PBTS: System Model and Properties

Outline

• System model
  • Synchronized clocks
  • Message delays
• Problem Statement
• Protocol Analysis - Timely Proposals
  • Timely Proof-of-Locks
  • Derived Proof-of-Locks
• Temporal Analysis
  • Safety
  • Liveness

System Model

[PBTS-CLOCK-NEWTON.0]

There is a reference Newtonian real-time t.

No process has direct access to this reference time, used only for specification purposes. The reference real-time is assumed to be aligned with the Coordinated Universal Time (UTC).

Synchronized clocks

Processes are assumed to be equipped with synchronized clocks, aligned with the Coordinated Universal Time (UTC).

This requires processes to periodically synchronize their local clocks with an external and trusted source of the time (e.g. NTP servers). Each synchronization cycle aligns the process local clock with the external source of time, making it a fairly accurate source of real time. The periodic (re)synchronization aims to correct the drift of local clocks, which tend to pace slightly faster or slower than the real time.

To avoid an excessive level detail in the parameters and guarantees of synchronized clocks, we adopt a single system parameter PRECISION to encapsulate the potential inaccuracy of the synchronization mechanisms, and drifts of local clocks from real time.

[PBTS-CLOCK-PRECISION.0]

There exists a system parameter PRECISION, such that for any two processes p and q, with local clocks C_p and C_q:

• If p and q are equipped with synchronized clocks, then for any real-time t we have |C_p(t) - C_q(t)| <= PRECISION.

PRECISION thus bounds the difference on the times simultaneously read by processes from their local clocks, so that their clocks can be considered synchronized.

Accuracy

A second relevant clock parameter is accuracy, which binds the values read by processes from their clocks to real time.

[PBTS-CLOCK-ACCURACY.0]

For the sake of completeness, we define a parameter ACCURACY such that:

• At real time t there is at least one correct process p which clock marks C_p(t) with |C_p(t) - t| <= ACCURACY.

As a consequence, applying the definition of PRECISION, we have:

• At real time t the synchronized clock of any correct process p marks C_p(t) with |C_p(t) - t| <= ACCURACY + PRECISION.

The reason for not adopting ACCURACY as a system parameter is the assumption that PRECISION >> ACCURACY. This allows us to consider, for practical purposes, that the PRECISION system parameter embodies the ACCURACY model parameter.

Message Delays

The assumption that processes have access to synchronized clocks ensures that proposal times assigned by correct processes have a bounded relation with the real time. It is not enough, however, to identify (and reject) proposal times proposed by Byzantine processes.

To properly evaluate whether the time assigned to a proposal is consistent with the real time, we need some information regarding the time it takes for a message carrying a proposal to reach all its (correct) destinations. More precisely, the maximum delay for delivering a proposal to its destinations allows defining a lower bound, a minimum time that a correct process assigns to proposal. While minimum delay for delivering a proposal to a destination allows defining an upper bound, the maximum time assigned to a proposal.

[PBTS-MSG-DELAY.0]

There exists a system parameter MSGDELAY for end-to-end delays of proposal messages, such for any two correct processes p and q:

• If p sends a proposal message m at real time t and q receives m at real time t', then t <= t' <= t + MSGDELAY.

Notice that, as a system parameter, MSGDELAY should be observed for any proposal message broadcast by correct processes: it is a worst-case parameter. As message delays depends on the message size, the above requirement implicitly indicates that the size of proposal messages is either fixed or upper bounded.

Problem Statement

In this section we define the properties of Tendermint consensus (cf. the arXiv paper (opens new window)) in this system model.

[PBTS-PROPOSE.0]

A proposer proposes a consensus value v that includes a proposal time v.time.

We then restrict the allowed decisions along the following lines:

[PBTS-INV-AGREEMENT.0]

• [Agreement] No two correct processes decide on different values v.

This implies that no two correct processes decide on different proposal times v.time.

[PBTS-INV-VALID.0]

• [Validity] If a correct process decides on value v, then v satisfies a predefined valid predicate.

With respect to PBTS, the valid predicate requires proposal times to be monotonic over heights of consensus:

[PBTS-INV-MONOTONICITY.0]

• If a correct process decides on value v at the height h of consensus, thus setting decision[h] = v, then v.time > decision[h'].time for all previous heights h' < h.

The monotonicity of proposal times, and external validity in general, implicitly assumes that heights of consensus are executed in order.

[PBTS-INV-TIMELY.0]

• [Time-Validity] If a correct process decides on value v, then the proposal time v.time was considered timely by at least one correct process.

PBTS introduces a timely predicate that restricts the allowed decisions based on the proposal time v.time associated with a proposed value v. As a synchronous predicate, the time at which it is evaluated impacts on whether a process accepts or reject a proposal time. For this reason, the Time-Validity property refers to the previous evaluation of the timely predicate, detailed in the following section.

Protocol Analysis - Timely proposals

For PBTS, a proposal is a tuple (v, v.time, v.round), where:

• v is the proposed value;
• v.time is the associated proposal time;
• v.round is the round at which v was first proposed.

We include the proposal round v.round in the proposal definition because a value v and its associated proposal time v.time can be proposed in multiple rounds, but the evaluation of the timely predicate is only relevant at round v.round.

Considering the algorithm in the arXiv paper (opens new window), a new proposal is produced by the getValue() method, invoked by the proposer p of round round_p when starting its proposing round with a nil validValue_p. The first round at which a value v is proposed is then the round at which the proposal for v was produced, and broadcast in a PROPOSAL message with vr = -1.

[PBTS-PROPOSAL-RECEPTION.0]

The timely predicate is evaluated when a process receives a proposal. More precisely, let p be a correct process:

• proposalReceptionTime(p,r) is the time p reads from its local clock when p is at round r and receives the proposal of round r.

[PBTS-TIMELY.0]

The proposal (v, v.time, v.round) is considered timely by a correct process p if:

1. proposalReceptionTime(p,v.round) is set, and
2. proposalReceptionTime(p,v.round) >= v.time - PRECISION, and
3. proposalReceptionTime(p,v.round) <= v.time + MSGDELAY + PRECISION.

A correct process at round v.round only sends a PREVOTE for v if the associated proposal time v.time is considered timely.

Considering the algorithm in the arXiv paper (opens new window), the timely predicate is evaluated by a process p when it receives a valid PROPOSAL message from the proposer of the current round round_p with vr = -1.

Timely Proof-of-Locks

A Proof-of-Lock is a set of PREVOTE messages of round of consensus for the same value from processes whose cumulative voting power is at least 2f + 1. We denote as POL(v,r) a proof-of-lock of value v at round r.

For PBTS, we are particularly interested in the POL(v,v.round) produced in the round v.round at which a value v was first proposed. We call it a timely proof-of-lock for v because it can only be observed if at least one correct process considered it timely:

[PBTS-TIMELY-POL.0]

If

• there is a valid POL(v,r) with r = v.round, and
• POL(v,v.round) contains a PREVOTE message from at least one correct process,

Then, let p is a such correct process:

• p received a PROPOSAL message of round v.round, and
• the PROPOSAL message contained a proposal (v, v.time, v.round), and
• p was in round v.round and evaluated the proposal time v.time as timely.

The existence of a such correct process p is guaranteed provided that the voting power of Byzantine processes is bounded by 2f.

Derived Proof-of-Locks

The existence of POL(v,r) is a requirement for the decision of v at round r of consensus.

At the same time, the Time-Validity property establishes that if v is decided then a timely proof-of-lock POL(v,v.round) must have been produced.

So, we need to demonstrate here that any valid POL(v,r) is either a timely proof-of-lock or it is derived from a timely proof-of-lock:

[PBTS-DERIVED-POL.0]

If

• there is a valid POL(v,r), and
• POL(v,r) contains a PREVOTE message from at least one correct process,

Then

• there is a valid POL(v,v.round) with v.round <= r which is a timely proof-of-lock.

The above relation is trivially observed when r = v.round, as POL(v,r) must be a timely proof-of-lock. Notice that we cannot have r < v.round, as v.round is defined as the first round at which v was proposed.

For r > v.round we need to demonstrate that if there is a valid POL(v,r), then a timely POL(v,v.round) was previously obtained. We observe that a condition for observing a POL(v,r) is that the proposer of round r has broadcast a PROPOSAL message for v. As r > v.round, we can affirm that v was not produced in round r. Instead, by the protocol operation, v was a valid value for the proposer of round r, which means that if the proposer has observed a POL(v,vr) with vr < r. The above operation considers a correct proposer, but since a POL(v,r) was produced (by hypothesis) we can affirm that at least one correct process (also) observed a POL(v,vr).

Considering the algorithm in the arXiv paper (opens new window), v was proposed by the proposer p of round round_p because its validValue_p variable was set to v. The PROPOSAL message broadcast by the proposer, in this case, had vr > -1, and it could only be accepted by processes that also observed a POL(v,vr).

Thus, if there is a POL(v,r) with r > v.round, then there is a valid POL(v,vr) with v.round <= vr < r. If vr = v.round then POL(vr,v) is a timely proof-of-lock and we are done. Otherwise, there is another valid POL(v,vr') with v.round <= vr' < vr, and the above reasoning can be recursively applied until we get vr' = v.round and observe a timely proof-of-lock.

Temporal analysis

In this section we present invariants that need be observed for ensuring that PBTS is both safe and live.

In addition to the variables and system parameters already defined, we use beginRound(p,r) as the value of process p's local clock when it starts round r of consensus.

Safety

The safety of PBTS requires that if a value v is decided, then at least one correct process p considered the associated proposal time v.time timely. Following the definition of timely proposals and proof-of-locks, we require this condition to be asserted at a specific round of consensus, defined as v.round:

[PBTS-SAFETY.0]

If

• there is a valid commit C for a value v
• C contains a PRECOMMIT message from at least one correct process

then there is a correct process p (not necessarily the same above considered) such that:

• beginRound(p,v.round) <= proposalReceptionTime(p,v.round) <= beginRound(p,v.round+1) and
• proposalReceptionTime (p,v.round) - MSGDELAY - PRECISION <= v.time <= proposalReceptionTime(p,v.round) + PRECISION

That is, a correct process p started round v.round and, while still at round v.round, received a PROPOSAL message from round v.round proposing v. Moreover, the reception time of the original proposal for v, according with p's local clock, enabled p to consider the proposal time v.time as timely. This is the requirement established by PBTS for issuing a PREVOTE for the proposal (v, v.time, v.round), so for the eventual decision of v.

Liveness

The liveness of PBTS relies on correct processes accepting proposal times assigned by correct proposers. We thus present a set of conditions for assigning a proposal time v.time so that every correct process should be able to issue a PREVOTE for v.

[PBTS-LIVENESS.0]

If

• the proposer of a round r of consensus is correct
• and it proposes a value v for the first time, with associated proposal time v.time

then the proposal (v, v.time, r) is accepted by every correct process provided that:

• min{p is correct : beginRound(p,r)} <= v.time <= max{p is correct : beginRound(p,r)} and
• max{p is correct : beginRound(p,r)} <= v.time + MSGDELAY + PRECISION <= min{p is correct : beginRound(p,r+1)}

The first condition establishes a range of safe proposal times v.time for round r. This condition is trivially observed if a correct proposer p sets v.time to the time it reads from its clock when starting round r and proposing v. A PROPOSAL message sent by p at local time v.time should not be received by any correct process before its local clock reads v.time - PRECISION, so that condition 2 of [PBTS-TIMELY.0] is observed.

The second condition establishes that every correct process should start round v.round at a local time that allows v.time to still be considered timely, according to condition 3. of [PBTS-TIMELY.0]. In addition, it requires correct processes to stay long enough in round v.round so that they can receive the PROPOSAL message of round v.round. It assumed here that the proposer of v broadcasts a PROPOSAL message at time v.time, according to its local clock, so that every correct process should receive this message by time v.time + MSGDELAY + PRECISION, according to their local clocks.

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