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[The main aspect which separates many approaches to mechanized mathematics is the importance (or lack thereof) of types. Most of mathematics requires dependent types at some level or another, making this an interesting testing ground for types members]

 Programming Languages for Mechanized Mathematics Workshop

As part of Calculemus 2007 <http://www.risc.uni-linz.ac.at/about/conferences/Calculemus2007/>

Hagenberg, Austria


The intent of this workshop is to examine more closely the intersection between programming languages and mechanized mathematics systems (MMS). By MMS, we understand computer algebra systems (CAS), [automated] theorem provers (TP/ATP), all heading towards the development of fully unified systems (the MMS), sometimes also called universal mathematical assistant systems (MAS) (see Calculemus 2007 <http://www.risc.uni-linz.ac.at/about/conferences/Calculemus2007/>).

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 <http://www.haskell.org/docon/> being typical). There are
     some promising languages on the horizon (Epigram
     <http://www.e-pig.org/>, Omega
     <http://web.cecs.pdx.edu/%7Esheard/Omega/index.html>) as well as
     some hybrid systems (Agda <http://agda.sourceforge.net/>, Focal
     <http://focal.inria.fr/site/index.php>), 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 <http://www.cs.chalmers.se/%7Eaugustss> [Credit Suisse]
Wieb Bosma <http://www.math.ru.nl/%7Ebosma/>[Radboud University Nijmegen, Netherlands] Jacques Carette <http://www.cas.mcmaster.ca/%7Ecarette> (co-Chair) [McMaster University, Canada]
David Delahaye <http://cedric.cnam.fr/%7Edelahaye/> [CNAM, France]
Jean-Christophe Filliâtre <http://www.lri.fr/%7Efilliatr/> [CNRS and Université de Paris-Sud, France]
John Harrison <http://www.cl.cam.ac.uk/%7Ejrh13/> [Intel Corporation, USA]
Markus (Makarius) Wenzel <http://www4.in.tum.de/%7Ewenzelm/> [Technische Universität München, Germany] Freek Wiedijk <http://www.cs.ru.nl/%7Efreek/> (co-Chair) [Radboud University Nijmegen, Netherlands] Wolfgang Windsteiger <http://www.risc.uni-linz.ac.at/people/wwindste/> [University of Linz, Austria]

   Location and Registration

Location and registration information can be found on the Calculemus <http://www.risc.uni-linz.ac.at/about/conferences/Calculemus2007/> web site.

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