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Abstract data types What does ‘ abstract ’ mean? From Latin: to ‘ pull out ’— the essentials To defer or hide the details Abstraction emphasizes essentials and defers the details, making engineering artifacts easier to use

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abstract data types
Abstract data types
  • What does ‘abstract’ mean?
  • From Latin: to ‘pull out’—the essentials
    • To defer or hide the details
    • Abstraction emphasizes essentials and defers the details, making engineering artifacts easier to use
  • I don’t need a mechanic’s understanding of what’s under a car’s hood in order to drive it
    • What’s the car’s interface?
    • What’s the implementation?
floating point numbers
Floating point numbers
  • You don't need to know how much about floating point arithmetic works to use float
    • Indeed, the details can vary depending on processor, even virtual coprocessor
    • But the compiler hides all the details from you--some numeric ADTs are built-in
    • All you need to know is the syntax and meaning of operators, +, -, *, /, etc.
  • Hiding the details of implementation is called encapsulation (data hiding)
  • See multimedia: ADT for digits (properties)
adt properties operations
ADT = properties + operations
  • An ADT describes a set of objects sharing the same properties and behaviors
    • The properties of an ADT are its data (representing the internal state of each object
      • double d; -- bits representing exponent & mantissa are its data or state
    • The behaviors of an ADT are its operations or functions (operations on each instance)
      • sqrt(d) / 2; //operators & functions are its behaviors
  • Thus, an ADT couples its data and operations
    • OOP emphasizes data abstraction
formal language independent adts
Formal, language-independent ADTs
  • An ADT is a formal description, not code; independent of any programming language
    • Why is code independence a good idea?
  • Promotes design by contract:
    • Specify responsibilities of suppliers and clients explicitly, so they can be enforced, if necessary
generic queue adt
Generic Queue ADT
  • An ADT specification has six parts
    • The first three dealing with syntax… NAME, SETS and SIGNATURES

NAME Queue<I>

SETS

I set of all items (generic type)

Q set of all Queues

B set of Boolean (elements T and F)

N set of natural numbers, including 0

  • The NAME specifies the name of a type
    • Generic parameter, <I>, to specify elements of collection types
  • SETS specifies all the types of parameters in SIGNATURES section
signatures section see umprobso you are adding
SIGNATURES section (see umprobso—“you are adding”)

SIGNATURES

Queue() -> Q front(Q) -/-> I

isEmpty(Q) -> B enqueue(Q, I) -> Q

length(Q) -> N dequeue(Q) -/-> Q

  • SIGNATURES specifies the operations or services provided by ADT
  • Notation of mathematical functions, with one or more inputs and producing one result:
    • isEmpty(Q) -> B: Given a Queue (domain), produces a Boolean (range)
  • Functions have no side effects at all:
    • front(Q): given a Q, returns an item (no change)
    • enqueue(Q, I): returns a NEW Queue
    • dequeue(Q): returns another Queue
  • Functional approach may seem inefficient, but facilitates semantics
    • Implementation should preserve the abstract behavior of ADT
  • Syntax is relatively easy to specify; semantics is a bit harder….
full vs partial functions
Full vs. partial functions

SIGNATURES

Queue() -> Q front(Q) -/-> I

isEmpty(Q) -> B enqueue(Q, I) -> Q

length(Q) -> N dequeue(Q) -/-> Q

  • -> denotes a full function over the set Q, always producing the specified type of output
  • ‑/‑> denotes a partial function over the set Q, which may not always produce the output
    • Instead its result may be undefined
  • When is front undefined? When is enQueue undefined?
  • Answering these questions about partial functions is semantics
  • Specifically, the preconditions
    • A partial function is undefined if any of its preconditions do not hold
semantics of adts
Semantics of ADTs
  • Three sections for semantics of ADTs: variables, preconditions, and postconditions

VARIABLES

i:I; q, r:Q; n:N; b:B

PRECONDITIONS

front(q) -> isEmpty (q) = false

dequeue(q) -> isEmpty (q) = false

  • VARIABLES -- declares instances of SETS, needed in PRE- and POST-CONDITIONS
  • How are the variables used in PRECONDITIONS?
  • What is the scope of these variables?
preconditions
Preconditions
  • Specify constraints on any partial functions, indicating when they fail
  • front(q) -> isEmpty (q) = false //What does this constraint tell you?
    • PRECONDITIONs very explicit about the when a partial function fails
  • Formalizes design by contract: analogous to written business contracts
    • Inspire confidence between clients and suppliers of products
  • E.g., a contract for building a house, or a contract to write a book
    • A contract specifies the product that the supplier will produce
    • A contract also specifies the price the client will pay and other terms
    • Such as constraints on a contract—installments, liability, etc.
  • ADT specifies a contract between the supplier and client of an ADT
    • Supplier warrants that ADT will produce the specified behavior
    • so long as client provides the expected inputs
    • so long as client doesn’t violate the pre-conditions, behaviors will work
    • if the client violates the contract (any pre-condition), the behavior fails
    • Yet even the failure is predictable and can be handled predictably
  • Thus PRECONDITIONS also set up exception handling
  • Note: no need to include trivial preconditions, e.g., isEmpty(q) -> true.
postconditions
Postconditions
  • Define effects of functions, i.e., what they accomplish

POSTCONDITIONS

Queue() = (qList = List()) isEmpty(q) = null(qList)

length(q) = length(qList)

front(q) = head(qList)

enqueue(q,i) = (qList = append(qList, i))

dequeue(q,i) = (qList = tail(qList, i))

  • First postcondition defines constructor in terms of List
    • Reusing List implies a constructive semantics, building from other ADTs, already defined
  • Why is constructive semantics a good fit for OOP?
  • What does second postcondition tell you?
axiomatic semantics
Axiomatic semantics
  • Defines relations between operations strictly in terms of universal axioms (self-evident truths)
    • Axioms define an ADT independently of other ADTs
    • Constructive approach builds new ADTs based on knowledge of existing ones
    • The buck has to stop somewhere: e.g., the List ADT uses an axiomatic semantics
    • Book has an optional section on universal axioms
list postconditions axioms
List postconditions (axioms)
  • null(List()) = true //How self-evident?
  • null(prepend(list1,i)) = null(append(list1,i)) = false //Explain?
  • length(List()) = 0
  • length(append(list1, i)) = length(list1)+1
  • tail(append(list1,i)) = if null(list1) then []

else append(tail(list1),i))

constructive semantics see umprobso queue of football
Constructive semantics(See umprobso, “Queue of football”)
  • Explain rest of Queue’s postconditions:

front(q) = head(qList)

enqueue(q,i) = (qList = append(qList, i))

dequeue(q,i) = (qList = tail(qList, i))

  • Why would this be harder with axioms?
    • (See umprobso, “axiomatic”)
inheritance and adts
Inheritance and ADTs
  • See Employee example
  • How does inheritance affect name section?
    • NAME Employee SUPERTYPES Person
  • How does inheritance affect other sections?
    • Employee inherits functions for name, address, etc, from Person
    • Inherits both syntax (SIGNATURES) and semantics
    • No need to redefine functions in Employee unless they do something different
      • So Employee just supplies new constructor, GROSS_PAY, TAX_DUE
    • Semantics can benefit further from reuse implied by inheritance
    • Constructor for Employee invokes constructor for Person
  • Could add notation {abstract} to specify abstract functions
fruit adt assignment
Fruit ADT assignment
  • Your assignment (on Blackboard):
    • Improve your UML analysis
      • Per my comments
      • You may want to improve my analysis!
    • Develop an ADT design
      • Think of it as a contract that you could hand over to a programmer (that would be you!)
    • Extra credit: design a user interface ADT(s)
      • loosely coupled to problem domain ADT