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Where I last left off in my discussion of the event calculus, I was discussing the idea of "encoding" vs. "embedding." In some discussions, these terms map onto the alternate terms "deep embedding" and "shallow embedding." If I understand them correctly, Alexiev's program gives a

*deep embedding*of the event calculus: the core constructs of the calculus, fluents and events, are modeled as

*terms*in the underlying logic program.

I was curious what it would mean to give a (shallow) embedding of the event calculus in linear logic, or indeed if this were even a possible task. On one level, EC and LL have entirely different purposes: the event calculus enables introspection on event specifications to determine intervals, and linear logic enables deduction that models reachability between state configurations.

On another level, EC and LL "do the same thing," i.e. model the same type of domain. They model evolving state, and they enable reasoning about propositions that hold in some configurations but not in others.

So the shallowest possible embedding seems like it would be one that populates a signature S for which S; D |- A iff D @ T |-_{EC} A @ T' for T < T'. In other words, the preorder given by logical entailment in linear logic should map to the preorder given by time in the event calculus.

To achieve this mapping, we would model

- fluents (time-indexed propositions) as linear propositions (or types)

- events as linear logic rules

An example domain (the same Yale Shooting Problem as from the previous post) is given below.

%% first take: fluents as types; events as rules.

%% "terminates E Ts" & "initiates E Is" |--> ruleE : Ts -o {Is}.

% fluents as types.

loaded : type.

empty : type.

dead : type.

alive : type.

% events as rules.

load : empty -o {loaded}.

unload : loaded -o {empty}.

shoot : loaded * alive -o {empty * dead}.

wait : {1}.

#trace * {empty * alive}.

This encoding does in some sense "simulate" the domain, but what it fails to take account for is control over "when things happen." The programmer of the domain has no means by which to dictate which rules proof search will select in which order, so unfortunately we cannot give an adequate encoding in this scheme.

We could try to fix it by explicitly modeling a happens E predicate and using it as a guard on our rules, but without explicitly modeling

*time*as well, we still cannot control the order in which events take place.

Thus,

**take 2**, in which events are data and fluents are boolean-indexed types. This version of the code uses a clock that gets incremented after each event "happens".

time : type.

z : time.

s : time -> time.

infty : time.

happens : event -> time -> type.

clock : time -> type.

bool : type.

tt : bool.

ff : bool.

loaded : bool -> type.

dead : bool -> type.

happens/load : clock T * happens load T * loaded ff

-o {loaded tt * clock (s T)}.

happens/unload : clock T * happens unload T * loaded tt

-o {loaded ff * clock (s T)}.

happens/shoot : clock T * happens shoot T * loaded tt * dead ff

-o {clock (s T) * loaded ff * dead tt}.

happens/wait : clock T * happens wait T -o {clock (s T)}.

init : type.

- : init -o {loaded ff * dead ff}.

#trace * {init

* clock z * happens wait z * happens wait (s (s z)) * happens shoot

(s (s (s z))) * happens load (s z)}.

Results: this correctly models and simulates the domain, including control over event occurrences. But it fails a basic property of the event calculus: filling in the "gaps" between events logically by way of inertia. In this setup, we have to manually specify what happens at each and every timestep.

In the

**final take**, I give a program that maintains events in a queue, which is centrally managed by a single rule:
happens : event -> time -> type.

holds : fluent -> type.

elist : type.

nil : elist.

cons : event -> time -> elist -> elist.

queue : elist -> type.

% domain things

load : event.

unload : event.

wait : event.

shoot : event.

loaded : fluent.

dead : fluent.

% clock management

tick : clock T -o {clock (s T)}.

stop : queue nil * clock _ -o {1}.

% key rule: "happens" management

happen : clock T * queue (cons E T' Es) * leq T' T

-o {queue Es * happens E T}.

% action cause/effects

happens/load : happens load T -o {holds loaded * clock T}.

happens/unload : happens unload T * holds loaded -o {clock T}.

happens/shoot : happens shoot T * holds loaded -o {holds dead * clock T}.

happens/wait : happens wait T -o {clock T}.

init : type.

- : init -o {clock z

* queue (cons load z (cons wait (s (s z)) (cons shoot (s (s (s z))) nil)))

}.

#trace * {init}.

This program more or less gives me what I was initially after: a forward-chaining program that simulates a sequence of events specified in event calculus terms, filling in the gaps logically. It's still not *quite* an adequate encoding, because we don't have control over when the tick rule fires, so it might happen any time -- note that the happen rule invokes "less than or equal" to catch the cases where we've incremented past the next event's scheduled time. But we're able to keep them in the correct order nonetheless, via the queue as well as the synchronization technique of revoking the clock token until after the event's effects have been incurred.

After carrying out this iterative design experiment, I conclude that the distinction I should have made previously was not between encoding and embedding, but rather between a program analysis and execution. And perhaps that's what I've learned about the event calculus in the process: it isn't really

*for*writing specifications we can*run*, just ones we can analyze. My proofs-as-programs training led me to faulty expectations.
I'm being a bit skimmy here, but I guess on one level the domain of the event calculus isn't quite as obvious as might initially appear? So we don't get what we expect by 'running' it, and in particular we don't get a neat schedule out without further work?

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