Chris Warburton  Katya Komendantskaya 
cmwarburton@dundee.ac.uk  e.komendantskaya@dundee.ac.uk 
University of Dundee  

$(\Sigma, V) \overset{TE}{\rightarrow} \text{Terms}(\Sigma, V)$
Theory Exploration describes any system for turning a "theory" (signature and variables) into terms of that theory which are wellformed, provable and interesting.
Mathematical theory exploration proceeds by applying, under the guidance of a human user, various algorithmic reasoners for producing new formulae from given ones and aims at building up (large) mathematical knowledge bases in an efficient, reliable, wellstructured, reusable, and flexible way.  (Buchberger 2004)
Example: Theorema
Mathematical^{Automated} theory exploration proceeds by applying,under the guidance of a human user, various algorithmic reasoners for producing new formulae from given ones and aims at building up (large) mathematical knowledge bases in an efficient, reliable, wellstructured, reusable, and flexible way.
Examples: IsaCoSy, Hipster, HipSpec
Strengths:
Weaknesses:
Problem: Doesn't seem to align very well with general Mathematics
Strengths:
Weaknesses:
Solution: Aligns well with programming
Haskell code often follows algebraic laws which Haskell is unable to express!
HipSpec is stateoftheart. How might it be scaled up?
inc x = x + 1
Thank you to the HipSpec team at Chalmers University (Moa Johansson, Koen Claessen, Nick Smallbone, Dan Rosén and Irene Lobo Valbuena) for useful discussions of these ideas, and Ouanis Seddaoui for help with our implementation.