The Vienna Development Method (VDM) is one of the longest-established Formal Methods for the development of computer-based systems. Originating in work done at IBM's Vienna Laboratory in the 1970s, it has grown to include a group of techniques and tools based on a formal specification language - the VDM Specification Language (VDM-SL). It has an extended form, VDM++, which supports the modeling of object-oriented and concurrent systems. Support for VDM includes commercial and academic tools for analyzing models, including support for testing and proving properties of models and generating program code from validated VDM models. There is a history of industrial usage of VDM and its tools and a growing body of research in the formalism has led to notable contributions to the engineering of critical systems, compilers, concurrent systems and in logic for computer science.
Computing systems may be modeled in VDM-SL at a higher level of abstraction than is achievable using programming languages, allowing the analysis of designs and identification of key features, including defects, at an early stage of system development. Models that have been validated can be transformed into detailed system designs through a refinement process. The language has a formal semantics, enabling proof of the properties of models to a high level of assurance. It also has an executable subset, so that models may be analyzed by testing and can be executed through graphical user interfaces, so that models can be evaluated by experts who are not necessarily familiar with the modeling language itself.
The origins of VDM-SL lie in the IBM Laboratory in Vienna where the first version of the language was called the Vienna Definition Language (VDL). The VDL was essentially used for giving operational semantics descriptions in contrast to the VDM - Meta-IV which provided denotational semantics
«Towards the end of 1972 the Vienna group again turned their attention to the problem of systematically developing a compiler from a language definition. The overall approach adopted has been termed the "Vienna Development Method"... The meta-language actually adopted ("Meta-IV") is used to define major portions of PL/1 (as given in ECMA 74 - interestingly a "formal standards document written as an abstract interpreter) in BEKIČ 74.»
So Meta-IV, was "used to define major portions of" the PL/I programming language. Other programming languages described, or partially described, using Meta-IV and VDM-SL include the BASIC programming language, FORTRAN, the APL programming language, ALGOL 60, the Ada programming language and the Pascal programming language. Meta-IV evolved into several variants, generally described as the Danish, English and Irish Schools.
The "English School" derived from work by Cliff Jones on the aspects of VDM not specifically related to language definition and compiler design (Jones 1980, 1990). It stresses modelling persistent state through the use of data types constructed from a rich collection of base types. Functionality is typically described through operations which may have side-effects on the state and which are mostly specified implicitly using a precondition and postcondition. The "Danish School" (Bjørner et al. 1982) has tended to stress a constructive approach with explicit operational specification used to a greater extent. Work in the Danish school led to the first European validated Ada compiler.
An ISO Standard for the language was released in 1996 (ISO, 1996).
The VDM-SL and VDM++ syntax and semantics are described at length in the VDMTools language manuals and in the available texts. The ISO Standard contains a formal definition of the language’s semantics. In the remainder of this article, the ISO-defined interchange (ASCII) syntax is used. Some texts prefer a more concise mathematical syntax.
A VDM-SL model is a system description given in terms of the functionality performed on data. It consists of a series of definitions of data types and functions or operations performed upon them.
Basic Types | ||
---|---|---|
bool
| Boolean datatype | false, true |
nat
| natural numbers (including zero) | 0, 1, 2, 3, ... |
nat1
| natural numbers (excluding zero) | 1, 2, 3, 4, ... |
int
| integers | ..., -3, -2, -1, 0, 1, 2, 3, ... |
rat
| rational numbers | a/b, where a and b are integers, b is not 0 |
real
| real numbers | ... |
char
| characters | A, B, C, ... |
token
| structureless tokens | ... |
| the quote type containing the value
| ... |
Data types are defined to represent the main data of the modelled system. Each type definition introduces a new type name and gives a representation in terms of the basic types or in terms of types already introduced. For example, a type modelling user identifiers for a log-in management system might be defined as follows:
types
UserId = nat
For manipulating values belonging to data types, operators are defined on the values. Thus, natural number addition, subtraction etc. are provided, as are Boolean operators such as equality and inequality. The language does not fix a maximum or minimum representable number or a precision for real numbers. Such constraints are defined where they are required in each model by means of data type invariants -- Boolean expressions denoting conditions that must be respected by all elements of the defined type. For example a requirement that user identifiers must be no greater than 9999 would be expressed as follows (where <=
is the “less than or equal to” Boolean operator on natural numbers):
UserId = nat
inv uid == uid <= 9999
Since invariants can be arbitrarily complex logical expressions, and membership of a defined type is limited to only those values satisfying the invariant, type correctness in VDM-SL is not automatically decidable in all situations.
The other basic types include char for characters. In some cases, the representation of a type is not relevant to the model’s purpose and would only add complexity. In such cases, the members of the type may be represented as structureless tokens. Values of token types can only be compared for equality – no other operators are defined on them. Where specific named values are required, these are introduced as quote types. Each quote type consists of one named value of the same name as the type itself. Values of quote types (known as quote literals) may only be compared for equality.
For example, in modelling a traffic signal controller, it may be convenient to define values to represent the colours of the traffic signal as quote types:
, , ,
The basic types alone are of limited value. New, more structured data types are built using type constructors.
Basic Type Constructors | |
---|---|
T1 $>$ T2 $|$ ... $|$ Tn
| Union of types T1,...,Tn |
T1*T2*...*Tn
| Cartesian product of types T1,...,Tn |
T :: f1:T1 ... fn:Tn
| Composite (Record) type |
The most basic type constructor forms the union of two predefined types. The type (A|B)
contains all elements of the type A and all of the type B
. In the traffic signal controller example, the type modelling the colour of a traffic signal could be defined as follows:
SignalColour =| | |
Enumerated types in VDM-SL are defined as shown above as unions on quote types.
Cartesian product types may also be defined in VDM-SL. The type (A1*…*An)
is the type composed of all pairs of values, the first element of which is from the type A1
and the second from the type A2
and so on. The composite or record type is a Cartesian product with labels for the fields. The type
T :: f1:A1
f2:A2
...
fn:An
is the Cartesian product with fields labelled f1,…,fn
. An element of type T
can be composed from its constituent parts by a constructor, written mk_T
. Conversely, given an element of type T
, the field names can be used to select the named component. For example, the type
Date :: day:nat1
month:nat1
year:nat
inv mk_Date(d,m,y) == day <=31 and month<=12
models a simple date type. The value mk_Date(1,4,2001)
corresponds to 1 April 2001. Given a date d
, the expression d.month
is a natural number representing the month. Restrictions on days per month and leap years could be incorporated into the invariant if desired. Combining these:
mk_Date(1,4,2001).month = 4
Collection types model groups of values. Sets are finite unordered collections in which duplication between values is suppressed. Sequences are finite ordered collections (lists) in which duplication may occur and mappings represent finite correspondences between two sets of values.
The set type constructor (written set of T
where T
is a predefined type) constructs the type composed of all finite sets of values drawn from the type T
. For example, the type definition
UGroup = set of UserId
defines a type UGroup
composed of all finite sets of UserId
values. Various operators are defined on sets for constructing their union, intersections, determining proper and non-strict subset relationships etc.
Main Operators on Sets (s, s1, s2 are sets) | |
---|---|
{a, b, c}
| Set enumeration: the set of elements a , b and c |
{ x $>$ x:T & P(x) $\}$
| Set comprehension: the set of x from type T such that P(x) |
{i, ..., j}
| The set of integers in the range i to j |
e in set s
| e is an element of set s |
e not in set s
| e is not an element of set s |
s1 union s2
| Union of sets s1 and s2 |
s1 inter s2
| Intersection of sets s1 and s2 |
s1 s2
| Set difference of sets s1 and s2 |
dunion s
| Distributed union of set of sets s |
s1 psubset s2
| s1 is a (proper) subset of s2 |
s1 subset s2
| s1 is a (weak) subset of s2 |
card s
| The cardinality of set s |
The finite sequence type constructor (written seq of T
where T
is a predefined type) constructs the type composed of all finite lists of values drawn from the type T
. For example, the type definition
String = seq of char
Defines a type String
composed of all finite strings of characters. Various operators are defined on sequences for constructing concatenation, selection of elements and subsequences etc. Many of these operators are partial in the sense that they are not defined for certain applications. For example, selecting the 5th element of a sequence that contains only three elements is undefined.
The order and repetition of items in a sequence is significant, so [a, b]
is not equal to [b, a]
, and [a]
is not equal to [a, a]
.
Main Operators on Sequences (s, s1,s2 are sequences) | |
---|---|
[a, b, c]
| Sequence enumeration: the sequence of elements a , b and c |
[f(x)$>$ x:T & P(x)]
| Sequence comprehension: sequence of expressions f(x) for each x of (numeric) type T such that P(x) holds (x values taken in numeric order) |
hd s
| The head (first element) of s |
tl s
| The tail (last element) of s |
len s
| The length of s |
elems s
| The set of elements of s |
s(i)
| The i ^{th} element of s |
s1^s2
| the sequence formed by concatenating sequences s1 and s2 |
A finite mapping is a correspondence between two sets, the domain and range, with the domain indexing elements of the range. It is therefore similar to a finite function. The mapping type constructor in VDM-SL (written map T1 to T2
) where T1
and T2
are predefined types) constructs the type composed of all finite mappings from sets of T1
values to sets of T2
values. For example, the type definition
Birthdays = map String to Date
Defines a type Birthdays
which maps character strings to Date
. Again, operators are defined on mappings for indexing into the mapping, merging mappings, overwriting extracting sub-mappings.
Main Operators on Mappings | |
---|---|
{a $>$-> r, b $|$-> s} | Mapping enumeration: a maps to r , b maps to s |
{x $>$-> f(x) $|$ x:T & P(x)}
| Mapping comprehension: x maps to f(x) for all x for type T such that P(x) |
dom m
| The domain of m |
rng m
| The range of m |
m(x)
| m applied to x |
m1 munion m2
| Union of mappings m1 and m2 (m1 , m2 must be consistent where they overlap) |
m1 ++ m2
| m1 overwritten by m2 |
The main difference between the VDM-SL and VDM++ notations are the way in which structuring is dealt with. In VDM-SL there is a conventionel modular extension whereas VDM++ has a traditional object-oriented structuring mechanism with classes and inheritance.
In the ISO standard for VDM-SL there is an informative annex that contains different structuring principles. These all follow traditional information hiding principles with modules and they can be explained as:
module
followed by the name of the module. At the end of a module the keyword end
is written followed again by the name of the module. imports
and followed by a sequence of imports from different modules. Each of these module imports are started with the keyword from
followed by the name of the module and a module signature. The module signature can either simply be the keyword all
indicating the import of all definitions exported from that module, or it can be a sequence of import signatures. The import signatures are specific for types, values, functions and operations and each of these are started with the corresponding keyword. In addition these import signatures name the constructs that there is a desire to get access to. In addition optional type information can be present and finally it is possible to rename each of the constructs upon import. For types one needs also to use the keyword struct
if one wish to get access to the internal structure of a particular type.exports
followed by an exports module signature. The exports module signature can either simply consist of the keyword all
or as a sequence of export signatures. Such export signatures are specific for types, values, functions and operations and each of these are started with the corresponding keyword. In case one wish to export the internal structure of a type the keyword struct
must be used.In VDM++ structuring are done using classes and multiple inheritance. The key concepts are:
class
followed by the name of the class. At the end of a class the keyword end
is written followed again by the name of the class. is subclass of
followed by a comma-separated list of names of superclasses.private
, public
and protected
.In VDM-SL, functions are defined over the data types defined in a model. Support for abstraction requires that it should be possible to characterize the result that a function should compute without having to say how it should be computed. The main mechanism for doing this is the implicit function definition in which, instead of a formula computing a result, a logical predicate over the input and result variables, termed a postcondition, gives the result's properties. For example, a function SQRT
for calculating a square root of a natural number might be defined as follows:
SQRT(x:nat)r:real
post r*r = n
Here the postcondition does not define a method for calculating the result r
but states what properties can be assumed to hold of it. Note that this defines a function that returns a valid square root; there is no requirement that it should be the positive or negative root. The specification above would be satisfied, for example, by a function that returned the negative root of 4 but the positive root of all other valid inputs. Note that functions in VDM-SL are required to be deterministic so that a function satisfying the example specification above must always return the same result for the same input.
A more constrained function specification is arrived at by strengthening the postcondition. For example the following definition constrains the function to return the positive root.
SQRT(x:nat)r:real
post r*r = n and r>=0
All function specifications may be restricted by preconditions which are logical predicates over the input variables only and which describe constraints that are assumed to be satisfied when the function is executed. For example, a square root calculating function that works only on positive real numbers might be specified as follows:
SQRTP(x:real)r:real
pre x >=0
post r*r = n and r>=0
The precondition and postcondition together form a contract that to be satisfied by any program claiming to implement the function. The precondition records the assumptions under which the function guarantees to return a result satisfying the postcondition. If a function is called on inputs that do not satisfy its precondition, the outcome is undefined (indeed, termination is not even guaranteed).
VDM-SL also supports the definition of executable functions in the manner of a functional programming language. In an explicit function definition, the result is defined by means of an expression over the inputs. For example, a function that produces a list of the squares of a list of numbers might be defined as follows:
SqList: seq of nat -> seq of nat
SqList(s) == if s = [] then [] else [(hd s)**2] ^ SqList(tl s)
This recursive definition consists of a function signature giving the types of the input and result and a function body. An implicit definition of the same function might take the following form:
SqListImp(s:seq of nat)r:seq of nat
post len r = len s and
forall i in set inds s & r(i) = s(i)**2
The explicit definition is in a simple sense an implementation of the implicitly specified function. The correctness of an explicit function definition with respect to an implicit specification may be defined as follows.
Given an implicit specification:
f(p:T_p)r:T_r
pre pre-f(p)
post post-f(p, r)
and an explicit function:
f:T_p -> T_r
we say it satisfies the specification iff:
forall p in set T_p & pre-f(p) => f(p):T_r and post-f(p, f(p))So, "
f
is a correct implementation" should be interpreted as "f
satisfies the specification".
In VDM-SL, functions do not have side-effects such as changing the state of a persistent global variable. This is a useful ability in many programming languages, so a similar concept exists; instead of functions, operations are used to change state variables (AKA globals).
For example, if we have a state consisting of a single variable someStateRegister : nat
, we could define this in VDM-SL as:
state Register of
someStateRegister : nat
end
In VDM++ this would instead be defined as:
instance variables
someStateRegister : nat
An operation to load a value into this variable might be specified as:
LOAD(i:nat)
ext wr someStateRegister:nat
post someStateRegister = i
The externals clause (ext
) specifies which parts of the state can be accessed by the operation; rd
indicating read-only access and wr
being read/write access.
Sometimes it is important to refer to the value of a state before it was modified; for example, an operation to add a value to the variable may be specified as:
ADD(i:nat)
ext wr someStateRegister : nat
post someStateRegister = someStateRegister~ + i
Where the ~
symbol on the state variable in the postcondition indicates the value of the state variable before execution of the operation.
max(s:set of nat)r:nat
pre true
post r in set s and
forall r' in set s & r' <= r
multp(i,j:nat)r:nat
pre true
post r = i*j
Applying the proof obligation forall p:T_p & pre-f(p) => f(p):T_r and post-f(p, f(p))
to an explicit definition of multp
:
multp(i,j) ==
if i=0
then 0
else if is-even(i)
then 2*multp(i/2,j)
else j+multp(i-1,j)
Then the proof obligation becomes:
forall i, j : nat & multp(i,j):nat and multp(i, j) = i*j
This can be shown correct by:
This is a classical example illustrating the use of implicit operation specification in a state-based model of a well-known data structure. The queue is modelled as a sequence composed of elements of a type Qelt
. The representation is Qelt
is immaterial and so is defined as a token type.
types
Qelt = token;
Queue = seq of Qelt;
state TheQueue of
q : Queue
end
operations
ENQUEUE(e:Qelt)
ext wr q:Queue
post q = q~ ^ [e];
DEQUEUE()e:Qelt
ext wr q:Queue
pre q <> []
post q~ = [e]^q;
IS-EMPTY()r:bool
ext rd q:Queue
post r <=> (len q = 0)
As a very simple example of a VDM-SL model, consider a system for maintaining details of customer bank account. Customers are modelled by customer numbers (CustNum), accounts are modelled by account numbers (AccNum). The representations of customer numbers are held to be immaterial and so are modelled by a token type. Balances and overdrafts are modelled by numeric types.
AccNum = token;
CustNum = token;
Balance = int;
Overdraft = nat;
AccData :: owner : CustNum
balance : Balance
state Bank of
accountMap : map AccNum to AccData
overdraftMap : map CustNum to Overdraft
inv mk_Bank(accountMap,overdraftMap) == forall a in set rng accountMap & a.owner in set dom overdraftMap and
a.balance >= -overdraftMap(a.owner)
With operations: NEWC allocates a new customer number:
operations
NEWC(od : Overdraft)r : CustNum
ext wr overdraftMap : map CustNum to Overdraftpost r not in set dom ~overdraftMap and overdraftMap = ~overdraftMap ++ { r |-> od};
NEWAC allocates a new account number and sets the balance to zero:
NEWAC(cu : CustNum)r : AccNum
ext wr accountMap : map AccNum to AccData
rd overdraftMap map CustNum to Overdraft
pre cu in set dom overdraftMappost r not in set dom accountMap~ and accountMap = accountMap~ ++ {r|-> mk_AccData(cu,0)}
ACINF returns all the balances of all the accounts of a customer, as a map of account number to balance:
ACINF(cu : CustNum)r : map AccNum to Balance
ext rd accountMap : map AccNum to AccDatapost r = {an |-> accountMap(an).balance | an in set dom accountMap & accountMap(an).owner = cu}
A number of different tools support VDM:
VDM has been applied widely in a variety of application domains. The most well-known of these applications are:
Data reification develops the abstract data types into more concrete data structures, while operation decomposition develops the (abstract) implicit specifications of operations and functions into algorithms that can be directly implemented in a computer language of choice.
Specification | Implementation | |
---|---|---|
Abstract data type | ––– Data reification → | Data structure |
Operations | ––– Operation decomposition → | Algorithms |
Data reification (stepwise refinement) involves finding a more concrete representation of the abstract data types used in a specification. There may be several steps before an implementation is reached. Each reification step for an abstract data representation ABS_REP
involves proposing a new representation NEW_REP
. In order to show that the new representation is accurate, a retrieve function is defined that relates NEW_REP
to ABS_REP
, i.e. retr : NEW_REP -> ABS_REP
. The correctness of a data reification depends on proving adequacy, i.e.
forall a:ABS_REP & exists r:NEW_REP & a = retr(r)
Since the data representation has changed, it is necessary to update the operations and functions so that they operate on NEW_REP
. The new operations and functions should be shown to preserve any data type invariants on the new representation. In order to prove that the new operations and functions model those found in the original specification, it is necessary to discharge two proof obligations:
forall r: NEW_REP & pre-OPA(retr(r)) => pre-OPR(r)
forall ~r,r:NEW_REP & pre-OPA(retr(~r)) and post-OPR(~r,r) => post-OPA(retr(~r,), retr(r))
INIT()
initialises the system, assumes that the factory is emptyENTER(p : Person)
records that a worker is entering the factory; the workers' details are read from the ID card)EXIT(p : Person)
records that a worker is exiting the factoryIS-PRESENT(p : Person) r : bool
checks to see if a specified worker is in the factory or notFormally, this would be:
types
Person = token;
Workers = set of Person;
state AWCCS of
pres: Workers
end
operations
INIT()
ext wr pres: Workerspost pres = {};
ENTER(p : Person)
ext wr pres : Workers
pre p not in set prespost pres = pres~ union {p};
EXIT(p : Person)
ext wr pres : Workers
pre p in set prespost pres = pres~{p};
ISPRESENT(p : Person) r : bool
ext rd pres : Workers
post r <=> p in set pres~
As most programming languages have a concept comparable to a set (often in the form of an array), the first step from the specification is to represent the data in terms of a sequence. These sequences must not allow repetition, as we do not want the same worker to appear twice, so we must add an invariant to the new data type. In this case, ordering is not important, so [a,b]
is the same as [b,a]
.
The Vienna Development Method is valuable for model-based systems. It is not appropriate if the system is time-based. For such cases, the calculus of communicating systems (CCS) is more useful.