CafeOBJ Hidden Sorts

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CafeOBJ Hidden Sorts

• Formal Methods se304/cs407
• Thanks to Joseph Goguen, Ataru T. Nakagawa,
  Toshimi Sawada, Kokichi Futatsugi

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• I saw a man on a hill with a telescope

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Hidden sorts

• Weve been covering visible sorts
• Another type
• Hidden sorts
• Exist for specifying object oriented (OO) systems
• Hard maths

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Hidden Sorts

• They effectively handle
• Concurrency,
• Distribution,
• Non-determinism,
• Classes, sub-classes (inheritance),
• Attributes and methods,
• Parameterised programming
• ...

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Hidden Algebra

• Based on
• Observational abstraction, aka
• Behavioural satisfaction
• New type of equivalence
• Behavioural equivalence
• Two terms are equal iff they behave identically
  under the given set of observations

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Syntax for sort signature

• Visible sorts
• list
• Hidden sorts
• list
• This makes it impossible to have
• Visible sub-sorts of hidden sorts, or
• Hidden sub-sorts of visible sorts
• They are disjoint sets

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Operators

• Behavioural operators
• bop name list_of_sorts -gt sort
• Arity contains exactly one hidden sort

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Hidden algebra

• Hidden sorts represent objects
• Behavioural operators represent methods
• This is why bops must have exactly one hidden
  operand
• That is the object on which the method is being
  called

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Example

• Account
• bop deposit Account Nat -gt Account
• bop withdraw Account Nat -gt Account
• bop balance Account -gt Nat
• op deposit Account Nat -gt Account

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Remember

• Behavioural equivalence
• Two terms are equal iff they behave identically
  under the given set of observations
• bops are those observations

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Behavioural equations

• beq term term .
• bceq term term if boolean_term .
• If either term has an operator with a hidden sort
  in its arity
• That operator must be behavioural

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example

• S a hidden sort and T a visible sort
• Which of the following operators would be allowed
  in a behavioural equation using S and T?

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example

• bop f S -gt T
• bop g S T -gt S
• op c -gt T
• op d -gt S
• op h T -gt T
• op k T -gt S
• op u S -gt T
• op v S T -gt T

Behavioural
Dont have S in arity
Have S in arity
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Ordinary equations

• Most behavioural equality is defined with
  ordinary equations
• Behavioural equations should be reserved for
  cases where explicit statements are required
• Typically when stating theorems

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Equivalence relations

• Reflexivity
• X X
• Symmetry
• X Y gt Y X
• Transitivity
• X Y, Y Z gt X Z

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Partial equivalence

• Take out one of those three attributes
• Partial equivalence
• Take out symmetry
• Transitions
• trans term gt term .
• ctrans term gt term if boolean term .
• btrans term gt term .
• bctrans term gt term if boolean term .

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Transition vs. equation

• module CHOICE
• State
• ops a b -gt State
• op __ State State -gt State
• vars X Y State
• trans X Y gt X .
• trans X Y gt Y .

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Transition vs. equation

• module CHOICE
• State
• ops a b -gt State
• op __ State State -gt State
• vars X Y State
• eq X Y X .
• eq X Y Y .

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Todays lab

• Write modules specifying the
• Admin and
• Student
• Classes from lab 1
• Revision of previous labs

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Marking scheme

• Next weeks lab is for revision
• Read over previous labs and ask questions
• Optional lab-exam in last week of semester
• If you dont take it, your CA mark will not
  change
• If you do take it, it will be worth 40 of your
  CA mark
• It will be based on the labs you have done
• Because of the exam, todays lab wont be worth
  as much as I said previously.