Formal Methods: Z

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Formal Methods Z

• CS 415, Software Engineering II
• Mark Ardis, Rose-Hulman Institute
• March 18, 2003

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Outline

• Types of Formal Methods
• Introduction to Z
• Examples

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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

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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

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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

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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

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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)

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Set Theory

• Membership x ? S, x ? T
• Union S ? T
• Intersection S ? T

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Functions and Relations

• element mapping x y
• domain, range dom(R), ran(R)
• overriding R ? S
• partial function x y

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Sequences

• definition ltgt, lta, bgt
• concatenation lta, bgt ? ltx, ygt
• 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

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Schema Operators

• conjunction S ? T
• disjunction S ? T
• hiding S \ (v1, , vn)
• hiding S \ T
• overriding S ? T

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Names

• Variables
• input name?
• output name!
• postcondition name'
• Schema
• changes state ?Name
• constant state ?Name

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Schemas

• Name
• declarations
• predicates

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Birthday Book Spivey 92

• Example of use of schemas
• Describes a calendar with birthdates

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• BirthdayBook
• known P NAME
• birthday NAME DATE
• known dom birthday

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Examples

• known Mark, Cheryl, Eric, Paul
• birthday Mark April 7,
• Cheryl July 9,
• Eric July 14,
• Paul April 30

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• AddBirthday
• ? BirthdayBook
• name? NAME
• date? DATE
• name? ? known
• birthday' birthday ?
• name? date?

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• FindBirthday
• ? BirthdayBook
• name? NAME
• date! DATE
• name? ? known
• date! birthday(name?)

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• Remind
• ? BirthdayBook
• today? DATE
• cards! P NAME
• cards! n known
• birthday(n) today?

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Initialization

• InitBirthday
• BirthdayBook
• known Ø

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Deriving Properties

• known' dom birthday'
• dom ( birthday ?
• name? date? )
• dom birthday ? dom
• name? date?
• dom birthday ? name?
• known ? name?

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Cartoon of the Day
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Cartoon of the Day (cont.)
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Symbol Table Hayes 87

• Describes a relation between symbols and values
• Illustrates use of schema operators

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Initial Definitions

• ST ? SYM VAL
• st ? ST
• st0 ? Ø

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• Retrieve
• ? ST
• s? SYM
• v! VAL
• s? ? dom(st)
• v! st(s?)

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• Declare
• ? ST
• s? SYM
• v? VAL
• st' st ? s? v?

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• NotPresent
• ? ST
• s? SYM
• rep! REPORT
• s? ? dom(st)
• rep! "Symbol not present"

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• Success
• rep! REPORT
• rep! "OK"

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Combining Schemas

• STRetrieve ? ( Retrieve ? Success) ?
  NotPresent
• STDeclare ? Declare ? Success

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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

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Block-Structured Symbol Tables

• BST ? seq ST
• ?/ seq ST ? ST
• ?/ ltgt Ø
• ?/ ( s ? lt t gt ) (?/ s ) ? t
• bst0 ? lt gt

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• BStart0
• ? BST
• bst' bst ? lt st0 gt

BEnd0 ? BST bst ? lt gt bst' front( bst )
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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

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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.