From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1666 Path: news.gmane.org!not-for-mail From: Dusko Pavlovic Newsgroups: gmane.science.mathematics.categories Subject: Re: coinduction Date: Thu, 26 Oct 2000 16:49:12 -0700 Message-ID: <39F8C2F8.3AF5D614@kestrel.edu> References: <003301c03a95$49f73610$448a0dd8@main> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018006 32326 80.91.229.2 (29 Apr 2009 15:13:26 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:13:26 +0000 (UTC) To: "Categories@Mta. Ca" Original-X-From: rrosebru@mta.ca Fri Oct 27 14:47:45 2000 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id e9RH4NT26134 for categories-list; Fri, 27 Oct 2000 14:04:23 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Mozilla 4.5 [en] (X11; U; SunOS 5.5.1 sun4u) X-Accept-Language: en Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 24 Original-Lines: 121 Xref: news.gmane.org gmane.science.mathematics.categories:1666 Archived-At: Al Vilcius asks: > > can anyone explain coinduction? > > in what sense it dual to induction? and most people reply along the lines of: > The principle of induction says that initial algebras have no > non-trivial subalgebras. > Coinduction says, dually, that the final coalgebra has no non-trivial > quotients. but wouldn't it be nice to keep the standard definition of induction, which says that the equations f(0) = c f(n+1) = g(f(n)) determine a unique f, given c and g? this is the definition that most first year students study, and many programmers use every day, usually without the slightest idea about subalgebras, or triviality. do we really gain anything by replacing this simple scheme by the higher-order statement about all subalgebras? ok, we do get the appearance of symmetry between induction and coinduction. but is the higher-order statement about quotients really the friendliest form of coinduction, suitable to be taken as the definition? i think barwise and moss somewhere describe coinduction as *induction without the base case*. the coinduction principle says that the equation X = a.X + bc.X determines a unique CCS-process, or a unique {a,b,c}-labelled hyperset. it also guarantees that the integral equation E(x) = 1 + \int_0^x E dt has a unique solution. (note that this is equivalent to the differential equation E(0) = 1 E = E' where the initial condition gives the base case for the power series solution. so the no-base-case slogan is just a matter of form? but note that E' = E, and more generally E' = h(E), for an analytic h, identifies two *infinite* streams, and is not a mere inductive step up the stream.) in any case, i contend that the practice of coinduction primarily consists of solving *guarded equations*, like the two above --- surely if you count the areas where people extensively use coinduction, without calling it by name, viz math analysis. ((it is true that computer scientists became aware of coinduction while working out the bisimulations etc, which is why the quotients came in light. but now it seems that the bisimulations were just the way to construct final coalgebras; whereas the coinduction, in analogy with the induction, should be a practically useful logical principle, available over the *given* final coalgebras.)) anyway, the heart of the matter probably lies in peter aczel's insight that the statement * V ---> PV is a final coalgebra for P = (finite) powerset functor equivalently means that * every accessible pointed graph has a unique decoration in V. presenting the graph definitions as systems of guarded equations (like in barwise and moss' book "vicious circles"), the above equivalence generalizes to the statement that * V --d--> FV is a final coalgebra for F iff * every system of guarded equations over V has a unique solution. where the notion of guarded is determined by F and d. this is, of course, just barwise and moss' solution lemma, lifted to F. or more succinctly : **V is a final fixpoint iff each guarded operation g:V-->V has a unique fixpoint** this, of course, supports the view that ***coinduction is solving guarded equations***... i was hoping to say more, but will have to leave it for later, and post this hoping to motivate discussion. unguardedly yours, -- dusko pavlovic PS i have a great appreciation for jan rutten's work on coinduction, but it seems to me that the title of *universal coalgebra* is a bit exaggerated. universal algebra does not dualize. by design, it is meant to be model theory of algebraic theories. what would be the "coalgebraic theories", leading to coalgebras as their models? what would their functorial semantics look like? (implicitly, these questions are already in manes' book.) and then, moving from coalgebraic theories to comonads, like from algebraic theories to monads, should presumably yield an HSP-like characterization for the categories of (comonad) coalgebras. but coalgebras for a comonad over a topos usually form just another topos. i may be missing something essential here, but then i do stand to be corrected.