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In terms of the solution we have presented these properties are: (PI) every card in the left deck precedes X in alphabetical order; (P2) none of the cards in the right deck precedes X in alphabetical order; (P3) the original deck can be retrieved by recombining the left and right decks in that order The proof of correctness hinges on two aspects of the repeated process, the so-called bound function and the invariant In general, the bound function is an integer function of the data being manipulated It measures, in some sense, the 'size' of the problem remaining to be solved We use it to prove that the repeated process will always terminate, ie that the program will not 'get into a loop' and repeat the same sequence of actions indefinitely.

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The invariant is a property of the data being manipulated that holds irrespective of the number of repetitions that have been executed We use the invariant to establish so-called 'conditional correctness', meaning that the program meets its specification under the assumption of (or 'conditional' on) a proof of termination (a) Proof of Termination For the bound function, we take the number of cards in the middle deck To use it to prove that the process of repeatedly executing (Rl) or (R2) will always terminate, we make two observations (Tl) There is a lower bound (zero) on the number of cards in the middle deck In this case, the lower bound is dictated by the physical characteristics of a deck of cards In general, we look to the condition for terminating the repeated process to provide such a bound.

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(T2) Every time the repeated part of the search process is executed, the number of cards in the middle deck decreases by at least one This is because we have been careful to ensure that the inspected card is always removed from the middle deck and added to either the left or right deck Barcode barcode library in net Using Barcode scanner for Together, these two observations make it obvious that the umber of times the repeated part of the algorithm is executed is at most equal to the number of cards in the original deck It is worth noting in passing that this proof of termination does not depend on how the middle deck is split at each iteration It is allowed, for example, to always split it so that the lower deck is empty This is correct but inefficient.

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A more efficient strategy is to always split the middle deck into two (as near as possible) equal parts, since this means that its size is (approximately) halved at each repetition We return to the issue of efficiency in 4 but, for the moment, correctness is our only concern (b) Conditional Correctness The second part of the proof of correctness is given the name conditional correctness1 A proof of conditional correctness assumes that the execution of a program terminates and concentrates on establishing that its specification is met The properties we use to establish the conditional correctness of our searching algorithm have already been stated.

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They are: (11) every card in the left deck precedes X in alphabetical order; (12) none of the cards in the right deck precedes X in alphabetical order; (13) the original deck may be retrieved by recombining the left, middle and right decks, in that order Note that these properties hold no matter how often the splitting process has been executed, and so we refer to them as invariants Note, also, that properties (I1)-(I3) generalize the properties (P1)-(P3) above Indeed, (II) and (PI) are identical, as are (12) and (P2) Properties (13) and (P3) are, however, different; they are the same only when the middle deck is empty It is in this sense that (13) generalizes (P3) The invention of invariants is a crucial process in program construction Typically, as this example illustrates, invariants are generalizations of the required postcondition.

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We use the principle of mathematical induction to prove that these three properties really are invariant Generally, the principle of mathematical induction is used to prove that some property Pn is true for all natural numbers n (The natural numbers are the whole numbers beginning at 0, thus 0,1, 2, 3 and so on The notation Tn' indicates that P is a function of n) In this case, n refers to the number of times the middle deck has been split and property P is property (II) and property (12) and property (13) In other words, what we wish to prove is that, for any natural number n, each of properties (II), (12) and (13) hold after n iterations of the repeated processRelated: .

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