Final Activity Report Summary - SYSTEMATHEX (Systematic Mathematical Theory Exploration within the Theorema System: Case Studies)
The general scientific objectives of the project were carrying out case studies in computer supported exploration of mathematical theories, using an exploration model proposed by Bruno Buchberger, based on knowledge schemes.
Two major case studies were proposed, exploration of the theory of tuples, and exploration of the Groebner bases theory.
Following a period of training for the newly hired master students, where they were introduced to topics in logic and foundations of mathematics, notions of automated theorem proving, in particular using the Theorema mathematical assistant, we have identified the need to carry out an additional case study: exploration of the theory of natural numbers. This theory is needed further for both of the case studies proposed.
Bottom up exploration of natural numbers and tuple theory was carried out jointly by the researcher and the master students. This involved:
- Implementation of new version of provers for the respective theories,
- Build-up of a library of knowledge schemes, and a prototype implementation of a mechanism to use these schemes.
The result of the exploration is encouraging, as compared to well established textbooks, and we have reported this in a paper accepted at a workshop dedicated to mathematical theory exploration, which will appear in proceedings by an international publisher (IEEE Computer Society Press).
In parallel, the researcher has extended an implementation of the lazy thinking method for algorithm synthesis, in the Theorema system, to synthesize an algorithm for Groebner base computation. Together with ongoing work to address improvements of the algorithm (critical pair criteria, reduced Groebner bases), this is a significant part of the forthcoming PhD thesis (due this November, in the frame of the PhD programme at the Research Institute for Symbolic Computation, Linz, Austria). This work was reported at a workshop in the frame of the Special Semester on Groebner Bases, in Linz, Austria, and in the Calculemus 2006 conference.
As a result of the work carried out, we have identified the need for a framework to support the exploration process. We are now working on a specification for this framework, and believe that its implementation, together with the on-going case studies, will lead to a powerful system to do mathematical theory exploration on computers, something that is not available today.
We consider that the work carried out so far in our project shows the versatility of the scheme based exploration model:
- We have invented systematically, using an implementation in the Theorema system, fairly large portions of the theory of natural numbers and tuples, compared to well established textbooks;
- We have used an implementation of the lazy thinking method to synthesise an algorithm for Groebner base computation, so far outside of the scope of any synthesis method;
- The case studies carried out led to new research into a framework to support the exploration activity;
- We have presented the work in workshops and have a paper accepted for publication.
Two major case studies were proposed, exploration of the theory of tuples, and exploration of the Groebner bases theory.
Following a period of training for the newly hired master students, where they were introduced to topics in logic and foundations of mathematics, notions of automated theorem proving, in particular using the Theorema mathematical assistant, we have identified the need to carry out an additional case study: exploration of the theory of natural numbers. This theory is needed further for both of the case studies proposed.
Bottom up exploration of natural numbers and tuple theory was carried out jointly by the researcher and the master students. This involved:
- Implementation of new version of provers for the respective theories,
- Build-up of a library of knowledge schemes, and a prototype implementation of a mechanism to use these schemes.
The result of the exploration is encouraging, as compared to well established textbooks, and we have reported this in a paper accepted at a workshop dedicated to mathematical theory exploration, which will appear in proceedings by an international publisher (IEEE Computer Society Press).
In parallel, the researcher has extended an implementation of the lazy thinking method for algorithm synthesis, in the Theorema system, to synthesize an algorithm for Groebner base computation. Together with ongoing work to address improvements of the algorithm (critical pair criteria, reduced Groebner bases), this is a significant part of the forthcoming PhD thesis (due this November, in the frame of the PhD programme at the Research Institute for Symbolic Computation, Linz, Austria). This work was reported at a workshop in the frame of the Special Semester on Groebner Bases, in Linz, Austria, and in the Calculemus 2006 conference.
As a result of the work carried out, we have identified the need for a framework to support the exploration process. We are now working on a specification for this framework, and believe that its implementation, together with the on-going case studies, will lead to a powerful system to do mathematical theory exploration on computers, something that is not available today.
We consider that the work carried out so far in our project shows the versatility of the scheme based exploration model:
- We have invented systematically, using an implementation in the Theorema system, fairly large portions of the theory of natural numbers and tuples, compared to well established textbooks;
- We have used an implementation of the lazy thinking method to synthesise an algorithm for Groebner base computation, so far outside of the scope of any synthesis method;
- The case studies carried out led to new research into a framework to support the exploration activity;
- We have presented the work in workshops and have a paper accepted for publication.