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Comment by adamnemecek | Hacker News Reader

GAN is based on a minimax algorithm (as mentioned in the paper). Linear logic has very minimaxy semantics.

I was actually more interested in a reference making an explicit construction of a probability theory using linear logic, or if that's unavailable, then some more on the LL -> games -> probability route.

Edit: I found this comment [1] by you 29 days ago. Would you mind expanding on the "more connections" you mention? Thanks.

[1] https://news.ycombinator.com/item?id=17432612

I'd actually like to see a coherent explanation, e.g. paper reference, of how the Minimax [1] algorithm relates to Girard's linear logic.

[1] https://en.wikipedia.org/wiki/Minimax

Now we are talking! Check out the paper "Stochastic Interaction and Linear Logic" by Lincoln, Mitchell, Scedrov

As far as I understand, that paper talks only about MALL, which is a small propositional fragment of LL's, in particular, exponentials are missing. Extending models for small fragments to full LL has always been difficult. Note that the Lincoln et al paper is > 2 decades old. Has anyone extended it to full LL?

Note also that this paper doesn't seem to be talking about Minimax.

What is the semantics of Linear Logic?

It has several interpretations. One is close to game theory, one is related to ownership, one is related to probability, one is related to interfaces. These are all the same idea.

Are they for full LL or just small fragments (like MALL)? And what kind of completeness properties are known to hold or not to hold for those models?

You have normal additive (and/with, or/plus) and multiplicative operators (forall/times, nonsuch/par) as well as exponential ones (ofcourse/ever, whynot/sometimes).

The final two are the linear part as they cause the logic to be positional.

The closest CS equivalent are categorial grammars (including combinatorial ones), the difference is that pure linear logic is commutative while these are not in general.

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