362 lines
12 KiB
Plaintext
362 lines
12 KiB
Plaintext
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Article 343 of eunet.jokes:
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Xref: puukko junk:7310 sci.math:1574 eunet.jokes:343
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Path: puukko!santra!tut!enea!mcvax!cernvax!ethz!heiser
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From: heiser@ethz.UUCP (Gernot Heiser)
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Newsgroups: rec.humor,sci.math,eunet.jokes
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Subject: Re: Math Jokes
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Message-ID: <464@ethz.UUCP>
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Date: 4 Jun 88 12:08:44 GMT
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References: <3440@pasteur.Berkeley.Edu> <2932@phoenix.Princeton.EDU> <1155@bentley.UUCP> <1156@bentley.UUCP> <546@osupyr.mast.ohio-state.edu> <583@picuxa.UUCP>
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Reply-To: heiser@iis.UUCP (Gernot Heiser)
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Organization: ETH Zuerich, Switzerland
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Lines: 184
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The following is from a book whose title I don't recall. The book is in German
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but the article is actually a translation from the original by H. Petard which
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appared in the American Monthly 54, 466 (1938). Unfortunately our library is
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lacking some years of this journal around WW 2, so I had to re-translate the
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stuff into English. (That will make you people share the experience of reading
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German translations of books on Einstein which also usually re-translate
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Einstein's words :-) ).
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A Contribution to the Mathematical Theory of Big Game Hunting
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=============================================================
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Problem: To Catch a Lion in the Sahara Desert.
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1. Mathematical Methods
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1.1 The Hilbert (axiomatic) method
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We place a locked cage onto a given point in the desert. After that we
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introduce the following logical system:
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Axiom 1: The set of lions in the Sahara is not empty.
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Axiom 2: If there exists a lion in the Sahara, then there exists a lion in
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the cage.
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Procedure: If P is a theorem, and if the following is holds:
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"P implies Q", then Q is a theorem.
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Theorem 1: There exists a lion in the cage.
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1.2 The geometrical inversion method
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We place a spherical cage in the desert, enter it and lock it from inside. We
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then performe an inversion with respect to the cage. Then the lion is inside
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the cage, and we are outside.
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1.3 The projective geometry method
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Without loss of generality we can view the desert as a plane surface. We
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project the surface onto a line and afterwards the line onto an interiour point
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of the cage. Thereby the lion is mapped onto that same point.
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1.4 The Bolzano-Weierstrass method
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Divide the desert by a line running from north to south. The lion is then
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either in the eastern or in the western part. Lets assume it is in the eastern
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part. Divide this part by a line running from east to west. The lion is either
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in the northern or in the southern part. Lets assume it is in the northern
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part. We can continue this process arbitrarily and thereby constructing with
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each step an increasingly narrow fence around the selected area. The diameter
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of the chosen partitions converges to zero so that the lion is caged into a
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fence of arbitrarily small diameter.
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1.5 The set theoretical method
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We observe that the desert is a separable space. It therefore contains an
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enumerable dense set of points which constitutes a sequence with the lion as
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its limit. We silently approach the lion in this sequence, carrying the proper
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equipment with us.
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1.6 The Peano method
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In the usual way construct a curve containing every point in the desert. It has
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been proven [1] that such a curve can be traversed in arbitrarily short time.
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Now we traverse the curve, carrying a spear, in a time less than what it takes
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the lion to move a distance equal to its own length.
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1.7 A topological method
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We observe that the lion possesses the topological gender of a torus. We embed
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the desert in a four dimensional space. Then it is possible to apply a
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deformation [2] of such a kind that the lion when returning to the three
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dimensional space is all tied up in itself. It is then completely helpless.
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1.8 The Cauchy method
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We examine a lion-valued function f(z). Be \zeta the cage. Consider the integral
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1 [ f(z)
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------- I --------- dz
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2 \pi i ] z - \zeta
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C
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where C represents the boundary of the desert. Its value is f(zeta), i.e. there
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is a lion in the cage [3].
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1.9 The Wiener-Tauber method
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We obtain a tame lion, L_0, from the class L(-\infinity,\infinity), whose
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fourier transform vanishes nowhere. We put this lion somewhere in the desert.
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L_0 then converges toward our cage. According to the general Wiener-Tauner
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theorem [4] every other lion L will converge toward the same cage.
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(Alternatively we can approximate L arbitrarily close by translating L_0
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through the desert [5].)
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2 Theoretical Physics Methods
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2.1 The Dirac method
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We assert that wild lions can ipso facto not be observed in the Sahara desert.
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Therefore, if there are any lions at all in the desert, they are tame. We leave
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catching a tame lion as an execise to the reader.
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2.2 The Schroedinger method
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At every instant there is a non-zero probability of the lion being in the cage.
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Sit and wait.
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2.3 The nuclear physics method
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Insert a tame lion into the cage and apply a Majorana exchange operator [6] on
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it and a wild lion.
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As a variant let us assume that we would like to catch (for argument's sake) a
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male lion. We insert a tame female lion into the cage and apply the Heisenberg
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exchange operator [7], exchanging spins.
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2.4 A relativistic method
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All over the desert we distribute lion bait containing large amounts of the
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companion star of Sirius. After enough of the bait has been eaten we send a
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beam of light through the desert. This will curl around the lion so it gets all
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confused and can be approached without danger.
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3 Experimental Physics Methods
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3.1 The thermodynamics method
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We construct a semi-permeable membrane which lets everything but lions pass
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through. This we drag across the desert.
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3.2 The atomic fission method
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We irradiate the desert with slow neutrons. The lion becomes radioactive and
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starts to diintegrate. Once the disintegration process is progressed far enough
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the lion will be unable to resist.
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3.3 The magneto-optical method
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We plant a large, lense shaped field with cat mint (nepeta cataria) such that
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its axis is parallel to the direction of the horizontal component of the
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earth's magnetic field. We put the cage in one of the field's foci. Throughout
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the desert we distribute large amounts of magnetized spinach (spinacia
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oleracea) which has, as everybody knows, a high iron content. The spinach is
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eaten by vegetarian desert inhabitants which in turn are eaten by the lions.
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Afterwards the lions are oriented parallel to the earth's magnetic field and
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the resulting lion beam is focussed on the cage by the cat mint lense.
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[1] After Hilbert, cf. E. W. Hobson, "The Theory of Functions of a Real
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Variable and the Theory of Fourier's Series" (1927), vol. 1, pp 456-457
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[2] H. Seifert and W. Threlfall, "Lehrbuch der Topologie" (1934), pp 2-3
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[3] According to the Picard theorem (W. F. Osgood, Lehrbuch der
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Funktionentheorie, vol 1 (1928), p 178) it is possible to catch every lion
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except for at most one.
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[4] N. Wiener, "The Fourier Integral and Certain of itsl Applications" (1933),
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pp 73-74
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[5] N. Wiener, ibid, p 89
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[6] cf e.g. H. A. Bethe and R. F. Bacher, "Reviews of Modern Physics", 8
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(1936), pp 82-229, esp. pp 106-107
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[7] ibid
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--
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Gernot Heiser <heiser@iis.UUCP> Phone: +41 1/256 23 48
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Integrated Systems Laboratory CSNET/ARPA: heiser%ifi.ethz.ch@relay.cs.net
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ETH Zuerich EARN/BITNET: GRIDFILE@CZHETH5A
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CH-8092 Zuerich, Switzerland EUNET/UUCP: {uunet,...}!mcvax!ethz!iis!heiser
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Article 426 of eunet.jokes:
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Path: puukko!santra!tut!enea!mcvax!steven
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From: steven@cwi.nl (Steven Pemberton)
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Newsgroups: eunet.jokes
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Subject: Re: Catching a Lion with computer science
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Message-ID: <388@piring.cwi.nl>
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Date: 6 Jul 88 11:24:18 GMT
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References: <2024@sics.se>
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Reply-To: steven@cwi.nl (mcvax!steven.uucp)
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Distribution: eunet
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Organization: CWI, Amsterdam
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Lines: 66
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Linear search:
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Stand in the top left hand corner of the Sahara Desert. Take one step
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east. Repeat until you have found the lion, or you reach the right
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hand edge. If you reach the right hand edge, take one step
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southwards, and proceed towards the left hand edge. When you finally
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reach the lion, put it the cage. If the lion should happen to eat you
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before you manage to get it in the cage, press the reset button, and
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try again.
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Dijkstra approach:
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The way the problem reached me was: catch a wild lion in the Sahara
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Desert. Another way of stating the problem is:
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Axiom 1: Sahara elem deserts
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Axiom 2: Lion elem Sahara
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Axiom 3: NOT(Lion elem cage)
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We observe the following invariant:
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P1: C(L) v not(C(L))
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where C(L) means: the value of "L" is in the cage.
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Establishing C initially is trivially accomplished with the statement
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;cage := {}
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Note 0.
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This is easily implemented by opening the door to the cage and
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shaking out any lions that happen to be there initially.
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(End of note 0.)
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The obvious program structure is then:
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;cage:={}
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;do NOT (C(L)) ->
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;"approach lion under invariance of P1"
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;if P(L) ->
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;"insert lion in cage"
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[] not P(L) ->
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;skip
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;fi
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;od
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where P(L) means: the value of L is within arm's reach.
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Note 1.
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Axiom 2 ensures that the loop terminates.
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(End of note 1.)
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Exercise 0.
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Refine the step "Approach lion under invariance of P1".
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(End of exercise 0.)
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Note 2.
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The program is robust in the sense that it will lead to abortion if
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the value of L is "lioness".
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(End of note 2.)
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Remark 0.
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This may be a new sense of the word "robust" for you.
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(End of remark 0.)
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Note 3.
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From observation we can see that the above program leads to the
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desired goal. It goes without saying that we therefore do not have to
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run it.
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(End of note 3.)
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(End of approach.)
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Steven Pemberton, CWI, Amsterdam; steven@cwi.nl
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Article 426 of eunet.jokes:
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Path: puukko!santra!tut!enea!mcvax!steven
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From: steven@cwi.nl (Steven Pemberton)
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Newsgroups: eunet.jokes
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Subject: Re: Catching a Lion with computer science
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Message-ID: <388@piring.cwi.nl>
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Date: 6 Jul 88 11:24:18 GMT
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References: <2024@sics.se>
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Reply-To: steven@cwi.nl (mcvax!steven.uucp)
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Distribution: eunet
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Organization: CWI, Amsterdam
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Lines: 66
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Linear search:
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Stand in the top left hand corner of the Sahara Desert. Take one step
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east. Repeat until you have found the lion, or you reach the right
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hand edge. If you reach the right hand edge, take one step
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southwards, and proceed towards the left hand edge. When you finally
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reach the lion, put it the cage. If the lion should happen to eat you
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before you manage to get it in the cage, press the reset button, and
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try again.
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|
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Dijkstra approach:
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The way the problem reached me was: catch a wild lion in the Sahara
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Desert. Another way of stating the problem is:
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Axiom 1: Sahara elem deserts
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Axiom 2: Lion elem Sahara
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Axiom 3: NOT(Lion elem cage)
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We observe the following invariant:
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P1: C(L) v not(C(L))
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where C(L) means: the value of "L" is in the cage.
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Establishing C initially is trivially accomplished with the statement
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;cage := {}
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Note 0.
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This is easily implemented by opening the door to the cage and
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shaking out any lions that happen to be there initially.
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(End of note 0.)
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The obvious program structure is then:
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;cage:={}
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;do NOT (C(L)) ->
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;"approach lion under invariance of P1"
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;if P(L) ->
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;"insert lion in cage"
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[] not P(L) ->
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;skip
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;fi
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;od
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where P(L) means: the value of L is within arm's reach.
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Note 1.
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Axiom 2 ensures that the loop terminates.
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(End of note 1.)
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Exercise 0.
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Refine the step "Approach lion under invariance of P1".
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(End of exercise 0.)
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Note 2.
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The program is robust in the sense that it will lead to abortion if
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the value of L is "lioness".
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(End of note 2.)
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Remark 0.
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This may be a new sense of the word "robust" for you.
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(End of remark 0.)
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Note 3.
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From observation we can see that the above program leads to the
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desired goal. It goes without saying that we therefore do not have to
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run it.
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(End of note 3.)
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(End of approach.)
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Steven Pemberton, CWI, Amsterdam; steven@cwi.nl
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