[TYPES] Semantics of modes in logic programming?

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Hello,

I'm interested in the semantics of mode systems in logic programming, 
especially (a) mode systems more sophisticated than simple input or 
output modes, and (b) denotational semantics of logic programs with modes.

Uday Reddy's 1992 paper "A typed foundation for directional logic 
programming" (https://urldefense.com/v3/__https://www.cs.bham.ac.uk/*udr/papers/directional.pdf__;fg!!IBzWLUs!RIsOJV9x2aZaYUcdxebrICDz_vREN4r14OX46bY1vpRhC6gHz6PZnsDR9JcDg9vQSGbGNvsoOfl3Kks_ocdKMUcJyqKwVlveYJZj8Q$ ) 
gives a beautiful mode system based on classical linear logic, and he 
shows it has a lot of really great operational properties.

I'd be especially interested in a denotational (especially categorical) 
semantics for this language, but want to cast the net broadly in this 
question.

Thanks!

Best,
Neel