A Typology of Elaborations

There are many kinds of elaborations a person can tolerate, and they pose different problems to different logical formalizations. Here are some kinds of elaborations.

irrelevant actors, actions and objects

Sentences establishing the existence of such entities should not vitiate the reasoning leading to a solution.

adding preconditions, actions, effects of actions and objects

The example of the oars adds a precondition to rowing and adds the action of picking up the oars. Several situation calculus and event calculus formalisms allow this--assuming sentences are added before the nonmonotonic reasoning is done. Tolerating added preconditions is a criterion for good solutions of the qualification problem, and tolerating adding effects relates similarly to the ramification problem.

changing a parameter

This is needed when the numbers of missionaries and cannibals are changed from 3 to 4. In English, this is accomplished by an added sentence. Doing it that way in logic requires a suitable belief revision method as part of the basic logical formalism. At present we must use minor brain surgery to replace certain occurrences of the number 3.

making a property situation dependent

Whether $x$ is a missionary is not situation dependent in MCP0, but we can elaborate to a missionary becoming a cannibal. It is tempting to say that all properties should be situation dependent from the beginning, and such a formalism would admit this elaboration easily. I think this might lead to an infinite regress, but I can't formulate the problem yet.

specialization

In one situation calculus formalization we have the action $Move(b1,b2)$. If there are guaranteed to be exactly two places, we can replace this action by $Move(b)$, regarding this as $Move(b,Opp(b))$, where $Opp(b)$ designates the opposite bank and satisfies $Opp(Opp(b)) = b$. We regard this kind of specialization as an easy kind of elaboration.

generalization

Some of our elaborations can be composed of an generalization of the language--replacing a function by a function of more arguments, e.g. making whether a person is a cannibal or missionary situation dependent or replacing going from a bank $b$ to the opposite bank $Opp(b)$ by going from $b1$ to $b2$. Many elaborations consist of a generalization followed by the addition of sentences, e.g. adding preconditions or effects to an action.

unabbreviation

This is a particular case of generalization. Suppose we write $(\forall a \in Actions)Abbreviates[a,Does(person,a)]$. We mean to use it in elaborating $Result(a,s)$ to $Result(Does(person,a),s)$ in sentences where it is somehow clear which person is referred to. The square brackets mean that this is a metalinguistic statement, but I don't presently understand precisely how unabbreviation is to work.

going into detail

An event like the action of crossing the river is made up of subactions. However, the relation between an event and its subevents is not often like the relation between a program and its subroutines, because asserting that the action occurs does not imply specific subactions. Rowing is a detail of crossing a river when rowboats are used, but rowing is not a part of the general notion of crossing a river.. Bailing if necessary is another detail. Getting oars or a bailing can are associated details. There is more about this apparently controversial point in [McC95].

m...s and c...s as actors

MCP and almost all of the elaborations we have considered take a god-like view of the actions, e.g. we send a cannibal to get an oar. We can also elaborate in the direction of supposing that the actions of cannibals and missionaries are sometimes determined by the situation. In this case, it may be convenient to use a predicate $Occurs(event,s)$ and let one possible event be $Does(C_1, Enter\mbox{-}Boat)$. The situation calculus treatment has to be altered and looks more like event calculus.

simple parallel actions

If one of the missionaries is Jesus Christ, we can transport 4 missionaries and 4 cannibals. We get 3 cannibals on the far bank and one on the initial bank. Then two ordinary missionaries and Jesus cross, the ordinaries in the boat and Jesus walking on water. The rest of the solution is essentially the same as in MCP0. A missionary and a cannibal row back and now the remaining two missionaries cross. We then send a cannibal to ferry the remaining two cannibals. We haven't tackled the problem of being able to say ``essentially the same'' in logic.

The formalization must permit Jesus to cross in parallel with the other missionaries so that the missionaries are never outnumbered. This isn't the same as having Jesus cross as a separate action.

full parallelism

This is what permits requiring that the boat be bailed.

events other than actions

The simple $Result(a,s)$ doesn't allow for events other than actions. To handle them we have used [McC95] a predicate $Occurs(e,s)$ asserting that the event $e$ occurs in the situation $s$. Then $Result(e,s)$ can be used.

comparing different situations

This works ok in situation calculus, but some other formalisms don't allow it or make it awkward. Thus we can have $s <_{better} Result(e,s)$ to say that the situation is better after event $e$ occurs. We may also want $Result(a1,s) <_{better} Result(a2,s)$, comparing the result of doing $a1$ with the result of doing $a2$.

splitting an entity

Sometimes an entity, e.g. a node in a graphy, an edge, or a concept needs to be split into two entities of the same type so that separate properties can be assigned to each subentity. Thus we may split cannibals into strong and weak cannibals.

continuous time and discrete time

If Achilles runs enough faster than the tortoise, there is a time when Achilles catches up. We use the fluent $Future(\pi,s)$ to assert that the situation $s$ will be followed in the future by a situation satisfying $\pi$. We do not require formalizing real numbers to express the Achilles catching up sentence

$$\begin{array}[l]{l} Future((\lambda s) (Value(Distance\mb... ...stance\mbox{-}covered\mbox{-}by(Tortoise),s)), S0). \end{array}$$