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Formal Methods: Z

Formal Methods: Z. CS 415, Software Engineering II Mark Ardis, Rose-Hulman Institute March 18, 2003. Outline. Types of Formal Methods Introduction to Z Examples. Formal Methods. Specification and verification methods Have formal (mathematical) semantics unambiguous

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Formal Methods: Z

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  1. Formal Methods: Z CS 415, Software Engineering II Mark Ardis, Rose-Hulman Institute March 18, 2003

  2. Outline • Types of Formal Methods • Introduction to Z • Examples

  3. Formal Methods • Specification and verification methods • Have formal (mathematical) semantics • unambiguous • facilitate proofs of correctness • In use since late 1970s • more popular in Europe than US • still only a niche market

  4. Types of Formal Methods • Model-theoretic • VDM, Z • Algebraic • ACT One, Larch , OBJ • Concurrent processes • CCS, CSP, Petri Nets • Finite State Machines • Esterel, Statecharts • Hybrid • LOTOS, SDL

  5. Model-theoretic Methods • Vienna Development Method (VDM) • invented at IBM Vienna lab in late 1970s • used for compilers (Denmark, Germany) and for information processing (England) • Z • Invented by Jean-Raymond Abrial (France) • Developed by Programming Research Group (PRG) at Oxford • Used at IBM Hursley in mid 1980s

  6. Foundations of Z • Model theoretic method • abstract model is constructed • properties of the model are proven • Set theory (and other discrete math) • First order predicate calculus • Schema calculus provides incrementality

  7. Predicate Logic • Variables ranging over arbitrary sets • Predicates: assertions about variables • Operators: • conjunction: A  B • disjunction: A  B • negation:  A • implication: A  B • Quantifiers • universal:  x: T  R(…x…) • existential:  x: T  R(…x…)

  8. Set Theory • Membership: x  S, x  T • Union: S  T • Intersection: S  T

  9. Functions and Relations • element mapping: x y • domain, range: dom(R), ran(R) • overriding: R  S • partial function: x y

  10. Sequences • definition: <>, <a, b> • concatenation: <a, b>  <x, y> • length: #S • functions: head(S) first element tail(S) all but the first element last(S) last element front(S) all but the last element

  11. Schema Operators • conjunction: S  T • disjunction: S  T • hiding: S \ (v1, …, vn) • hiding: S \ T • overriding: S  T

  12. Names • Variables • input: name? • output: name! • postcondition: name' • Schema • changes state: Name • constant state: Name

  13. Schemas Name declarations predicates

  14. Birthday Book [Spivey 92] • Example of use of schemas • Describes a calendar with birthdates

  15. BirthdayBook known: P NAME birthday: NAMEDATE known = dom birthday

  16. Examples known = { Mark, Cheryl, Eric, Paul } birthday = { Mark April 7, Cheryl July 9, Eric July 14, Paul April 30 }

  17. AddBirthday  BirthdayBook name? : NAME date? : DATE name?  known birthday' = birthday  {name? date?}

  18. FindBirthday  BirthdayBook name? : NAME date! : DATE name?  known date! = birthday(name?)

  19. Remind  BirthdayBook today? : DATE cards! : P NAME cards! ={ n: known | birthday(n) = today? }

  20. Initialization InitBirthday BirthdayBook known = Ø

  21. Deriving Properties known' =dom birthday' =dom ( birthday  {name? date?} ) =dom birthday dom {name? date?} =dom birthday { name? } = known { name? }

  22. Cartoon of the Day

  23. Cartoon of the Day (cont.)

  24. Symbol Table [Hayes 87] • Describes a relation between symbols and values • Illustrates use of schema operators

  25. Initial Definitions ST SYM VAL st  ST st0  Ø

  26. Retrieve  ST s? : SYM v! : VAL s? dom(st) v! = st(s?)

  27. Declare  ST s? : SYM v? : VAL st' = st  { s?v? }

  28. NotPresent  ST s? : SYM rep! : REPORT s? dom(st) rep! = "Symbol not present"

  29. Success rep! : REPORT rep! = "OK"

  30. Combining Schemas STRetrieve ( Retrieve  Success)  NotPresent STDeclare  Declare  Success

  31. Overriding Definitions • Introduce a new symbol table for each level of scope • Need to override the previous definitions of symbols: { s v }  { s w } • Need to introduce a distributed override operator for sequences of symbol tables

  32. Block-Structured Symbol Tables BST seq ST / : seq ST  ST / <> = Ø / ( s  < t > ) = (/ s )  t bst0  < >

  33. BStart0  BST bst' = bst  < st0 > BEnd0  BST bst  < > bst' = front( bst )

  34. Z Method • Introduce basic sets • Define an abstract state in terms of sets, functions, relations, sequences, etc. • Specify the initial state • Define pre- and post-conditions of operations • State and prove theorems

  35. References Ian Hayes (editor), Specification Case Studies, Prentice-Hall International, 1987, ISBN 0-13-826579-8. J.M. Spivey, The Z Notation: A Reference Manual, Prentice-Hall International, 1992, ISBN 0-13-978529-9.

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