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 1970smore popular in Europe than USstill 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 methodabstract model is constructedproperties 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: <>, <a, b> concatenation: <a, b> <x, y> length: #S functions:

head(S) first elementtail(S) all but the first elementlast(S) last elementfront(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'

Schemachanges state: Nameconstant 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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BirthdayBookknown: P NAMEbirthday: 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 BirthdayBookname? : NAMEdate? : DATE

name? knownbirthday' = birthday

{name? date?}

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FindBirthday

BirthdayBook

name? : NAME

date! : DATE

name? known

date! = birthday(name?)

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Remind BirthdayBooktoday? : DATEcards! : P NAME

cards! = { n: known |

birthday(n) = today? }

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InitBirthday

BirthdayBook

known = Ø

Initialization

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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

/ <> = Ø

/ ( s < t > ) = (/ s ) t

bst0 < >

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BStart0

BST

bst' = bst < st0 >

BEnd0

BST

bst < >

bst' = front( bst )

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Z Method

1. Introduce basic sets2. Define an abstract state in terms of sets,

functions, relations, sequences, etc.3. Specify the initial state4. Define pre- and post-conditions of

operations5. 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.