Florian, As I mentioned, over the break I'm trying to climb the substructural logic hill. It looks particularly interesting.
Axiom is based on group theory. It splits into two parts "categories" (not category theory) which are collections of signatures and "domains" which are implementations of signatures. So in Axiom you start with Set or Type and then define things like Group (which requires more singatures), Rings, (more required signatures), Fields (e.g. a division signature). Once the tower of categories exists you can define a domain that inherits from some (sub-)tower. Now you have to write code to implement each of the inherited signatures. This gives Axiom a very useful way of organizing along the lines of group theory. Using my own form of reasoning (unreasonable analogy) it must be possible to organize substructural logic into a set of categories that can be organized in a similar way. So, by analogy, it should be possible to start a root category with a "reflexive signature" A |- A Then a set of categories can be defined, one each, to allow structural rules like Weakening which you might or might not allow. Then a set of categories with specific introduction and elimination rules (e.g. one for ->I, ->E for implication) Organized in this granular fashion one can then create something like a LinearLogic, OrderedLogic, or a Gentzen domain that inherits only the categories that apply. Clearly Halleck's work http://www.cc.utah.edu/~nahaj/logic/structures/index.html both supports this idea and shows that it is too simplistic to really cover all of logic. But for a reasonable subset such as I will need while proving Axiom it seems that the naive organization might be sufficient. Just after our attempted phone call two books showed up "An Introduction to Substructural Logics" and "Abstract Algeraic Logic: An Introductory Textbook". I'm part way through the first book and it looks like it supports the organization I'm postulating. The second book, at first glance, looks like Font has already done the organizational work I need. It is, however, a much slower read. Even more interesting is that Avigad's Proof Theory course showed that there are logics of equivalent power but different levels of expression. This seems to me to imply that there is a form of automated coercion that must exist between various logics that can be constructed. Axiom does automatic coercions in its group theory setting. These really complicate the Type Theory reasoning I'm trying to use. At the moment I'm quite confused about coercion so I may have to do a survey paper to try to get a grip on the topic. Tim On Tue, Dec 12, 2017 at 9:30 AM, Florian Rabe <florian.r...@gmail.com> wrote: > Hi Tim, > > I missed you yesterday, but I've added jguidry899 on skype now. 3pm my > time usually works. > > Best, > Florian > > > On 11.12.2017 15:12, Tim Daly wrote: > >> I tried to Skype but it failed. >> I tried to call but it failed. >> Lets use email. >> >> I'll write later today. >> >> Tim >> >> >> On Sun, Dec 10, 2017 at 10:13 PM, Tim Daly <axiom...@gmail.com> wrote: >> >> There is a 6 hour time difference between Pittsburgh and >>> Berlin. I'll try to skype with you at 3pm your time (9am mine). >>> >>> I'll be using the skype id jguidry899 as skype doesn't seem >>> to like any of my prior passwords for my normal id. >>> >>> If this is not convenient, please let me know. >>> >>> Tim >>> >>> >>> On Thu, Dec 7, 2017 at 1:51 PM, Florian Rabe <florian.r...@gmail.com> >>> wrote: >>> >>> Yes, I'll be around. >>>> >>>> Skype: florian.rabe >>>> WhatsApp/cell phone: +49 1520 5321277 >>>> >>>> On 07.12.2017 17:47, Tim Daly wrote: >>>> >>>> I'm a bit buried at the moment. I printed out your "top 5" papers. >>>>> I plan to read them over the weekend so I understand what you're doing. >>>>> Are you available some time next week? I have the whole week free. >>>>> >>>>> Tim >>>>> >>>>> >>>>> On Thu, Dec 7, 2017 at 6:53 AM, Florian Rabe <florian.r...@gmail.com >>>>> <mailto:florian.r...@gmail.com>> wrote: >>>>> >>>>> Hi Tim, >>>>> >>>>> I'm exactly the right person to contact about this. >>>>> I'm CC'ing Michael Kohlhase with whom I ran the LATIN project a few >>>>> years ago, whose goal was to form at atlas of logics in this style. >>>>> >>>>> The logical framework LF was developed for that purpose and my MMT >>>>> framework significantly abstracts from and generalizes the LF results. >>>>> You can find the most relevant papers at >>>>> https://uniformal.github.io/doc/philosophy/papers.html < >>>>> https://uniformal.github.io/doc/philosophy/papers.html> >>>>> >>>>> >>>>> We strongly commend any work on Axion that takes these logical >>>>> framework ideas into account. >>>>> An integration of our systems (which are good for defining the >>>>> logics) with Axiom (which needs to use the logics to verify results) >>>>> would >>>>> be very interesting. >>>>> >>>>> However, as your second email indicates you have found out already >>>>> yourself by now, the problem is quite hard. >>>>> Therefore, depending on your interests and available resources, I >>>>> have wildly different advice on how to proceed. >>>>> Maybe we should have a phone call about this? >>>>> >>>>> Best, >>>>> Florian >>>>> >>>>> >>>>> On 06.12.2017 14:47, Tim Daly wrote: >>>>> >>>>> James Davenport suggested I forward this question to you. >>>>> ... >>>>> >>>>> I'm trying to prove Axiom correct. (http://axiom-developer.org >>>>> ) >>>>> >>>>> Axiom draws its power from organizing algorithms based on group >>>>> theory, adding propositions as the complexity increases. This >>>>> helps >>>>> limit the assumptions needed to provide algorithms. >>>>> >>>>> Recently I've learned a bit about substructural and (super?) >>>>> structural >>>>> logics. So, for example, given the propositions >>>>> (W) Weakening >>>>> (C) Contraction >>>>> (E) Exchange >>>>> they can be combined to form (Crary): >>>>> W & C & E ==> Persistant logic. >>>>> W & ~C & E ==> Affine logic >>>>> ~W & C & E ==> Strict logic >>>>> ~W & ~C & E ==> Linear logic >>>>> W & C & ~E ==> ? >>>>> W & ~C & ~E ==> ? >>>>> ~W & C & ~E ==> ? >>>>> ~W & ~C & ~E ==> Ordered logic >>>>> >>>>> In addition, there appears to be a group theory-like >>>>> organization of >>>>> logics above the substructural e.g. predicate, simply-typed >>>>> lambda, >>>>> and lambda calculus (Avigad). >>>>> >>>>> Organizing these logics by the gradual introduction of >>>>> propositions >>>>> and properties would fit exactly into Axiom's structure and >>>>> make >>>>> proof assumptions clear. I want to organize these in Axiom so >>>>> the >>>>> proofs inherit the required logic. >>>>> >>>>> Can you give me any references to a group-theory-like >>>>> organization >>>>> of the various logics, similar to the above? I have done a >>>>> literature >>>>> search but have not found anything yet. >>>>> >>>>> Thank you. >>>>> Tim Daly >>>>> >>>>> >>>>> >>>>> >>>>> >>> >>
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