Equational correspondence and equational embedding

[Philip Wadler <wadler@inf.ed.ac.uk>]

The notion of *equational correspondence* was defined by Sabry and
Felleisen in their paper:

    Amr Sabry and Matthias Felleisen
    Reasoning about programs in continuation-passing style
    LISP and Symbolic Computation, 6(3--4):289--360, November 1993.

We lay out below the definition, along with a related notion of
*equational embedding*.  We've seen relatively little about equational
correspondence in the literature, and nothing about equational
embedding.  Given that these seem to be fundamental concepts, we
suspect we've been looking in the wrong places.  Any pointers to
relevant literature would be greatly apprectiated.

An *equational theory* T is a set of terms t of T, and a set of equations
t =_T t' (where t, t' are terms of T).

Let S, T be equational theories.  We say that f : S -> T and g : T -> S
constitute an *equational correspondence* between S and T if

   1.  g(f(s)) =_S s,  for all terms s in S
   2.  f(g(t)) =_T t,  for all terms t in T
   3.  s =_S s'  implies  f(s) =_T f(s'),  for all terms s, s' in S
   4.  t =_T t'  implies  g(t) =_S g(t'),  for all terms t, t' in T

Let S, T be equational theories.  We say that f : S -> T and
g : f(S) -> S (where f(S) is the image of S under f, a subset of T)
constitute an *equational embedding* between S and T if

   1.  g(f(s)) =_S s,  for all s in S
   2.  s =_S s'  implies  f(s) =_T f(s'),  for all s, s' in S

It is easy to see that there is an equational embedding of S into T if
and only if there is a function from S to T that preserves and
reflects equations.

Clearly, f and g constitute an equational correspondence between S and
T if f and g constitute an equational embedding of S into T and g and
f constitute an equational embedding of T into S.

We conjecture that whenever there is an equational embedding of S into
T and of T into S that there is an equational correspondence between S
and T.  This is not immediate, because one equational embedding might
be given by f and g and the other by h and k, with no obvious relation
between the two.

As I said, any pointers to relevant literature would be greatly
appreciated!

Yours,  -- Sam Lindley, Philip Wadler, Jeremy Yallop

-- 
  \ Philip Wadler, Professor of Theoretical Computer Science
  /\ School of Informatics, University of Edinburgh
/  \ http://homepages.inf.ed.ac.uk/wadler/

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