Classes and Objects

Classes and Objects Aims and Objectives • Define: • Class schemas to specify object classes (i.e.“types” of objects). » models are possible object instances of a class. • Bag-schemas to specify collections of object instances of a given class. • » models are collections of object instances  i.e. bags of models of a class schema. • Illustrate these two types of schema using a Library example. Unit 3: Classes and Objects

Classes • Class, in object-oriented approach, specifies: • object’s variables – “state” (i.e. “type”) of objects; • methods – “operations” on objects of a given type. • Overall system is more complicated: • a set of objects • each object has a type  is a model of its class schema • system operations can operate on individual objects • … or can change collection of objects (e.g. create, destroy) • Class schemas for types of objects • Operation schemas for operations on objects. • “Bag-schemas” for bags or multisets of objects Unit 3: Classes and Objects

Bags … half way between sets and lists. • In a list: order and multiplicity all matter. • [3,5], [5,3], [3,3,5] are all different lists • In a set: order and multiplicity make no difference • {3,5}, {5,3}, {3,3,5}are all the same set • In a bag: multiplicity matters, order doesn’t. • {|3,5|}, {|5,3|} are the same bag, • {|3,3,5|} is different • The system has bag of models of class schemas, as collection of different objects of the same class. Unit 3: Classes and Objects

How to define a bag e.g. A bag of integers: Define a set X (“indexing set”) and a family of integers indexed by X. e.g. • a={|3,5|}: X any 2-element set, e.g. {Harry, Tom} aHarry = 3, aTom = 5, where “a” denotes the bag • b={|3,3,5|}: X any 3-element set, e.g.{Tom, John, Harry} bTom= bJohn= 3, bHarry= 5, where “b” denotes the bag • Why the funny X’s? Why not just {1,2} or {1,2,3}? • Just as example to emphasize that X does not have to have any intrinsic ordering – if it did we’d just have lists. Unit 3: Classes and Objects

About the indexing set • Indexing set X provides “individual identities” for elements of bag. • cf. addresses of objects: X = {addresses used} • Two different objects may have same type, but they’ll still be distinguished by their different addresses. • Finite bag – when X is finite. Unit 3: Classes and Objects

FinBag(Y) [X] X0: FX f: X  Y x:X. xX0 Bag(Y) [X] f: X Y Schemas whose models are bags Assume a set of elements Y and an indexing set X, then: Model = bag of elements Y. Model = finite bag of elements Y. • What’s X0 for? • It’s just a trick to make sure X is finite: • X0 is declared as a finite subset of X; • the axiom says X is a subset of X0. • Hence X0 = X and both are finite. Unit 3: Classes and Objects

Bags of models of schemas • What we saw before was bags of elements of sets. • Given one schema S, we can define a new “bag-schema” whose models are bags of models of S. How to define them • Simple, when S does not include any base sort. • Add a new base sort X denoting the indexing set. • Constants become functions from X. • Functions and predicates gain an extra argument of sort X. Unit 3: Classes and Objects

Example: a library Making many simplifying assumptions: • Books are described by ISBNs (International Standard Book Numbers) • Library may have more than one copy of the same book. • Each copy has a location comprising a library (e.g. central or departmental) and a shelf-mark. • Some copies may be identical twins of each other (same book, same location) but they are still distinct. • Borrowers have id codes and upper limit on no. books they can borrow. Unit 3: Classes and Objects

Copy isbn: seqCHAR location: LOCATION status: COPYSTATUS Copies as objects in a class schema • LOCATION = LIBRARYSHELFMARK • (e.g. Library = {Maths, Physics, …} • SHELFMARK = … whatever. • We don’t need to know the details.) • COPYSTATUS = {in, out} Unit 3: Classes and Objects

NewCopy Copy isbn?: seqCHAR location?: LOCATION isbn = isbn? location = location? status = in MoveCopy DCopy to?: LOCATION isbn' = isbn status' = status location' = to? Initialization, operations on copies Unit 3: Classes and Objects

Collection [stock] stock0: F(stock) isbn: stock  seqCHAR location: stock  LOCATION status: stock  COPYSTATUS "x:stock. xstock0(forces stock to be finite) Bag of copies • Elements of stock identify individual copies held. • For each xstock, values isbn(x), location(x) and status(x) are the values needed to define a model of Copy. • Collection is FinBag(Copy): a model of Collection is a finite bag of models of Copy. Unit 3: Classes and Objects

MoveOneCopy DCollection moving?: stock to?: LOCATION stock' = stock isbn' = isbn  status' = status "x:stock. (location'(x)=location(x)  x = moving?) location'(moving?) = to? Promotion Operations on individual copies are “promoted” to give operations on the entire collection. e.g.:MoveCopy - moving one copy but in a collection Unit 3: Classes and Objects

OneNewCopy DCollection isbn?: seqCHAR location?: LOCATION new!: stock' new!  stock stock' = stock{new!} "x:stock. (isbn'(x) = isbn(x)  location'(x) = location(x)  status'(x) = status(x)) isbn'(new!) = isbn? location'(new!) = location? status'(new!) = in Creation NewCopy: creating one new copy in a collection Unit 3: Classes and Objects

Other classes Borrower id: seqCHAR max: N (max = max. no. books to be borrowed) • Suppose it also has • a class schema Borrower: • a corresponding bag-schema • Borrowers = FinBag(Borrower), with indexing set “borrower”. System is more than just a collection of objects from a single class schema. • Library • Collection • Borrowers • borrowed: F(borrowerstock) • "x: stock. (status(x) = out $y:borrower. (y,x)  borrowed) • "y:borrower. |{x:stock | (y,x)  borrowed}|  max(y) Unit 3: Classes and Objects

Borrow DLibrary x?:stock y?:borrower status(x?) = in |{x:stock. (y?,x)  borrowed}| < max(y?) stock' = stock  borrower' = borrower isbn' = isbn  location' = location id' = id  max' = max "x:stock. (status'(x) = status(x)  x = x?) status'(x?) = out borrowed' = borrowed{(y?,x?)} Example Operation Unit 3: Classes and Objects

Classes and Objects