jack@cs.glasgow.ac.UK (Mr Jack Campin) (10/03/88)
smryan@garth.UUCP (Steven Ryan) wrote: >> "Insufficient attention has been paid to the problem of continuous >> actions." Now, a question that immediately comes to mind is "What problem?" > Continuous systems are computable using calculus, but is this `effective > computation?' Calculus uses a number of existence theorems which prove some > point or set exists, but provide no method to effectively compute the value. > It is not clear that all natural phenomena can be modelled on the discrete > and finite digital computer. If not, what computer could we use? I brought up this same point in the Usenet sci.logic newsgroup a short while ago. There is a precise sense in which analogue computers are more powerful than digital ones - i.e. there are continuous phenomena unsimulatable on a Turing machine. Most of the work on this has been done by Marian Pour-El and her coworkers. An early paper is "A computable ordinary differential equation which possesses no computable solution", Annals of Mathematical Logic, volume 17, 1979, pages 61-90. This result is a bit of a cheat (the way the equation is set up has little relation to anything in the physical world) but I believe later papers tighten it up somewhat (one uses the wave equation, which you'd expect to be a powerful computing device given that interferometers can calculate Fourier transforms in constant time). I haven't seen these later articles, though. -- Jack Campin, Computing Science Department, Glasgow University, 17 Lilybank Gardens, Glasgow G12 8QQ, SCOTLAND. 041 339 8855 x6045 wk 041 556 1878 ho ARPA: jack%cs.glasgow.ac.uk@nss.cs.ucl.ac.uk USENET: jack@glasgow.uucp JANET: jack@uk.ac.glasgow.cs PLINGnet: ...mcvax!ukc!cs.glasgow.ac.uk!jack