[TYPES] Types as theories

[Rosu, Grigore]

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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