Showcasing AlgebraicABMs features

This file is an incomplete showcase of some less obvious features in AlgebraicABMs. It is not an intro-tutorial (the other demos are more geared towards that). To begin, we want to load our packages with `using

using AlgebraicABMs, Catlab, AlgebraicRewriting
ENV["JULIA_DEBUG"] = "AlgebraicABMs"; # turn off @debug messages for this package

Basis

Rewrote rules have associated timers which are scheduled to fire in the future. For normal rules, at any point in time, there is exactly one such timer per match morphism from the pattern $L$ of the rule into the present state of the world, $X$. This is often what we want, though in some circumstances, we want the 'per-what?' of the rule to be something different from the pattern. Thus we have the ability to explicitly give a basis for the rule, given as a map $B \rightarrowtail L$ into the rule's pattern. The rule will have timers per every 'match' $B \rightarrow X$ (which is just a partial match $L \nrightarrow X$). Once it's time to fire, the rest of the match is completed at random (if possible, otherwise nothing happens).

For example, we may want to associate the timers of an infection process with simply the infected people, even though the pattern of an infection rule is a suceptible + an infected person. E.g. "every infected person tries to infect someone once per day", which would not be affected by how many susceptible people there are. Let's demonstrate this:

@present SchSI(FreeSchema) begin
  (S,I)::Ob
end

@acset_type SI(SchSI)

si   = @acset SI begin S=1; I=1 end
i    = @acset SI begin I=1 end
ii   = @acset SI begin I=2 end
init = @acset SI begin S=10000;I=1 end
inf_rule = Rule(homomorphism(i, si), homomorphism(i, ii; any=true))
basis_hom = homomorphism(i, si) # our actual basis is just a single infected

abm_rule = ABMRule(inf_rule, DiscreteHazard(1); basis = basis_hom)
abm = ABM([abm_rule])
run!(abm, init; maxevent=3);

┌ Debug: Step 0: Event 1 | Fired @ t = 1.0 (0 queued)
└ @ AlgebraicABMs.ABMs ~/work/AlgebraicABMs.jl/AlgebraicABMs.jl/src/ABMs.jl:521
┌ Debug: Step 1: Events 1, 1 | Fired @ t = 2.0 (0 queued)
└ @ AlgebraicABMs.ABMs ~/work/AlgebraicABMs.jl/AlgebraicABMs.jl/src/ABMs.jl:521
┌ Debug: Step 3: Events 1, 1, 1, 1 | Fired @ t = 3.0 (0 queued)
└ @ AlgebraicABMs.ABMs ~/work/AlgebraicABMs.jl/AlgebraicABMs.jl/src/ABMs.jl:521

We can see that after the first timestep there is 1 infection, then 2 on the next timestep, then four on the next one.

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