Symmetries and transitions of bounded Turing machines
Abstract
We consider the structures given by repeatedly generalising the definition of finite state automata by symmetry considerations, and constructing analogues of transition monoids at each step. This approach first gives us nondeterministic automata, then (nondeterministic) twoway automata and bounded Turing machines  that is, Turing machines where the read / write head is unable to move past the end of the input word. In the case of twoway automata, the transition monoids generalise to endomorphism monoids in compact closed categories. These use Girard's resolution formula (from the Geometry of Interaction representation of linear logic) to construct the images of singleton words. In the case of bounded Turing machines, the transition homomorphism generalises to a monoid homomorphism from the natural numbers to a monoid constructed from the union of endomorphism monoids of a compact closed category, together with an appropriate composition. These use Girard's execution formula (also from the Geometry of Interaction representation of linear logic) to construct images of singletons.
 Publication:

arXiv eprints
 Pub Date:
 December 1998
 DOI:
 10.48550/arXiv.cs/9812019
 arXiv:
 arXiv:cs/9812019
 Bibcode:
 1998cs.......12019H
 Keywords:

 Logic in Computer Science;
 Category Theory;
 F.1.1;
 f.4.1
 EPrint:
 21 pages, submitted