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Formal Modelling in VDM

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The VDM Specification Language

The complete VDM-10 Language Manual can be downloaded from here.

VDM Features

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: numeric, character, token and quote types

VDM-SL includes basic types modelling numbers and characters as follows:

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
<A> the quote type containing the value <A>

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:

<Red>, <Amber>, <FlashingAmber>, <Green>

Type Constructors: Union, Product and Composite Types

The basic types alone are of limited value. New, more structured data types are built using type constructors.

Basic Type Constructors

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 = <Red> | <Amber> | <FlashingAmber> | <Green>

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 tuples 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) == d <=31 and m<=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)


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 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


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 conventional modular extension whereas VDM++ has a traditional object-oriented structuring mechanism with classes and inheritance.

Structuring in VDM-SL

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:

Structuring in VDM++

In VDM++ structuring are done using classes and multiple inheritance. The key concepts are:

The VDM-RT extension of VDM++

The VDM-RT extensions can be summarised as:

Modelling functionality

Functional modelling

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 = x

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 = x 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 = x 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”.

State-based modelling

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 

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:

	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:

	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.