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A good place to start is http://math.ucr.edu/home/baez/week115.htmlAnonymoushttps://www.blogger.com/profile/02376760053977600605noreply@blogger.comtag:blogger.com,1999:blog-33803276.post-48763640253350264342007-08-05T15:32:00.000-04:002007-08-05T15:32:00.000-04:00Interesting, I'll definitely have to study this so...Interesting, I'll definitely have to study this some more, like I said I'm no expert, but the idea popped into my head, and I figured it'd be an interesting idea to share.<BR/><BR/>I am a little confused, I think that you can still use the DFA I described towards the end as a kind of a type-checker, because it is designed to specifically deal with composition of arrows. You certainly seem to be more knowledgable about this than me. I suppose the idea is just this:<BR/><BR/>a valid function created by:<BR/>(f . g . h . i) <BR/>has a signature that looks like <BR/><BR/>a -> b<BR/> b -> c<BR/> c -> d<BR/> d -> e<BR/>or<BR/>a -> e<BR/><BR/>So if the states are the types, then the functions are arrows between types, composition is then implicitly defined, I think.<BR/><BR/>Anyway,thanks for the info. <BR/><BR/>~~JoeJoe Fredettehttps://www.blogger.com/profile/07170469679136904832noreply@blogger.comtag:blogger.com,1999:blog-33803276.post-48681714247827838422007-08-05T14:17:00.000-04:002007-08-05T14:17:00.000-04:00I think your definition of Category is a little of...I think your definition of Category is a little off. A category has a class of objects, and for each pair (A,B) of objects, a set Hom(A,B), of arrows from A to B. (There may even be infinitely many such arrows.)<BR/><BR/>Further, if f: A -> B is an arrow in Hom(A,B), and g: B -> C is an arrow in Hom(B,C), then their composite g.f exists and is an arrow in Hom(A,C).<BR/><BR/>Additionally, for every object A, there is an arrow id_A: A -> A such that f . id_A = f and id_A . g = g for any f and g such that the composition is defined.<BR/><BR/>Also, whenever the compositions are defined, we have (f . g) . h = f . (g . h).<BR/><BR/>These laws are actually rather constraining in terms of which graph structures can actually be made into categories. The fact that compositions must be defined implies a certain kind of closure property, and that each object has at least one self-loop is also fairly constraining.<BR/><BR/>For example, if you want a single DFA to be a category, with state transitions as arrows, it must also have all compositions of state transitions as arrows as well. You'd also have to have identity arrows, which in the case of a DFA, wouldn't correspond to real transitions. So the process of turning a DFA into a category will in some sense modify its alphabet into strings on the original alphabet, add transitions on the empty string to every state, and then a transition for every composition of transitions whose symbol is the corresponding string of symbols. As a result, we could read off the language that the DFA accepts simply by collecting up the strings on each of the arrows from the start state to each of the accept states.<BR/><BR/>Another natural sort of category to form involving DFAs would be the category which has as objects all DFAs, and as arrows, has DFA-morphisms, where a DFA-morphism f:(Q,S,A,q_0,delta) -> (Q',S',A',q_0',delta'), consists of a pair of functions, f_Q: Q -> Q', assigning states of the source DFA to states of the target, and f_S: S -> S', input symbols of the source DFA to input symbols of the target, such that transitions are preserved: <BR/><BR/>delta'(f_S(s),f_Q(q)) = f_Q(delta(s,q))<BR/><BR/>initial state is preserved:<BR/><BR/>f_Q(q_0) = q_0'<BR/><BR/>and accept states are preserved:<BR/><BR/>If q is in A, then f_Q(q) is in A'.<BR/><BR/>One can form similar definitions in order to form, for example, a category of Moore machines, or a category for most any other kind of automaton.Cale Gibbardhttps://www.blogger.com/profile/02239068589033148700noreply@blogger.com