From mboxrd@z Thu Jan 1 00:00:00 1970 X-Spam-Checker-Version: SpamAssassin 3.4.4 (2020-01-24) on polar.synack.me X-Spam-Level: X-Spam-Status: No, score=-1.9 required=5.0 tests=BAYES_00 autolearn=ham autolearn_force=no version=3.4.4 X-Google-Thread: 103376,93a8020cc980d113 X-Google-Attributes: gid103376,public X-Google-Language: ENGLISH,ASCII-7-bit Newsgroups: comp.lang.ada Subject: Re: What is wrong with Ada? References: <1176150704.130880.248080@l77g2000hsb.googlegroups.com> <461B52A6.20102@obry.net> <461BA892.3090002@obry.net> <82dgve.spf.ln@hunter.axlog.fr> <1176226291.589741.257600@q75g2000hsh.googlegroups.com> <4eaive.6p9.ln@hunter.axlog.fr> <1rbtw92apxpl1.1ednvo8v6oiq8$.dlg@40tude.net> <13tcswu59l28h.zxb26cabf9a0.dlg@40tude.net> <15k5b4j6za8ag.tpkuccinvzbd.dlg@40tude.net> From: Markus E Leypold Organization: N/A Date: Sun, 22 Apr 2007 23:50:52 +0200 Message-ID: User-Agent: Some cool user agent (SCUG) Cancel-Lock: sha1:Z/npDLUMYyL3eylcoXz+syNtmPs= MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii NNTP-Posting-Host: 88.74.33.156 X-Trace: news.arcor-ip.de 1177278180 88.74.33.156 (22 Apr 2007 23:43:00 +0200) X-Complaints-To: abuse@arcor-ip.de Path: g2news1.google.com!news3.google.com!feeder3.cambrium.nl!feeder5.cambrium.nl!feed.tweaknews.nl!68.142.88.75.MISMATCH!hwmnpeer01.ams!news.highwinds-media.com!feed10.multikabel.net!multikabel.net!feed20.multikabel.net!newsfeed.freenet.de!news.unit0.net!newsfeed.arcor-ip.de!news.arcor-ip.de!not-for-mail Xref: g2news1.google.com comp.lang.ada:15236 Date: 2007-04-22T23:50:52+02:00 List-Id: "Dmitry A. Kazakov" writes: > On Sun, 15 Apr 2007 18:01:10 +0200, Markus E Leypold wrote: > >> Now I'd like you to close this loophole for arbitrary hand waving and >> define NON-TRIVIAL in a way suitable to you purposes (but keep it >> convincing, still -- defining it to FALSE won't wash with me) and >> perhaps try to prove the central assertion above. > > OK, here is a formalization of "non-trivial." Let me use a more or less > standard notation: > > IN is the set of input states (the language over a finite alphabet A) > S is the set of states > s1 is the initial state > T : S x A -> S is the transition function > OUT = the set of output states (a subset of S, which we don't care) > > def: Closure of T > ---------------------- > Let a=(a' a'' a''' a'''' ... a*) be a finite input from IN. > > P(a)=T(a*, ... T(a''', T(a'', T(a', s1)))) > > Informally P(a) is the state to which a would bring the machine. > > P : IN -> S > > def: Equivalent input states (strings) > --------------------------------------------- > a, b of IN are called equivalent iff P(a)=P(b). > > Let's denote non-equivalent states as a#b > > (P was defined on finite strings of IN. Defining it in some reasonable way > for infinite cases would require efforts, which I don't want to run into.) > > def: Non-trivial input (language) > -------------------------------------- > IN is non-trivial iff for any finite subset {a1, a2, a3,..., aN} of IN > there exits an input string b in IN such that forall i=1..N b#ai. > > From this definition immediately follows that any machine handling > non-trivial input will necessarily have infinite S. > > A classical example > > ab > aabb > aaabbb > aaaabbbb > ... > > can serve as one of non-trivial input. > > Other three I have mentioned in my previous posts: counting integers, > recognizing irrational numbers, summation of infinite series 1-1+1-1+1... > (aka Tompson's lamp). > > P.S. This does not apply to Robert's example, because he considered a set > of machines with individual machines parametrized by some number which > influences the number of states. You don't feel the need to justify the use of "trivial" for any actually existing program? Regards -- Markus