[TYPES] Equational correspondence and equational embedding

[Matthias Felleisen]

Phil, Amr responded with the bare facts. Here is my path of  
discovery, with two additional lit ptrs:

John Gately was an MS student at IU working in Indiana. I suggested  
to him that he work out a cbv theory of combinators in the spirit of  
Gordon Plotkin's CBN/CBV paper in TCS(1974). I also suggested that he  
read Barendregt (chapter 7, with an overview in chapter 2) because it  
summarizes the equational correspondence I was after. John dropped  
out from IU but rejoined me at Rice and published a paper on his CBV  
CL ~ lambda_v correspondence paper at some workshop (mathematical  
foundations of programming languages?).

Around the same time, I started Eric Crank on a correspondence  
between cps(\lambda_v) and lambda with the hope that a reduction rule  
Bruce Duba and I had played with since 1985: (\x.E[x]) M  = E[M] for  
x not in fv(E) [beta_Omega]. While Eric preferred to switch to  
equational characterizations of imperative parameter passing, Amr  
came along and solved this puzzle.

As Amr said, we tried to stay as close as possible to the source  
(Curry) with our write-up for LFP 1992.

-- Matthias


On Apr 15, 2008, at 8:07 AM, Philip Wadler wrote:
> 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/
>
> The University of Edinburgh is a charitable body, registered in
> Scotland, with registration number SC005336.
>