The object of this section is to give narratives illustrating the treatment of concurrent events in two cases. The first is when two sub-narratives do not interact, and the second is when they do. The first sub-narrative is ordinary block stacking (as discussed in many situation calculus papers), and we suppose the stacking to be done by a person called Daddy in New York.
In the second sub-narrative, the actor is named Junior, and he wants to fly from Glasgow to Moscow via London. The story is taken from earlier web-published but widely circulated manuscripts [McC92] discussing how circumscription could be used to treat an unexpected obstacle to a plan, and [McC95] how narratives should be represented. This story is also used by Shanahan in [Sha97] in Chapter 10 as an example to motivate a use of context.
These two sub-narratives do not interact, and thus give an example of our first goal, a treatment of non-interacting narratives that can be conjoined consistently.
Because we want to treat interacting events, we make life more complicated for Junior. If he loses his ticket, he must wire Daddy in New York for money. Daddy, who normally indulges Junior, has to interrupt his block stacking and sell a block in order to get the money to send Junior. In this part of the narrative we have an example of adding details to an event. We state the event of Junior getting money occurs, we also give a sequence of events, Daddy stacking blocks until block3 is clear, then selling block 3, receiving money and sending it to Junior. The sequence realizes the single event of getting money. We show that both statements are consistent with each other, and the explanation can be consistently conjoined onto the narrative that mentions only the first event.
The following uses the axiomatizations of traveling and commerce and blocks-world in the Appendix. In the text we only give those axioms that are particular to the story.
We give axiomatizations of both the narratives where Junior loses his ticket, and contacts Daddy who sends him money, that Daddy raises by selling a block (In New York, blocks are made of Gold). Naturally Daddy has to clear the block before selling it, so the narratives interact in a non-trivial way.
In this narrative Junior doesn't lose his tickets, and and gets to Moscow without asking for help. Daddy stacks blocks in New York. There is no interaction, and nothing is said about the time relations between the two sub-narratives.
We should be able to infer:
To infer this we need to know that,
That is, no event occurs that would cause Junior to lose his ticket before he has to give it to the air-hostess. We actually state the following stronger fact, that no events that would cause Junior to no longer have a ticket occur, save of course flying from Glasgow.
When Junior is in London, inertia, and the instance of the above axiom with , gets us that Junior still has the ticket to Moscow. As for the ticket to London, we would infer that he does not have it as we brought up the fact that a ticket is used up when one takes the flight the ticket is for. That is certainly a part of the knowledge of anyone who travels using tickets. Thus someone who had traveled by bus would infer it about airplane travel. Indeed it could be inferred from more general principles about commerce, e.g. that a seller doesn't want to allow the buyer to get an arbitrary number of what he has a paid for one of. However, anyone who travels has the more specific information and doesn't need to infer it from general principles about commerce. Indeed he may never have formulated any general principles about commerce.
We wish to infer,
Again we need to know that no bad events occur, that is, Junior doesn't lose any tickets.
We call these sentences Nar1J, that is the sentences from 4 to 7. Now we begin Daddy's life as a block stacker. We have no relation between the situations S0 and S0' and know nothing of their temporal relations. If we asserted S0 < S0' < S1, then we could conclude that Junior still had the tickets in S0'. Also asserting S0' = S0 would do no harm to the conclusions drawn about either sub-narrative.
We also need to know that no other actions that would interrupt the block stacking occur.
We call the sentences from 8 to 9 Nar1D We now notice that if B is the axiomatization of blocksworld in the Appendix, and T is the axiomatization of traveling, then
Thus we can derive the obvious conclusions of our narrative. We further note that the two narratives are consistent.
In this narrative Junior loses the ticket and sends a telegram to Daddy asking for money. Daddy, who normally indulges Junior, sells a block and sends Junior the money, who buys a London-Moscow ticket and goes on to Moscow.
We chose a telegram rather than a telephone call, because we would not want to tell about a telephone call as a sequence of statements by Junior and Daddy but rather to regard its result as a joint action, e.g. an agreement that Junior and Daddy would do certain actions.
Note also we haven't treated what Daddy now knows as the result of the telegram. It seems that treating knowledge and treating agreement are similar in their requirement for treating intentional entities. The intentional state that Junior has requested that Daddy send him the money is not merely that Daddy knows that Junior wants Daddy to send him the money. Also the agreement is likely to have something like a bit of narrative as an argument, e.g. a set of actions that Junior and Daddy will do with only partial time relations between the actions.
Here we include sentences 4 and 5. Up to here, narrative 2 is the same as narrative 1. We will also need the sentences 8 and 9.
This contradicts 7, which stated that no event that lost the ticket happened before S2. We want to regard losing the ticket as something that happens to Junior rather than as something he does. That's why we don't write . The bad consequences of doing the latter would arise when we get around to writing laws that quantify over voluntary actions. We will use some of the same names now for situations that are different than in narrative 1.
Here we intend to have two explanations for what happens next. One is the simple observation that Daddy does sell a block, and send the money to Junior. This is a simple sequence of events, like we detailed earlier.
However we also know that as a dutiful father, Daddy will get money to Junior. Thus we can predict the event of Daddy getting money to Junior. Here we are treating Daddy as if his actions were determined by our inputs. Sometimes it is useful to describe people in that way. In more elaborate narratives we would need to reason about Daddy mental processes, but for this case we can treat him as an automaton.
The following axiom characterizes what Daddy does when he receives a request from Junior.
This is an example of a triggered action, as we have the defining rule for ,
We now state that the money arrives before S3, when Junior buys the ticket.
We now give the other facts about occurrences.
We also need to know that no events occur that would divert the money in the meantimes between these events and the result of the previous events.
We now consider the consequences of narrative two. Let Nar2 be those sentences directly above and the sentences from 4 to 5 and 8 and 9 and 11 to 15.
Most interestingly we can derive the occurrence of a triggered action:
We have two explanations for Junior receiving the money, the gets event, and the send event. We cannot tell which happens first, or if they happen simultaneously. Thus our formalism allows us to add detail of an event without contradiction.