294 lines
9.9 KiB
Plaintext
294 lines
9.9 KiB
Plaintext
To: humourous-backbone@rascal.ics.utexas.edu
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Subject: [Rich Kulawiec: GSP Digest #186]
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Date: Tue, 25 Jul 89 00:34:23 EST
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From: Gene Spafford <spaf@cs.purdue.edu>
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The classic bit about how to catch a lion. Added to the end are some
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new ways for physicists & computer scientists. I especially like
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Dijkstra's method...
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--spaf
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------- Forwarded Message
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==========
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Item 3:
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==========
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From: ji@walkuere.altair.fr (John Ioannidis)
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Date: 19 Jul 89 10:30:04 GMT
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Subject: how to catch a lion
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I found this some time ago in my mailbox. I'm not certain of its
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origins, but it's a good one. It looks like something that might have
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appeared in the JIR (Journal of Irreproducible Results) but I don't
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have all the back issues with me to check.
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A Contribution to the Mathematical Theory of Big Game Hunting
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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
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we 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
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lion in 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
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inside. We then performe an inversion with respect to the cage. Then
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the lion is inside 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.
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We project the surface onto a line and afterwards the line onto an
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interiour point of the cage. Thereby the lion is mapped onto that same
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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
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then either in the eastern or in the western part. Let's assume it is
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in the eastern part. Divide this part by a line running from east to
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west. The lion is either in the northern or in the southern part.
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Let's assume it is in the northern part. We can continue this process
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arbitrarily and thereby constructing with each step an increasingly
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narrow fence around the selected area. The diameter of the chosen
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partitions converges to zero so that the lion is caged into a fence of
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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
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contains an enumerable dense set of points which constitutes a
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sequence with the lion as its limit. We silently approach the lion in
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this sequence, carrying the proper 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
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desert. It has been proven [1] that such a curve can be traversed in
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arbitrarily short time. Now we traverse the curve, carrying a spear,
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in a time less than what it takes the lion to move a distance equal to
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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.
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We embed the desert in a four dimensional space. Then it is possible
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to apply a deformation [2] of such a kind that the lion when returning
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to the three dimensional space is all tied up in itself. It is then
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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
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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),
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i.e. there 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),
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whose fourier transform vanishes nowhere. We put this lion somewhere
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in the desert. L_0 then converges toward our cage. According to the
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general Wiener-Tauner theorem [4] every other lion L will converge
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toward the same cage. (Alternatively we can approximate L arbitrarily
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close by translating L_0 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
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desert. Therefore, if there are any lions at all in the desert, they
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are tame. We leave 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
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the cage. 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
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operator [6] on it and a wild lion.
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As a variant let us assume that we would like to catch (for argument's
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sake) a male lion. We insert a tame female lion into the cage and
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apply the Heisenberg 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
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of the companion star of Sirius. After enough of the bait has been
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eaten we send a beam of light through the desert. This will curl
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around the lion so it gets all confused and can be approached without
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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
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pass 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
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radioactive and starts to disintegrate. Once the disintegration
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process is progressed far enough 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)
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such that its axis is parallel to the direction of the horizontal
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component of the earth's magnetic field. We put the cage in one of the
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field's foci. Throughout the desert we distribute large amounts of
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magnetized spinach (spinacia oleracea) which has, as everybody knows,
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a high iron content. The spinach is eaten by vegetarian desert
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inhabitants which in turn are eaten by the lions. Afterwards the
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lions are oriented parallel to the earth's magnetic field and the
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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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4 Contributions from Computer Science.
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4.1 The search method
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We assume that the lion is most likely to be found in the direction to
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the north of the point where we are standing. Therefore the REAL
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problem we have is that of speed, since we are only using a PC to
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solve the problem.
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4.2 The parallel search method.
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By using parallelism we will be able to search in the direction to the
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north much faster than earlier.
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4.3 The Monte-Carlo method.
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We pick a random number indexing the space we search. By excluding
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neighboring points in the search, we can drastically reduce the number
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of points we need to consider. The lion will according to probability
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appear sooner or later.
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4.4 The practical approach.
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We see a rabbit very close to us. Since it is already dead, it is
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particularly easy to catch. We therefore catch it and call it a lion.
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4.5 The common language approach.
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If only everyone used ADA/Common Lisp/Prolog, this problem would be
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trivial to solve.
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4.6 The standard approach.
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We know what a Lion is from ISO 4711/X.123. Since CCITT have specified
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a Lion to be a particular option of a cat we will have to wait for a
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harmonized standard to appear. $20,000,000 have been funded for
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initial investigastions into this standard development.
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4.7 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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4.8 The 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 shaking
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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 esnures 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
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abortion if the value of L is "lioness".
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(End of note 2.)
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Remark 0: 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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