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let/if-then-else

let/if-then-else. let In the let-expression ( let x1 == E1 ; … ; xn == En • E), the variables x1 … xn are local; their scope includes the expression E ( let r == iroot(a) • r*r < a < (r+1)*(r+1 ))

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let/if-then-else

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  1. let/if-then-else let In the let-expression (let x1 == E1; … ; xn == En •E), the variables x1 … xnare local; their scope includes the expression E (let r == iroot(a) • r*r < a < (r+1)*(r+1)) if then else - conditional expression: if P then E1 else E2 – E1 and E2 type should be the same | x | = if x > 0 then x else -x

  2. Z Schemas • Schema: math in a box, with a name attached • Schema calculus builds large schemas from smaller ones. • Model states and operations. • Can be used as declarations, predicates, expressions…

  3. Z Schema • The Z schema is a 2-dimensional graphical notation for describing: – state spaces – operations • Definition: A vertical-form schema is either of the form

  4. Z Schema Or • In the latter case, the predicate part is assumed to be ‘true’.

  5. Z Schema • The declarations part of the schema will contain: – a list of variable declarations; and – references to other schemas (this is called schema inclusion) • Variable declarations have the usual form: x1; x2; : : : ; xn : T; • The predicate part of a schema contains a list of predicates, separated either by semi-colons or new lines.

  6. Schema

  7. State and Operation Schema States A state is a variable assignment A schema describes a set of states Operations To describe an operation, a schema must describe pairs of states (pre/post)

  8. State Schema • The value of tel will be a partial function from NAME to PHONE, i.e., it will associate names with phone numbers

  9. Operation Schemas • In specifying a system operation, we must consider: • the objects that are known to remain unchanged by the operation • the objects that may be altered by the operation • pre-conditions • post-conditions

  10. Example: Phonebook Lookup Analyze the problem: • Accesses the PhoneBookschema • It does not change it • Takes a single ‘input’ —a name • Produces a single output —a phone number • Pre-condition: the name is known to the database.

  11. Example: Phonebook Lookup

  12. Schema calculus: Conjunction

  13. Example - Conjunction  Quotient n, d, q, r: N d  0 n = q * d + r Remainder r, d: N r < d Division = Quotient  Remainder Division n, d, q, r: N d  0 n = q * d + r r < d

  14. Schema calculus : Disjunction

  15. Example - Disjunction  Division n, d, q, r: N d  0 n = q * d + r r < d DivideByZero d, q, r: N report!: TEXT d = 0  q = 0  r = 0 report! = Error T_Division = Division  DivideByZero T_Division n, d, q, r: N report!: TEXT (d  0  r < d  n = q * d + r )  (d = 0  q = 0  r = 0  report! = Error)

  16. Z Specification Yes Yes No Yes Yes Yes Yes NL: newline Ident: variable/type names, e.g, ch?

  17. Z Specification Process

  18. Modeling Large Systems Large systems are composed of components collected into subsystems. Complex components can be built up from simpler ones using schema inclusion.

  19. Case Study A: Using Z to define an abstract data type state space (a set of variables and their constrains), initial state, operations State space: Initial State: Operation:

  20. Case Study B: Birthday book Model a system which records people's birthdays, and is able to issue a reminder when the day comes round. Step 1 Define given sets and types [NAME] [DATE] Step 2 Define state space • knownis the set of names with birthdays recorded • birthdayis a function which, when applied to certain names, gives the birthdays associated with them

  21. Case Study B: Birthday book Step 3 initial state Step 4 operations Add, find, reminder

  22. Case Study B: Birthday book Add Birthday

  23. Case Study B: Birthday book Find Birthday

  24. Case Study B: Birthday book Remind

  25. Case Study C: Editor Example Natural language (English) description: • User can type in text • User can move cursor forward and backward through the text • User can delete the character in front of the cursor

  26. State schemas Editor left, right: TEXT # ( left  right ) maxsize left is before the cursor, right is after

  27. Initialization schemas InitEditor Editor left = right = <> conventional name begins with Init Editor is included left and right come from Editor

  28. Operation schemas printing: P Char Insert  Editor ch?: CHAR ch?  printing left’ = left  <ch?> right’ = right global declaration of printing characters “” means that the state will change “?” means input (“!” means output) precondition left’ is the value after the operation

  29. Schema Inclusion InitEditor Editor left = right = <> InitEditor left, right: TEXT # ( left  right )  maxsize left = right = <> Editor left, right: TEXT # ( left  right )  maxsize

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