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7.2 Practical Themes
As shown by (H
¨
ofner and Struth, 2007; Foster and
Struth, 2012), the level of abstraction that semirings
and Kleene algebra provide is very well-suited for au-
tomation. It would therefore be very interesting to
mechanise and automate the algebraic theory of this
paper. With the aid of such a mechanisation, more
elaborate applications of the algebra would become
feasible.
I have emphasised earlier, that the abstract algebra
is the pivot of the framework presented in this paper –
not a model (a conrete algebra). The axioms were jus-
tified by common sense, and the relational model was
only used for proving technical results. Reasoning in
this framework is therefore genuinely about pushing
symbols, especially since the operators in Kleene al-
gebra and semirings all have a low, fixed arity.
4
This
means that a robot, say, that would be constructed on
the basis of the theory in this paper, would not have a
sole, involved model as its theoretical underpinning,
but just the axioms of a very simple abstract algebra,
and with them the whole associated class of models
(such as sets with only a few members, the relational
model, etc.). In combination with the tools avail-
able for automated proof and counterexample genera-
tion, this could make for very efficient and pragmatic
knowledge engineering.
ACKNOWLEDGEMENTS
Most of this work was done at the Department of Phi-
losophy, Uppsala University, but some was done at the
Computing Department, Macquarie University, Syd-
ney, and a significant part at the University of Queens-
land. A grant from Sven och Dagmar Sal
´
ens Stif-
telse made the stay in Sydney possible. The work
in Queensland was supported by the Australian Re-
search Council (ARC) Discovery Grant DP0987452.
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