[TYPES] Types as theories

[PATRICK BROWNE]

Grigore,
Thanks for your clarification.
Yes, MONOID2 is convoluted. It was the simplest example that I could think
of that illustrated the sub-model idea and was similar to Goguen's example
(he actually used monoids and groups).
Thanks,
Pat


On 13 May 2014 15:17, Rosu, Grigore <grosu at illinois.edu> wrote:

> Hi Pat,
>
> Your two claims are correct, but note that in the second case,  model M
> containing another model M' can mean many different things, depending upon
> the importing mode.  For example, in your second example, you are stating
> that you protect TRIV, which actually makes your MONOID2 spec inconsistent:
> indeed, there is no model of MONOID2 whose reduct to the signature of TRIV
> is the initial model of TRIV, because the former will always contain at
> least one element (the unit) while the latter is the empty model.  So yes,
> you should use extending instead of protecting mode.
>
> However, both your algebraic specifications are unusual for the
> OBJ/CafeOBJ/Maude language families.  The most natural definition of a
> monoid is to just have a sort Monoid, the unit, the binary operation, and
> the three axioms, without getting the TRIV theory or the Elt sort involved
> at all.
>
> Grigore
>
>
>
> ________________________________________
> From: Types-list [types-list-bounces at lists.seas.upenn.edu] on behalf of
> PATRICK BROWNE [patrick.browne at dit.ie]
> Sent: Tuesday, May 13, 2014 7:09 AM
> To: types-list at lists.seas.upenn.edu
> Subject: [TYPES] Types as theories
>
> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list]
>
> Dear list
> I am studying Goguen's paper Types as theories [1].
> Below are two examples coded in CafeOBJ (any other algebraic specification
> language would do).
> Based on Goguen's paper, are the following true?
>
> 1) Subsort inheritance provides a classification of values, every value of
> the sub-sort is a value of the super-sort.(MONOID1)
>
> 2) Module import inheritance provides classification of models, each model
> of the class of models of the importing module contains a model of the
> imported module.(MONOID2)
>
>
> My intention is that MONOID1 illustrates 1) and 2) MONOID2 illustrates 2).
> Assumptions
>
>     the CafeOBJ module is the software realization of a theory
>
>     loose semantics, so any model of TRIV will do
>
>     the term type is synonym for sort
>
>     the < symbol indicates subsort
>
>     a monoid is associative and has a  neutral element.
>
> mod* MONOID1 {
> [  Monoid < Elt ]
> -- Signature in terms of super type
> op e : -> Elt
> op _._ : Elt Elt -> Elt
> -- equations in terms of subtype
> vars A B C : Group
> eq A . e = A .
> eq e . A = A .
> eq A . (B . C) = ((A . B) . C) .
>  }
>
>
> mod* MONOID2 {
> --
> -- The TRIV theory represented by the CafeOBJ
> -- built in loose theory TRIV which has Elt as its principal sort and no
> operations.
> -- I used protecting mode for import, perhaps it should be extending?
> pr(TRIV)
> op e : -> Elt
> op _._ : Elt Elt -> Elt
> vars A B C : Elt
> eq A . e = A .
> eq e . A = A .
> eq A . (B . C) = ((A . B) . C) . -- Associativity could be specified as a
> property
> }
>
> Regards, Pat
>
> [1]Goguen, J. (1991). Types as Theories. Topology and Category in Computer
> Science. G. M. Reed, A. W. Roscoe and R. F. Wachter, Oxford University
> Press: 357-390.
> http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.16.2241
>
> --
>
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Tá ITBÁC ag aistriú go Gráinseach Ghormáin – DIT is on the move to 
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