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Introduction to Distributed Algorithms

Gerard Tel

Chapter 2

The Model - all with Video Answers

Educators

Chapter Questions

Problem 1

Define a model of dastributer zustems dihere messages rum be passed both synchronously and asynchronotisly.
The following three exercises investigate the difference between an invariant and an always-true predicate, the disjunctivity and conjunctivity of unvariants and derivates, and how invariants behave under parallel program composition.

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01:59

Problem 2

Give a transition system $S$ and an assertion $P$ such that $P$ is always true in $S$, but 25 not an inenriant of $S$.
(Hint: An example with three configurations is given in [GT90]. Carn an example with two configurations be found?)

Adriano Chikande
Adriano Chikande
Numerade Educator
04:00

Problem 3

Assume $P_1$ and $P_2$ are tnvarants of a system $S$. Prove that $\left(P_1 \vee P_2\right)$ and $\left(P_1 \wedge P_2\right)$ are invariants.
Assume $P_1$ and $P_2$ are $Q$-derivates of a system $S$. Prove that $\left(P_1 \vee P_2\right)$ and $\left(P_1 \wedge P_2\right)$ are $Q$-derivates of a system $S$.
Transition systems with the same set of configuratious and initial configurations can be composed by combining their transition relations. Thus, the parallel composition of $S_1=\left(\mathcal{C}, \rightarrow_1, \mathcal{I}\right)$ and $S_2=\left(\mathcal{C}, \rightarrow_2, I\right)$ is the system $S=(\mathcal{C}, \rightarrow, I)$, where $\rightarrow=(\rightarrow 1 \cup \rightarrow 1)$.

Yujie Wang
Yujie Wang
College of San Mateo

Problem 4

Let $S$ be the parallel composition of $S_1$ and $S_2$.
Prove that if $P$ is an invariant of $S_1$ and $S_3$, then $P$ is an muariant of $S$. Prove that if $P$ is a $Q$-derivate of $S_1$ and $S_2$, then $P$ is a $Q$-dervate of $S$. Give an example where $P$ is always true in both $S_1$ and $S_2$, but not in $S$.
The next exercise concerns well-founded sets. A set can be well-founded, even when it has an element smaller than which there exist infinitely many
elements in the set. The lericographic order on vectors $a=\left(a_1, a_2, \ldots, a_n\right)$ and $b=\left(b_1, b_2, \ldots, b_n\right)$ is defined by the relation
$$
a<1 b \Longleftrightarrow a_1<b_1 \vee\left(a_1=b_1 \wedge\left(a_2, \ldots, a_n\right)<1\left(b_2, \ldots, b_n\right)\right) .
$$

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02:21

Problem 5

Prove that $\left(\mathbb{N}^n . \leq_1\right)$ (where $n \geq 2$ ) has elements smaller than which there exist infinttely many elements.
Prove that $\left(N^{\circ}, \leq_l\right)$ is well-founded.

Nick Johnson
Nick Johnson
Numerade Educator
05:31

Problem 6

Label the events of Figure 2.1 unth the clock values assigned by Lamport', logical clack.
Eabel the cuents wath the clock values assigned by Mattern's vector cluck. Identify some pairs of concurrent avents and check whether the labels asstgned to these events are ordered.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator

Problem 7

Give a datributed algorithm (simalar to Algorithm 8.9) that tabieis the events with the clock values assigned by Mattern's vector clock.

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Problem 8

Can one prove Theorem 2.21 by repeatedly applying Theorem 2.19 to cuents of $E$ ?

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Problem 9

Prove Theorem 2.24.

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Problem 10

Define the causal order for the transitions of a system with synchronous communication. Define clocks for such systems, and give a distributed algorithm for computing a clack.

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