The article seems to imply both. The basis is that wave propagate signals throughout the fungal network. The signals are made of electricity, which acts in a manner where they assume it represents numbers such as 1. Then they define logic. This is to me a pretty safe assumption if the most we have to deal with is a two-state automata that takes in 0 or 1. We can observe from its natural behavior and physical shape from experiments that there's clearly some mathematical optimization going on for well-known computer science problems that define its characteristics. I think of this way: if it looks like math, its probably computing how it forms. If it exhibits a great deal of complexity that can just be because of how it interacts with itself like on a cellular automata basis or a Belousov-Zhabotinsky reaction-diffusion basis. You can get a lot of complex and chaotic behavior from really simple rules. They tell you how fast the system works and its clock speed basically, so you can compare and relate that to your computer as well.
Nature is the highest programmer because nature can write pretty fluently in just binary. All nature needs sometimes is something activating and deactivating, which can be things living or dying next to each other. And when nature arrives at a proof that a method is optimal, it arrives at it in a probable amount of time/space. When programs are valid and the proof exists, we have that we find it in nature since it gives a natural advantage to its biological host. We do the same thing but way slower and it requires a bunch of chalkboards and silicon sometimes.
The author, has a number of publications that are biological computation models. You can substitute a computer made of billiard-balls for soldier crabs.
https://en.wikipedia.org/wiki/Andrew_Adamatzky