Lambda the Ultimate

There are various ways in which these two subjects of
programming languages and systems for mathematics meet:

• Many systems for mathematics contain a dedicated programming
  language. For instance, most computer algebra systems contain a
  dedicated language (and are frequently built in that same language);
  some proof assistants (like the Ltac language for Coq) also have
  an embedded programming language. Note that in many instances
  this language captures only algorithmic content, and
  declarative or representational issues are
  avoided.
• The mathematical languages of many systems for mathematics
  are very close to a functional programming language. For
  instance the language of ACL2 is just Lisp, and the language
  of Coq is very close to Haskell. But even the mathematical
  language of the HOL system can be used as a functional
  programming language that is very close to ML and Haskell.
  On the other hand, these languages also contain very rich
  specification capabilities, which are rarely available in
  most computation-oriented programming languages. And even
  then, many specification languages ((B, Z, Maude, OBJ3, CASL, etc)
  can still teach MMSes a trick or two regarding representational
  power.
• Conversely, functional programming languages have been getting
  "more mathematical" all the time. For instance, they seem to have
  discovered the value of dependent types rather recently. But
  they are still not quite ready to 'host' mathematics (the non-success
  of docon being typical).
  There are some promising languages on the horizon (Epigram, Omega) as
  well as some hybrid systems (Agda, Focal), although it is
  unclear if they are truly capable of expressing the full range of ideas
  present in mathematics.
• Systems for mathematics are used to prove programs correct.
  (One method is to generate "correctness conditions" from a
  program that has been annotated in the style of Hoare logic
  and then prove those conditions in a proof assistant.) An
  interesting question is what improvements are needed for
  this both on the side of the mathematical systems and on the
  side of the programming languages.

We are interested in all these issues. We hope that a certain synergy
will develop between those issues by having them explored in parallel.

These issues have a very colourful history. Many programming language
innovations first appeared in either CASes or Proof Assistants, before
migrating towards more mainstream languages. One can cite (in no particular
order) type inference, dependent types, generics, term-rewriting, first-class
types, first-class expressions, first-class modules, code extraction, and so
on. However, a number of these innovations were never aggressively pursued by
system builders, letting them instead be developped (slowly) by
programming language researchers. Some, like type inference and generics
have flourished. Others, like first-class types and first-class expressions,
are not seemingly being researched by anyone.

We want to critically examine what has worked, and what has not.
Why are all the current ``popular'' computer algebra systems untyped? Why
are the (strongly typed) proof assistants so much harder to use than a
typical CAS? But also look at question like what
forms of polymorphism exists in mathematics? What forms of dependent types
exist in mathematics? How can MMS regain the upper hand on issues of
'genericity'? What are the biggest barriers to using a more mainstream
language as a host language for a CAS or an ATP?

This workshop will accept two kinds of submissions: full research
papers as well as position papers. Research papers should be nore more than
15 pages in length, and positions papers no more than 3 pages.
Submission will be through EasyChair. An informal
version of the proceedings will be available at the workshop, with a more
formal version to appear later. We are
looking into having the best papers completed into full papers
and published as a special issue of a Journal (details to follow).

Important Dates
April 25, 2007: Submission Deadline
June 29-30, 2007: Workshop

Program Committee
Lennart Augustsson [Credit Suisse]
Wieb Bosma[Radboud University Nijmegen, Netherlands]
Jacques Carette (co-Chair) [McMaster University, Canada]
David Delahaye [CNAM, France]
Jean-Christophe Filliâtre [CNRS and Université de Paris-Sud, France]
John Harrison [Intel Corporation, USA]
Markus (Makarius) Wenzel [Technische Universität München, Germany]
Freek Wiedijk (co-Chair) [Radboud University Nijmegen, Netherlands]
Wolfgang Windsteiger [University of Linz, Austria]

Location and Registration
Location and registration information can be found on the
Calculemus web site.