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proves G (-o A B) :- proves (A :: G) B.
proves (-o A B :: G) C :- append G1 G2 G, proves G1 A, proves (B :: G2) C.
Since the specification logic in Abella is not linear, you cannot write a spec that depends on assumptions being used linearly.[1]If you remain in the reasoning logic to define your provability relation, like the ll-meta proofs so far, then there is no problem at all with linearly. ILL will be conceptually no harder than two-sided classical LL which is already done a bit in the ll-meta proofs.In the future Abella is probably going to evolve to have some kind of linear specification langauge. However, I am not sure what the timeline on that is. I know of multiple people who are working on it.Kaustuv[1] At least there is no simple way to do it. Reed and Pfenning have proposed an alternative way to encode substructural logics using vanilla intuitionistic logic by reifying the substructural properties as side conditions. Frank may be able to tell you what is the best thing to read regarding the current status of this work.
On Sun, Jan 27, 2019 at 1:31 AM Siva Somayyajula <siva.som...@gmail.com> wrote:
--Hi all,The work on formalizing classical linear logic in Abella (see here) presents it as a sequent calculus, with contexts as lists of hypotheses, in the reasoning logic.On the other hand, if I were to encode e.g. some fragment of intuitionistic linear logic, I could do it Twelf-style as a linear lambda calculus enforcing linear/exact-once use of hypotheses as a relation over abstractions.Is there an approach similar to the LJ case study, where the proof system is written in the specification logic, but `{L |- conc C}` "uses" hypotheses in `L` linearly? I'm still trying to wrap my head around how contexts can be manipulated in Abella.Thanks,Siva
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I kept the provability relation in the spec logic, but I thread through a context as a list of hypotheses, like this (e.g. linear implication):
proves G (-o A B) :- proves (A :: G) B.
proves (-o A B :: G) C :- append G1 G2 G, proves G1 A, proves (B :: G2) C.
Thanks for the response; I'm looking forward to linear specs! I kept the provability relation in the spec logic, but I thread through a context as a list of hypotheses, like this (e.g. linear implication):
Dan DaCosta is in fact developing a dissertation at the University of Minnesota that is aimed at enabling the use of linear logic as a specification logic within the Abella system. He will probably have more to say on the topic.-Gopalan