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>I'd like to know why you consider it abhorrent 1. I am a Turing fan-boy. 2. I am also a big advocate of the fact that Software IS hardware. And in reality, al

>I'd like to know why you consider it abhorrent 1. I am a Turing fan-boy. 2. I am also a big advocate of the fact that Software IS hardware. And in reality, all computation is software ('hardware' is merely the materials engineering that implements the computational 'logic' e.g. Electrical Engineering for modern-day computers). Classifying a Turing Machine as a 'hardware' abstraction is just so not true. It's an Abstract Computational Machine -- it abstracts the concept of computation (not the fact that it uses a Tape, etc.) and as I said earlier: computation IS software. >I'm not sure what the concern is with every object having a morphism Every program is a function type, and every type is an object. Thus, (type-theoretically at least) every object within the category is also a morphism and vice-versa. Considering Turing Completeness, we can create any program and thus any type from other types (it itself being a type). Thus, the Category can be achieved as a whole as a 'program-type (function type)' since everything within the category is a type itself. This creates strange paradoxes, where the category (and all sub-categories and functors between them) are inside the category itself. (Side-thought: I wonder though if you could address decision problems with this craziness.) >typed linear lambda theories Symmetric Monoidal Categories give rise to Linear Type Theories. However, Braided Monoidal Categories (and especially Monoidal Categories) allow for more. The Rosetta Stone diagram in the conclusion section doesn't restrict the category class (if I remember correctly). >I tend to find this paper really illuminating and instructive Me too. :) >but instead an exploration of lambda theories. hehe, I'm not a fan of Lambda Theories (but please excuse my bias). EDIT: An explanation for the downvote would be appreciated. Thanks. :)

deleted 12y ago
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tel 12y ago
Regardless of your belief about the relationship between software and hardware, I think it's reasonable to compare TM and LC by considering TM as "concrete" with its notion of the tape and the algorithm and LC as "abstract" with its notion of reduction and equivalences. Every program (for a suitable definition of program) is a morphism in a category with objects as types. The types themselves are not (programmatic) objects at all—they're merely the "objects" of the category. The arrows from 1 to a type are the values of that type, though. As stated, Completeness would allow you to embed an interpreter which means some of those arrows ought to look a bit like `(a -> b) -> (a -> b)`, the simplest of which being the `id` arrow on `(a -> b)`. That said, not all theories of typed lambda calculus are Complete. There's nothing that says a Category has to be well-founded, either, for that matter. If you use set comprehension to deal with the components of your category then you might end up with a Russel's paradox. For mathematics involving "large" categories you use a Universe formalism which lets you dodge those issues for a while. > However, Braided Monoidal Categories (and especially Monoidal Categories) allow for more. The Rosetta Stone diagram in the conclusion section doesn't restrict the category class (if I remember correctly). Yep. And there are many kinds of LC. A lot of interesting work comes out of studying them all and comparing them all. Edit: Note, I wasn't the downvoter.
- StandardFuture 12y ago
  >compare TM and LC You can, but Turing already did such in 1937 and showed equivalence of expressiveness. :p >The types themselves are not (programmatic) objects at all—they're merely the "objects" of the category. Do you mean 'programmatic' in the computational sense or categorical sense? Because, I was referring to the fact that every program (morphism) also has an equivalent function type (internal hom). Function types are objects within the Category as well. :) >If you use set comprehension to deal with the components of your category then you might end up with a Russel's paradox. It seems that this paper is using a set-theoretic comprehension for the concepts. Although, you could consider 'more powerful' constructions like Universes. I know a lot has been put into the study of LCs and I have nothing against studying them. I just have a personal preference to use other 'methods' to study computational theory. :p
  - tel 12y ago
    I think we're just talking around one another. I don't understand how your criticisms are anything besides style, I suppose. If you dislike LC and distrust the mechanics of CT then of course a paper which uses CT to unify LC with math and physics won't float your boat—but it's still perfectly cromulent.

deleted 12y ago

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tel 12y ago

Regardless of your belief about the relationship between software and hardware, I think it's reasonable to compare TM and LC by considering TM as "concrete" with its notion of the tape and the algorithm and LC as "abstract" with its notion of reduction and equivalences. Every program (for a suitable definition of program) is a morphism in a category with objects as types. The types themselves are not (programmatic) objects at all—they're merely the "objects" of the category. The arrows from 1 to a type are the values of that type, though. As stated, Completeness would allow you to embed an interpreter which means some of those arrows ought to look a bit like `(a -> b) -> (a -> b)`, the simplest of which being the `id` arrow on `(a -> b)`. That said, not all theories of typed lambda calculus are Complete. There's nothing that says a Category has to be well-founded, either, for that matter. If you use set comprehension to deal with the components of your category then you might end up with a Russel's paradox. For mathematics involving "large" categories you use a Universe formalism which lets you dodge those issues for a while. > However, Braided Monoidal Categories (and especially Monoidal Categories) allow for more. The Rosetta Stone diagram in the conclusion section doesn't restrict the category class (if I remember correctly). Yep. And there are many kinds of LC. A lot of interesting work comes out of studying them all and comparing them all. Edit: Note, I wasn't the downvoter.

StandardFuture 12y ago
>compare TM and LC You can, but Turing already did such in 1937 and showed equivalence of expressiveness. :p >The types themselves are not (programmatic) objects at all—they're merely the "objects" of the category. Do you mean 'programmatic' in the computational sense or categorical sense? Because, I was referring to the fact that every program (morphism) also has an equivalent function type (internal hom). Function types are objects within the Category as well. :) >If you use set comprehension to deal with the components of your category then you might end up with a Russel's paradox. It seems that this paper is using a set-theoretic comprehension for the concepts. Although, you could consider 'more powerful' constructions like Universes. I know a lot has been put into the study of LCs and I have nothing against studying them. I just have a personal preference to use other 'methods' to study computational theory. :p
- tel 12y ago
  I think we're just talking around one another. I don't understand how your criticisms are anything besides style, I suppose. If you dislike LC and distrust the mechanics of CT then of course a paper which uses CT to unify LC with math and physics won't float your boat—but it's still perfectly cromulent.