Hi, consider this picture from a Message Sequence Chart. Identify the events in this diagram.

In this Message Sequence Chart (MSC), the visible events are the message send and message receive actions on the instance lifelines.

Instances

• i1

• i2

• i3

Messages

• m1 goes from i1 to i2

• m2 goes from i3 to i2

Events in the diagram

On instance i1

1. Send of m1

   • Event: send(m1)

   • This is the point where i1 sends message m1 to i2.

On instance i2

2. Receive of m1

   • Event: receive(m1)

   • This is where i2 receives m1 from i1.

3. Receive of m2

   • Event: receive(m2)

   • This is where i2 receives m2 from i3.

On instance i3

4. Send of m2

   • Event: send(m2)

   • This is the point where i3 sends m2 to i2.

A compact event list

You could label the events as:

• e1 = send(m1) on i1

• e2 = receive(m1) on i2

• e3 = send(m2) on i3

• e4 = receive(m2) on i2

Ordering relations visible in the chart

Because time flows top to bottom on each lifeline:

• e1 < e2 (sending m1 happens before receiving m1)

• e3 < e4 (sending m2 happens before receiving m2)

• On i2, e2 < e4 (the receive of m1 occurs before the receive of m2)

So the main ordering is:

send(m1) < receive(m1) < receive(m2)

send(m2) < receive(m2)

``

and there is no order shown between:

• send(m1) on i1

• send(m2) on i3

unless an additional constraint is given.

Final answer

The diagram contains four main events:

1. i1 sends m1

2. i2 receives m1

3. i3 sends m2

4. i2 receives m2

If you want, I can also express the same MSC as a partial order, a trace, or in MSC/UML notation.

Question:

What is the correct ordering between sending m1 and sending m2?

Answer:

The correct ordering between sending m1 and sending m2 is:

They are not ordered

There is no causal or visual ordering constraint in the MSC between:

• send(m1) on i1

• send(m2) on i3

Because these two events occur on different instance lifelines and there is no message or vertical order relation connecting them directly, they are incomparable in the partial order of the chart.

What is ordered in the diagram?

From the MSC, the following orderings are clear:

1. send(m1) < receive(m1)

2. send(m2) < receive(m2)

3. On lifeline i2: receive(m1) < receive(m2)

So the constraints are:

send(m1) < receive(m1) < receive(m2)

send(m2) < receive(m2)

``

But there is no constraint saying either:

send(m1) < send(m2)

``

or

send(m2) < send(m1)

``

Interpretation

So, in MSC semantics:

• send(m1) and send(m2) are concurrent / independent / unordered

• either one may happen first in a valid linearization, as long as the other ordering constraints are respected

Example valid total orders

Both of these are valid:

send(m1), receive(m1), send(m2), receive(m2)

``

and

send(m2), send(m1), receive(m1), receive(m2)

``

The second one is valid because the only requirement on i2 is:

receive(m1) before receive(m2)

``

Final answer

There is no fixed ordering between sending m1 and sending m2; they are unordered (concurrent) in this MSC.

If you want, I can also draw the partial order (Hasse-style dependency graph) for the four events.

Question:

What is the correct ordering between receiving m1 and receiving m2?

The correct ordering is:

receive(m1) < receive(m2)

Why?

Both events occur on the same instance lifeline i2, and in a Message Sequence Chart, events on a single lifeline are ordered from top to bottom.

In the diagram:

• the reception of m1 on i2 is drawn higher

• the reception of m2 on i2 is drawn lower

So the ordering is:

receive(m1) before receive(m2)

``

Final answer

m1 is received before m2 is received.

If you want, I can now list all event order constraints in the MSC as one complete partial order.

Question:

Based on the diagram above, generate a message sequence chart in which message m1 is abstracted away. Please produce the resulting MSC after abstracting from m1.

Answer:

Yes — if we abstract away message m1, then both of its associated events are removed:

• send(m1) on instance i1

• receive(m1) on instance i2

What remains is only the communication for m2:

• send(m2) on i3

• receive(m2) on i2

Resulting MSC after abstracting from m1

Informally

• i1 remains as an instance but has no events

• i2 only contains the reception of m2

• i3 only contains the sending of m2

ASCII-style MSC

        i1              i2              i3

        |               |               |

        |               |               |

        |               |<--------------|   m2

        |               |               |

        |               |               |

``

Since MSC arrows are usually drawn from sender to receiver, the same chart can also be written more explicitly as:

        i1              i2              i3

        |               |               |

        |               |               |

        |               |<----- m2 -----|

        |               |               |

Event structure of the abstracted MSC

The remaining events are:

• e1 = send(m2) on i3

• e2 = receive(m2) on i2

with the ordering:

send(m2) < receive(m2)

There are no other event-order constraints left after removing m1.

Compact formal representation

If the original MSC had event set

{ send(m1), receive(m1), send(m2), receive(m2) }

``

then after abstraction from m1, the resulting event set is:

{ send(m2), receive(m2) }

``

with partial order:

send(m2) < receive(m

Final resulting MSC

msc abstract_m1 {

  instance i1;

  instance i2;

  instance i3;

  i3 -> i2 [label="m2"];

}

``

If you want, I can also produce the same abstracted MSC in standard graphical notation or explain whether i1 should be removed entirely under a particular MSC semantics.

Please draw the image of the MSC.

Question:

Consider the original MSC given below. Merge instance i2 and i3; produce the message sequence chart after merging. Please draw the resulting MSC.

Answer:

Drawing merged MSC and considering vertical order

I need to draw the resulting message sequence chart (MSC) based on merging instances i2 and i3. After merging, m2 becomes an internal message, so I'll represent it as an internal event on the merged lifeline. I'll also ensure proper vertical order: i2's m1 receive is above m2 receive.

Clarifying message order after merge

After merging, m2 becomes both a send and a receive on the same lifeline. The order might be: receive m1, send m2, then receive m2. For the merged lifeline, the message from i3 to i2 could have some interleaving, with the option for non-deterministic ordering between events. I’ll need to draw this sequence out.

Creating MSC drawing with merged instance

I’ll create the MSC after merging, drawing i1 and the merged i23. m1 will go from i1 to the merged instance, and m2 will appear as a self-message below. I could generate the diagram using Python, making it as a PNG and providing a link to the resulting file.

Generating MSC after instance merge

bash -lc python3 - <<'PY'

import matplotlib.pyplot as plt

from matplotlib.patches import Rectangle, FancyArrowPatch, PathPatch

from matplotlib.path import Path

fig, ax = plt.subplots(figsize=(7.5,5.0))

ax.set_xlim(0, 10)

ax.set_ylim(0, 7)

ax.axis('off')

# outer frame

ax.add_patch(Rectangle((0.7,0.55), 8.6, 5.9, fill=False, linewidth=1.1, color='0.25'))

ax.text(5.0, 6.15, r'\bf msc $merged$', ha='center', va='center', fontsize=18, family='serif')

# positions

x1=2.4; x23=6.6

yt=5.2; yb=1.0

# labels and heads/foots

for x,label in [(x1,'i1'),(x23,'i2/i3')]:

ax.text(x, 5.55, label, ha='center', va='bottom', fontsize=18, family='serif', fontstyle='italic')

ax.add_patch(Rectangle((x-0.75,4.78),1.5,0.38, fill=False, linewidth=1.0, edgecolor='0.35'))

ax.plot([x,x],[4.78,1.33], color='0.45', linewidth=1.0)

ax.add_patch(Rectangle((x-0.75,0.95),1.5,0.38, facecolor='black', edgecolor='black'))

# m1 i1 -> merged

ym1=4.0

ax.plot([x1, x1], [4.78, ym1], color='0.45', linewidth=1.0)

ax.add_patch(FancyArrowPatch((x1,ym1),(x23,ym1), arrowstyle='-|>', mutation_scale=18, linewidth=1.1, color='black'))

ax.text((x1+x23)/2, ym1+0.18, 'm1', ha='center', va='bottom', fontsize=16, family='serif', fontstyle='italic')

# self message m2 on merged instance, below m1 receive

# draw loop to right with arrow back to lifeline lower down

ys=3.15; yr=2.5

verts=[(x23,ys),(x23+1.25,ys),(x23+1.25,yr),(x23,yr)]

codes=[Path.MOVETO, Path.CURVE4, Path.CURVE4, Path.LINETO]

# use simple polyline + arrow final segment

ax.plot([x23,x23+1.3,x23+1.3],[ys,ys,yr], color='black', linewidth=1.1)

ax.add_patch(FancyArrowPatch((x23+1.3,yr),(x23,yr), arrowstyle='-|>', mutation_scale=18, linewidth=1.1, color='black'))

ax.text(x23+1.45, (ys+yr)/2, 'm2', ha='left', va='center', fontsize=16, family='serif', fontstyle='italic')

# small event ticks on merged lifeline? no

plt.savefig('/mnt/data/merged_msc_i2_i3.png', dpi=200, bbox_inches='tight', facecolor='white')

PY

Question:

Consider the original MSC given below. Please produce the set of all of its traces.

For the original MSC, let the events be:

• s₁ = send(m1)on i1

• r₁ = receive(m1)on i2

• s₂ = send(m2)on i3

• r₂ = receive(m2)on i2

1) Partial-order constraints from the MSC

From the diagram, the ordering constraints are:

1. s₁ < r₁(a message must be sent before it is received)

2. s₂ < r₂

3. r₁ < r₂(both receptions are on lifeline i2, and m1is received above m2)

So the partial order is:

s₁ < r₁ < r₂, s₂ < r₂

This implies that r₂must occur after all the other events.

2) All traces (all linearizations)

A trace is any total ordering of the four events that respects the above constraints.

Since r₂must be last, we only need to order:

• s₁

• r₁

• s₂

subject to the constraint s₁ < r₁.

That gives exactly three possibilities:

$$\begin{matrix} t_{1} & = \langle s_{1},\ r_{1},\ s_{2},\ r_{2}\rangle \\ t_{2} & = \langle s_{1},\ s_{2},\ r_{1},\ r_{2}\rangle \\ t_{3} & = \langle s_{2},\ s_{1},\ r_{1},\ r_{2}\rangle \end{matrix}$$

3) Final set of traces

$$\boxed{Traces(MSC) = \left\{ \langle send(m1),\ receive(m1),\ send(m2),\ receive(m2)\rangle,\ \langle send(m1),\ send(m2),\ receive(m1),\ receive(m2)\rangle,\ \langle send(m2),\ send(m1),\ receive(m1),\ receive(m2)\rangle \right\}}$$

4) Compact notation

If we abbreviate:

• s₁ = !m1

• r₁ = ?m1

• s₂ = !m2

• r₂ = ?m2

then:

$$\boxed{\left\{ \langle!m1,\ ?m1,\ !m2,\ ?m2\rangle,\ \langle!m1,\ !m2,\ ?m1,\ ?m2\rangle,\ \langle!m2,\ !m1,\ ?m1,\ ?m2\rangle \right\}}$$

If you want, I can also give the Hasse diagram / dependency graph for this MSC, or explain how these traces change under synchronous versus asynchronous message semantics.

Please draw a labelled transition system that has the same traces as the MSC. Please produce one static picture, no interactive visualisation.