Vienna Development Method
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 the IBM Laboratory Vienna 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. Philosophy. 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. History. 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.» There is no connection between Meta-IV, and Schorre's Meta-II language, or its successor Tree Meta; these were compiler-compiler systems rather than being  suitable for formal problem descriptions. So Meta-IV was "used to define major portions of" the PL/I programming language. Other programming languages retrospectively 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). 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: 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: 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 codice_1 is the "less than or equal to" Boolean operator on natural numbers): 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: 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. The most basic type constructor forms the union of two predefined types. The type codice_2 contains all elements of the type A and all of the type codice_3. In the traffic signal controller example, the type modelling the colour of a traffic signal could be defined  as follows: 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 codice_4 is the type composed of all tuples of values, the first element of which is from the type codice_5 and the second from the type codice_6 and so on. The composite or record type is a Cartesian product with labels for the fields. The type is the Cartesian product with fields labelled codice_7. An element of type codice_8 can be composed from its constituent parts by a constructor, written codice_9 . Conversely, given an element of type codice_8, the field names can be used to select the named component. For example, the type models a simple date type. The value codice_11 corresponds to 1 April 2001. Given a date codice_12, the expression codice_13 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: Collections. 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. Sets. The set type constructor (written codice_14 where codice_8 is a predefined type) constructs the type composed of all finite sets of values drawn from the type codice_8. For example, the type definition defines a type codice_17 composed of all finite sets of codice_18 values. Various operators are defined on sets for constructing their union, intersections, determining proper and non-strict subset relationships etc. Sequences. The finite sequence type constructor (written codice_19 where codice_8 is a predefined type) constructs the type composed of all finite lists of values drawn from the type codice_8 . For example, the type definition Defines a type codice_22 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 codice_23 is not equal to codice_24, and codice_25 is not equal to codice_26. Maps. 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 codice_27 where codice_28 and codice_29 are predefined types) constructs the type composed of all finite mappings from sets of codice_28 values to sets of codice_29 values. For example, the type definition Defines a type codice_32 which maps character strings to codice_33. Again, operators are defined on mappings for indexing into the mapping, merging mappings, overwriting extracting sub-mappings. Structuring. 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: 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 codice_49 for calculating a square root of a natural number might be defined as follows: Here the postcondition does not define a method for calculating the result codice_50 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. 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: 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: 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: 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: and an explicit function: we say it satisfies the specification iff: So, "codice_51 is a correct implementation" should be interpreted as "codice_51 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 (also known as "globals"). For example, if we have a state consisting of a single variable codice_53, we could define this in VDM-SL as: In VDM++ this would instead be defined as: An operation to load a value into this variable might be specified as: The "externals" clause (codice_54) specifies which parts of the state can be accessed by the operation; codice_55 indicating read-only access and codice_56 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: Where the codice_57 symbol on the state variable in the postcondition indicates the value of the state variable before execution of the operation. Examples. The "max" function. This is an example of an implicit function definition. The function returns the largest element from a set of positive integers: The postcondition characterizes the result rather than defining an algorithm for obtaining it. The precondition is needed because no function could return an r in set s when the set  is empty. Natural number multiplication. multp(i,j:nat)r:nat Applying the proof obligation codice_58 to an explicit definition of codice_59: Then the proof obligation becomes: This can be shown correct by: Queue abstract data type. 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 codice_60. The representation is codice_60 is immaterial and so is defined as a token type. Bank system example. 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. With operations: "NEWC" allocates a new customer number: "NEWAC" allocates a new account number and sets the balance to zero: "ACINF" returns all the balances of all the accounts of a customer, as a map of account number to balance: Tool support. A number of different tools support VDM: Industrial experience . VDM has been applied widely in a variety of application domains. The most well-known of these applications are: Refinement. Use of VDM starts with a very abstract model and develops this into an implementation. Each step involves "data reification", then "operation decomposition". 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. Data reification. 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 codice_62 involves proposing a new representation codice_63. In order to show that the new representation is accurate, a "retrieve function" is defined that relates codice_63 to codice_62, i.e. codice_66. The correctness of a data reification depends on proving "adequacy", i.e. Since the data representation has changed, it is necessary to update the operations and functions so that they operate on codice_63. 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)) Example data reification. In a business security system, workers are given ID cards; these are fed into card readers on entry to and exit from the factory. Operations required: Formally, this would be: 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 codice_72 is the same as codice_73. 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.