Impossibility of FLP
Foundations of Blockchain
Suyash Bagad
Impossibility of FLP
- No deterministic protocol solves consensus in a message-relaying asynchronous system in which at most one process may fail by crashing
consensus
message-relaying
asynchronous
crashing
- Unbounded delay in processing and delivering messages
- Undetectable faulty process
- Agreement
- Termination
- Validity
- No byzantine failures
\((p,m)\)
- \(\texttt{send}(e), \ e=(p,m)\)
Impossibility of FLP
- No deterministic protocol solves consensus in a message-relaying asynchronous system in which at most one process may fail by crashing
consensus
message-relaying
asynchronous
crashing
- Unbounded delay in processing and delivering messages
- Undetectable faulty process
- Agreement
- Termination
- Validity
\((p,m)\)
- \(\texttt{send}(e), \ e=(p,m)\)
- \(\texttt{receive}(p)\)
- No byzantine failures
Impossibility of FLP
- No deterministic protocol solves consensus in a message-relaying asynchronous system in which at most one process may fail by crashing
consensus
message-relaying
asynchronous
crashing
- Unbounded delay in processing and delivering messages
- Undetectable faulty process
- Agreement
- Termination
- Validity
\(p\)
- \(\texttt{send}(e), \ e=(p,m)\)
- \(\texttt{receive}(p)\)
\(m\)
- No byzantine failures
Impossibility of FLP
- No deterministic protocol solves consensus in a message-relaying asynchronous system in which at most one process may fail by crashing
consensus
message-relaying
asynchronous
crashing
- Unbounded delay in processing and delivering messages
- Undetectable faulty process
- Agreement
- Termination
- Validity
\(p\)
- \(\texttt{send}(e), \ e=(p,m)\)
- \(\texttt{receive}(p)\)
\((p,m)\)
\(\perp\)
- No byzantine failures
Impossibility of FLP
- No deterministic protocol solves consensus in a message-relaying asynchronous system in which at most one process may fail by crashing
consensus
message-relaying
asynchronous
crashing
- Unbounded delay in processing and delivering messages
- Undetectable faulty process
- Agreement
- Termination
- Validity
\(p\)
- \(\texttt{send}(e), \ e=(p,m)\)
- \(\texttt{receive}(p)\)
\((p,m)\)
\(\perp\)
deterministic
- No byzantine failures
Model
\(i_1, o_1\)
\(i_2, o_2\)
\(i_3, o_3\)
\(i_4, o_4\)
\(i_5, o_5\)
\(i_6, o_6\)
- Configuration \(C = (s, M)\)
\(M\)
- Internal state \(s = \Big\{I_k, i_k, o_k\Big\}_{k \in [N]} \)
- Each \(i_k \in \{0,1\}, o_k \in \{b,0,1\}\)
- Step \(C \longrightarrow C' = (s', M')\) where
- Event \(e = (p,m)\)
- \(C' = e(C)\)
\((p_3, m)\)
\(C\)
Model
\(i_1, o_1\)
\(i_2, o_2\)
\(i_3, o_3\)
\(i_4, o_4\)
\(i_5, o_5\)
\(i_6, o_6\)
- Configuration \(C = (s, M)\)
\(M\)
- Internal state \(s = \Big\{I_k, i_k, o_k\Big\}_{k \in [N]} \)
- Each \(i_k \in \{0,1\}, o_k \in \{b,0,1\}\)
- Step \(C \longrightarrow C' = (s', M')\) where
- Event \(e = (p,m)\)
- \(C' = e(C)\)
\(m\)
\(C\)
Model
\(i_1, o_1\)
\(i_2, o_2\)
\(i_3, o_3^{\prime}\)
\(i_4, o_4\)
\(i_5, o_5\)
\(i_6, o_6\)
- Configuration \(C = (s, M)\)
\(M\)
- Internal state \(s = \Big\{I_k, i_k, o_k\Big\}_{k \in [N]} \)
- Each \(i_k \in \{0,1\}, o_k \in \{b,0,1\}\)
- Step \(C \longrightarrow C' = (s', M')\) where
- Event \(e = (p,m)\)
- \(C' = e(C)\)
\(m\)
\(C\)
Model
\(i_1, o_1\)
\(i_2, o_2\)
\(i_3, o_3^{\prime}\)
\(i_4, o_4\)
\(i_5, o_5\)
\(i_6, o_6\)
- Configuration \(C = (s, M)\)
\(M\)
- Internal state \(s = \Big\{I_k, i_k, o_k\Big\}_{k \in [N]} \)
- Each \(i_k \in \{0,1\}, o_k \in \{b,0,1\}\)
- Step \(C \longrightarrow C' = (s', M')\) where
- Event \(e = (p,m)\)
- \(C' = e(C)\)
\(C\)
\((p_1,m_1),\)
\((p_4,m_4)\)
\(\texttt{send}\big\{\)
\(\big\}\)
Model
\(i_1, o_1\)
\(i_2, o_2\)
\(i_3, o_3^{\prime}\)
\(i_4, o_4\)
\(i_5, o_5\)
\(i_6, o_6\)
- Configuration \(C = (s, M)\)
\(M\)
- Internal state \(s = \Big\{I_k, i_k, o_k\Big\}_{k \in [N]} \)
- Each \(i_k \in \{0,1\}, o_k \in \{b,0,1\}\)
- Step \(C \longrightarrow C' = (s', M')\) where
- Event \(e = (p,m)\)
- \(C' = e(C)\)
\(C'\)
\((p_1,m_1),\)
\((p_4,m_4)\)
- Schedule \(\sigma = (e_1, e_2, \dots, e_J)\) - finite or infinite sequence of events
- \(\sigma(C) = e_J\bigg(\dots\Big(e_2\big(e_1(C)\big)\Big)\bigg)\)
- Run \(R = (C,\sigma)\)
Commutativity of Schedules
\(C\)
\(C\)
\(C_1\)
\(C_2\)
\(C_3\)
\(\sigma_1\)
\(\sigma_2\)
\(\sigma_2\)
\(\sigma_1\)
- If the steps in schedules \(\sigma_1\) and \(\sigma_2\) are mutually disjoint, then
Proof Sketch
- For a protocol \(P\), there is an initial configuration that is bivalent.
- On applying some event to a bivalent initial configuration, it is possible to reach another bivalent configuration.
- Continue to another bivalent configuration... \(\implies\) infinite undecided run!
\(C\)
\(C_1\)
\(C_M\)
\(C_2\)
\(\sigma_1\)
\(\sigma_M\)
\(\sigma_2\)
\(o_p = 0\)
\(\vdots\)
\(o_p = 0\)
\(o_p = 0\)
\(o_p = 0\)
\(C\)
\(C_1\)
\(\sigma_1\)
\(C_2\)
\(\sigma_2\)
\(C_M\)
\(\sigma_M\)
\(\vdots\)
\(o_p = 0\)
\(o_p = 1\)
\(o_p = 1\)
\(0\)-valent
Bivalent/Undecided
Bivalent Initial Configuration
- Protocol \(P\) has a bivalent initial configuration.
\(\rightarrow\) Suppose not. By validity, all initial configurations must be \(0\)-valent or \(1\)-valent.
\(p_1\)
\(p_2\)
\(p_3\)
\(p_4\)
\(p_5\)
\(p_6\)
\(0\)
\(0\)
\(0\)
\(0\)
\(0\)
\(0\)
\(C_0\)
\(0\)
\(C_1\)
\(1\)
\(0\)
\(0\)
\(0\)
\(0\)
\(1\)
\(0\)
\(0\)
\(C_2\)
\(0\)
\(0\)
\(0\)
\(1\)
\(0\)
\(0\)
\(0\)
\(C_i\)
\(0\)
\(1\)
\(0\)
\(1\)
\(0\)
\(1\)
\(0\)
\(C_{i+1}\)
\(0\)
\(1\)
\(0\)
\(1\)
\(1\)
\(0\)
\(\vdots\)
\(\vdots\)
\(\vdots\)
\(\vdots\)
\(\vdots\)
\(\vdots\)
Contradiction!
\(1\)
Procrastination Lemma
- Let \(C\) be bivalent, and let \(e\) be a step applicable to \(C\). Then, there is a (possibly empty) schedule \(\sigma\) not containing \(e\) s.t. \(e\big(\sigma(C)\big)\) is bivalent
\(\rightarrow\) We'll prove this too by contraction.
\(C\)
\(\sigma_1\)
\(C_1\)
\(C_2\)
\(\sigma_2\)
\(C_i\)
\(\sigma_i\)
\(\vdots\)
\(\vdots\)
\(C_{M}\)
\(\sigma_{M}\)
\(D_1\)
\(\mathbb{C}\)
\(D_2\)
\(D_i\)
\(D_{M}\)
\(e\)
\(e\)
\(e\)
\(e\)
\(\vdots\)
\(\vdots\)
\(\mathbb{D}\)
\(C_i\)
\(D_i\)
- Assume \(\mathbb{D}\) has no bivalent configurations, so it must contain both \(0\)-valent and \(1\)-valent configurations
Procrastination Lemma
- Let \(C\) be bivalent, and let \(e\) be a step applicable to \(C\). Then, there is a (possibly empty) schedule \(\sigma\) not containing \(e\) s.t. \(e\big(\sigma(C)\big)\) is bivalent
\(C\)
\(\sigma(\bar{e})\)
\(E_0\)
\(D_0\)
\(e\)
Assume \(\mathbb{D}\) has no bivalent configurations, so it must contain both \(0\)-valent and \(1\)-valent configurations
\(\rightarrow\)
\(D_0\)
\(E_0\)
\(\sigma(\bar{e})\)
\(e\)
\(0\)
\(C\) is bivalent \(\implies \exists E_0, E_1 \) reachable from \(C\) which is \(0,1\)-valent resp.
\(\mathbb{C}\)
\(0\)
\(\mathbb{D}\)
\(\mathbb{D}\)
\(0\)
\(0\)
\(D_0 \in \mathbb{D}\) is \(0\)-valent!
Similarly, we can show that \(\exists D_1 \in \mathbb{D}\) is \(1\)-valent!
Procrastination Lemma
- Let \(C\) be bivalent, and let \(e\) be a step applicable to \(C\). Then, there is a (possibly empty) schedule \(\sigma\) not containing \(e\) s.t. \(e\big(\sigma(C)\big)\) is bivalent
Consider neighbours \(C_0, C_1 \in \mathbb{C}\) s.t. \(C_1 = e'(C_0), \ e' = (p',m')\)
\(\rightarrow\)
\(C\)
\(\sigma(\bar{e})\)
\(C_0\)
\(C_{1}\)
\(D_0\)
\(D_{1}\)
\(e\)
\(e\)
\(e'\)
\(\mathbb{C}\)
\(\mathbb{D}\)
Let \(D_0, D_1 \in \mathbb{D}\) be \(0,1\)-valent resp.
Case I: \(p' \neq p\)
- Apply \(e'\) to \(D_0\)
- By commutativity, we have \(e'(D_0) = D_1\)
- But \(D_0\) was \(0\)-valent! Contradiction!
\(0\)
\(1\)
\(e'\)
Procrastination Lemma
- Let \(C\) be bivalent, and let \(e\) be a step applicable to \(C\). Then, there is a (possibly empty) schedule \(\sigma\) not containing \(e\) s.t. \(e\big(\sigma(C)\big)\) is bivalent
Consider neighbours \(C_0, C_1 \in \mathbb{C}\) s.t. \(C_1 = e'(C_0), \ e' = (p',m')\)
\(\rightarrow\)
\(C\)
\(\sigma(\bar{e})\)
\(C_0\)
\(C_{1}\)
\(D_0\)
\(D_{1}\)
\(e\)
\(e\)
\(e'\)
\(\mathbb{C}\)
\(\mathbb{D}\)
Let \(D_0, D_1 \in \mathbb{D}\) be \(0,1\)-valent resp.
Case II: \(p' = p\)
\(0\)
\(1\)
- Let \(R\) be a deciding run from \(C_0\) in which \(p\) fails
- Apply \(R\) to \(D_0, D_1\)
- Apply \(e, (e, e')\) to A to reach \(E_0, E_1\)
- Thus, \(A\) is bivalent. Contradiction!
\(R(\bar{p})\)
\(A\)
\(E_0\)
\(E_1\)
\(R(\bar{p})\)
\(R(\bar{p})\)
\(e\)
\(e,e'\)
\(0\)
\(1\)
Circumventing FLP
- No deterministic protocol solves consensus in a message-relaying asynchronous system in which at most one process may fail by crashing
deterministic
asynchronous
one
crashing
- Use randomization to terminate with arbitrarily high probability
- Use failure detectors
- Agreement within a range of values with some tolerance
- \(\ldots\)