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: <>, concatenation:  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 \ (v 1, …, v n ) 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: NAME DATE 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 InitBirthday BirthdayBook known = Ø Initialization

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 st 0  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  ) = (  / s )  t bst 0 

33 BStart 0  BST bst' = bst  BEnd 0  BST bst  bst' = front( bst )

34 Z Method 1. Introduce basic sets 2. Define an abstract state in terms of sets, functions, relations, sequences, etc. 3. Specify the initial state 4. Define pre- and post-conditions of operations 5. State and prove theorems

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