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Formal Methods and Models

Z Specification of the Library System

Boumediene BelkhoucheFall 2012

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Library System Description The library has many books on different topics The library has registered readers A book may have several copies Each copy in the library has a unique ID Each reader has a unique ID At any time a certain number of copies of books

are on loan to readers A reader cannot borrow more than a certain

maximum Available books are kept on the library shelves A book can be borrowed or returned When a book is returned, it is put back on the shelf

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Library System Operations

Issue a copy of a book to a reader (borrow) Return a copy of a borrowed book by the

reader Add a copy of a book to the library Remove a copy of a book from the library Enquire which books are borrowed by a

given reader Enquire which reader has a given book Register a new reader Remove an existing reader

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A Z Specification Consists of 4 parts

Given sets, data types and constants

State definition Initial state Operations

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Z Specification of the Library1. Given Sets [Book, Copy, Reader]

NB. Symbols used:

⇸ ↦ Ξ ∀ × → Δ ≠ ∧ ∨ ∉ ∈ ∃ ℤ ℕ ℙ ℝ ∅ ≙

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Z Specification of the Library2. State definition

Librarystock: Copy ⇸ Bookissued: Copy ⇸ Readershelved: Copyreaders: Reader

shelved dom issued = dom stockshelved dom issued = ran issued readers∀ r : readers #(issued {r}) maxloans

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Explanationsstock records copies of books in the libraryissued records which copies are on loan and

to whomshelved is a subset of the copies on the

shelves and available for borrowingreaders is the set of registered readers

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Z Specification of the Library3. Initial State Library_Init

ΔLibrary

shelved’ = Ø

issued’ = Ø

readers’ = Ø

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Symbols used:

⇸ ↦ Ξ ∀ × → Δ ≠ ∧ ∨ ∉ ∈ ∃ ℤ ℕ ℙ ℝ ∅ ≙ : Teacher-Course {ITBP120} tuples whose second element is “ITBP120” are

returned. : { bb } Teacher-Course tuples whose first element is “bb” are returned. : dir {num?}tuples whose second element is “num?” are

removed. : {name?} dir tuples whose first element is “name?” are removed.

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Function override operator

Function override adds pairs from f1 to f2 such that it:

acts as a domain subtraction and a set union, if any element of f2 is mapped anywhere by f1,

{Noura ΠprojectA } {Noura ΠnewProj}

=

({Noura } studentproject) {Noura ΠnewProj}

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Z Specification of the Library4. Operations

WcopiesOK Library

c! : Copy r? : Reader r? ∈ readers

c! = dom (issued {r?})

Restricts range to only those elements of the argument

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Z Specification of the Library4. Operations

RemoveCopyOK Library c? : Copy

c? ∈ shelved shelved’ = shelved \ {c?}stock’ = {c?} stock

Restricts the domain to only those elements not equal to the argument

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Z Specification of the Library4. Operations

Borrow Library

c? : Copy r? : Reader c? ∈ shelved r? ∈ readers

# (issued {r?}) maxloansissued’ = issued {c? ↦ r?}

shelved’ = shelved {c?}stock’ = stock; readers’ = readers

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Z Specification of the Library4. Operations

Return Library

c? : Copy r? : Reader c? ∈ dom issued r? ∈ readers

issued’ = issued {c? ↦ r?} shelved’ = shelved {c?}

stock’ = stock; readers’ = readers

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A Z Specification for the Library IS4. Operations

WhoHasCopy Ξ Library

c? : Copy; r! : Reader

c? ∈ dom issued r! = issued (c?)

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Z Specification of the Library4. Operations

RegisterReader Library r? : Reader

r? ∉ readers readers’ = reader {r?}

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Z Specification of the Library4. Operations

RemoveReader Library r? : reader

r? ∈ readers readers’ = reader {r?}

What if r? has borrowed books?

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Z Specification of the Library4. Operations

AddCopyReport Ξ Library stock: Copy ⇸ Book b? : Book rep! : Report

b? ∈ ran stock rep! = CopyAdded b? ∉ ran stock rep! = NewTitleAdded

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Z Specification of the Library4. Operations

EnterNewCopy Library b? : Book

c: Copy c ∉ dom stock

stock’ = stock {c ↦ b?} shelved’ = shelved {c}issued’ = issued; readers’ = readers

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Z Specification of the Library4. Operations

AddCopy ≙ EnterNewCopy ∧ CopyReport

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Z Specification of the Library4. Operations

Execise: do

1. Which books a reader has2. Remove a copy from the library