPOCHS_PER_SLASHINGS_VECTOR] += validator.effective_balance
    slashing_penalty = validator.effective_balance // MIN_SLASHING_PENALTY_QUOTIENT_BELLATRIX  # [Modified in Bellatrix]
    decrease_balance(state, slashed_index, slashing_penalty)

    # Apply proposer and whistleblower rewards
    proposer_index = get_beacon_proposer_index(state)
    if whistleblower_index is None:
        whistleblower_index = proposer_index
    whistleblower_reward = Gwei(validator.effective_balance // WHISTLEBLOWER_REWARD_QUOTIENT)
    proposer_reward = Gwei(whistleblower_reward * PROPOSER_WEIGHT // WEIGHT_DENOMINATOR)
    increase_balance(state, proposer_index, proposer_reward)
    increase_balance(state, whistleblower_index, Gwei(whistleblower_reward - proposer_reward))
```

## Beacon chain state transition function

### Execution engine

The implementation-dependent `ExecutionEngine` protocol encapsulates the execution sub-system logic via:

* a state object `self.execution_state` of type `ExecutionState`
* a notification function `self.notify_new_payload` which may apply changes to the `self.execution_state`

*Note*: `notify_new_payload` is a function accessed through the `EXECUTION_ENGINE` module which instantiates the `ExecutionEngine` protocol.

The body of this function is implementation dependent.
The Engine API may be used to implement this and similarly defined functions via an external execution engine.

#### `notify_new_payload`

```python
def notify_new_payload(self: ExecutionEngine, execution_payload: ExecutionPayload) -> bool:
    """
    Return ``True`` if and only if ``execution_payload`` is valid with respect to ``self.execution_state``.
    """
    ...
```

### Block processing

*Note*: The call to the `process_execution_payload` must happen before the call to the `process_randao` as the former depends on the `randao_mix` computed with the reveal of the previous block.

```python
def process_block(state: BeaconState, block: BeaconBlock) -> None:
    process_block_header(state, block)
    if is_execution_enabled(state, block.body):
        process_execution_payload(state, block.body.execution_payload, EXECUTION_ENGINE)  # [New in Bellatrix]
    process_randao(state, block.body)
    process_eth1_data(state, block.body)
    process_operations(state, block.body)
    process_sync_aggregate(state, block.body.sync_aggregate)
```

#### Execution payload

##### `process_execution_payload`

```python
def process_execution_payload(state: BeaconState, payload: ExecutionPayload, execution_engine: ExecutionEngine) -> None:
    # Verify consistency of the parent hash with respect to the previous execution payload header
    if is_merge_transition_complete(state):
        assert payload.parent_hash == state.latest_execution_payload_header.block_hash
    # Verify prev_randao
    assert payload.prev_randao == get_randao_mix(state, get_current_epoch(state))
    # Verify timestamp
    assert payload.timestamp == compute_timestamp_at_slot(state, state.slot)
    # Verify the execution payload is valid
    assert execution_engine.notify_new_payload(payload)
    # Cache execution payload header
    state.latest_execution_payload_header = ExecutionPayloadHeader(
        parent_hash=payload.parent_hash,
        fee_recipient=payload.fee_recipient,
        state_root=payload.state_root,
        receipts_root=payload.receipts_root,
        logs_bloom=payload.logs_bloom,
        prev_randao=payload.prev_randao,
        block_number=payload.block_number,
        gas_limit=payload.gas_limit,
        gas_used=payload.gas_used,
        timestamp=payload.timestamp,
        extra_data=payload.extra_data,
        base_fee_per_gas=payload.base_fee_per_gas,
        block_hash=payload.block_hash,
        transactions_root=hash_tree_root(payload.transactions),
    )
```

### Epoch processing

#### Slashings

*Note*: The function `process_slashings` is modified to use `PROPORTIONAL_SLASHING_MULTIPLIER_BELLATRIX`.

```python
def process_slashings(state: BeaconState) -> None:
    epoch = get_current_epoch(state)
    total_balance = get_total_active_balance(state)
    adjusted_total_slashing_balance = min(
        sum(state.slashings) * PROPORTIONAL_SLASHING_MULTIPLIER_BELLATRIX,  # [Modified in Bellatrix]
        total_balance
    )
    for index, validator in enumerate(state.validators):
        if validator.slashed and epoch + EPOCHS_PER_SLASHINGS_VECTOR // 2 == validator.withdrawable_epoch:
            increment = EFFECTIVE_BALANCE_INCREMENT  # Factored out from penalty numerator to avoid uint64 overflow
            penalty_numerator = validator.effective_balance // increment * adjusted_total_slashing_balance
            penalty = penalty_numerator // total_balance * increment
            decrease_balance(state, ValidatorIndex(index), penalty)
```

## Testing

*Note*: The function `initialize_beacon_state_from_eth1` is modified for pure Bellatrix testing only.
Modifications include:
1. Use `BELLATRIX_FORK_VERSION` as the previous and current fork version.
2. Utilize the Bellatrix `BeaconBlockBody` when constructing the initial `latest_block_header`.
3. Initialize `latest_execution_payload_header`.
  If `execution_payload_header == ExecutionPayloadHeader()`, then the Merge has not yet occurred.
  Else, the Merge starts from genesis and the transition is incomplete.

```python
def initialize_beacon_state_from_eth1(eth1_block_hash: Hash32,
                                      eth1_timestamp: uint64,
                                      deposits: Sequence[Deposit],
                                      execution_payload_header: ExecutionPayloadHeader=ExecutionPayloadHeader()
                                      ) -> BeaconState:
    fork = Fork(
        previous_version=BELLATRIX_FORK_VERSION,  # [Modified in Bellatrix] for testing only
        current_version=BELLATRIX_FORK_VERSION,  # [Modified in Bellatrix]
        epoch=GENESIS_EPOCH,
    )
    state = BeaconState(
        genesis_time=eth1_timestamp + GENESIS_DELAY,
        fork=fork,
        eth1_data=Eth1Data(block_hash=eth1_block_hash, deposit_count=uint64(len(deposits))),
        latest_block_header=BeaconBlockHeader(body_root=hash_tree_root(BeaconBlockBody())),
        randao_mixes=[eth1_block_hash] * EPOCHS_PER_HISTORICAL_VECTOR,  # Seed RANDAO with Eth1 entropy
    )

    # Process deposits
    leaves = list(map(lambda deposit: deposit.data, deposits))
    for index, deposit in enumerate(deposits):
        deposit_data_list = List[DepositData, 2**DEPOSIT_CONTRACT_TREE_DEPTH](*leaves[:index + 1])
        state.eth1_data.deposit_root = hash_tree_root(deposit_data_list)
        process_deposit(state, deposit)

    # Process activations
    for index, validator in enumerate(state.validators):
        balance = state.balances[index]
        validator.effective_balance = min(balance - balance % EFFECTIVE_BALANCE_INCREMENT, MAX_EFFECTIVE_BALANCE)
        if validator.effective_balance == MAX_EFFECTIVE_BALANCE:
            validator.activation_eligibility_epoch = GENESIS_EPOCH
            validator.activation_epoch = GENESIS_EPOCH

    # Set genesis validators root for domain separation and chain versioning
    state.genesis_validators_root = hash_tree_root(state.validators)

    # Fill in sync committees
    # Note: A duplicate committee is assigned for the current and next committee at genesis
    state.current_sync_committee = get_next_sync_committee(state)
    state.next_sync_committee = get_next_sync_committee(state)

    # [New in Bellatrix] Initialize the execution payload header
    # If empty, will initialize a chain that has not yet gone through the Merge transition
    state.latest_execution_payload_header = execution_payload_header

    return state
```
