Formal Methods of Systems Specification Logical Specification of Hard- and Software Prof. Dr. Holger Schlingloff Institut für Informatik der Humboldt Universität and Fraunhofer Institut für Rechnerarchitektur und Softwaretechnik

Slide 2 H. Schlingloff, Logical Specification Question from last week Q.: Is there anything which can not be specified with Z?  anything = a function from Nat to Nat A.: The number of possible Z specifications is at most countable, while there are uncountably many such functions  More general, the number of possible specifications in any well-defined specification formalism is at most countable  yet, it is hard to find a „natural“ function which is not specifiable, since the formalism is designed especially to formulate all „natural“ functions…

Slide 3 H. Schlingloff, Logical Specification repetition: Z Basic building blocks: Z schemes  declarations (signature)  predicates (formulas constraining variable values) High expressiveness by set theory and logic Possibility of under-specification in Z Modularity (but no object orientation) Well-suited for program verification Not well-suited for refinement (transformational program development) and/or test generation

Slide 4 H. Schlingloff, Logical Specification B-method Tries to overcome this shortcoming Aiming at program development and proof  refinement, implementation, code generation  generalized substitution The B method was developed by J.-R. Abrial (co- author of Z) as an extension of Z Several commercial and university tools Various case-studies in safety-critical systems

Slide 5 H. Schlingloff, Logical Specification Substitution in Predicate Logic Needs distinction between free and bound variable occurrences  occurrences of x in t(… x…) and p(… x …) are free  every occurrence of x in  is bound in  x   a variables can have free and bound occurrences in a formula  [x:=t] is the formula derived from  by replacing every free occurrence of x with term t  substitution may not create new bound occurrences of variables; e.g. ((  x  )  [x:=t])  x  y p(x,y)  (  y p(x,y))[x:=y]  x  y p(x,y)   y p(y,y)  inductive definition of „t is free for y in  “

Slide 6 H. Schlingloff, Logical Specification Substitutions in B Written in prefix notation  [x:=t]  instead of  [x:=t]  [x:=2](x  5) is (2  5), a true statement Substitution as assignment („predicate transformer“)  read as: if x is assigned 2, then x is less or equal to 5  [  ]  can be interpreted as “  establishes  ”  Derived from E.Dijkstra‘s wp- (weakest precondition-) calculus Program specification  admissible starting states specified by formula , desired final states specified by formula   a program is a generalized substitution  such that (  [  ]  )

Slide 7 H. Schlingloff, Logical Specification B Based on abstract machines  don‘t confuse with abstract state machines (ASM) Predicates define properties of abstract machines (states of interest)  a state is a valuation of variables Substitutions define value changes, allow to transform predicates  generalized substitutions can be seen as operations on states Invariants are properties which hold in all states  similar to Z formulas  often used to define type constraints Initialization is by a special substitution Refinements define transformations between abstract machines

Slide 8 H. Schlingloff, Logical Specification Global View Reference: Most of the following material is taken from Moreira, Deharbe, Software Engineering with the B method; 8th Braz. Sym. on FM,

Slide 9 H. Schlingloff, Logical Specification Basic Structure of an Abstract Machine MACHINE Name (Parameters) VARIABLES list of variables INVARIANT invariant predicate INITIALISATION initialization substitution OPERATIONS outputs  name(inputs) ≙ substitution END

Slide 10 H. Schlingloff, Logical Specification Example MACHINE BirthdayAgenda (NAME, DATE) VARIABLES known, birthday INVARIANT known  NAME  birthday  known  DATE INITIALISATION known, birthday := ,  OPERATIONS Register (n, d) ≙ PRE n  NAME  d  DATE  n  known THENknown := known  {n} || birthday := birthday  {n ↦ d} END d  FindBirthday (n) ≙ PRE n  NAME  n  known THEN d := birthday(n) END; END a  FindParty (d) ≙ PRE d  DATE THEN a := birthday -1 (d) END

Slide 11 H. Schlingloff, Logical Specification Verification of a Specification The initialization shall establish the invariant  The machine shall initiate in a valid state  Proof obligation: [  ] , where  is the initialization substitution, and  is the invariant predicate The operations shall preserve the invariant  The operations of the machine shall not take it into an invalid state, assuming that their pre-conditions are respected  Proof obligation: (     [  ]  ), where -  is the invariant predicate, -  is the pre-condition of the operation, and -  is the substitution of the operation

Slide 12 H. Schlingloff, Logical Specification Example

Slide 13 H. Schlingloff, Logical Specification Generalized Substitutions [  1 ;  2 ]  is [  2 ][  1 ]  [  1 ||  2 ]  is [  1 ][  2 ]  (disjoint sets of variables) [x,y:=s,t]  is [tmp:=t][x:=s][y:=tmp]  [IF  THEN  1 ELSE  2 END]  is ((  [  1 ]  )  (¬  [  2 ]  )) [SELECT  1 THEN  1 WHEN  2 THEN  2 END]  is ((  1  [  1 ]  )  (  2  [  2 ]  )) [SKIP]  is  [ANY x WHERE  THEN  END]  is  x (  [  ]  ) [CHOICE  1 OR  2 END]  is ([  1 ]   [  2 ]  ) [PRE  THEN  END]  is (   [  ]  ) …

Slide 14 H. Schlingloff, Logical Specification Modularization An abstract B machine can  USE  SEE  INCLUDE  PROMOTE  EXTEND other abstract machines That way, it is possible to build complex libraries of abstract machines Rich libraries are available for most basic types